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A Historical Overview of

Symmetry Methods for

Di!erential Equations:

From Lie to Noether

to Birkho! to Ovsiannikov

to Bluman and beyond

Peter J. Olver

University of Minnesota

http://www.math.umn.edu/! olver

Vancouver, 2014

A Brief History

• Abel, Galois

• Lie

• Backlund, Vessiot

• Noether

• Birkho!

• Ovsiannikov

• Bluman, Cole

Symmetry Groups of

Di!erential Equations

System of di!erential equations

"(x, u(n)) = 0

G — Lie group acting on the space of independentand dependent variables (point transformations)

(!x, !u) = g · (x, u) = (#(x, u),$(x, u))

G acts on functions u = f(x) by transforming their graphs:

Definition. G is a symmetry group of the system" = 0 if !f = g · f is a solution whenever f is.

Infinitesimal Generators

Vector field:

v|(x,u) =d

d!g! · (x, u)|!=0

In local coordinates:

"i(x, u)#

$"(x, u)#

generates the one-parameter group

d!= "i(x, u)

d!= $"(x, u)

Jet Spaces=" Ehresmann (1953) — Lie pseudo-groups

x = (x1, . . . , xp) — independent variables

u = (u1, . . . , uq) — dependent variables

#xj1 . . . #xk— partial derivatives

(x, u(n)) = ( . . . xi . . . u" . . . u"J . . . ) # Jn

— jet coordinates

dim Jn = p+ q(n) = p+ q

Prolongation to Jet Space

Since G acts on functions, it acts on their derivatives,leading to the prolonged group action on the jet space:

(!x, !u(n)) = pr(n) g · (x, u(n))

=" formulas provided by implicit di!erentiation.

Prolonged vector field or infinitesimal generator:

pr v = v +"

$"J(x, u

The Prolongation FormulaRecursive version:

$"J,k = Dk $

(Dk"i)u"

=" Lie, Eisenhart, OvsiannikovExplicit version:

$"J = DJ Q

"i u"J,i

Q = (Q1, . . . , Qq) — characteristic of v

Q"(x, u(1)) = $" $p"

=" Olver (1976)

Infinitesimal Symmetry Criterion

Theorem. (Lie) A connected group of transformationsG is a symmetry group of a nondegenerate system of di!erentialequations " = 0 if and only if

pr v(") = 0 on solutions (%)for every infinitesimal generator v of G.

(%) are the determining equations (or defining equations) for thesymmetry group.

For nondegenerate systems, (%) is equivalent to

pr v(") = A ·" ="

Nondegeneracy Conditions

Maximal Rank:

· · ·#"#

#xi· · ·

· · ·$

Local Solvability: Any each point (x0, u(n)0 ) such that

"(x0, u(n)0 ) = 0

there exists a solution u = f(x) with

u(n)0 = pr(n) f(x0)

Nondegenerate = maximal rank + locally solvable

Calculation of Symmetry Groups

Example. Potential Burgers’ equation.

ut = uxx + u2x

Prolonged symmetry generator:

pr v = "(x, t, u)#

#x+ %(x, t, u)

#t+ $(x, t, u)

+ $x #

+ $t #

+ $xx #

+ $xt #

+ $tt #

+ · · ·

Infinitesimal symmetry criterion

$t = $xx + 2ux$x on solutions

Prolongation formulae:

Q = $$ "ux $ %ut

$x = DxQ+ "uxx + %uxt

= $x + ($u $ "x)ux $ %xut $ "uu2x $ %uuxut

$t = DtQ+ "uxt + %utt

= $t $ "tux + ($u $ %t)ut $ "uuxut $ %uu2t

$xx = D2xQ+ "uxxt + %uxtt

= $xx + (2$xu $ "xx)ux $ %xxut

+ ($uu $ 2"xu)u2x $ 2%xuuxut $ "uuu

$ %uuu2xut + ($u $ 2"x)uxx $ 2%xuxt

$ 3"uuxuxx $ %uutuxx $ 2%uuxuxt

Determining equations:

Coe!cient Monomial

0 = $2%u uxuxt

0 = $2%x uxt

$%u = $%u u2xx

$2%u = $%uu $ 3%u u2xuxx

$"u = $2%xu $ 3"u $ 2%x uxuxx

$u $ %t = $%xx + $u $ 2"x uxx

$%u = $%uu $ 2%u u4x

$"u = $2%xu $ "uu $ 2%x $ 2"u u3x

$u $ %t = $%xx + $uu $ 2"xu + 2$u $ 2"x u2x

$"t = 2$xu $ "xx + 2$x ux

$t = $xx 1

General solution:

" = c1 + c4x+ 2c5t+ 4c6xt

% = c2 + 2c4t+ 4c6t2

$ = c3 $ c5x$ 2c6t$ c6x2 + &(x, t)e"u

where &t = &xx

Symmetry algebra:

v1 = #x

v2 = #t

v3 = #u

v4 = x#x + 2t#t

v5 = 2t#x $ x#u

v6 = 4xt#x + 4t2#t $ (x2 + 2t)#u

v" = &(x, t)e"u#u where &t = &xx

& Lie pseudo-group

' Hopf-Cole transformation: w = eu maps to heat equation.

=" Bluman–Kumei

Symmetry algebra:

v1 = #x

v2 = #t

v3 = #u

v4 = x#x + 2t#t

v5 = 2t#x $ x#u

v6 = 4xt#x + 4t2#t $ (x2 + 2t)#u

v" = &(x, t)e"u#u where &t = &xx

& Lie pseudo-group

' Hopf-Cole transformation: w = eu maps to heat equation.

=" Bluman–Kumei

Determining the Structure

of the Symmetry Group

from the Determining Equations

• Reid (1991), Lisle–Reid (1998)— Taylor expansions and numerical linear algebra.

• Olver–Pohjanpelto (2005)— restriction of the structure equations for the

di!eomorphism pseudo-group to the determining equations.

Symmetry Groups ofFree Boundary Value Problems

=" Benjamin–Olver (1982)

Theorem. In the absence of surface tension, the two-dimensionalwater wave problem admits a nine parameter symmetry groupand, making use of Zakharov’s Hamiltonian structure,exactly eight local conservation laws.

' Surface tension reduces both counts by one.

Contact Structure on Jet Space

Definition. A di!erential form ( on jet space Jn is called acontact form if

(jnf)%( = 0 for all smooth functions u = f(x).

Contact Ideal:

C =%(&&& (jnf)

%( = 0 for all f'

In local coordinates, C is generated bythe basic contact one-forms

("J = du"J $

u"J,i dx

=" The variational bicomplex— Vinogradov, Tsujishita, I. Anderson

The Simplest Example.

M = R2, x, u # R

Horizontal formdx

Contact (vertical) forms

( = du$ ux dx

(x = Dx( = dux $ uxx dx

(xx = D2x( = duxx $ uxxx dx

The Variational Bicomplex... ... ... ... ...

%0,3 dH" %1,3 dH" · · · dH" %p"1,3 dH" %p,3 %" F3

%0,2 dH" %1,2 dH" · · · dH" %p"1,2 dH" %p,2 %" F2

%0,1 dH" %1,1 dH" · · · dH" %p"1,1 dH" %p,1 %" F1

####$E

R' %0,0 dH" %1,0 dH" · · · dH" %p"1,0 dH" %p,0

conservation laws Lagrangians PDEs (Euler–Lagrange) Helmholtz conditions

Variational problems

L[u ] =(

!L(x, u(n)) dx

Euler-Lagrange equations

" = E(L) = 0

Euler operator (variational derivative)

E"(L) =)L

($D)J#L

Local Exactness of the Variational Bicomplex

=" Vinogradov, Tsujishita, I. Anderson

Null Lagrangians/Conservation Laws

Theorem.

E(L) ( 0 if and only if L = DivP

Helmholtz Conditions

Theorem. " = 0 is the Euler-Lagrange equations for somevariational problem, so " = E(L), if and only if its Frechetderivative D" is self-adjoint:

D%" = D".

Frechet derivative

Given P (x, u(n)), its Frechet derivative (or formal lineariza-tion) is the di!erential operator DP defined by

DP [Q ] =d

d!P [u+ !Q ]

&&&&&! = 0= pr vQ(P )

(Formal) Adjoint

AJDJ D% =

($D)J · AJ

Conservation Laws

Definition. A conservation law of a system of partial di!eren-tial equations " = 0 is a divergence expression

DivP = 0

which vanishes on all solutions to the system.

Proposition. Every conservation law of a nondegeneratesystem of partial di!erential equations is equivalent to aconservation law in characteristic form

DivP = Q ·" ="

TheKey Formula

DivP = Q ·"

if and only if

0 = E(Q ·") = D%"(Q)$D%

ThusD%

"(Q) = 0 on solutions to " = 0

=" Olver (1980, 1986), Anco–Bluman (1997)

Example. Burgers’ equation: ut = uxx + uux

" = ut $ uxx $ uux

D" = Dt $D2x $ uDx $ ux

D%" = $Dt $D2

x + uDx

If Q = Q(t, x, u, ux, uxx, . . . , un), then

D%"Q =

($un,t $ un+2) + · · · = $2#Q

un+2 + · · ·

on solutions, which implies #Q/#un = 0, and hence,by induction, Q = q(t, x), whence

D%"Q = $ qt $ qxx + u qx = 0

implies that Q = constant, and hence the only conservation lawis, up to multiple, the equation itself:

Dtu$Dx(ux +12 u

2 ) = 0

Generalized Variational Symmetries

Definition. A generalized vector field is a variational symme-try if it leaves the variational problem invariant up to a di-vergence:

pr v(L) + LDiv " = DivB

' v is a variational symmetry if and only if its evolutionaryform vQ is.

pr vQ(L) = Div)B

Theorem. If v is a variational symmetry, then it is asymmetry of the Euler-Lagrange equations.

Proof :

Integration by parts:

pr vQ(L) = QE(L) + DivA

for some A depending on Q,L.

Therefore

0 = E(pr vQ(L)) = E(QE(L)) = E(Q")

= D%"Q+D%

Q" = D"Q+D%Q"

= pr vQ(") +D%Q"

establishing the infinitesimal symmetry conditions. Q.E.D.

Noether’s (First) Theorem. Let " = 0 be a normal

system of Euler-Lagrange equations. Then there

is a one-to-one correspondence between (equiva-

lence classes of) nontrivial conservation laws and

(equivalence classes of) nontrivial variational sym-

metries. The characteristic of the conservation law

is the characteristic of the associated symmetry.

The Kepler Problem

utt +µu

r3= 0 L = 1

2 u2t $

rr = )u )

Generalized symmetries:

v = (u · utt) #u + ut (u · #u)$ 2u (ut · #u)

Conservation lawpr v(L) = DtR

whereR = ut * (u * ut)$

r=" Runge-Lenz vector

Contact Transformations

=" Lie, Backlund

A local di!eomorphism on jet space

$ : Jn $' Jn

is a contact transformation if it preserves the contact ideal:

$%C + C

Prolongation: Given a contact transformation $ on Jn, there isa unique contact transformation pr(k)$ on Jn+k for k > 0which projects back down to $.

Example

The nonlinear di!usion equation

ut =umx

admits only a finite dimensional group of point symmetries, but has aninfinite-dimensional abelian algebra of contact symmetries generated by

vF = F (t, ux) #u

whose characteristic F (t, p) satisfies the linear parabolic equation

Ft = pmFpp

As with the potential Burgers’ equation, this implies linearizability. Thecontact transformation

y = ux, t = t, v = u$ xux, vy = $x, vt = ut

maps the equation to the linear di!usion equation

vt = ymvyy

Backlund’s Theorem

Theorem. Every contact transformation $ on Jn is the(n$ 1)st prolongation of a first order contact transformationon J1.

Moreover, if the number of dependent variables is more thanone, q > 1, then every contact transformation on Jn is thenth prolongation of a point transformation.

Proposition. A first order generalized symmetry is equivalentto an infinitesimal contact transformation if and only ifits characteristic Q = (Q1, . . . , Qq) satisfies the contactconditions

+ "i )"& = 0, &, * = 1, . . . , q, i = 1, . . . , p.

• If q > 1, the integrability conditions for these equations implythat the symmetry does not depend on the derivatives u&

and so is equivalent to a point transformation.

• If q = 1, these conditions serve to define the x-components "i

of the contact transformation, while the u component is

$" = Q" +p"

Internal Symmetries

=" Anderson–Kamran–Olver (1993)

Definition. An internal symmetry of a system of di!erentialequations R + Jn is an invertible transformation & : R ' Rwhich maps R to itself and which preserves the restrictionof the contact ideal on R :

&%(C | R) + C | R

The restrictions of Backlund’s Theorem do not apply, andthere are examples of internal symmetries which are notprolongations of first order transformations.

TheHilbert–Cartan Equation

The under-determined ordinary di!erential equation

vx = (uxx)2

was originally proposed by Hilbert as an example of asystem whose general solution cannot be written in termsof an arbitrary function and its derivatives.

• It was shown by Cartan to possess an internal symmetrygroup isomorphic to the 14 dimensional exceptional Liegroup G2.

Theorem. Every first order generalized symmetry of the Hilbert–Cartanequation is a linear constant coe'cient combination of the followingfourteen generalized vector fields

v1 = #u, v2 = #v,

v3 = x#u, v4 = x2#u + 4ux#v,

v5 = x3#u + 12(xux $ u)#v, v6 = u#u + 2v#v,

v7 = 3v#u + 4v3/2x #v, v8 = ux#u + vx#v,

v9 = (2xux $ 3u)#u + 2xvx#v,

v10 = (x2ux $ 3xu)#u + (x2vx $ 4u2x)#v,

v11 = (3xv $ 4u2x)#u + (4xv3/2x $ 8uxvx)#v,

v12 = (3x2v $ 8xu2x + 12uux)#u + (12uxv + 12uvx + 4x2v3/2x $ 16xuxvx)#v,

v13 = (3x3v $ 12x2u2x + 36xuux $ 36u2)#u +

+(36xuxv $ 36uv + 4x3v3/2x $ 24x2uxvx + 36xuvx $ 16u3x)#v,

v14 = (9uv $ 4u3x)#u + (9v2 $ 12u2

xvx + 12uv3/2x )#v.

Each generalized symmetry can be identified with an internal symmetry:

Theorem. Every internal symmetry is equivalent to afirst order generalized symmetry. Conversely, every first ordergeneralized symmetry satisfying the contact conditions

#u&x+ " )"& =

+"'#"'

#u&n, on R, &,* = 1, . . . , q (%)

for some functions "i,+"' is equivalent to an internal symmetry.

Except for systems of ordinary di!erential equations, thecontact conditions (%) are rather restrictive, and, except forsuitably “degenerate” systems of partial di!erential equations,imply that every internal symmetry is an ordinary (external)contact or point symmetry.

Generalized Vector Fields

and Generalized Symmetries

' Due to Noether (1918) ' NOT Lie or Backlund!

Key Idea: Allow the coe'cients of the infinitesimal generatorto depend on derivatives of u, but drop the requirementthat the (prolonged) vector field define a geometricaltransformation on any finite order jet space:

"i(x, u(k))#

$"(x, u(k))#

Characteristic :

Q"(x, u(k)) = $" $

Evolutionary vector field:

vQ =q"

Q"(x, u(k))

Prolongation formula:

pr v = pr vQ +p"

pr vQ ="

DJ = Dj1 · · ·Djk

=" total derivative

Generalized Flows

• The one-parameter group generated by the evolutionaryvector field

vQ =q"

Q"(x, u(k))

is found by solving the Cauchy problem for an associatedsystem of evolution equations

#!= Q"(x, u

(n)) u|!=0 = f(x)

& Existence/uniqueness?

, Ill–posedness?

Example. v =#

#xgenerates the one-parameter group of

translations:(x, y, u) -$' (x+ !, y, u)

Evolutionary form:

vQ = $ux

#xCorresponding group:

#!= $ux

Solutionu = f(x, y) -$' u = f(x$ !, y)

Generalized Symmetriesof Di!erential Equations

Determining equations :

pr v(") = 0 whenever " = 0

For totally nondegenerate systems, this is equivalent to

pr v(") = D" ="

' v is a generalized symmetry if and only if its evolutionaryform vQ is.

• A generalized symmetry is trivial if its characteristic vanisheson solutions to ". Two symmetries are equivalent if theirevolutionary forms di!er by a trivial symmetry.

Example. Burgers’ equation.

ut = uxx + uux

Symmetries:ux

uxx + uux

uxxx +32uuxx +

uxxxx + 2uuxxx + 5uxuxx +32u

2ux + 3uu2x +

12u3ux

Equations with one higher order symmetry almost always haveinfinitely many.

Equations with higher order generalized symmetries are called“integrable”.

Linearizable “C integrable” — Burgers’

Solvable by inverse scattering “S integrable” — KdV

See Mikhailov–Shabat and J.P. Wang’s thesis for long lists ofequations with higher order symmetries.

Beukers–Sanders–Wang use number theoretic techniques toclassify all integrable evolution equations, of a prescribedtype, e.g. polynomial with linear leading term

Bakirov’s Example:

The “triangular system” of evolution equations

ut = uxxxx + v2 vt =15 vxxxx

has one sixth order generalized symmetry,but no further higher order symmetries.

• Bakirov (1991)

• Beukers–Sanders–Wang (1998)

• van der Kamp–Sanders (2002)

Recursion operators

=" Olver (1977)

Definition. An operator R is called a recursion operator forthe system " = 0 if it maps symmetries to symmetries, i.e.,if vQ is a generalized symmetry (in evolutionary form), and!Q = RQ, then v!Q is also a generalized symmetry.

Note a recursion operator generates infinitely many symmetrieswith characteristics

Q, RQ, R2Q, R3Q, . . .

Theorem. Given the system " = 0 with Frechet derivative(linearization) D", if

[D",R] = 0

on solutions, then R is a recursion operator.

Example. Burgers’ equation.

ut = uxx + uux

D" = Dt $D2x $ uDx $ ux

R = Dx +12u+ 1

2uD"1x

D" · R = DtDx $D3x $

12(5ux + u2)Dx +

12ut $

$ 32uxx $

32uux +

12(uxt $ uxxx $ uuxx $ u2

x)D"1x ,

R ·D" = DtDx $D3x $

12(5ux + u2)Dx $ uxx $ uux

[D",R] = 12(ut $ uxx $ uux) +

12(uxt $ uxxx $ uuxx $ u2

x)D"1x

which vanishes on solutions.

Symmetries:ux

uxx + uux

uxxx +32uuxx +

uxxxx + 2uuxxx + 5uxuxx +32u

2ux + 3uu2x +

Second recursion operator:

)R = tR+ 12 x = tDx +

12 t u+ 1

2x+ 12 t uD

Linear Equations

Theorem. Let"[u] = 0

be a linear system of partial di!erential equations. Then anysymmetry vQ with linear characteristic Q = D[u] determines arecursion operator D.

[D,"] = !D ·"Therefore, if D1, . . . ,Dm determine linear symmetries vQ1

, . . . ,vQm,

then any polynomial in the Dj’s also gives a linear symmetry.

Question: Given a linear system, when are all symmetriesa) linear? b) generated by first order symmetries?

Evolution Equations

Consider a kth order evolution equation

ut = K[u] = K(x, u(k))

A generalized vector field vQ determines a symmetry if and onlyif the flows commute

us = Q[u] = Q(x, u(n))

Note, on solutions,

Qt = pr vK(Q) = DQ(K)

Infinitesimal criterion:

pr vQ(K)$ pr vK(Q) = 0

orQt $Ks = DK(Q)$DQ(K) = 0

Linearize: first note that

DQt= (DQ)t +DQ ·DK

Thus we get the linear symmetry condition

(DQ)t $ (DK)s $ [DK, DQ] = 0

degrees: n k n+ k $ 1

For n . k, the dominant terms are

(DQ)t $ [DK,DQ]

which must vanish, modulo a di!erential operator of degree k.

Formal Symmetries

=" Mikhailov–Shabat (1987)

A (pseudo-)di!erential operator D of degree m is called a formalsymmetry of order n of the kth order evolution equation

ut = K[u] = K(x, u(k))

ifdeg(Dt $ [DK,D] ) / m+ k $ n

Proposition. If Q(x, u(n)) is the characteristic of an nth ordergeneralized symmetry, then its Frechet derivative DQ is aformal symmetry of order n.

A formal symmetry of order 0 is a recursion operator for theequation. Integrable equations are characterized by theexistence of a recursion operator.

Integrable second order equations

Theorem. Every integrable second order evolution equationis equivalent, under a contact transformation, to one of thefollowing:

ut = uxx + q(x)u

ut = uxx + uux + h(x)

ut = (u"2ux + axu+ bu)x

ut = (u"2ux)x + 1

Second order equations:

A second order evolution equation is integrable if andonly if it has a formal symmetry of order 5.

Third order equations:

A third order evolution equation is integrable if andonly if it has a formal symmetry of symmetry oforder 8.

Symmetry–Based Solution Methods

Ordinary Di!erential Equations

• Lie’s method

• Solvable groups

• Variational and Hamiltonian systems

• Potential symmetries

• Exponential symmetries = +–symmetries

• Generalized symmetries

Solvable Groups

Theorem. (Bianchi) If an nth order o.d.e. has a(regular) r-parameter solvable symmetry group,then its solutions can be found by quadraturefrom those of the (n$r)th order reduced equation.

Example

x2 u$$ = f(xu$ $ u)

Symmetry group:

v = x #u, w = x #x, [ v,w ] = $v.

Reduction with respect to v: set z = xu$ $ u. The reduced equation

x z$ = h(z)

remains invariant under w = x #x, and hence can be solved by quadrature.

Wrong way reduction with respect to w:

y = u, z = z(y) = xu$

Reduced equation:z(z$ $ 1) = h(z $ y)

• No remaining symmetry; not clear how to integrate directly.

Generalized Symmetries

Theorem. If an nth order ordinary di!erential equation is integrable byquadrature, it possesses n independent commuting generalized symme-tries.

One uses x, I1(x, u(n)), . . . , In(x, u

(n)), where the I’s are the first integralsof the equation, as new coordinates on the jet space. The vector fieldsvj = #/#Ij , when rewritten in the standard derivative coordinates

(x, u(n)) are generalized symmetries.

, Not e!ective: the determining equation for the generalized symmetries isan nth order linear partial di!erential equation, which is considerablymore complicated than the original o.d.e.

Exponential Symmetries= C% Symmetries = + Symmetries

=" Olver (1986), Muriel–Romero (2001), Cicogna–Gaeta–Morando (2004)

• Integration of ordinary di!erential equations without Lie symmetries.

Nonlocal exponential symmetry generator:

v = e*(dx v( = e

*(dx[ "((x, u)#x + $((x, u)#u ]

Prolongation:

pr v = e*(dx pr v( where pr v( = v( +

$n((x, u

$n+1( = (Dx + +)$n

( $ un(Dx + +)"

Theorem. Every reduction in order can be associated withsuch an exponential symmetry!

Example. The ordinary di!erential equation

u(ux $ 1) = h(u$ x)

has the exponential symmetry

v = e*(u"1)dx #

#ui.e.

#uwith + = u$ 1

Potential Symmetries

=" Bluman, Kumei, Reid, Anco, Cheviakov, . . .

Substitute u = f(x, v, vx): conservative form

DxH(x, v(n)) = 0

Local symmetries of the equation

H(x, v(n)) = c

may give non-local symmetries of the original equation.

Example. Nonlinear heat conduction

2[(1 + u2)ux]x + xux = 0

Substitution u = vx leads to

2(1 + v2x)vxx + xvx $ v = 0

The latter ordinary di!erential equation is SO(2)–invariant. Tointegrate, set

, = d(/dr

whereby2,r + (4r $ 1 + r), + (2r + r2),3 = 0

=" Bernoulli equation

Partial Di!erential Equations

• Group-invariant solutions

• Non-classical method

• Weak symmetry groups

• Clarkson-Kruskal direct method

• Partially invariant solutions

• Di!erential constraints

• Nonlocal symmetries

• Separation of variables

Group Invariant Solutions=" Lie (1895), Ovsiannikov (1958)

System of partial di!erential equations

"(x, u(n)) = 0

Assume the symmetry group G acts regularly on M with r-dimensional orbits intersecting the vertical fibers transversally.

Definition. u = f(x) is a G-invariant solution if

g · f = f for all g # G.

i.e. the graph (f = { (x, f(x))} is a (locally) G-invariant subset.

• Similarity solutions, travelling waves, . . .

• Non-transversal version: Anderson–Fels (1997)

Proposition. Let G have infinitesimal generators v1, . . . ,vr

with associated characteristics Q1, . . . , Qr. A functionu = f(x) is G-invariant if and only if it is a solution tothe system of first order partial di!erential equations

Q#(x, u(1)) = 0, - = 1, . . . , r.

Theorem. (Lie) If G has r-dimensional orbits, and actstransversally to the vertical fibers {x = const.}, then allthe G-invariant solutions to " = 0 can be found by solvinga reduced system of di!erential equations "/G = 0 in rfewer independent variables.

Proposition. Let G have infinitesimal generators v1, . . . ,vr

with associated characteristics Q1, . . . , Qr. A functionu = f(x) is G-invariant if and only if it is a solution tothe system of first order partial di!erential equations

Q#(x, u(1)) = 0, - = 1, . . . , r.

Theorem. (Lie) If G has r-dimensional orbits, and actstransversally to the vertical fibers {x = const.}, then allthe G-invariant solutions to " = 0 can be found by solvinga reduced system of di!erential equations "/G = 0 in rfewer independent variables.

Method 1: Invariant Coordinates.

The new variables are given by a complete set of function-ally independent invariants of G:

."(g · (x, u)) = ."(x, u) for all g # G

Infinitesimal criterion:

vk[."] = 0, k = 1, . . . , r.

New independent and dependent variables:

y1 = .1(x, u), . . . , yp"r = .p"r(x, u)

w1 = /1(x, u), . . . , wq = /q(x, u)

Invariant functions:

w = .(y), i.e. /(x, u) = h[.(x, u)]

Reduced equation:"/G(y,w(n)) = 0

Every solution determines a G-invariant solution to the originalp.d.e.

Example. The heat equation ut = uxx

Scaling symmetry: x #x + 2 t #t + a u #u

Invariants: y =x1t, w = t"au

u = taw(y), ut = ta"1($ 12 y w

$ + aw ), uxx = taw$$.

Reduced equation

w$$ + 12yw$ $ aw = 0

Solution: w = e"y2/8U( 2 a+ 12, y/

=" parabolic cylinder function

Similarity solution:

u(x, t) = tae"x2/8tU( 2 a+ 12, x/

12 t )

Example. The heat equation ut = uxx

Galilean symmetry: 2 t #x $ xu #uInvariants: y = t w = ex

u = e"x2/4tw(y), ut = e"x2/4t

w$ +x2

uxx = e" x2/4t

Reduced equation: 2 y w$ + w = 0

Source solution: w = k y"1/2, u =k1tex

Method 2: Direct substitution:

Solve the combined system as an overdetermined system of p.d.e.:

"(x, u(n)) = 0, Qk(x, u(1)) = 0, k = 1, . . . , r

where the Qj’s are the characteristics of the infinitesimal generators(invariant surface condition)

For a one-parameter group, we solve

Q(x, u(1)) = 0

for#u"

#xp=$"

Rewrite in terms of derivatives with respect to x1, . . . , xp"1. The reducedequation has xp as a parameter. Dependence on xp can be found bysubstituting back into the characteristic equation.

Non-Classical Method

=" Bluman–Cole (1969)

Here we require not invariance of the original partial di!erentialequation, but rather invariance of the combined system

"(x, u(n)) = 0, Qk(x, u(1)) = 0, k = 1, . . . , r

• Nonlinear determining equations.

• Which nonlinear equations admit non-classical symmetrieswhose solutions cannot be derived using ordinary groupinvariance?

Weak (Partial) Symmetry Groups

=" Olver–Rosenau (1987)

Here we require invariance of

"(x, u(n)) = 0, Qk(x, u(1)) = 0, k = 1, . . . , r

and all the associated integrability conditions

• Every group is a weak symmetry group.

• Every solution can be derived in this way.

• Compatibility of the combined system?

• Overdetermined systems of partial di!erential equations.

The Boussinesq Equation

utt +12(u

2)xx + uxxxx = 0

Classical symmetry group:

v1 = #x v2 = #t v3 = x #x + 2 t #t $ 2u #u

For the scaling group

$Q = xux + 2 t ut + 2u = 0

Invariants:

y =x1t

w = t u u =1

Reduced equation:

w$$$$ + 12 (w

2)$$ + 14 y

2w$$ + 74 y w

$ + 2w = 0

utt +12(u

2)xx + uxxxx = 0

Non-classical: Galilean group

v = t #x + #t $ 2 t #u

Not a symmetry, but the combined system

utt +12(u

2)xx + uxxxx = 0 $Q = t ux + ut + 2 t = 0

does admit v as a symmetry. Invariants:

y = x$ 12 t

2, w = u+ t2, u(x, t) = w(y)$ t2

Reduced equation:

w$$$$ + ww$$ + (w$)2 $ w$ + 2 = 0

utt +12(u

2)xx + uxxxx = 0

Weak Symmetry: Scaling group: x #x + t #tNot a symmetry of the combined system

utt +12(u

2)xx + uxxxx = 0 Q = xux + t ut = 0

Invariants: y =x

tu Invariant solution: u(x, t) = w(y)

The Boussinesq equation reduces to

t"4w$$$$ + t"2[(w + 1$ y)w$$ + (w$)2 $ y w$] = 0

so we obtain an overdetermined system

w$$$$ = 0 (w + 1$ y)w$$ + (w$)2 $ y w$ = 0

Solutions: w(y) = 23 y

2 $ 1, or w = constant

Similarity solution: u(x, t) =2 x2

3 t2$ 1

DirectMethod

=" Clarkson–Kruskal (1989)

Basic similarity ansatz:

u(x, t) = U(x, t, w(z)) z = z(x, t)

' Goal: reduce the p.d.e. to an o.d.e. for w(z)

• Subsumed by nonclassical method(Levi–Winternitz, Clarkson–Nucci)

Boussinesq equation:

utt +12(u

2)xx + uxxxx = 0

Direct Ansatz:

u(x, t) = a(x, t) + b(x, t)w(z) z = z(x, t)

Plug into the equation:

0 = bz4xw$$$$ + (6bz2xzxx + 4bxz

$$$ + b2z2xww$$

+ b (3z2xx + 4zxzxxx + · · · + z2t )w$$ + · · ·

For this to be an ordinary di!erential equation for w(z), thecoe'cients of the di!erent monomials must be functions of z.There are some freedoms in the definitions of a, b, which can beused to advantage:

bz4x = b2 z2xg(z)

so, up to a function of z we can set b = z2x.

Final result:

u(x, t) = q(t)2w(z)$ q(t)$ 2(xq + s)2

z(x, t) = xq(t) + s(t)

q$$ = Aq5 s$$ = q4(As+B)

w$$$$ + ww$$ + w$2 + (Az +B)w$ + 2Aw = 2(Az + B)2

=" Painleve equation

Di!erential Constraints

=" Meleshko (1983), Sidorov–Shapeev–Yanenko (1984),Olver–Rosenau (1986)

Append extra “side conditions” or di!erential constraints to theoriginal system:

"(x, u(n)) = 0

Qj(x, u(k)) = 0 j = 1, . . . ,m

2 Includes (almost) all methods, including group-invariantsolutions, non-classical and weak solutions,partially invariant solutions, separation of variables, etc.

' “Generalized weak symmetries”

, Too general: which constraints are compatible??Reasonable ansatzes??

Example. Separation of variables

uxx + uyy = 0

Additive:

uxy = 0 =" u(x, y) = v(x) + w(y)

Multiplicative:

uuxy $ uxuy = 0 =" u(x, y) = v(x)w(y)

Example. Boussinesq equation

utt +12(u

2)xx + uxxxx = 0

Di!erential constraint

uuxt $ uxut +14 uxt = 0

or, equivalently,u = v(t)w(x)$ 1

Reduced system

vtt = + v2, 12 (w

2)xx = + (w $ wxx).

Elliptic solutions, including

u =(x+ c)2 + 3

(t+ d)2 $ 1

Partially Invariant Solutions

=" Ovsiannikov (1958)

System of partial di!erential equations

"(x, u(n)) = 0

G — symmetry group

r — dimensional orbits

(f — graph of a solution dim(f = p

Defect:dimG · (f = p+ d 0 / d / min{r, q}

• d = 0 =" f is G-invariant

• d < r =" f is partially invariant

Suppose the infinitesimal generators of G

v1, . . . ,vr

have characteristics

Qj(x, u(1)) = (Q1

j , · · · , Qqj), j = 1, . . . , r

Form the r 3 q “characteristic matrix”

Q(x, u(1)) = (Q"j (x, u

(1)) )

Theorem. u = f(x) is a partially invariant solution to " = 0of rank d if and only if

rankQ(x, u(1)) = d.

• Most partially invariant solutions are invariant under somesubgroup.

This is always true if the system is elliptic and d = 1 — Ondich

Example. (Ovsiannikov)

Equations for (steady y = t) trans-sonic gas flow

ut = vx vt + uux = 0

or, combined,utt +

2)xx = 0

Symmetry generators

v1 = x#x + t#t

v2 = x#x + 2u#u + 3v#v

v3 = #v

v4 = (tu2 $ xv)#x $ (xu+ 2tv)#t + 2uv#u + (32v2 $ 2

3u3)#v

v% = "(u, v)#x + %(u, v)#t where "v = %u, "u = $u%v

Partially invariant solutions for the subgroup generated by

v0 = #t v3 = #v

Characteristic matrix:

#ut vt0 1

For defect d = 1, we require

rankQ = 1 4" ut = 0

Overdetermined system:

ut = vx vt + uux = 0 ut = 0

Invariants:u x

Equation for G · (:u = f(x)

Substitute into system:

ut = vx vt = $uux = $ff $

Integrability conditions:

[ ff $ ]$ = 0

u = f =1ax+ b v = 1

1a t+ c

v1 = x#x + t#t v3 = #vCharacteristic matrix:

#xux + tut xvx + tvt

rankQ = 1 4" xux + tut = 0

Overdetermined system:

ut = vx vt + uux = 0 xux + tut = 0

Invariants:u z =

Equation for G · (:u = f(z) = f(x/t)

Substitute into system:

vx = ut = $z

tf $ vt = $uux = $ff $

Integrability conditions:

[ (z2 + f)f $ ]$ = 0 =" (z2 + f)f $ = c

=" Abel equation

Example. Ondich (1989)

Boundary layer (Prandtl) equations

uyy = ut + uux + vuy + px ux + vy = 0 = py

Symmetry generators

v1 = #t

v2 = 2t#t + 2x#x + y#y $ v#v

v3 = x#x + u#u + 2p#p

va = a(t)#x + a$(t)#u $ a$$(t)x#p

vb = b(t)#y + b$(t)#v

v0 = #t vb=1 = #y v3 = x#x + u#u + 2p#p

Characteristic matrix:

,-ut vt ptuy vy 0

xux $ u xvx xpx $ 2p

rankQ = 2

Group Splitting

• Lie (1895)

• Vessiot (1904)

• H.H. Johnson (1962)

• Ovsiannikov (1969)

• Nutku–Sheftel, Martina–Sheftel–Winternitz (2001)

• R. Thompson (2013)

Key idea: Given a system of partial di!erential equations

"(x, u(n)) = 0

with symmetry (pseudo-)group G, “split” the system into:

• An automorphic system

A(x, u(n)) = 0

with symmetry group G and the property that if u0 is onesolution, then the most general solution is u = g · u0. Inother words, all solutions can be obtained from a singlesolution by applying group transformations.

• A resolving systemR(y, v(n)) = 0

(hopefully simpler than the original equation)with no residual symmetry inherited from G.

More correctly, each particular solution to the resolving sys-tem determines a corresponding automorphic system, fromwhich can be reconstructed solutions to the original system.The resolving system is expressed in terms of the associateddi!erential invariants and their sygygies (di!erential iden-tities) which can be systematically constructed using thetheory of equivariant moving frames.

Thompson’s approach to solving the automorphic systemrelies on the reconstruction equations obtained from thecorresponding moving frame.

Group splitting provides a means of systematically constructingnon-invariant and partially invariant solutions.

A Simple Example

Original system: ut = uxx $u2x

uSymmetry group: (t, x, u) -$' (t, x,+u)

Di!erential invariants:

t, x, I =ux

u, J =

Resolving system:

J = DxI, DxJ = DtI, or, equivalently It = Ixx.

Given a solution I = f(t, x), J = fx(t, x), to the resolvingsystem, the corresponding automorphic system is

ux = f(t, x)u, ut = fx(t, x)u.

Example:

I = x, J = 1, u(t, x) = c ex2/2+t.

ahistoricaloverviewof symmetry methods for diﬀerential …cheviakov/bluman2014/talks/olver.pdf ·...

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