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INDEX

Aabbreviations, 1627Abel equation, 443, 525active functions, 1628adding integration constants, 1665additive separable solution, 1374, 1459additively separable, 1651, 1710adiabatic equation, 1262adiabatic exponent, 1262adiabatic nonisentropic gas flow, 1204adsorption coefficients, 1231Airy equation, 861Airy functions, 1193algebraic equations, 1441algebraic equations with even powers, 1441algebraic systems of equations, 1441alternative definitions of functions, 1628analysis of some ordinary differential

equations, 1447analytical derivation of numerical methods,

1680, 1729analyticalnumerical solutions, 1680, 1728analytical solutions

of nonlinear systems, 1659, 1719visualizations, 1629, 1691

anonymous functions, 1741ansatz methods for constructing traveling

wave solutions, 1641, 1700antikink, 1636, 1695antisoliton, 1636, 1695application program interface, 1736applied shear stress at boundary, 1252arbitrary functions (included in PDEs)

dependent ondifferent arguments, 1164linear combination of unknowns, 1133,

1157, 1173, 1178, 1185product of powers of unknowns, 1145,

1162ratio of unknowns, 1137, 1159, 1175,

1181, 1187sum or difference of squares of unknowns,

1146, 1163unknowns in complex way, 1148

arithmetic operators, 1627, 1738array, 1629assignment/unassignment operators, 1627associativity equations, 972autoBacklund transformation, 681, 1413,

1635, 1695axial flows, 1303axisymmetric fluid flows, 1274axisymmetric jet, 910

boundary layer approximation, 1335

axisymmetric steady laminar hydrodynamicboundary layer equations, 909

axisymmetric unsteady laminar boundarylayer equations, 929

BBacklund transformations, 1413, 1635, 1695

autoBacklund transformations, 681, 1413,1635, 1695

canonic for evolution equations, 1420constructed via RF pairs, 1415for secondorder equations, 1413

basic arithmetic operators, 1689basic relations used in symmetry analysis of

systems of equations, 1527basic steps of Painleve test for nonlinear

equations, 1570basic transformation rules, 1689BBM equation, 958Bellman type equations and related equa

tions, 805Beltarmi surface, 253Benney moment equations, 764Bernoulli equation, 394, 421Bessel functions, 1160, 1255, 1259bilinear functional equations, 1775Blasius problem, 904, 1333block, 1691blowup regimes, 216boolean, 1739boolean expression, 1628, 1689Born–Infeld equation, 766, 1401boundary conditions, 1752boundary layer

approximation for axisymmetric jet, 1335axisymmetric steady laminar hydrody

namic, 909axisymmetric unsteady laminar, 929equations, 903for nonNewtonian fluid, 915nonlinear diffusion equations, 346, 402nonlinear thermal equations, 346on flat plate, 1333on Vshaped body, 1334steadystate equations, 903steadystate equations for Newtonian

fluid, 903steadystate equations for nonNewtonian

fluids, 911steadystate laminar, 903symmetries of steadystate equations, 1528unsteady, 917unsteady equations for Newtonian

fluid, 917

Page: 1841

1841

1842 Polyanin & Zaitsev, Handbook of Nonlinear PDEs, 2nd Edition, 2012

boundary layer (continued)unsteady equations for nonNewtonian

fluids, 930with pressure gradient, 907

boundary layer equation for nonNewtonianfluid of general form, 917

boundary layer equation for powerlaw fluidwith pressure gradient, 913

boundary layer equations (Prandtl equations),1333

boundary value problem, 1679, 1727Boussinesq equation, 368, 1504

in canonical form, 987modifications, 987

Boussinesqtype equation, 1028Boussinesq transformation, 1289breaking soliton equation, 1030breather, 492brief introduction to Maple, 1625brief introduction to Mathematica, 1687brief introduction to MATLAB, 1735builtin functions, 1740Burgers equation, 186, 1409, 1414, 1572,

1637, 1673, 1677, 1697, 1723, 1725,1743, 1761

Burgers–Huxley equation, 189Burgers–Korteweg–de Vries equation, 884Burgers vortex, 1290

CCalogero equation, 745, 746Calogero typeequation, 1400Camassa–Holm equation, 968Camassa–Holm multipeakon solutions, 965canonic Backlund transformations of

evolution equations, 1420canonical form

Boussinesq equation, 987for systems of equations, 1611Kadomtsev–Petviashvili equation, 1000Korteweg–de Vries equation, 857modified Korteweg–de Vries equation, 867of elliptic equations, 1392of gas dynamics systems, 1611of hyperbolic equations, 1390–1392of parabolic equations, 1390

Cauchy–Lagrange integral, 1201Cauchy problem, 1357, 1379, 1611

for Hamilton–Jacobi equation, 1380qualitative features of solutions, 1611

centered rarefaction wave, 1359Chaplygin equations, 1202Chaplygin gas equation, 765characteristic, 1393characteristic curves, 1647characteristic equation, 1389, 1393, 1647,

1706characteristic system, 1355characteristic velocity, 1615Charpit equations, 1647, 1706

checking whether nonlinear systems ofequations of mathematical physics passPainleve test, 1576

chromatography equations, 1231Clairaut equation, 118, 152, 172

in implicit form, 122partial differential, 1375

Clarkson–Kruskal direct method, 1503classic worksheet, Maple, 1626classical Cauchy problem, 1647, 1707, 1709,

1756classical Couette–Poiseuille solution, 1263classical hodograph transformation, 1400classical method of separation of variables,

1651, 1711classical method of symmetry reductions,

1513classical selfsimilar solutions, 1435classical travelingwave solutions, 1431classification of secondorder nonlinear

equations, 1389classification results for nonlinear first and

secondorder equations, 1565cnoidal waves, 858coefficient of reflection, 1588coefficients of equations contain

exponential functions, 11hyperbolic functions, 14logarithmic functions, 16powerlaw functions, 3trigonometric functions, 17

Cole–Hopf transformation, 1573for Burgers equation, 1637, 1696

combined KdVmKdV equation, 871commutator of operators, 1579compacton, 889comparing exact solution generation

methods, 1662comparison of asymptotic and numerical

solutions, 1684, 1733compatibility condition, 1581complete elliptic integral, 858complete integral, 1373, 1647, 1650, 1706,

1709complete Lin–Reissner–Tsien equation, 757complete solution, 1647, 1706complex separation of variables in nonlinear

partial differential equations, 1462composite data types, 1741condition

compatibility, 1581condition, for integrability of Pfaffian

equation by single relation, 1601Courant–Friedrichs–Levy, 1763invariance, 1516, 1527, 1595invariant, 1527invariant surface, 1533jump, 1362, 1363, 1369Lax, 1616, 1617noslip, 1253Rankine–Hugoniot, 1126, 1128, 1759Rankine–Hugoniot jump, 1363, 1369, 1615

Page: 1842

INDEX 1843

condition (continued)stability, 1369stability, for generalized solution, 1369strip, 1651, 1710transversality, 1380

conditionsboundary, 1752evolutionary, 1616

conical vortex flow, 1290connection between method of differential

constraints and other methods, 1561conservation form, 1678, 1725, 1762, 1763conservation law at discontinuity, 1363conservation laws, 1365, 1593

for some nonlinear equations of mathematical physics, 1593

constructing analytical solutions in terms ofpredefined functions, 1629

constructing complete integral using two firstintegrals, 1377

constructing exact solutions using symboliccomputation, 1662

constructing finitedifference approximations,1677, 1725, 1760

constructing generalized separable solutions,1661, 1721

constructing numerical solutions, 1683, 1731in terms of predefined functions, 1672,

1722constructing selfsimilar solutions, 1645,

1705constructing separable solutions, 1651, 1710constructing solutions along characteristics,

1647, 1706constructing solutions of nonlinear equations

that fail Painleve test, using truncatedexpansions, 1577

constructing solutions using predefinedfunctions, 1692

constructing solutions via transformations,1635, 1694

constructing travelingwave solutions, 1637,1659, 1697, 1719

constructive solid geometry, 1754contact transformations, 1403

for ordinary differential equations, 1403continuity equation, 1197, 1247, 1610continuous point group, 1526control structures, 1741convective fluid motions, 1329coordinates of first and second prolongations,

1515Couette–Poiseuille plane flow, 1251Couette–Poiseuille type solution, 1283Courant–Friedrichs–Levy condition, 1763Crocco transformation, 906, 909, 1422Crocco variables, 748, 1107cylindric Burgers equation, 191cylindrical Korteweg–de Vries equation, 866cylindrical vortex flow, 1290

Ddata structures, 1741data types, 1690decomposed geometry, 1752Degasperis–Procesi equation, 968description of simplified scheme for

constructing solutions based onpresetting one system of coordinatefunctions, 1465

desktop, 1737determining equation, 1555development environment, 1736different types of cells, 1688differential constraints, 1539, 1545

of arbitrary order, 1542differential equation

adiabatic, 1262Airy, 861BBM, 958Bernoulli, 394, 421Born–Infeld, 766, 1401boundary layer, for nonNewtonian

fluid, 917boundary layer, for powerlaw fluid with

pressure gradient, 913Boussinesq, 368, 987, 1504Boussinesq, in canonical form, 987Boussinesqtype, 1028breaking soliton, 1030Burgers, 186, 1409, 1414, 1572, 1637,

1673, 1677, 1697, 1723, 1725, 1743,1761

Burgers–Huxley, 189Burgers–Korteweg–de Vries, 884Calogero, 745, 746Calogerotype, 1400Camassa–Holm, 968Chaplygin, 765characteristic, 1389, 1393, 1647, 1706Clairaut, 118, 152, 172, 1375Clairaut, in implicit form, 122combined KdVmKdV, 871complete Lin–Reissner–Tsien, 757continuity, 1197, 1247, 1610cylindric Burgers, 191cylindrical Korteweg–de Vries, 866Degasperis–Procesi, 968determining, 1555double sineGordon, 492double sinhGordon, 485Dym, 878elliptic, 1392Emden–Fowler, 248, 442, 444, 447, 462,

464, 647, 648, 655Enneper, 491Euler–Lagrange, 1595evolution, nonlinear in second derivative,

346fast diffusion, 211firstorder quasilinear, 1355first Painleve, 861, 989

Page: 1843


differential equation (continued)Fisher, 176, 1639, 1699, 1744FitzHugh–Nagumo, 179, 1746fKdV, 1034Fokas–Yortsos, 237fourthorder, Liouville type, 1030fourth Painleve, 988Fujita–Storm, 212Gardner, 871Gauss, 1772general fifthorder KdV, 1034generalized Burgers, 288generalized Burgers–Korteweg–de

Vries, 1045, 1503generalized Calogero, 748generalized Emden–Fowler, 327, 477, 511,

512, 513, 516, 527, 528, 697, 698generalized Kadomtsev–Petviashvili, 1029generalized Khokhlov–Zabolotskaya, 751generalized Korteweg–de Vries, 871generalized Kuramoto–Sivashinsky, 983generalized Landau–Ginzburg, 360generalized Liouville, 1108Gibbons–Tsarev, 764Goursat, 549Grad–Shafranov, 678, 691Hamilton–Jacobi, 1379Hamilton–Jacobi, Cauchy problem, 1380Harry Dym, 878Helmholtz, 380, 397, 399, 420, 423, 558,

570, 601, 938, 1289homogeneous Monge–Ampere, 775, 1430Hopf, 5Hunter–Saxton, 745hydrodynamic boundary layer, with

pressure gradient, 907hyperbolic, first canonical form, 1391hyperbolic, second canonical form, 1392hypergeometric, 1012integrating factor of Pfaffian, 1601inviscid Burgers, 1674, 1763inviscid Burgers (explicit method), 1678,

1725inviscid Burgers (initial value problem),

1762inviscid Burgers (shock waves and

rarefaction waves), 1757inviscid Burgers (shock waves and weak

solutions), 1759Ito, 1037Kadomtsev–Petviashvili, 1000, 1587Kadomtsev–Petviashvili, in canonical

form, 1000Kadomtsev–Petviashvili type, 1002Karman–Howarth, 253Kaup–Kupershmidt, 1036Kawahara, 1032KdV, 857Khokhlov–Zabolotskaya, 749KK, 1036Klein–Gordon, 1641, 1656, 1701, 1716

k(n,n), 891Kolmogorov–Petrovskii–Piskunov

(KPP), 279, 1633, 1694Korteweg–de Vries, 857, 1574, 1580,

1583, 1638, 1698Korteweg–de Vries–Burgers, 884Korteweg–de Vries, in canonical form, 857Korteweg–de Vries type, with exponential

nonlinearity, 874Korteweg–de Vries type, with logarithmic

nonlinearity, 874Korteweg–de Vries type, with powerlaw

nonlinearity, 872KPPtype, 1633, 1694Kuramoto–Sivashinsky, 982Laplace, 394, 409, 421, 556, 558, 579,

646, 938, 1392Lax, 1035, 1068Lax, with forcing term, 1037Lin–Reissner–Tsien, 756, 759linear determining, 1556, 1557linear heat, 807, 1252linear Schrodinger, 865, 1186linear wave, 1392Liouville, 144, 165, 470, 540, 1117, 1414Liouville type, nthorder, 1109mKdV, 867model, for dynamics of nonlinear

strings, 534model, of gas dynamics, 5, 25, 1360model, of nonlinear waves with damping,

6modification of coupled KdV, 1172modified Burgers, 191modified Burgers–Korteweg–de Vries, 886modified Harry Dym, 891modified k(n,n), 892modified Korteweg–de Vries, 862, 866,

869, 1583modified Korteweg–de Vries–Burgers, 886modified Korteweg–de Vries, in canonical

form, 867Monge–Ampere, 774, 1393ndimensional modified Schrodinger, with

cubic nonlinearity, 431ndimensional Schrodinger, with cubic

nonlinearity, 430Newell–Whitehead, 1534ninthorder KdVtype, 1041nonlinear diffusion, 1645, 1705nonlinear diffusion, with cubic source,

1577nonlinear firstorder partial differential,

1373nonlinear Klein–Gordon, 499, 1494nonlinear Kolmogorov type, 377, 398nonlinear Poisson, 1679, 1727, 1750nonlinear Poisson–Boltzmann, 1755nonlinear Schrodinger, 348, 425, 1582nonlinear wave, 1400, 1519, 1536, 1673,

1678, 1723, 1727, 1764

Page: 1844

INDEX 1845

differential equation (continued)nthorder Liouville type, 1109of axisymmetric steady hydrodynamic

boundary layer, 1508of Burgers hierarchy, 1067of complete Korteweg–de Vries hierarchy,

1069of helical surface, 8of Korteweg–de Vries hierarchy, 1068of light rays, 63of minimal surfaces, 768, 1597of modified Korteweg–de Vries hierarchy,

1069of steady transonic gas flow, 657of surface of bodies of revolution, 8of unsteady transonic gas flows, 756onesoliton solution, mKdV, 867onesoliton solution, potential KdV, 876Ostrovsky, 968palindromic, 1442Pfaffian, 1600Pfaffian, condition for integrability by

single relation, 1601Plebanski, first heavenly, 802Plebanski, second heavenly, 803Poisson, 379, 397, 423, 580, 588, 938,

1008porous medium, 210potential, of onedimensional flow of

compressible gas, 768potential Korteweg–de Vries (KdV), 876potential modified Korteweg–de Vries

(mKdV), 877Rayleigh–Zababakhin–Plesset, 1262Riccati, 140, 142, 162, 164, 423, 425, 453,

482, 520, 1012, 1277Sawada–Kotera, 1035Sawada–Kotera, with forcing term, 1038Schrodinger, of general form, 358, 360,

363, 364Schrodinger, with cubic nonlinearity, 348,

350, 351, 352, 355Schrodinger, with powerlaw nonlinearity,

351, 353Schwarzian Korteweg–de Vries, 870secondorder nonlinear partial differential,

1392secondorder quasilinear partial differential,

1393second Painleve, 861, 867, 988, 989seventhorder KdVtype, 1040Sharm–Tasso–Olver, 887, 1067sineGordon, 485, 490, 543, 1117, 1635,

1639, 1646, 1695, 1698, 1706sinhGordon, 485, 542, 1580SK, 1035spherical Korteweg–de Vries, 866stationary heat, with nonlinear source, 682stationary Khokhlov–Zabolotskaya, 649telegraph, 494, 538, 583, 585thin film, 985

thirdorder Liouville type, 972Thomas, 544threedimensional Khokhlov–Zabolotskaya,

752threedimensional nonlinear Schrodinger,

of general form, 429threedimensional Schrodinger, with cubic

nonlinearity, 428tractrix, 253Tricomi, 659twodimensional Khokhlov–Zabolotskaya,

749twodimensional linear heat, 1256twodimensional nonlinear Schrodinger, of

general form, 427twodimensional Schrodinger, with cubic

nonlinearity, 425twodimensional Schrodinger, with

powerlaw nonlinearity, 427Tzitzeica, 541unnormalized Boussinesq, 990unnormalized Burgers, 187unnormalized Kadomtsev–Petviashvili,

1002unnormalized Korteweg–de Vries, 865unnormalized modified Korteweg–de

Vries, 870variant of modified Burgers–Korteweg–de

Vries, 886vector Burgers, 417Weber, 1193Yermakov, 309, 518

differential equationsadmitting variational form, 1595associativity, 972axisymmetric steady laminar hydrodynamic

boundary layer, 909axisymmetric unsteady laminar boundary

layer, 929basic steps of Painleve test, 1570Bellman type, 805Benney moment, 764boundary layer (Prandtl equations), 1333elliptic, canonical form, 1392evolution, canonic Backlund transforma

tions, 1420Chaplygin, 1202Charpit, 1647, 1706chromatography, 1231coefficients contain exponential functions,

11coefficients contain hyperbolic functions,

14coefficients contain logarithmic functions,

16coefficients contain powerlaw functions, 3coefficients contain trigonometric

functions, 17cubic in derivatives, 134Euler, 938, 1197, 1681, 1729Euler, for barotropic gas flow, 1201

Page: 1845


differential equations (continued)Euler, in Gromeka–Lamb form, 1201Euler, in various coordinate systems, 1197evolution, nonlinear in second derivative,

346exact methods, 1353explicitly dependent on x and/or t, 266fifthorder, 1031firstorder with two independent variables

quadratic in derivatives, 43firstorder with three or more independent

variables, 125firstorder with two independent variables

of general form, 99firstorder quasilinear, 3for convective fluid motions, 1329fourthorder, 977heat and mass transfer, 697heat and mass transfer, in quiescent

or moving media with chemicalreactions, 390

heat and mass transfer, with complicatingfactors, 734

heat, with powerlaw or exponentialtemperaturedependent thermaldiffusivity, 409

hydrodynamic boundary layer, 903hydrodynamic type, order reduction, 1422hyperbolic, canonical form, 1390invariance under scaling transformations,

1431invariance under translations, 1429involving arbitrary differential operators,

1101involving arbitrary functions, 77, 279, 390,

499, 546, 682involving arbitrary functions of derivatives,

113involving arbitrary functions of four

variables, 123involving arbitrary functions of indepen

dent variables, 107involving arbitrary functions of one

variable, 19, 113involving arbitrary functions of three

variables, 120involving arbitrary functions of two

variables, 30, 116, 168involving arbitrary parameters, 43, 540involving arbitrary powers of derivatives,

102, 136involving cosine, 276involving cotangent, 278involving derivatives in radicands, 101involving exponential nonlinearities, 587involving first derivative in t, 811, 857,

977involving first derivative in t and linear in

highest derivative, 1031involving fourth powers of derivatives, 99involving hyperbolic cosine, 268

involving hyperbolic cotangent, 270involving hyperbolic sine, 269involving hyperbolic tangent, 270involving inverse trigonometric functions,

279involving mixed derivatives, 1000, 1103involving one arbitrary power of derivative,

107involving powerlaw functions of

derivatives, 156involving powerlaw nonlinearities, 583involving products of derivatives, 133involving products of derivatives with

respect to different variables, 131involving roots and moduli of derivatives,

136involving second derivative in t, 896, 987,

1088involving secondorder mixed derivatives,

949involving sine, 277involving squares of derivatives, 133involving squares of one or two deriva

tives, 125involving squares of three derivatives, 130involving tangent, 278involving thirdorder mixed derivatives,

958involving two or more second derivatives,

839involving two or three arbitrary powers of

derivatives, 111linear in mixed derivative, 745, 847mathematical physics, Painleve test, 1565mathematical physics, transformations,

1395Navier–Stokes, 1247Navier–Stokes, general properties, 1249Navier–Stokes, in Boussinesq approxima

tion, 1329Navier–Stokes, in various coordinate

systems, 1247nonlinear, involving arbitrary parameters,

99nonlinear diffusion boundary layer, 402nonlinear elliptic, geometrical models,

1754nonlinear elliptic, in two space dimensions,

1748nonlinear, in highest derivatives, 404nonlinear, in two independent variables,

1392nonlinear, involving arbitrary linear

differential operators, 1065nonlinear, of general form, 150, 169, 1392nonlinear, of thermal (diffusion) boundary

layer, 346nonlinear, parabolic, 1677, 1725nonlinear, secondorder, 1516nonlinear, simple separation of variables,

1459

Page: 1846

INDEX 1847

differential equations (continued)nonlinear telegraph, with two space

variables, 583nonlinear, with arbitrary number of

variables involving arbitrary functions,162

nonlinear, with arbitrary number ofvariables involving arbitrary parameters,158

nonlinear, with four independent variables,154

nonlinear, with three variables involvingarbitrary functions, 139

nonlinear, with three variables quadratic inderivatives, 125

nonstationary, 942nonstationary hydrodynamic (Navier–

Stokes equations), 1013nonstationary, of motion of viscous

incompressible fluid, 1013of atmospheric circulation in equatorial

region, 1225of breeze and monsoons, 1223of Burgers and Korteweg–de Vries

hierarchies, 1067of dynamic convection in the sea, 1227of flows in the baroclinic layer of the

ocean, 1229of general form, involving arbitrary

functions of single argument, 358of general form, involving arbitrary

functions of two arguments, 362of general form, involving first derivative

in t, 1070of heat and mass transfer in anisotropic

media, 412of higher orders, 1031of ideal incompressible fluid, 942of ideal plasticity, 1238of mass transfer in quiescent or moving

media with chemical reactions, 406of motion of ideal fluid (Euler equations),

938, 1197of nonlinear vibrations stratified medium,

1131of sixth to ninthorder, 1039of stationary transonic planeparallel gas

flow, 1122of viscous incompressible fluid, 1003ordinary differential, 1445ordinary differential, Painleve test, 1566Painleve, 1566parabolic, canonical form, 1390passing Painleve test, 1572Pfaffian, 1600Pfaffian, not satisfying integrability

condition, 1602Plebanski heavenly, 802quadratic in derivatives, 139, 154, 158, 162quadratic in highest derivatives, 772quasilinear, 1393, 1647, 1649, 1707, 1708

quasilinear, discontinuous solutions, 1360quasilinear, in conservative form, 1368quasilinear, of general form, 1368quasilinear, qualitative features, 1360quasilinear, with n independent variables,

1356reducible to Korteweg–de Vries equation,

875secondorder, classification, 1389secondorder, elliptic, with three or more

space variables, 713secondorder, elliptic, with two space

variables, 641secondorder, evolution, 1547secondorder, hyperbolic, 1551secondorder, hyperbolic, with one space

variable, 433secondorder, hyperbolic, with two or more

space variables, 553secondorder, involving mixed derivatives

and some other, 745secondorder, involving real functions of

real variables, 1165secondorder, nonlinear, for laser systems,

1170secondorder, of general form, 811, 1553secondorder, parabolic, with one space

variable, 175secondorder, parabolic, with two or more

space variables, 367semilinear, in two independent variables,

1389shallow water, 1124stationary, 938stationary hydrodynamic, (Navier–Stokes

equations), 1003steady boundary layer, for nonNewtonian

fluids, 911steady hydrodynamic boundary layer, for

Newtonian fluid, 903steady hydrodynamic boundary layer,

symmetries, 1528steadystate hydrodynamic boundary

layer, 903thirdorder, 857threeargument functional, 1497threedimensional, 1240threedimensional, involving arbitrary

functions, 730threedimensional, of ideal plasticity, 1240three and ndimensional, 428with arbitrary dependence on derivatives,

148with arbitrary number of independent

variables, 39with cubic nonlinearities involving

arbitrary functions, 355with exponential nonlinearities, 254, 469,

662with hyperbolic nonlinearities, 268, 485,

675

Page: 1847


differential equations (continued)with logarithmic nonlinearities, 271, 388,

486, 677with n independent variables, 739, 853with n space variables, 417with powerlaw nonlinearities, 175, 433,

641, 808with powerlaw nonlinearity in derivatives,

161, 167with power nonlinearity in derivatives, 146with quadratic nonlinearities, 805, 896with three independent variables, 35, 838with three or more space variables, 406with three or more space variables

involving arbitrary functions, 624with three space variables involving

arbitrary parameters, 604with three space variables involving

exponential nonlinearities, 722with three space variables involving

powerlaw nonlinearities, 713with trigonometric nonlinearities, 276, 389,

680with two independent variables, 1355with two independent variables involving

arbitrary functions, 19with two independent variables involving

arbitrary parameters, 3with two independent variables, nonlinear

in two or more highest derivatives, 849with two space variables involving

arbitrary functions, 589with two space variables involving

exponential nonlinearities, 384, 574with two space variables involving

powerlaw nonlinearities, 367, 553twodimensional, 425, 1238twodimensional Euler, for incompressible

ideal fluid (plane flows), 1198unsteady boundary layer, for Newtonian

fluid, 917unsteady boundary layer, for non

Newtonian fluids, 930von Karman (fourthorder elliptic

equations), 1192differential substitutions, 1409differentiation method, 1492direct and inverse scattering problems, 1587direct method of symmetry reductions and

differential constraints, 1562direct method of symmetry reductions of

nonlinear equations, 1503doubleprecision floating point, 1740double sineGordon equation, 492double sinhGordon equation, 485Dym equation, 878dynamic output, 1691dynamics of shallow water, 1610

Eeditor, MATLAB, 1737

Eigen–Schuster model, 1137Ekman flow, 1282elementary symmetric polynomial, 1444elementary theory of using invariants for

solving equations, 1439ellipsoidal vortex, 1278elliptic equation, 1392Emden–Fowler equation, 248, 442, 444, 447,

462, 464, 647, 648, 655Enneper equation, 491equation

adiabatic, 1262Airy, 861BBM, 958Bernoulli, 394, 421Born–Infeld, 766, 1401boundary layer, for nonNewtonian



1673, 1677, 1697, 1723, 1725, 1743,1761

Burgers–Huxley, 189Burgers–Korteweg–de Vries, 884Calogero, 745, 746Calogerotype, 1400Camassa–Holm, 968Chaplygin, 765characteristic, 1389, 1393, 1647, 1706Clairaut, 118, 152, 172, 1375Clairaut, in implicit form, 122combined KdVmKdV, 871complete Lin–Reissner–Tsien, 757continuity, 1197, 1247, 1610cylindric Burgers, 191cylindrical Korteweg–de Vries, 866Degasperis–Procesi, 968determining, 1555double sineGordon, 492double sinhGordon, 485Dym, 878elliptic, 1392Emden–Fowler, 248, 442, 444, 447, 462,

464, 647, 648, 655Enneper, 491Euler–Lagrange, 1595evolution, nonlinear in second derivative,

346fast diffusion, 211firstorder quasilinear, 1355first Painleve, 861, 989Fisher, 176, 1639, 1699, 1744FitzHugh–Nagumo, 179, 1746fKdV, 1034Fokas–Yortsos, 237fourthorder, Liouvilletype, 1030

Page: 1848

INDEX 1849

equation (continued)fourth Painleve, 988Fujita–Storm, 212functional, 1498, 1500, 1767, 1769functional, describing travelingwave

solutions, 1431Gardner, 871Gauss, 1772Gelfand–Levitan–Marchenko, integral, 863,

868, 990, 1002Gelfand–Levitan–Marchenko type,

integral, 1584general fifthorder KdV, 1034generalized Burgers, 288generalized Burgers–Korteweg–de

Vries, 1045, 1503generalized Calogero, 748generalized Emden–Fowler, 327, 477, 511,

512, 513, 516, 527, 528, 697, 698generalized Kadomtsev–Petviashvili, 1029generalized Khokhlov–Zabolotskaya, 751generalized Korteweg–de Vries, 871generalized Kuramoto–Sivashinsky, 983generalized Landau–Ginzburg, 360generalized Liouville, 1108generalized reciprocal polynomial, 1443Gibbons–Tsarev, 764Goursat, 549Grad–Shafranov, 678, 691Hamilton–Jacobi, 1379Hamilton–Jacobi, Cauchy problem, 1380Harry Dym, 878Helmholtz, 380, 397, 399, 420, 423, 558,


pressure gradient, 907hyperbolic, first canonical form, 1391hyperbolic, second canonical form, 1392hypergeometric, 1012inviscid Burgers, 1674, 1763inviscid Burgers, explicit method, 1678,

1725inviscid Burgers, initial value problem,

1762inviscid Burgers, shock waves and

rarefaction waves, 1757inviscid Burgers, shock waves and weak

solutions, 1759Ito, 1037Kadomtsev–Petviashvili, 1000, 1587Kadomtsev–Petviashvili, in canonical

form, 1000Kadomtsev–Petviashvili type, 1002Karman–Howarth, 253Kaup–Kupershmidt, 1036Kawahara, 1032KdV, 857Khokhlov–Zabolotskaya, 749

KK, 1036Klein–Gordon, 1641, 1656, 1701, 1716k(n,n), 891Kolmogorov–Petrovskii–Piskunov


1583, 1638, 1698Korteweg–de Vries, in canonical form, 857Korteweg–de Vries–Burgers, 884Korteweg–de Vries type, with exponential




646, 938, 1392Lax, 1035, 1068Lax, with forcing term, 1037Lin–Reissner–Tsien, 756, 759linear determining, 1556, 1557linear heat, 807, 1252linear Schrodinger, 865, 1186linear wave, 1392Liouville, 144, 165, 470, 540, 1117, 1414mKdV, 867model, for dynamics of nonlinear


6modification of coupled KdV, 1172modified k(n,n), 892modified Burgers, 191modified Burgers–Korteweg–de Vries, 886modified Harry Dym, 891modified Korteweg–de Vries, 862, 866,

869, 1583modified Korteweg–de Vries, in canonical

form, 867modified Korteweg–de Vries–Burgers, 886Monge–Ampere, 774, 1393ndimensional modified Schrodinger, with


nonlinearity, 430Newell–Whitehead, 1534ninthorder KdVtype, 1041nonlinear diffusion, 1645, 1705nonlinear diffusion, with cubic source,

1577nonlinear firstorder partial differential,

1373nonlinear Klein–Gordon, 499, 1494nonlinear Kolmogorov type, 377, 398nonlinear Poisson, 1679, 1727, 1750nonlinear Poisson–Boltzmann, 1755nonlinear Schrodinger, 348, 425, 1582

Page: 1849


equation (continued)nonlinear wave, 1400, 1519, 1536, 1673,

1678, 1723, 1727, 1764nthorder Liouville type, 1109of axisymmetric steady hydrodynamic



1069of steady transonic gas flow, 657of surface of bodies of revolution, 8of unsteady transonic gas flows, 756onesoliton solution, mKdV, 867onesoliton solution, potential KdV, 876Ostrovsky, 968palindromic, 1442Pexider, 1771Pfaffian, 1600Pfaffian, condition for integrability by

single relation, 1601Pfaffian, integrating factor, 1601Plebanski first heavenly, 802Plebanski second heavenly, 803Poisson, 379, 397, 423, 580, 588, 938,



(mKdV), 877Rayleigh–Zababakhin–Plesset, 1262reciprocal, 1442Riccati, 140, 142, 162, 164, 423, 425, 453,

482, 520, 1012, 1277Sawada–Kotera, 1035Sawada–Kotera, with forcing term, 1038Schrodinger, of general form, 358, 360,



351, 353Schwarzian Korteweg–de Vries, 870secondorder nonlinear partial differential,

1392secondorder quasilinear partial differential,

1393second Painleve, 861, 867, 988, 989seventhorder KdVtype, 1040Sharm–Tasso–Olver, 887, 1067sineGordon, 485, 490, 543, 1117, 1635,

1639, 1646, 1695, 1698, 1706sinhGordon, 485, 542, 1580SK, 1035

spherical Korteweg–de Vries, 866stationary heat, with nonlinear source, 682stationary Khokhlov–Zabolotskaya, 649telegraph, 494, 538, 583, 585thin film, 985thirdorder Liouville type, 972Thomas, 544threedimensional, Khokhlov–Zabolotskaya,

752threedimensional, Schrodinger, nonlinear,

of general form, 429threedimensional, Schrodinger, with cubic

nonlinearity, 428tractrix, 253Tricomi, 659twodimensional, heat, linear, 1256twodimensional, Khokhlov–Zabolotskaya,

749twodimensional, Schrodinger, nonlinear, of

general form, 427twodimensional, Schrodinger, with cubic

nonlinearity, 425twodimensional, Schrodinger, with




Vries, 886vector Burgers, 417Weber, 1193Yermakov, 309, 518

equationsadmitting variational form, 1595algebraic, 1441algebraic, with even powers, 1441associativity, 972axisymmetric steady laminar hydrodynamic


layer, 929basic steps of Painleve test, 1570Bellman type, 805Benney moment, 764bilinear functional, 1775boundary layer (Prandtl equations), 1333elliptic, canonical form, 1392evolution, canonic Backlund transforma

tions, 1420Chaplygin, 1202Charpit, 1647, 1706chromatography, 1231cubic in derivatives, 134Euler, 938, 1197, 1681, 1729Euler, for barotropic gas flow, 1201Euler, in Gromeka–Lamb form, 1201Euler, in various coordinate systems, 1197

Page: 1850

INDEX 1851

equations (continued)evolution, nonlinear in second derivative,

346exact methods, 1353explicitly dependent on x and/or t, 266fifthorder, 1031firstorder, quasilinear, 3firstorder, with three or more independent

variables, 125firstorder, with two independent variables

of general form, 99firstorder, with two independent variables

quadratic in derivatives, 43for convective fluid motions, 1329fourthorder, 977functionaldifferential, reducible to bilinear

equation, 1776Gelfand–Levitan–Marchenko type, 1584heat, with powerlaw or exponential

temperaturedependent thermaldiffusivity, 409

heat and mass transfer, 697heat and mass transfer, in quiescent















highest derivative, 1031involving fourth powers of derivatives, 99

involving hyperbolic cosine, 268involving hyperbolic cotangent, 270involving hyperbolic sine, 269involving hyperbolic tangent, 270involving inverse trigonometric functions,



derivatives, 156involving powerlaw nonlinearities, 583involving products of derivatives, 133involving products of derivatives with











systems, 1247nonlinear, diffusion boundary layer, 402nonlinear, elliptic, geometrical models,

1754nonlinear, elliptic, in two space dimen

sions, 1748nonlinear, functional, containing complex

argument, 1776nonlinear, functional, reducible to

bilinear, 1775nonlinear, functional, solutions, 1496nonlinear, in highest derivatives, 404nonlinear, in two independent variables,


differential operators, 1065nonlinear, involving arbitrary parameters,

99nonlinear, of general form, 150, 169, 1392

Page: 1851


equations (continued)nonlinear, of thermal (diffusion) boundary


1459nonlinear, telegraph, with two space


variables containing arbitrary functions,162

nonlinear, with arbitrary number ofvariables containing arbitrary parameters,158


nonlinear, with three variables containingarbitrary functions, 139

nonlinear, with three variables quadratic inderivatives, 125

nonstationary, 942nonstationary, hydrodynamic (Navier–











938, 1197of nonlinear vibrations, stratified medium,


flow, 1122of viscous incompressible fluid, 1003ordinary differential, 1445ordinary differential, Painleve test, 1566Painleve, 1566parabolic, canonical form, 1390passing Painleve test, 1572Pfaffian, 1600

Pfaffian, not satisfying integrabilitycondition, 1602

Plebanski heavenly, 802pseudodifferential, 1483, 1484quadratic in derivatives, 139, 154, 158, 162quadratic in highest derivatives, 772quasilinear, 1393, 1647, 1649, 1707, 1708quasilinear, discontinuous solutions, 1360quasilinear, in conservative form, 1368quasilinear, of general form, 1368quasilinear, qualitative features, 1360quasilinear, with n independent variables,












1389shallow water, 1124stationary, 938stationary, hydrodynamic (Navier–Stokes


fluids, 911steady hydrodynamic boundary layer, 903steady hydrodynamic boundary layer, for


symmetries, 1528thirdorder, 857threeargument, functional, 1497threedimensional, 1240threedimensional, involving arbitrary

functions, 730threedimensional, of ideal plasticity, 1240three and ndimensional, 428twodimensional, 425, 1238twodimensional, Euler, for incompressible

ideal fluid (plane flows), 1198

Page: 1852

INDEX 1853

equations (continued)unsteady boundary layer, for Newtonian



equations), 1192with arbitrary dependence on derivatives,


variables, 39with coefficients involving exponential

functions, 11with coefficients involving hyperbolic

functions, 14with coefficients involving logarithmic

functions, 16with coefficients involving powerlaw

functions, 3with coefficients involving trigonometric

functions, 17with cubic nonlinearities involving



675with logarithmic nonlinearities, 271, 388,



161, 167with power nonlinearity in derivatives, 146with quadratic nonlinearities, 805, 896with three independent variables, 35, 838with three or more space variables, 406with three or more space variables,

involving arbitrary functions, 624with three space variables, involving

arbitrary parameters, 604with three space variables, involving

exponential nonlinearities, 722with three space variables, involving


680with two independent variables, 1355with two independent variables, involving

arbitrary functions, 19with two independent variables, involving


in two or more highest derivatives, 849with two space variables, involving

arbitrary functions, 589with two space variables, involving

exponential nonlinearities, 384, 574with two space variables, involving

powerlaw nonlinearities, 367, 553error function, 187errors in constructing solutions by symbolic

computations, 1668Euler equations, 938, 1197, 1681, 1729

for barotropic gas flow, 1201in Gromeka–Lamb form, 1201in various coordinate systems, 1197

Euler–Lagrange equation, 1595Euler transformation, 122, 346, 769, 957,

1403, 1407Eulerian coordinates, 1680, 1728evaluation of function, 1628, 1740evolution equations nonlinear in second

derivative, 346evolutionary conditions, 1616exact methods for nonlinear partial differen

tial equations, 1353exact nonlinear problem, 1681, 1729exact solutions of nonlinear partial differen

tial equations, 1exact solutions of various types of nonlinear

PDEs, 1634exact solutions with simple separation of

variables, 1459examples of constructing conservation laws

using Noetherian symmetries, 1596examples of constructing exact generalized

separable solutions, 1467examples of constructing exact solutions,

1503, 1534examples of constructing functional separable

solutions, 1492examples of constructing invariant solutions

to nonlinear equations, 1522examples of constructing invariant solutions

to nonlinear partial differentialequations, 1451

examples of finding exact solutions ofsecond and thirdorder equations, 1465

examples of finding symmetries of nonlinearequations, 1516

examples of Lax pairs for nonlinearequations of mathematical physics,1580

examples of selfsimilar solutions tomathematical physics equations, 1432

examples of solutions having movablesingularities, 1565

examples of solutions of some specificfunctional equations, 1770, 1771

examples of solutions of specific functionalequations, 1773

examples of solving Cauchy problem, 1381examples of viscosity (nonsmooth) solutions,

1385existence and uniqueness theorem, 1357,

1359, 1379, 1380expfunction method, 1643, 1703explicit central difference method, 1678,

1727, 1764

Page: 1853


explicit finite difference scheme, 1761, 1763explicit method, 1677, 1678, 1725, 1727exponential isotherm, 1235export facilities, 1751external flow around cylinder, 1253

FFalkner–Skan problem, 907, 1334false, 1739fast diffusion, 251fast diffusion equation, 211fifthorder equations, 1031file name, 1740finding linear subspaces invariant under given

nonlinear operator, 1480finite element method, 1748first hodograph transformation, 1399firstorder chemical reaction, 1135, 1137firstorder differential constraints, 1539

for PDEs, 1547firstorder equations

classification results, 1565describing convective mass transfer with

volume reaction, 1602general solution, 1355hydrodynamic type, involving three or

more equations, 1197in one unknown, overdetermined system,

1599quasilinear, 3, 1355nonlinear, 1373, 1659, 1719, 1720nonlinear, methods for solving, 1373nonlinear, with three or more independent

variables, 125nonlinear, with two independent variables

of general form, 99quasilinear, hyperbolic, 1607quasilinear, methods for solving, 1355solution, general, 1355with three or more independent variables,

125with two independent variables of general

form, 99with two independent variables quadratic

in derivatives, 43firstorder hydrodynamic and other systems

involving three or more equations, 1197firstorder hyperbolic systems of quasilinear

equations, 1607firstorder nonlinear equations with three or

more independent variables, 125firstorder nonlinear equations with two

independent variables of generalform, 99

firstorder quasilinear equations, 3firstorder quasilinear partial differential

equation, 1355first Painleve equation, 861, 989first Stokes problem, 1251Fisher equation, 176, 1639, 1699, 1744

FitzHugh–Nagumo equation, 179, 1746fixed singularities of solutions, 1565fKdV equation, 1034flow

adiabatic nonisentropic gas, 1204around cylinder, external, 1253barotropic gas, Euler equations, 1201Couette–Poiseuille plane, 1251Ekman, 1282Hagen–Poiseuille, 1253Homann, 1302in confuser, 1261in diffuser, 1261in rotating layer, 1283irrotational gas, 1201Jeffery–Hammel, 1260Karman, 1290onedimensional, compressible gas, 768onedimensional, polytropic ideal gas,

1125potential, 1334potential gas, 1201quasiplane, 1285stationary transonic planeparallel

gas, 1122steady transonic gas, 657von Karman, 1290

flowsaxial, 1303axisymmetric, 1274conical vortex, 1290cylindrical vortex, 1290generalized Couette–Poiseuille plane, 1329in baroclinic layer of ocean, 1229onedimensional barotropic, ideal

compressible gas, 1127onedimensional rotation fluid, 1257plane, twodimensional Euler equations for

incompressible ideal fluid, 1198plane, twodimensional solutions in

cylindrical coordinates, 1270plane, twodimensional solutions in

rectangular Cartesian coordinates, 1263plane, unidirectional, 1251quasiplane, 1285threedimensional stagnationpoint

type, 1302unidirectional, in tubes of various cross

sections, 1253unidirectional, plane, 1251unsteady transonic gas, 756von Karmantype rotationally symmetric,

1316with twononzero velocity components,

1282fluid streamlines, 1253, 1257Fokas–Yortsos equation, 237forward difference method, 1677, 1725,

1761forward time backward space method, 1676fourthorder equations, 977fourthorder Liouville type equation, 1030

Page: 1854

INDEX 1855

fourth Painleve equation, 988front end, 1688Fuchs index, 1570Fuchs indices, 1567, 1568, 1570Fujita–Storm equation, 212function

error, 187handle, 1741Jacobi elliptic, 990mathematical library, 1736mfiles, 1737modified Bessel, 1260name of, 1740

functionaldifferential equations reducible tobilinear equation, 1776

functional equation, 1498, 1500, 1767, 1769describing travelingwave solutions, 1431Pexider, 1771

functional equationsbilinear, 1775nonlinear, containing complex argument,

1776nonlinear, reducible to bilinear, 1775nonlinear, solutions, 1496

functional separable solutions, 1487, 1652,1711

functional separation of variables, 1655,1715

functional separation of variables bydifferentiation, 1656, 1716

functionsactive functions, 1628anonymous, 1741Bessel, 1160, 1255, 1259builtin, 1740inert, 1628Kelvin, 1256nested, 1741predefined, 1628, 1740pure, 1690special, 1740written in mfiles, 1740

GGalileo transformation, 1514Gardner equation, 871gas dynamic type systems linearizable with

hodograph transformation, 1122Gauss equation, 1772Gelfand–Levitan–Marchenko integral

equation, 863, 868, 990, 1002Gelfand–Levitan–Marchenko type integral

equations, 1584general consistency method for two

equations, 1542general description of method of differential

constraints, 1546general fifthorder KdV equation, 1034general form

equations involving first derivativein t, 1070

for symmetry reduction, 1506of conservation laws, 1593of contact transformations for partial

differential equations, 1405of functional differential equations, 1464of point transformations, 1395of selfsimilar solutions, 1431of solutions, 1464of travelingwave solutions, 1429

general integral, 1373, 1374general properties of Navier–Stokes

equations, 1249general scheme

for analysis of nonlinear partial differentialequations, 1570

for constructing generalized travelingwavesolutions, 1489

of using invariants for solving mathematical equations, 1439, 1440

general solutionof characteristic system, 1355of firstorder PDE, 1355

generalizations to pseudodifferentialequations, 1483

generalized Blasius problem, 912generalized Burgers equation, 288generalized Burgers–Korteweg–de Vries

equation, 1045, 1503generalized Calogero equation, 748generalized Cauchy problem, 1649, 1650,

1708, 1709generalized classical method of characteris

tics, 1385generalized Couette–Poiseuille plane

flows, 1329generalized Emden–Fowler equation, 327,

477, 511, 512, 513, 516, 527, 528, 697,698

generalized Falkner–Skan problem, 914generalized Kadomtsev–Petviashvili equation,

1029generalized Khokhlov–Zabolotskaya

equation, 751generalized Korteweg–de Vries equation,

871generalized Kuramoto–Sivashinsky equation,

983generalized Landau–Ginzburg equation, 360generalized Langmuir isotherm, 1233generalized Liouville equation, 1108generalized method of characteristics, 1647,

1650, 1709generalized porous medium equation with

nonlinear source, 340generalized reciprocal polynomial equation,

1443generalized Schlichting problem, 912generalized selfsimilar solutions, 1436generalized separable solutions, 1464, 1620,

1652, 1654, 1711, 1714

Page: 1855


generalized separable solutions (continued)by differentiation, nthorder nonlinear

PDE, 1652, 1711by splitting, 1653, 1712

generalized solution, 1364generalized travelingwave solutions, 1487generalized viscosity solutions and their

applications, 1382generalized/functional separation of variables

vs. differential constraints, 1561generalizing exact solutions, 1665generic system, 1754geometry description matrix, 1755Gibbons–Tsarev equation, 764global variables, 1739Goursat equation, 549Grad–Shafranov equation, 678, 691graphical user interface, 1736, 1748graphical user interface development

environment, 1738graphics system, 1736group generator, 1514, 1526group invariants, 1526

HHagen–Poiseuille flow, 1253Hamilton–Jacobi equation, 1379Hamiltonian, 1379Harry Dym equation, 878heat and mass transfer equations, 697, 730

in quiescent or moving media withchemical reactions, 390

with complicating factors, 734heat equations with powerlaw or exponential


HeleShaw cell, 984Helmholtz equation, 380, 397, 399, 420, 423,

558, 570, 601, 938, 1289Henry coefficients, 1231Hiemenz problem, 1264higherorder differential constraints, 1555Hill vortex, 1278hodograph transformation, 35, 212, 344,

1129, 1397Homann flow, 1302homogeneous Monge–Ampere equation, 775,

1430Hopf–Cole transformation, 187, 888, 1067,

1409, 1414Hopf equation, 5Hopf formula for generalized solution, 1366Hunter–Saxton equation, 745hydrodynamictype system of diagonal

form, 1236hydrodynamictype system of nondiagonal

form, 1236hydrodynamic boundary layer equation with

pressure gradient, 907hydrodynamic boundary layer equations, 903

hyperbolic equationfirst canonical form, 1391second canonical form, 1392

hyperbolic n × n systems of conservationlaws, 1614

hyperbolic sineGordon equation, 485hyperbolic system, 1607, 1608, 1615hypergeometric equation, 1012

Iideal (inviscid) fluid, 938, 1197ideal plasticity with von Mises yield

criterion, 1238implicit form of solution, 1648, 1708incorrect response, 1626, 1688, 1737inert functions, 1628infinitesimal generator, 1514infinitesimal operator, 1513, 1514, 1526initial value problem, 1383, 1611integral equation

Gelfand–Levitan–Marchenko, 863, 868,990, 1002

Gelfand–Levitan–Marchenko type, 1584integrals of motion, 1593integrating factor of Pfaffian equation, 1601introducing and removing additional arbitrary

constants, 1666invariance condition, 1516, 1527, 1595invariance of equations under scaling

transformations, 1431invariance of equations under translations,

1429invariance of solutions and equations under

translation transformations, 1430invariant, 1439, 1445, 1450, 1526

determination procedure, 1446of group, 1526of infinitesimal operator, 1514of operator, 1514, 1526of transformation, 1440

invariant condition, 1527invariant solution, 1431, 1435, 1451, 1513,

1520, 1527, 1602, 1620invariant surface condition, 1533inverse problems, 1455inverse scattering method, 990, 1002inverse scattering problem, 1587inviscid Burgers equation, 1674, 1763

explicit method, 1678, 1725initial value problem, 1762shock waves and rarefaction waves, 1757shock waves and weak solutions, 1759

irrotational gas flow, 1201isotherms of adsorption, 1231Ito equation, 1037

JJacobi elliptic function, 990Jacobi–Mayer bracket, 1377Jacobian elliptic cosine, 858

Page: 1856

INDEX 1857

Jeffery–Hamel flow, 1260Jeffery–Hamel problem, 1261Jeffery–Hamel solution, 1272jetsource problem, 1335jump condition, 1362, 1363, 1369

KKarman–Howarth equation, 253Kadomtsev–Petviashvili equation, 1587

in canonical form, 1000related equations, 1000

Kadomtsev–Petviashvili type equation, 1002Kaup–Kupershmidt equation, 1036Kawahara equation, 1032KdV equation, 857Kelvin functions, 1256Khokhlov–Zabolotskaya and related

equations, 749kinematic momentum, 1335kinematic viscosity, 1247kink, 1636, 1695KK equation, 1036Klein–Gordon equation, 1641, 1656, 1701,

1716k(n,n) equation, 891Kolmogorov–Petrovskii–Piskunov (KPP)

equation, 279, 1633, 1694Kolmogorov model, 1006Korteweg–de Vries–Burgers equation, 884Korteweg–de Vries equation, 857, 1574,

1580, 1583, 1638, 1698in canonical form, 857

Korteweg–de Vries type equationwith exponential nonlinearity, 874with logarithmic nonlinearity, 874with powerlaw nonlinearity, 872

Kovalevskaya–Painleve test, 1569KPPtype equation, 1633, 1694Kuramoto–Sivashinsky equation, 982

LLagrange–Charpit method, 1376Lagrangian, 1595Lagrangian coordinates, 1680, 1728laminar fluid flows, 1247Landau problem, 1012Landau solution, 1277Langmuir isotherm, 1231Laplace equation, 394, 409, 421, 556, 558,

579, 646, 938, 1392law of conservation of mass, 1610Lax condition, 1616, 1617Lax equation, 1035, 1068

with forcing term, 1037Lax method, 1678, 1726, 1763Lax pair, 1579leading terms of equation, 1567Legendre transformation, 122, 767, 768, 770,

1403, 1406with many variables, 1408

library functions, 1628, 1740Lie group, 1526limiting selfsimilar solution, 1454, 1456Lin–Reissner–Tsien equation, 756, 759linear determining equation, 1556, 1557linear heat equation, 807, 1252linear Schrodinger equation, 865, 1186linear subspaces invariant under nonlinear

operator, 1477linear transformations, 1396linear wave equation, 1392linearization of systems of gas dynamic type

by hodograph transformation, 1610Liouville equation, 144, 165, 470, 540, 1117,

1414lists, 1629, 1691local structure of generalized viscosity

solutions, 1383local transformations of derivatives, 1526local variables, 1739logical expression, 1739logical operators, 1628, 1689, 1738Loitsianskii decay law, 253Lorentz transformation, 1514Lotka–Volterra system, 1135

Mmfile, 1737Mach angle, 1203Mach number, 757main program, 1737Maple language, 1627Maple worksheets, 1626Maplet applications, 1627Maplet user interface, 1627Martin transformation, 775Mathematica language, 1689Mathematica objects, 1691mathematical constants, 1690mathematical function library, 1736Mathlink, 1688MATLAB language, 1738matrix, 1629, 1691, 1741mesh, 1677, 1725mesh points, 1677, 1725method

based on compatibility condition forsystems of linear equations, 1581

based on linear integral equations, 1584based on using Lax pairs, 1579classical, of separation of variables, 1651,

1711classical, of symmetry reductions, 1513Clarkson–Kruskal direct, 1503differentiation, 1492direct, of symmetry reductions and

differential constraints, 1562direct, of symmetry reductions of nonlinear

equations, 1503expfunction, 1643, 1703

Page: 1857


method (continued)explicit, 1677, 1678, 1725, 1727explicit central difference, 1678, 1727,

1764finite element, 1748for constructing stable generalized

solutions, 1370forward difference, 1677, 1725, 1761forward time backward space, 1676general consistency for two equations,

1542inverse scattering, 990, 1002inverse scattering, Cauchy problem, 1587inverse scattering, Cauchy problem for

nonlinear equations, 1589Lagrange–Charpit, 1376Lax, 1678, 1726, 1763of argument elimination by test functions,

1772of characteristics, 32, 35, 1355–1357,

1647, 1649, 1707, 1708, 1756of characteristics, generalized, 1385, 1647,

1650, 1709of conversion to equivalent system of

equations, 1398of differential constraints, 1539of differential constraints, general

description, 1546of differential constraints for ordinary

differential equations, 1539of differentiation in independent variables,

1771of differentiation in parameter, 1769of functional separation of variables, 1487of generalized separation of variables,

1459of invariants, 1439of lines, 1722, 1743of separation of variables, 1374of symmetry reductions, nonclassical,

1533, 1562of vanishing viscosity, 1366, 1383similarity, 1431sinecosine, 1642, 1702specifying singlestage numerical, 1674,

1676specifying twostage numerical, startup

method, 1676spectral collocation, 1680, 1729splitting, 1496tanhcoth, 1641, 1701Titov–Galaktionov, 1477

methodsexact, for nonlinear partial differential

equations, 1353for constructing travelingwave solutions,

1641, 1700for solving firstorder nonlinear equations,

1373for solving firstorder quasilinear equa

tions, 1355

numerical and graphical, 1672numerical, embedded in Maple, 1674of inverse scattering problem (soliton

theory), 1579Miura transformation, 862, 868, 1410mKdV equation, 867model equation for dynamics of nonlinear

strings, 534model equation of gas dynamics, 5, 25,

1360model equation of nonlinear waves with

damping, 6modification of coupled KdV equation, 1172modified k(n,n) equation, 892modified Bessel function, 1260modified Burgers equation, 191modified Burgers–Korteweg–de Vries

equation, 886modified Harry Dym equation, 891modified Korteweg–de Vries–Burgers

equation, 886modified Korteweg–de Vries equation, 862,

866, 869, 1583in canonical form, 867

module, 1629, 1691module system, 1741Monge–Ampere equation, 774, 1393Monge cone, 1647, 1706movable singularities of solutions, 1565

of ordinary differential equations, 1565mth prolongation of group generator, 1527multipeakon solutions, 968multiple statements in line, 1739multiplicative separable solution, 1374, 1459,

1651, 1710

Nndimensional modified Schrodinger equation

with cubic nonlinearity, 431ndimensional Schrodinger equation with

cubic nonlinearity, 430nsoliton solution, 859, 1591

of Korteweg–de Vries equation, 1591Nadai multiplicative separable solution, 1238Nadai solution, 1238name of symbol, 1689, 1690Navier–Stokes equations, 1247

general properties, 1249in Boussinesq approximation, 1329nonstationary, 1013stationary hydrodynamic, 1003

nested functions, 1741nested lists, 1691Newell–Whitehead equation, 1534Newtonian fluid, 1247Nijenhuis tensor, 1237ninthorder KdVtype equation, 1041noslip condition, 1253Noetherian symmetry, 1595

Page: 1858

INDEX 1859

nonNewtonian fluid, 911, 932, 933boundary layer equations, 915of general form, 915

nonclassical hodograph transformation, 345,1399

nonclassical method of symmetry reductions,1533

nonclassical method of symmetry reductionsand differential constraints, 1562

nonclassical selfsimilar solution, 500, 685,689, 1435

nonclassical symmetries, 1533nonclassical travelingwave solution, 679,

690, 706noninvariant selfsimilar solution, 1435nonlinear diffusion boundary layer equations,

402nonlinear diffusion equation, 1645, 1705

with cubic source, 1577nonlinear dispersive nondissipative waves,

857nonlinear elliptic PDEs (geometrical

models), 1754nonlinear elliptic PDEs in two space

dimensions, 1748nonlinear equation

adiabatic, 1262BBM, 958Bernoulli, 394, 421Born–Infeld, 766, 1401boundary layer, for nonNewtonian



1673, 1677, 1697, 1723, 1725, 1743,1761

Burgers–Huxley, 189Burgers–Korteweg–de Vries, 884Calogero, 745, 746Calogerotype, 1400Camassa–Holm, 968Chaplygin, 765characteristic, 1389, 1393, 1647, 1706Clairaut, 118, 152, 172, 1375Clairaut, in implicit form, 122combined KdVmKdV, 871complete Lin–Reissner–Tsien, 757cylindric Burgers, 191cylindrical Korteweg–de Vries, 866Degasperis–Procesi, 968diffusion, 1645, 1705diffusion, with cubic source, 1577double sineGordon, 492double sinhGordon, 485Dym, 878elliptic, 1392

Emden–Fowler, 248, 442, 444, 447, 462,464, 647, 648, 655

Enneper, 491Euler–Lagrange, 1595evolution, nonlinear in second derivative,

346fast diffusion, 211firstorder, 1355, 1373first Painleve, 861, 989Fisher, 176, 1639, 1699, 1744FitzHugh–Nagumo, 179, 1746fKdV, 1034Fokas–Yortsos, 237fourthorder, Liouville type, 1030fourth Painleve, 988Fujita–Storm, 212Gardner, 871Gauss, 1772general fifthorder KdV, 1034generalized Burgers–Korteweg–de

Vries, 1045, 1503generalized Burgers, 288generalized Calogero, 748generalized Emden–Fowler, 327, 477, 511,

512, 513, 516, 527, 528, 697, 698generalized Kadomtsev–Petviashvili, 1029generalized Khokhlov–Zabolotskaya, 751generalized Korteweg–de Vries, 871generalized Kuramoto–Sivashinsky, 983generalized Landau–Ginzburg, 360generalized Liouville, 1108Gibbons–Tsarev, 764Goursat, 549Grad–Shafranov, 678, 691Hamilton–Jacobi, 1379Hamilton–Jacobi, Cauchy problem, 1380Harry Dym, 878homogeneous Monge–Ampere, 775, 1430Hopf, 5Hunter–Saxton, 745hydrodynamic boundary layer, with

pressure gradient, 907hyperbolic, first canonical form, 1391hyperbolic, second canonical form, 1392hypergeometric, 1012inviscid Burgers, 1674, 1763inviscid Burgers, explicit method, 1678,

1725inviscid Burgers, initial value problem,

1762inviscid Burgers, shock waves and

rarefaction waves, 1757inviscid Burgers, shock waves and weak

solutions, 1759Ito, 1037k(n,n), 891Kadomtsev–Petviashvili, 1000, 1587Kadomtsev–Petviashvili, in canonical

form, 1000Kadomtsev–Petviashvili type, 1002Karman–Howarth, 253Kaup–Kupershmidt, 1036

Page: 1859


nonlinear equation (continued)Kawahara, 1032KdV, 857Khokhlov–Zabolotskaya, 749KK, 1036Klein–Gordon, 499, 1494, 1641, 1656,

1701, 1716Kolmogorov type, 377, 398Kolmogorov–Petrovskii–Piskunov





nonlinearity, 872KPPtype, 1633, 1694Kuramoto–Sivashinsky, 982Lax, 1035, 1068Lax, with forcing term, 1037Lin–Reissner–Tsien, 756, 759Liouville, 144, 165, 470, 540, 1117, 1414mKdV, 867model, for dynamics of nonlinear


6modification of coupled KdV, 1172modified Burgers, 191modified Burgers–Korteweg–de Vries, 886modified Harry Dym, 891modified k(n,n), 892modified Korteweg–de Vries, 862, 866,


form, 867Monge–Ampere, 774, 1393of axisymmetric steady hydrodynamic



1069of steady transonic gas flow, 657of surface of bodies of revolution, 8of unsteady transonic gas flows, 756onesoliton solution, mKdV, 867onesoliton solution, potential KdV, 876Ostrovsky, 968ndimensional modified Schrodinger, with


nonlinearity, 430Newell–Whitehead, 1534ninthorder KdVtype, 1041nthorder Liouville type, 1109palindromic, 1442Pfaffian, 1600Pfaffian, condition for integrability by

single relation, 1601Plebanski first heavenly, 802Plebanski second heavenly, 803Poisson, 1679, 1727, 1750Poisson–Boltzmann, 1755porous medium, 210potential, of onedimensional flow of


(mKdV), 877Rayleigh–Zababakhin–Plesset, 1262Riccati, 140, 142, 162, 164, 423, 425, 453,

482, 520, 1012, 1277Sawada–Kotera, 1035Sawada–Kotera, with forcing term, 1038Schrodinger, 348, 425, 1582Schrodinger, of general form, 358, 360,



351, 353Schwarzian Korteweg–de Vries, 870second Painleve, 861, 867, 988, 989seventhorder KdVtype, 1040Sharm–Tasso–Olver, 887, 1067sineGordon, 485, 490, 543, 1117, 1635,

1639, 1646, 1695, 1698, 1706sinhGordon, 485, 542, 1580SK, 1035spherical Korteweg–de Vries, 866stationary heat, with nonlinear source, 682stationary Khokhlov–Zabolotskaya, 649telegraph, 494, 538, 583, 585thin film, 985thirdorder Liouville type, 972Thomas, 544threedimensional Khokhlov–Zabolotskaya,

752threedimensional Schrodinger, of general

form, 429threedimensional Schrodinger, with cubic

nonlinearity, 428tractrix, 253Tricomi, 659twodimensional Khokhlov–Zabolotskaya,

749twodimensional Schrodinger, of general

form, 427twodimensional Schrodinger, with cubic

nonlinearity, 425

Page: 1860

INDEX 1861

nonlinear equation (continued)twodimensional Schrodinger, with




Vries, 886vector Burgers, 417wave, 1400, 1519, 1536, 1673, 1678, 1723,

1727, 1764Weber, 1193Yermakov, 309, 518

nonlinear equationsadmitting variational form, 1595algebraic, 1441algebraic, with even powers, 1441axisymmetric steady laminar hydrodynamic


layer, 929basic steps of Painleve test, 1570Bellman type, 805Benney moment, 764boundary layer (Prandtl equations), 1333evolution, canonic Backlund transforma

tions, 1420Chaplygin, 1202Charpit, 1647, 1706chromatography, 1231cubic in derivatives, 134diffusion boundary layer, 402elliptic, geometrical models, 1754elliptic, in two space dimensions, 1748Euler, 938, 1197, 1681, 1729Euler, for barotropic gas flow, 1201Euler, in Gromeka–Lamb form, 1201Euler, in various coordinate systems, 1197evolution, nonlinear in second derivative,

346exact methods, 1353explicitly dependent on x and/or t, 266fifthorder, 1031firstorder with three or more independent


of general form, 99firstorder with two independent variables

quadratic in derivatives, 43for convective fluid motions, 1329fourthorder, 977functional, containing complex argument,

1776functional, reducible to bilinear, 1775functional, solutions, 1496functionaldifferential, 1776




heat, with powerlaw or exponentialtemperaturedependent thermaldiffusivity, 409

hydrodynamic boundary layer, 903hydrodynamic type, order reduction, 1422hyperbolic, canonical form, 1390hyperbolic, first canonical form, 1391hyperbolic, second canonical form, 1392in highest derivatives, 404in two independent variables, 1392invariance under scaling transformations,









variables, 30, 116, 168involving arbitrary linear differential

operators, 1065involving arbitrary parameters, 43, 99, 540involving arbitrary powers of derivatives,



highest derivative, 1031involving fourth powers of derivatives, 99involving hyperbolic cosine, 268involving hyperbolic cotangent, 270involving hyperbolic sine, 269involving hyperbolic tangent, 270involving inverse trigonometric functions,



derivatives, 156involving powerlaw nonlinearities, 583

Page: 1861


nonlinear equations (continued)involving products of derivatives, 133involving products of derivatives with











systems, 1247nonstationary, 942nonstationary, hydrodynamic (Navier–



region, 1225of breeze and monsoons, 1223of Burgers hierarchies, 1067of diffusion (thermal) boundary layer, 346of dynamic convection in the sea, 1227of flows in the baroclinic layer of the

ocean, 1229of general form, 150, 169, 1392of general form, involving arbitrary




media, 412of higher orders, 1031of ideal incompressible fluid, 942of ideal plasticity, 1238of Korteweg–de Vries hierarchies, 1067of mass transfer in quiescent or moving


938, 1197of sixth to ninthorder, 1039of stationary transonic planeparallel gas

flow, 1122of thermal (diffusion) boundary layer, 346of vibrations, of stratified medium, 1131of viscous incompressible fluid, 1003ordinary differential, 1445ordinary differential, Painleve test, 1566Painleve, 1566parabolic, 1677, 1725parabolic, canonical form, 1390passing Painleve test, 1572Pfaffian, 1600Pfaffian, not satisfying integrability

condition, 1602Plebanski heavenly, 802quadratic in derivatives, 139, 154, 158, 162quadratic in highest derivatives, 772reducible to Korteweg–de Vries equation,











1389shallow water, 1124simple separation of variables, 1459stationary, 938stationary hydrodynamic (Navier–Stokes





layer, 903telegraph, with two space variables, 583thirdorder, 857three and ndimensional, 428

Page: 1862

INDEX 1863

nonlinear equations (continued)threeargument functional, 1497threedimensional, 1240threedimensional, involving arbitrary

functions, 730threedimensional, of ideal plasticity, 1240twodimensional, 425, 1238twodimensional Euler, for incompressible

ideal fluid (plane flows), 1198unsteady boundary layer, for Newtonian



equations), 1192with arbitrary dependence on derivatives,


variables, 39with arbitrary number of variables,

involving arbitrary functions, 162with arbitrary number of variables,

involving arbitrary parameters, 158with coefficients involving exponential







662with four independent variables, 154with hyperbolic nonlinearities, 268, 485,




146, 161, 167with quadratic nonlinearities, 805, 896with three independent variables, 35, 838with three or more space variables, 406with three or more space variables,




powerlaw nonlinearities, 713with three variables, involving arbitrary

functions, 139

with three variables, quadratic in derivatives, 125

with trigonometric nonlinearities, 276, 389,680

with two independent variables, 1355with two independent variables, involving



in two or more highest derivatives, 849with two space variables, involving

arbitrary functions, 589with two space variables, involving

exponential nonlinearities, 384, 574with two space variables, involving

powerlaw nonlinearities, 367, 553nonlinear firstorder partial differential

equation, 1373nonlinear functional equations containing

complex argument, 1776nonlinear functional equations reducible to

bilinear equations, 1775nonlinear Klein–Gordon equation, 499, 1494nonlinear Kolmogorov type equation, 377,

398nonlinear parabolic equations, 1677, 1725nonlinear partial differential equations, 1650,

1709with Maple, 1625with Mathematica, 1687with MATLAB, 1735

nonlinear Poisson–Boltzmann equation, 1755nonlinear Poisson equation, 1679, 1727,

1750nonlinear problems of suspension transport in

porous media, 1604nonlinear Schrodinger equation, 348, 425,

1582nonlinear Schrodinger equations and related

equations, 348nonlinear systems

of firstorder PDEs, 1659, 1719, 1720of many equations involving first

derivatives with respect to t, 1348of partial differential equations, 1599of secondorder PDEs, 1660, 1661, 1720,

1721of two equations involving first derivatives

with respect to t, 1337of two equations involving second

derivatives with respect to t, 1344nonlinear telegraph equations with two space

variables, 583nonlinear wave equation, 1400, 1519, 1536,

1673, 1678, 1723, 1727, 1764nonlocal transformation, 1412nonstationary equations, 942

of motion of viscous incompressiblefluid, 1013

Page: 1863


nonstationary hydrodynamic equations(Navier–Stokes equations), 1013

notation, 1630, 1692nthorder chemical reaction, 1142nthorder Liouville type equation, 1109number

Grashov, 1329Mach, 757, 1203Prandtl, 347, 1329Reynolds, 253, 1013, 1021, 1260, 1275,

1291, 1310–1312Reynolds, modified, 1280

numerical and graphical solutions, 1673,1723

by default methods, 1672specifying singlestage numerical method,

1674specifying singlestage numerical method

and numerical boundary condition(NBC), 1676

specifying twostage numerical method,startup method, and NBC, 1676

numerical approximations, 1627, 1689numerical methods embedded in Maple,

1674numerical solutions and their visualizations,

1672, 1722numerical solutions via predefined functions,

1741

OOlver–Rosenau solution, 1028onedimensional barotropic flows of ideal

compressible gas, 1127onedimensional case, 1205onedimensional polytropic ideal gas

flow, 1125onedimensional rotation fluid flows, 1257onedisk problem, 1290onemode solution, 1683, 1731oneparameter Lie groups of point transfor

mations, 1526oneparameter transformations, 1513

local properties, 1513onesoliton solution, 869, 1592, 1638, 1698

of mKdV equation, 867of potential KdV equation, 876

online help system, 1626, 1737operator

infinitesimal, 1513, 1514, 1526infinitesimal, invariant, 1514invariant, 1514, 1526second prolongation, 1516

operatorsarithmetic, 1627, 1738assignment/unassignment, 1627basic arithmetic, 1689commutator, 1579logical, 1628, 1689, 1738range, 1627relational, 1627, 1738, 1739

operators := and =, 1627order reduction of hydrodynamic type

equations, 1422order reduction procedure for equations with

n ≥ 2 (reduction to solvable form withn = 1), 1446

ordinary differential equationAiry, 861Emden–Fowler, 248, 442, 444, 447, 462,

464, 647, 648, 655Emden–Fowler, generalized, 327, 477, 511,

512, 513, 516, 527, 528, 697, 698hypergeometric, 1012Painleve, first, 861, 989Painleve, second, 861, 867, 988, 989Riccati, 140, 142, 162, 164, 423, 425, 453,

482, 520, 1012, 1277Yermakov, 309, 518

ordinary differential equations, 1445oscillatory motion

of flat plate, 1252of flat porous plate, 1265

Ostrovsky equation, 968overdetermined systems

of firstorder equations in one unknown,1599

of two equations, 1599

Ppackages (subpackages), 1627Painleve equations, 1566Painleve property, 1566Painleve test, 1569

for nonlinear equations of mathematicalphysics, 1565

for ordinary differential equations, 1566for systems of ordinary differential

equations, 1569palettes, 1627, 1688, 1691, 1737palindromic equation, 1442palindromic polynomial, 1442partial differential equation

adiabatic, 1262BBM, 958Bernoulli, 394, 421Born–Infeld, 766, 1401boundary layer, for nonNewtonian


pressure gradient, 913Boussinesq, 368, 987, 1504Boussinesq, in canonical form, 987Boussinesqtype, 1028Burgers, 186, 1409, 1414, 1572, 1637,

1673, 1677, 1697, 1723, 1725, 1743,1761

Burgers–Huxley, 189Burgers–Korteweg–de Vries, 884Calogero, 745, 746Calogerotype, 1400

Page: 1864

INDEX 1865

partial differential equation (continued)Camassa–Holm, 968Chaplygin, 765Clairaut, 118, 152, 172, 1375Clairaut, in implicit form, 122combined KdVmKdV, 871complete Lin–Reissner–Tsien, 757continuity, 1197, 1247, 1610cylindric Burgers, 191cylindrical Korteweg–de Vries, 866Degasperis–Procesi, 968determining, 1555double sineGordon, 492double sinhGordon, 485Dym, 878elliptic, 1392Enneper, 491Euler–Lagrange, 1595evolution, nonlinear in second derivative,

346fast diffusion, 211firstorder quasilinear, 1355Fisher, 176, 1639, 1699, 1744FitzHugh–Nagumo, 179, 1746fKdV, 1034Fokas–Yortsos, 237fourthorder, Liouville type, 1030fourth Painleve, 988Fujita–Storm, 212Gardner, 871Gauss, 1772general fifthorder KdV, 1034generalized Burgers, 288generalized Burgers–Korteweg–de

Vries, 1045, 1503generalized Calogero, 748generalized Kadomtsev–Petviashvili, 1029generalized Khokhlov–Zabolotskaya, 751generalized Korteweg–de Vries, 871generalized Kuramoto–Sivashinsky, 983generalized Landau–Ginzburg, 360generalized Liouville, 1108Gibbons–Tsarev, 764Goursat, 549Grad–Shafranov, 678, 691Hamilton–Jacobi, 1379Hamilton–Jacobi, Cauchy problem, 1380Harry Dym, 878Helmholtz, 380, 397, 399, 420, 423, 558,


pressure gradient, 907hyperbolic, first canonical form, 1391hyperbolic, second canonical form, 1392inviscid Burgers, 1674, 1763inviscid Burgers (explicit method), 1678,

1725inviscid Burgers (initial value problem),

1762

inviscid Burgers (shock waves andrarefaction waves), 1757

inviscid Burgers (shock waves and weaksolutions), 1759

Ito, 1037k(n,n), 891Kadomtsev–Petviashvili, 1000, 1587Kadomtsev–Petviashvili, in canonical

form, 1000Kadomtsev–Petviashvili type, 1002Karman–Howarth, 253Kaup–Kupershmidt, 1036Kawahara, 1032KdV, 857Khokhlov–Zabolotskaya, 749KK, 1036Klein–Gordon, 1641, 1656, 1701, 1716Kolmogorov–Petrovskii–Piskunov






646, 938, 1392Lax, 1035, 1068Lax, with forcing term, 1037Lin–Reissner–Tsien, 756, 759linear heat, 807, 1252linear Schrodinger, 865, 1186linear wave, 1392Liouville, 144, 165, 470, 540, 1117, 1414mKdV, 867model, for dynamics of nonlinear


6modification of coupled KdV, 1172modified Burgers–Korteweg–de Vries, 886modified Burgers, 191modified Harry Dym, 891modified k(n,n), 892modified Korteweg–de Vries, 862, 866,


form, 867Monge–Ampere, 774, 1393ndimensional modified Schrodinger, with


nonlinearity, 430

Page: 1865


partial differential equation (continued)Newell–Whitehead, 1534ninthorder KdVtype, 1041nonlinear diffusion, 1645, 1705nonlinear diffusion, with cubic source,

1577nonlinear firstorder, 1373nonlinear Klein–Gordon, 499, 1494nonlinear Kolmogorov type, 377, 398nonlinear Poisson, 1679, 1727, 1750nonlinear Poisson–Boltzmann, 1755nonlinear Schrodinger, 348, 425, 1582nonlinear wave, 1400, 1519, 1536, 1673,

1678, 1723, 1727, 1764nthorder Liouville type, 1109of axisymmetric steady hydrodynamic



1069of steady transonic gas flow, 657of surface of bodies of revolution, 8of unsteady transonic gas flows, 756Ostrovsky, 968Plebanski first heavenly, 802Plebanski second heavenly, 803Poisson, 379, 397, 423, 580, 588, 938,



(mKdV), 877Rayleigh–Zababakhin–Plesset, 1262Sawada–Kotera, 1035Sawada–Kotera, with forcing term, 1038Schrodinger, of general form, 358, 360,



351, 353Schwarzian Korteweg–de Vries, 870secondorder nonlinear, 1392secondorder quasilinear, 1393seventhorder KdVtype, 1040Sharm–Tasso–Olver, 887, 1067sineGordon, 485, 490, 543, 1117, 1635,

1639, 1646, 1695, 1698, 1706sinhGordon, 485, 542, 1580SK, 1035spherical Korteweg–de Vries, 866stationary heat, with nonlinear source, 682stationary Khokhlov–Zabolotskaya, 649

telegraph, 494, 538, 583, 585thin film, 985thirdorder Liouville type, 972Thomas, 544threedimensional Khokhlov–Zabolotskaya,

752threedimensional nonlinear Schrodinger,

of general form, 429threedimensional Schrodinger, with cubic

nonlinearity, 428Tricomi, 659twodimensional Khokhlov–Zabolotskaya,

749twodimensional linear heat, 1256twodimensional nonlinear Schrodinger, of

general form, 427twodimensional Schrodinger, with cubic

nonlinearity, 425twodimensional Schrodinger, with




Vries, 886vector Burgers, 417Weber, 1193

partial differential equationsadmitting variational form, 1595axisymmetric steady laminar hydrodynamic


layer, 929Bellman type, 805Benney moment, 764boundary layer (Prandtl equations), 1333elliptic, canonical form, 1392evolution, canonic Backlund transforma

tions, 1420Chaplygin, 1202Charpit, 1647, 1706chromatography, 1231cubic in derivatives, 134Euler, 938, 1197, 1681, 1729Euler, for barotropic gas flow, 1201Euler, in Gromeka–Lamb form, 1201Euler, in various coordinate systems, 1197evolution, nonlinear in second derivative,

346exact methods, 1353explicitly dependent on x and/or t, 266fifthorder, 1031firstorder with three or more independent


quadratic in derivatives, 43

Page: 1866

INDEX 1867

partial differential equations (continued)firstorder with two independent variables

of general form, 99firstorder quasilinear, 3for convective fluid motions, 1329fourthorder, 977Gelfand–Levitan–Marchenko type, 1584heat, with powerlaw or exponential

















highest derivative, 1031involving fourth powers of derivatives, 99involving hyperbolic cosine, 268involving hyperbolic cotangent, 270involving hyperbolic sine, 269involving hyperbolic tangent, 270involving inverse trigonometric functions,



derivatives, 156

involving powerlaw nonlinearities, 583involving products of derivatives, 133involving products of derivatives with








derivatives, 111linear in mixed derivative, 745, 847Navier–Stokes, 1247Navier–Stokes, general properties, 1249Navier–Stokes, in Boussinesq approxima


systems, 1247nonlinear, diffusion boundary layer, 402nonlinear, elliptic, geometrical models,

1754nonlinear, elliptic, in two space dimen

sions, 1748nonlinear, in highest derivatives, 404nonlinear, in two independent variables,


differential operators, 1065nonlinear, involving arbitrary parameters,

99nonlinear, of general form, 150, 169, 1392nonlinear, of thermal (diffusion) boundary


1459nonlinear, telegraph, with two space


variables involving arbitrary functions,162

nonlinear, with arbitrary number ofvariables involving arbitrary parameters,158


nonlinear, with three variables involvingarbitrary functions, 139

Page: 1867


partial differential equations (continued)nonlinear, with three variables quadratic in

derivatives, 125nonstationary, 942nonstationary, hydrodynamic (Navier–











938, 1197of nonlinear vibrations stratified medium,


flow, 1122of viscous incompressible fluid, 1003Plebanski heavenly, 802parabolic, canonical form, 1390passing Painleve test, 1572quadratic in derivatives, 139, 154, 158, 162quadratic in highest derivatives, 772quasilinear, 1393, 1647, 1649, 1707, 1708quasilinear, discontinuous solutions, 1360quasilinear, in conservative form, 1368quasilinear, of general form, 1368quasilinear, qualitative features, 1360quasilinear, with n independent variables,












1389shallow water, 1124stationary, 938stationary, hydrodynamic (Navier–Stokes





layer, 903thirdorder, 857three and ndimensional, 428threeargument functional, 1497threedimensional, 1240threedimensional, involving arbitrary

functions, 730threedimensional, of ideal plasticity, 1240transformations, 1395twodimensional, 425, 1238twodimensional Euler, for incompressible

ideal fluid (plane flows), 1198with arbitrary dependence on derivatives,


variables, 39with coefficients involving exponential









486, 677with n independent variables, 739, 853with n space variables, 417

Page: 1868

INDEX 1869

partial differential equations (continued)with powerlaw nonlinearities, 175, 433,


161, 167with power nonlinearity in derivatives, 146with quadratic nonlinearities, 805, 896with three independent variables, 35, 838with three or more space variables, 406with three or more space variables,





680with two independent variables, 1355with two independent variables, involving



in two or more highest derivatives, 849with two space variables involving

arbitrary functions, 589with two space variables involving

exponential nonlinearities, 384, 574with two space variables involving

powerlaw nonlinearities, 367, 553unsteady boundary layer, for Newtonian



equations), 1192pattern, 1689, 1739performing Painleve test and truncated

expansions for studying some nonlinearequations, 1572

persistent variables, 1739Pexider equation, 1771Pfaffian equations, 1600

not satisfying the integrability condition,1602

their solutions, 1600phase diagram, 1640, 1700phase portraits, 1639, 1699phase trajectories, 1640, 1700plane jet, 1335plane radially symmetric unsteady fluid

motion, 1261Plebanski heavenly equation

first, 802second, 803

Poincare phase plane, 1640, 1700point transformations: overview and

examples, 1395Poisson equation, 379, 397, 423, 580, 588,

938, 1008polar coordinates, 1198, 1248porous medium equation, 210potential, 1587potential equation of onedimensional flow of

compressible gas, 768potential flow, 1334potential gas flow, 1201potential Korteweg–de Vries (KdV) equation,

876potential modified Korteweg–de Vries

(mKdV) equation, 877powerlaw nonNewtonian fluid, 911power isotherm, 1234Prandtl–Meyer solutions, 1203Prandtl solution, 1238predefined constants, 1628, 1740predefined functions, 1628, 1740pressure flow in rotating layer, 1283pressure gradient, 1253previous results, 1626, 1688, 1737probability integral, 187problem

Blasius, 904, 1333Blasius, reversed statement, 1334boundary value, 1679, 1727Cauchy, 1357, 1379, 1611Cauchy, classical, 1647, 1707, 1709, 1756Cauchy, for Hamilton–Jacobi equation,

1380Cauchy, procedure of solving, 1358Cauchy, qualitative features of solutions,

1611Cauchy, solution, 1360Cauchy, solution by inverse scattering

problem method, 1587Cauchy, solution for nonlinear equations

by inverse scattering problem method,1589

direct scattering, 1587Falkner–Skan, 907, 1334first Stokes, 1251generalized Blasius, 912generalized Cauchy, 1649, 1650, 1708,

1709generalized Falkner–Skan, 914generalized Schlichting, 912Hiemenz, 1264initial value, 1383, 1611initial value, for inviscid Burgers equation,

1762inverse problem, 1455inverse scattering, 1587inverse scattering, methods of solving,

soliton theory, 1579Jeffery–Hammel, 1261jetsource, 1335Landau, 1012onedisk, 1290of motion of material point, 126of motion of rod, 129of propagation of signal, 1367

Page: 1869


problem (continued)of suspension transport in porous media,

1604on motion of two point bodies, 126without initial data, 1252, 1256Riemann, 1611Riemann, qualitative features of solutions,

1611Schlichting, 905second Stokes, 1252terminal value, 1383, 1386twobody, in celestial mechanics, 65twodisk, 1291

problemsboundary layer, selfsimilar solutions, 1333Cauchy, solutions, 1756Riemann, solutions, 1618simple inverse, determination of form of

equations from their properties, 1455procedure

for constructing exact solutions, 1450for constructing invariant solutions, 1520of solving Cauchy problem, 1358

prolongation of group generator, 1527prompt symbol, 1626, 1736pseudodifferential equations, 1484, 1483pure functions, 1690purely radial fluid motions, 1260

Qqualitative features and discontinuous

solutions of quasilinear equations, 1360quasiplane flow, 1285quasiplane flows (with fluid velocity

components independent of z), 1285quasilinear equations, 35, 1393, 1647, 1649,

1707, 1708discontinuous solutions, 1360firstorder, 3, 1355firstorder, hyperbolic systems, 1607firstorder, methods for solving, 1355hyperbolic systems, 1607in conservative form, 1368of general form, 1368PDEs, 1647, 1649, 1707, 1708qualitative features, 1360secondorder, 1393semiHamiltonian system, 1236with n independent variables, 1356

quotes, 1740

Rrange operator, 1627Rankine–Hugoniot condition, 1126, 1128,

1759Rankine–Hugoniot jump condition, 1363,

1369, 1615rarefaction wave, 6, 26, 1360, 1360, 1611,

1758

rational solution, 858Rayleigh–Zababakhin–Plesset equation, 1262reciprocal equation, 1442reciprocal polynomial, 1442reduction of systems to canonical form,

1611reduction to standard functional equation,

1496redundant solution, 1662reflectance, 1588reflection factor, 1588regular expression, 1739regular point of generalized solution, 1383related equations, 935relational operators, 1627, 1738, 1739remarks on Painleve test, 1567removing redundant exact solutions, 1665removing redundant solutions via predefined

functions, 1667resonance, Painleve test, 1570reversed statement of Blasius problem, 1334Reynolds number, 1021, 1260RF pairs, 1413

their use for constructing Backlundtransformations, 1415

Riccati equation, 140, 142, 162, 164, 423,425, 453, 482, 520, 1012, 1277

Riemann invariants, 1231, 1237, 1609, 1611,1612

Riemann problem, 1611qualitative features of solutions, 1611

Riemann simple waves, 1125, 1127, 1129rotation transformation, 1396rotationally symmetric motions of fluid,

1200rotationally symmetric motions of general

form, 1297round jet, 1277Ryzhov–Shefter solution, 759

SSawada–Kotera equation, 1035

with forcing term, 1038scalar nonlinear PDEs in one space

dimension, 1742scaling transformation, 1396, 1645, 1705,

1770scattering data, 1588scheme for constructing exact solutions by

nonclassical method, 1533Schlichting–Bickley problem, 1335Schlichting problem, 905Schrodinger equation

of general form, 358, 360, 363, 364with cubic nonlinearity, 348, 350, 351,

352, 355with powerlaw nonlinearity, 351, 353

Schwarzian Korteweg–de Vries equation,870

script, 1737, 1741mfiles, 1737

Page: 1870

INDEX 1871

second hodograph transformation, 1400secondorder differential constraints, 1553

for PDEs, 1553secondorder elliptic equations

with three or more space variables, 713with two space variables, 641

secondorder equationsBacklund transformations, 1413classification, 1389elliptic, 1191elliptic, with three or more space variables,

713elliptic, with two space variables, 641evolution, 1547hyperbolic, 1551hyperbolic, with one space variable, 433hyperbolic, with two or more space

variables, 553involving secondorder mixed derivatives,

949involving mixed derivatives and some

other, 745involving real functions of real variables,

1165Klein–Gordon type, 1173nonlinear, 1516nonlinear, classification, 1389, 1565nonlinear, for laser systems, 1170nonlinear partial differential, 1392nonlinear, symmetries, 1516of general form, 811, 1553of reactiondiffusion type, 1620parabolic, with one space variable, 175parabolic, with two or more space

variables, 367quasilinear, 1393

secondorder evolution equations, 1547secondorder hyperbolic equations, 1551

with one space variable, 433with two or more space variables, 553

secondorder nonlinear equations of lasersystems, 1170

secondorder parabolic equationswith one space variable, 175with two or more space variables, 367

secondorder quasilinear partial differentialequation, 1393

second Painleve equation, 861, 867, 988,989

second prolongation of operator, 1516second Stokes problem, 1252selfsimilar continuous solutions, 1607selfsimilar solutions, 1429, 1431, 1452

of some boundary layer problems, 1333selfsimilar variables, 1645, 1705semiHamiltonian system of quasilinear

equations, 1236semilinear equations in two independent

variables, 1389separation constant, 1459separation of variables, 1651, 1710

sequences, 1629set formula, 1755sets, 1629, 1691seventhorder KdVtype equation, 1040shallow water equations, 1124Sharm–Tasso–Olver equation, 887, 1067Shcheprov solution, 1278shock wave, 6, 26, 1675, 1126, 1128, 1362,

1615, 1616, 1758similarity and invariant solutions, 1646similarity method, 1431similarity reductions in equations with three

or more independent variables, 1510similarity transformation, 1645, 1705simple inverse problems (determination

of form of equations from theirproperties), 1455

simple nonlinear point transformations, 1396simple Riemann waves, 1609simple scheme for studying nonlinear partial

differential equations, 1570simple separation of variables in nonlinear

partial differential equations, 1459simple transformations, 1446simplest model of chemical reactor with

secondorder kinetic functions, 1115simplest plane purely radial fluid motion,

1260simplified scheme for constructing general

ized separable solutions, 1465simplifying exact solutions, 1665Simulink, 1736sinecosine method, 1642, 1702sineGordon equation, 490, 543, 1117, 1635,

1639, 1646, 1695, 1698, 1706singleprecision floating point, 1740singlesoliton solution, 491singular integral, 1373, 1374singular point, 1640, 1700

of generalized solution, 1383sinhGordon equation, 485, 542, 1580SK equation, 1035Slezkin solution, 1276slow diffusion, 251soliton, 857, 1636, 1695solution

additive separable, 1374, 1459by reduction to equations with quadratic

(or power) nonlinearities, 1487complete, 1647, 1706Couette–Poiseuille type, 1283general, firstorder PDEs, 1355generalized, 1364generalized, regular points, 1383generalized, singular points, 1383generalized, stability condition, 1369invariant, 1431, 1435, 1451, 1513, 1520,

1527, 1602, 1620Jeffery–Hamel, 1272Landau, 1277limiting selfsimilar, 1454, 1456methods, 1373

Page: 1871


solution (continued)Mises, 768multiplicative separable, 1374, 1459, 1651,

1710nsoliton, 859, 1591nsoliton, of Korteweg–de Vries equation,

1591Nadai, 1238Nadai, multiplicative separable, 1238nonclassical selfsimilar, 500, 685, 689,

1435nonclassical travelingwave, 679, 690, 706noninvariant selfsimilar, 1435of Cauchy problem, 1360of Cauchy problem by inverse scattering

problem method, 1587of Cauchy problem for nonlinear equations

by inverse scattering problem method,1589

of determining equations in form ofpolynomials in λ, 1582

of functionaldifferential equations bydifferentiation, 1467

of functionaldifferential equations bysplitting, 1471

Olver–Rosenau, 1028onemode, 1683, 1731onesoliton, 869, 1592, 1638, 1698onesoliton, mKdV equation, 867onesoliton, potential KdV equation, 876Prandtl, 1238rational, 858redundant, 1662Ryzhov–Shefter, 759selfsimilar, 1429, 1431, 1452Shcheprov, 1278singlesoliton, 491Slezkin, 1276threesoliton, KdV equation, 859travelingwave, 1429, 1602, 1615, 1620,

1637, 1659, 1697, 1720twosoliton, 491, 869twosoliton, KdV equation, 858twosoliton, periodic, 492viscosity, 1365, 1366Volosov, 179von Mises, 768weak, 1364, 1675, 1758

solutionsadditive separable, 1459along characteristics, constructing

method, 1647, 1706analytical, 1629, 1691analytical, nonlinear systems, 1659, 1719analyticalnumerical, 1680, 1728classical selfsimilar, 1435classical travelingwave, 1431describing shock waves, 1618discontinuous, quasilinear equations, 1360exact, using nonclassical method, 1533exact, using symmetries of equations, 1520

exact, verifying, 1667for fixed singularities, 1565for Pfaffian equations, 1600for Riemann problem, 1618functional separable, structure, 1487generalized separable, structure, 1464generalized separable, simplified scheme

for constructing, 1465generalized travelingwave, 1487generalized viscosity, 1382generalized viscosity, local structure, 1383generalizing exact, 1665invariant, 1646multipeakon, 968multiplicative separable, 1459numerical, their visualizations, 1672, 1722numerical, via predefined functions, 1741numerical and graphical, 1673, 1723numerical and graphical, by default

methods, 1672numerical and graphical, specifying

singlestage numerical method, 1674numerical and graphical, specifying single

stage numerical method and numericalboundary condition (NBC), 1676

numerical and graphical, specifying twostage numerical method, startup method,and NBC, 1676

of nonlinear equations that fail Painlevetest, using truncated expansions, 1577

of nonlinear functional equations and theirapplications, 1496

of partial differential equations withmovable pole, 1569

of simple functional equations and theirapplication, 1472

oneparameter, using symmetries ofequations, 1520

removing redundant, exact, 1665removing redundant, via predefined

functions, 1667selfsimilar continuous, 1607selfsimilar, for some boundary layer

problems, 1333similarity, 1646simplifying exact, 1665special functional separable, 1487stable generalized, method for construct

ing, 1370steadystate, 1329symbolic and numerical, with Maple,

Mathematica and MATLAB, 1623travelingwave, constructing method, 1637,

1659, 1697, 1719twodimensional, in cylindrical coordinates

(plane flows), 1270twodimensional, in rectangular Cartesian

coordinates (plane flows), 1263unsteady, 1331via predefined functions, method for

constructing, 1692

Page: 1872

INDEX 1873

solutions (continued)via transformations, constructing method,

1635, 1694viscosity, based on test functions and

differential inequalities, 1383viscosity, based on use of parabolic

equation, 1382with linear dependence of velocity

components on one space variable, 1323with linear dependence of velocity

components on two space variables,1303, 1317

with movable singularities, 1565with movable singularities, for ordinary

differential equations, 1565with one nonzero component of fluid

velocity, 1251with spiral symmetry, 1328with three nonzero fluid velocity com

ponents dependent on three spacevariables, 1302

with three nonzero fluid velocity components dependent on two spacevariables, 1285

with two nonzero components of fluidvelocity, 1263

solving Cauchy problems, 1756solving nonlinear ODEs, 1664some systems depending on arbitrary

parameters, 1150spatial discretization, 1743special form for symmetry reduction, 1505special functional separable and generalized

travelingwave solutions, 1487special functional separable solutions, 1487special functions, 1740spectral collocation method, 1680, 1729spherical Korteweg–de Vries equation, 866spherical radially symmetric fluid motion,

1261spherical radially symmetric unsteady fluid

motion, 1262spherical vortex, 1278splitting in derivatives, 1516splitting method, 1496splitting procedure, 1527stability condition, 1369

for generalized solution, 1369statement separator, 1628, 1739statements, 1628, 1739stationary anisotropic heat (diffusion)

equation, 702stationary equations, 938stationary heat equation with nonlinear

source, 682stationary hydrodynamic equations (Navier–

Stokes equations), 1003stationary Khokhlov–Zabolotskaya equation,

649steady boundary layer equations for non

Newtonian fluids, 911

steady hydrodynamic boundary layerequations for Newtonian fluid, 903

steady laminar hydrodynamic boundarylayer, 903

steadystate hydrodynamic boundary layerequations, 903

steadystate solutions, 1329stream function, 903, 909, 911, 915, 917,

929, 930, 932, 938, 1003, 1013, 1333strictly hyperbolic system, 1608, 1615string, 1628, 1690string variable, 1739strip condition, 1651, 1710structure, 1737

of functional separable solutions, 1487of generalized separable solutions, 1464

subfunctions, 1741surface tension coefficient, 1262symbol, 1689symbolic and numerical solutions of nonlin

ear PDEs with Maple, Mathematica, andMATLAB, 1623

symbolic math toolbox, 1736symmetric bivariate polynomial, 1444symmetries, 1439symmetries, of equations of steady hydrody

namic boundary layer, 1528of nonlinear secondorder equations, 1516of systems of equations of mathematical

physics, 1527symmetry reductions based on generalized

separation of variables, 1507system

algebraic, 1441characteristic, 1355generic, 1754graphics, 1736hydrodynamictype, diagonal form, 1236hydrodynamictype, nondiagonal form,

1236hyperbolic, 1607, 1608, 1615Lotka–Volterra, 1135module, 1741nonlinear, of firstorder PDEs, 1659, 1719,

1720nonlinear, of secondorder PDEs, 1660,

1720of coordinate functions, simplified scheme

for constructing solutions, 1465of gas dynamic type equations, 1403of nonlinear elliptic equations, 1753of two PDEs, 1402semiHamiltonian, of quasilinear equations,

1236strictly hyperbolic, 1608, 1615

systemsbasic relations used in symmetry analysis,

1527depending on arbitrary parameters, 1150describing fluid flows in atmosphere, seas

and oceans, 1223

Page: 1873


systems (continued)firstorder hydrodynamic, involving three

or more equations, 1197firstorder hyperbolic, of quasilinear

equations, 1607gas dynamic type, linearizable with

hodograph transformation, 1122, 1610gas dynamics, canonical form, 1611hydrodynamictype, 1236hyperbolic n × n, of conservation

laws, 1614in form of conservation laws, 1607involving arbitrary functions, 1117involving arbitrary parameters, 1115involving thirdorder evolution equations,

1172mathematical physics, symmetries, 1527nonlinear, analytical solutions, 1659, 1719nonlinear, of many equations involving

first derivatives with respect to t, 1348nonlinear, of partial differential equations,

1599nonlinear, of secondorder PDEs, 1660,

1661, 1720, 1721nonlinear, of two equations, 1347nonlinear, of two equations involving first

derivatives with respect to t, 1337nonlinear, of two equations involving

second derivatives with respect tot, 1344

nonlinear, Painleve test, 1576of algebraic equations symmetric with

respect to permutation of arguments,1444

of conservation laws of gas dynamictype, 1607

of firstorder equations describingconvective mass transfer with volumereaction, 1602

of general form, 1337of nonlinear elliptic PDEs in two space

dimensions, 1752of nonlinear equations, 1557of nonlinear PDEs in one space dimension,

1745of secondorder equations of reaction

diffusion type, 1620of two elliptic equations, 1185of two equations, 1122of two firstorder partial differential

equations, 1115of two parabolic equations, 1133of two quasilinear equations, 1165, 1607of two secondorder elliptic equations,

1191of two secondorder Klein–Gordon type

hyperbolic equations, 1173ordinary differential equations, Painleve

test, 1569overdetermined, in one unknown, 1600overdetermined, of firstorder equations in

one unknown, 1599overdetermined, of two equations, 1599reduction to canonical form, 1611reducible to ordinary differential equations,

1603various coordinate, Euler equations, 1197various coordinate, Navier–Stokes, 1247

Ttable, 1629, 1691tanhcoth method, 1641, 1701telegraph equation, 494, 538, 583, 585tensor, 1691terminal value problem, 1383, 1386thin film equation, 985thirdorder equations, 857, 949

evolution, 1172involving thirdorder mixed derivatives,

958examples of finding exact solutions, 1465

thirdorder Liouvilletype equation, 972Thomas equation, 544three and ndimensional equations, 428threeargument functional equations of

special form, 1497threedimensional equations, 1240

involving arbitrary functions, 730of ideal plasticity, 1240

threedimensional Khokhlov–Zabolotskayaequation, 752

threedimensional nonlinear Schrodingerequation of general form, 429

threedimensional Schrodinger equation withcubic nonlinearity, 428

threedimensional stagnationpoint typeflows, 1302

threesoliton solution, KdV, 859Titov–Galaktionov method, 1477toolboxes, 1738toroidal vortex, 1278tractrix equation, 253transformation

autoBacklund, 681, 1413, 1635, 1695Boussinesq, 1289classical hodograph, 1400Cole–Hopf, 1573Cole–Hopf, for Burgers equation, 1637,

1696Crocco, 906, 909, 1422Euler, 122, 346, 769, 957, 1403, 1407first hodograph, 1399Galileo, 1514hodograph, 35, 212, 344, 1129, 1397hodograph, for gas dynamic type systems,

1122, 1610hodograph, nonclassical, 345, 1399hodograph, second, 1400Hopf–Cole, 187, 888, 1067, 1409, 1414Legendre, 122, 767, 768, 770, 1403, 1406Legendre, with many variables, 1408Lorentz, 1514

Page: 1874

INDEX 1875

transformation (continued)Martin, 775Miura, 862, 868, 1410nonclassical hodograph, 345, 1399nonlocal, 1412rotation, 1396scaling, 1396, 1645, 1705, 1770second hodograph, 1400similarity, 1645, 1705translation, 1396von Mises, 761, 906, 909, 916, 937, 971,

1106, 1409, 1410transformations

Backlund, 1413, 1635, 1695Backlund, canonic for evolution equations,

1420Backlund, for secondorder equations,

1413based on conservation laws, 1425contact, 1403contact, for ordinary differential equations,

1403contact, for partial differential equations,

1405for constructing solutions, 1635, 1694in plane, 1514linear, 1396local, for derivatives, 1526of equations of mathematical physics, 1395oneparameter, 1513oneparameter, local properties, 1513oneparameter, point, Lie groups, 1526point, general form, 1395point, oneparameter Lie groups, 1526point, overview and examples, 1395point, simple, 1396preserving form of equations, 1439, 1445,

1450scaling, invariance of equations, 1431simple, 1446simple, nonlinear point, 1396translation, invariance of equations, 1430translation, invariance of solutions, 1430

transient fluid motion for whirlwind, 1258transient motion of flat plate, 1251translation transformation, 1396transversality condition, 1380travelingwave solutions, 1429, 1602, 1615,

1620, 1637, 1659, 1697, 1720ansatz methods, 1641, 1700classical, 1431functional equation, 1431general form, 1429generalized, 1487generalized, general scheme for construct

ing, 1487methods for constructing, 1641, 1700nonclassical, 679, 690, 706

Tricomi equation, 659truncated expansions, 1571twobody problem in celestial mechanics, 65

twodimensional equations, 425, 1238twodimensional Euler equations for

incompressible ideal fluid (planeflows), 1198

twodimensional Khokhlov–Zabolotskayaequation, 749

twodimensional linear heat equation, 1256twodimensional nonlinear Schrodinger

equation of general form, 427twodimensional Schrodinger equation with

cubic nonlinearity, 425twodimensional Schrodinger equation with

powerlaw nonlinearity, 427twodimensional solutions in cylindrical

coordinates (plane flows), 1270twodimensional solutions in rectangular

Cartesian coordinates (plane flows),1263

twodisk problem, 1291twosoliton periodic solution, 492twosoliton solution, 491, 869

KdV equation, 858two classes of functions, 1690two control structures, 1629, 1691types of brackets, 1628, 1690, 1739types of equations, 1389types of numbers, 1628, 1690, 1740types of quotes, 1628, 1690Tzitzeica equation, 541

Uunassignment of definitions, 1690unidirectional flows in tubes of various

crosssections, 1253unidirectional plane flows, 1251universal invariant, 1526unnormalized Boussinesq equation, 990unnormalized Burgers equation, 187unnormalized Kadomtsev–Petviashvili

equation, 1002unnormalized Korteweg–de Vries equation,

865unnormalized modified Korteweg–de Vries

equation, 870unsteady boundary layer equations

for Newtonian fluid, 917for nonNewtonian fluids, 930

unsteady hydrodynamic boundary layer, 917unsteady solutions, 1331upwind explicit finite difference scheme,

1762usage of several differential constraints,

1557userdefined function, 1628, 1691, 1737,

1740using integral relations for determining

generalized solutions, 1364using symmetries of equations for construct

ing oneparameter solutions, 1520using symmetries of equations for finding

exact solutions, 1520

Page: 1875


VVshaped body, 1334variable name, 1627, 1738variant of modified Burgers–Korteweg–de

Vries equation, 886various reserved keywords, 1627, 1738vector, 1629, 1691, 1741vector and array indices, 1741vector Burgers equation, 417verifying exact solutions, 1667viscosity solutions, 1365, 1366

based on test functions and differentialinequalities, 1383

based on use of parabolic equation, 1382viscous incompressible fluid, 1247Volosov solution, 179von Karman equations (fourthorder elliptic

equations), 1192von Karman flow, 1290von Karmantype rotationally symmetric

flows, 1316von Karmantype rotationally symmetric

motions, 1290von Mises solution, 768von Mises transformation, 761, 906, 909,

916, 937, 971, 1106, 1409, 1410von Mises variables, 1106von Mises yield criterion, 1238, 1239, 1240

Wwave

breaking, 1360breaking effect, 26centered rarefaction, 1359cnoidal, 858linear equation, 1392nonlinear, model equation with damping, 6nonlinear dispersive nondissipative, 857nonlinear equation, 1400, 1519, 1536,

1673, 1678, 1723, 1727, 1764overturn, 1360rarefaction, 6, 26, 1360, 1360, 1611, 1758rarefaction, for inviscid Burgers equation,

1757Riemann simple, 1125, 1127, 1129shock, 6, 26, 1675, 1126, 1128, 1362,

1615, 1616, 1758shock, for inviscid Burgers equation, 1757simple Riemann, 1609

weak solution, 1364, 1675, 1758Weber equation, 1193Weierstrass elliptic function, 176, 885, 988,

1122Whitham rule of equal areas, 1363

YYermakov equation, 309, 518

Page: 1876

index a b - eqworldeqworld.ipmnet.ru/en/info/npdehandbook/index.pdf · index a abbreviations, 1627...

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