1. V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York,

(1978).

2. 1. Perceive1 and D. Richards, Introduction to Dynamics, Cambridge University

Press, Cambridge (1982)

3. A. Somrnerfeld, Partial Differential Equations in Physics, Academic Press (1949).

4. R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. 1 and 2, Wiey

(1961).

5. M.J.Ablowitz and P.A.Clarkson, Solitons, Nonlinear Evolution Equations and In-

verse Scattering, Cambridge University Press, Cambridge (1991).

6. L.D. Faddeev, L. A. Takhtejan, Hamiltonian Methods in the Theory of Solitons.

Springer Verlag, Berlin 1987.

7. M. Toda, Nonlinear Waves and Solitons, Kluwer, Dordrecht, Holland (1989).

8. M. Lakshmanan and K. Mwali. Cham in Nonlinear Oscillators : Controlling and

Synchronization. World Scientific, Singapore (1996).

9. N. J . Zabusky and M. D. Kruskal, Interactions of "Solitons" in a Collisionless

Plasma and the Recurrence of the Initial States, Phys. Rev. Lett. 15, 240 (1965).

10. C. S. Gardner. J. M Greene, M. D. Kruslral and R. M Miwa, Method for Solving

the Korteweg-de Vries Equation, Phys. Rev. Lett. 19, 1095 (1967).

11. P. D. Lax, Integral of Nonlinear Equations of Evolution and Solitary Waves,

Comm. Pure. Appl. Math. 21, 467 (1968).

12. M. J. Ablowitz, D. J. Kaup, A. €. Newel! and H. Segur, Method for Solving the

Sine-Gordon Equation, Phys. Rev. Lett. SO, 1262 (1973).

13. V. E. Zdrharov and A. B. Shabat, Exact Theory of TmDimensional Self-Focusing

and O n e - D i i o n a l Self-Modulation of Waves in Nonlinear Media, Sov. Phys.

JETP S4,62 (1972).

14. R Hirota, Direct Method in Soliton Theory, Iwanami, Tokyo (1992). [in Japanese].

15. J.Matsuno, Bilinear Transformation Method, Academic Ptess, London (1984).

16. R Hirota, Soliton Solutions to the BKP Equations.1. The Pfaffian Technique, J.

P M . Soc. Jpn. 68,2285 (1989).

Soiiton S o l u h to the BKP Equation. II. The Integral Equation, J. Phys. Soc.

Jpn. 58, 2705 (1989).

17. R H i i and J.Sateuma, Soliton Solutions of a Coupled Korteweg-de Vriea Equa-

t h , Php. Lett 85A, 407 (1981).

18. J.Hietarinta, A Search for Bilinear Equations Passing Hirota's Three-Soliton Con-

dition.1. KdV-Type Bilinear Equations J. Math. Phys. 28, 1732 (1987).

A Search for Bilinear Equations Passing Hirota's Three-Soliton Condition.11. MKdV-

Type Bilinear Equations J. Math. Phys. 28, 2094 (1987).

A Search for Bilinear Equations Passing Hirota's ThreSoliton Condition.111. Sine

Gordon-Type Bilinear Equations J. Math. Phys. 28, 2586 (1987).

A Search for Bilinear Equations Passing Hirota's ThreeSoliton Condition.IV.

Complex Bilinear Equations J. Math. Phys. 29, 628 (1988).

19. J.Satsuma, N-Soliton Solution of the Two Dimensional Korteweg-de Vries Equa-

tion. J. Phys. Soc. Jpn. 40. 286 (1976).

20. J.Satsurna, A Wronskian Representation of N-Soliton Solutions of Nonlinear Eve

lution Equations, J. Phys.Soc. Jpn. 46, 359 (1979).

21. N.C.Reeman and J.J.C.Nimmo. Soliton Solutions of the Korteweg-de Vries and

Kadomtsev-Petviashvilli Equations : The Wronskian Technique, Phys. Lett. 95A,

1 (1983).

22. J . J. C. Nimrno and N. C. Freeman, A Method of Obtaining the Soliton Solution

of The Boussinesq Equation in Terms of a Wronskian. Phys. Lett. 95A, 4 (1983).

23. R. Hirota. Direct Methods in Soliton Theory, in Solitons, Eds. R. K. Bullough

and P. J . Caudrey, Springer Verlag, Berlin, 157 (1980).

24. S. Miydte, Y. Ohta and J. Satsuma, A Representation of Solutions for the KP

Hierarchy and Its Algebraic Structure, J. Phys. Soc. Jpn. 59, 48 (1990).

25. R Hirota. Y. Ohta and J . Satsuma, Solutions of the KP Equation and the Two

D i o n d Toda Equations, J . Phys. Soc. Jpn. 57. 1901 (1988).

26. R. Hirota, Y. Ohta and J. Satsuma, Wronskian Strucutwes of Soliton Equations,

Prog. Thaor. Phys. Suppl. 94, 59 (1988).

27. M. J. Ablowitz and H. Segur, Exact Lineariaation of a Painled hmcendent,

Phys. Rev. Lett. 38, 1103 (1977).

28. M. J. Ablowitz, A. Raaani and H. Segur, Nonlinear Evolution Equations and

O d h r y Di&asntid Equations of Painled Type, Lett. Nuovo Cim. 2S, 333

( 1 m .

29. A. Rammi, B. Gremmaticos and T. Bountis, The Paink6 Property and Singu-

h i t y Analysie of Integrable and Non-Integrable Syetems, Phys. Rep. 180, 159

(1984).

30. M. l,&hmmm and R Sahadevan, PainlwC Analysis, Lie Symmetries and Inte-

grabiity of Coupled Nonlinear Oscillators of Polynomid Type, Phys. Rep. 224,

1 (1993).

31. J. Web, M.Tabor and G.Carnede, The Painlev4 Property for Partial Differential

Eguations, J. Math. Phys. 24! 522 (1983).

32. R. Conk, Invariant Painlev6 Analysis of Partial Dierential Equations, Phys. Lett.

lWA, 100 (1988); 140A, 383 (1989).

33. D. Levi and P. Winternitz eds : Painlev6 ~ ~ d e n t s , Their Asymptotics and

Physical Applications, Plenum, New York (1992).

34. H. Wahlquist and F. B. Estabrook, Prolongation Structures of Nonlinear Evolution

Equations, J. Math. Phys. 16, 1 (1975).

35. F. B. Mabrook and H. Wahlquist, Prolongation Structures of Nonlinear Evolution

Equations 11, J. Math. Phys. 17, 1293 (1976).

36. P.J.Olver, Applications of Lie Groups to DifTerential Equations Springer-Verlag,

New York (1986).

37. A.S.Fokas and RL. Anderson, On the Use of Isospectral Eienvalue Problems for

Obtaining Hereditary Symmetries for Hamiltonian Systems, J. Math. Phys. 23.

loss (1982). 38. B.hduPsteiner and A.S.Fokas, Symplectic Structures, their W u n d 'handorma-

tions and Hereditary Symmetries, Physica 4D, 47 (1981).

39. A.S.Foh and P.M.Santini, The Recursion Operator of the Kadomtsev-Petviashvilli

E g W n and The Squared Eigenfunctions of the Schrodinger Operator, Stud.

Appl. Math. 76, 179 (1986).

40. B. G. Konopelchen&, N o w Integrable Equations, Lect. Notee in Phys. 270,

Springer Vcalag, Heidelberg (1b7).

41. H.H.Chen, Y.C.Lee and J.E.Liu, On a new Hierarchy of Symmetries for the

Kdasntrrv-Petviaahvili Equation, Physica BD, 439 (1983).

42. K. M. 'hmirhmani, Geometdcal, Group Theoretical and S i t y Strucutre Ae

pecb d Ccat.in Nonlinear Partirl Djfkentid -ions, Ph.D. Thesis, B h a r W

h Univanlty, TiruTiruchirrpdli, India (1986).

a. F. Magri, A Simple Model of the Integmble. Ehdtonian Equation, J. Math. Phys.

I@, 1156 (1978).

44. L. A. Dichey, Soliton Equations and Hamiltonian Systems, Adv. Ser. Math. Phys.

12, World Scientific, Singapore (1991).

45. W.Oeve1, H.Zhang and B.Fuchsateier, Symmetries, C o r n e d Quantities and

Hierarchies for some Lattice Systems with Soliton Structure, J. Math. Phys. 32,

1908 (1991).

46. W.Oevel, B.Fuchssteiner, H. Zhang and O.Ragnisco, Master Symmetries, Angle

Variables and Recursion Operator of the Relativistic Toda Lattice, J. Math. Phys.

SO, 2661 (1989).

47. W.Oevel, H.Zhang and B.Fu*er, Master Symmetries and Multi-Hamiltonian

Formulations for Some Integrable Lattice Systems, Prog. Theor. Phys. 81, 294

(1989).

48. A. S. F o b , Symmetries and Integrability, Stud. Appl. Math. 77. 253 (1987).

49. W. Oevel and B. Fuchssteiner. The Bi-Hamiltonian Structure of some Nonlinear

FiHh and Seventh-order D~fferential Equations and Recursion Formulas for their

Symmetries and Conserved Cowiants, J. Math. Phys. 23, 358 (1982).

Explicit Formulas for the Symmetries and Conservation Laws of the Kadomtsev-

Petviashvili Equation, Phys. Lett. 88A, 323 (1982).

New Hierarchies of Nonlinear Completely Integrable Systems related to a Change

of Variables for Evolution Parameten, Physica 1&A, 67 (1987).

Y. Koemann-Schwanbach, Lie Algebras of Symmetries Partial Differential Equa-

ti-, in Differential Geometric Methods in Mathematical Physics, Ed. S. Stern-

berg, D. Reidel Publishing Company 241 (1984).

W. Strampp, Lax-peirs, Spectral Problems, and Recursion Operators, J. Math.

Plays. 25,2905 (1984).

H. Zh.ng, Gui-Zhang 'h, W. Oevel and B. Fucbteiner, Symmetries, Conserved

Quantitiss and Hieardries of Some h t i c e Syetems with Soliton structure. J.

Muh. Plays. 31, 1908 (1991).

50. H. Stsphd, Diercmtid Equations. Tbek Solution Using Symmetries, Cambridge

UnivsSaty Pram, Cambridge (1989)

51. L V. OvrLnnihw, Group Amlysie of D h t i d Equations, Academic h,

N a Yo& (1982).

52. bA.Sato, Soliton Equatione ae Dynamical SyetemYl on an Wte Dimensional Grass

n r a ~ Manifold, RIMS, Kokyuroh 439, 30 (1981).

53. M.Sato and Y . b , Soliton Equatim as Dynamical Systems on Infinite Grass-

mann Manifold, in Nonlinear Partial DiEerential Egustions in Applied Science,

Eds. H.&jita, P.D.Lax and G.Strang, Kinokdp/North-Hohd, Tokyo, 259

(1983). 54. Y.Ohta, J.Satsuma, D.?Bkahashi and T.Tokihiro, An Elementary Introduction to

Sato Theory, Prog. Theor. Phys. Suppl. 94, 210 (1988).

55. E.Date, M.Jimbo, M.Kashiwara and T.Miwa, 'Ttsnsformation Groups for Soliton

Equations, "Proc. RIMS Symp. Nonlinear Integrable Systems, Classical and

Quantum Theory (Kyoto 1981)", 39, World Scientific, Singapore (1983).

56. M.Jimbo and T.Miwa, Solitons and Infinite Dimensional Lie Algebras, Publ. RIMS.

Kyoto Univ. 19, 943 (1983).

57. E.Date, M.Jimbo and T.Miwa, Method for h a t i n g Discrete Soliton Equations

I-V, J. Phys. Soc. Jpn. 51, 4116, 4125 (1982); 52, 388, 761, 766 (1983).

58. K.Ueno and K.Takasaki, Group Representations and systems of Differential equa-

tions. Adv. Stud. in Pure Math. V4, Kinokuniya. Tokyo, p.1 (1984).

59. J.Matsukidaira, J.Satsuma and W.Strampp, Conserved Quantitias and Symme-

tries of KP H'ierarchy, J. Math. Phys. 31. 1426 (1990).

60. K.Kajiwara, J.Matsukidaira and J.Satsuma, Conserved Quantities of Two Com-

ponent KP Hierarchy. Phys. Lett. 146A. 115 (1990).

61. K.Kajiwara and J.Sstsuma, The Conserved Quantities and Symmetries of the

Turr-Dimensional Toda Lattice Hierarchy, J. Math. Phys. 32, 506 (1991).

62. B. G. Konopelchenko and W. Oevel, Sato theory and Integrable huations in

2+1 Dimensions, In P m e d q s of the 7th Workshop on Nonlinear Evolution

Equations and Dynamid Systems (NEEDS191), Baia Verde, Italy, 19-29, June

(lQDl) 63. B. Konopelchenko and W. M., An r-matrix Approach to Nonstandard Clesses

of Intagratale Equations, Publ. RIMS. Kyoto Univ. 29. 581, (1993).

64 H . w The Toda Lattia.1 Existence of Integrals, Phys. Rev. 9B, 1924

(1974).

65. H.- On tbe Toda Lattia.11 Invarse Scattering Solution, Prog. Theor.

Phys. Suppl. 51, 703 (1974).

86. M. J.Ablowitz and J . F . M , Nonlinear ~ ~ M & a e m c e Equations, J . Math.

Phys. 16, 598 (1975).

67. M.J.Ablowitz and J.F.Ladik, Nonlinear DiEerential-Dierence Equations and Fourier

Anaysis, J. Math. Phys. 17, 1011 (1976).

68. M.Toda and M.Wadati, Biicklund Transformations for the Exponential Lattice, J.

Phys. Soc. Jpn. 39, 1196 (1975).

69. B.A.Kuperechmidt, Discrete Lax Equations and Differential-Difference Calculus,

h t h q u e , 123, 1 (1985)

70. R. Hirota, Exact N-Soliton Solutions of a Nonlinear Lumped Network Equation,

J. Phys. Soc. Jpn. 40, 286 (1976).

71. R Hirota, Exact N-Soliton Solutions of Nonlinear Self-dual Network Equations,

J. Phys. Soc. Jpn. 35, 289 (1973).

72. R. Hirota, Nonlinear Partial Difference Equations. I. A Difference Analogue of the

Korteweg-de Vries Equation, J. Phys. Soc. Jpn. 43, 1424 (1977).

73. R. Hirota, Nonlinear Partial Difference Equations. 11. Discretetime Toda Equa-

tion, J. Phys. Soc. Jpn. 43, 2074 (1977).

74. R Hirota. Nonlinear Partial Difference Equations. 111. Discrete Sine-Gordon

Equation, J. Phys. Soc. Jpn. 43, 2079 (1977).

75. R. Hirota, Nonlinear Partial Difference Equations. W. B a u n d aansformation

for the Discretetime Toda Equation, J. Phys. Soc. Jpn. 45, 321 (1977).

76. R Hirota. Nonlinear Partial Dierence Equations. V. Nonlinear Equations Re-

duable to Linear equations. J. Phys. Soc. Jpn. 46, 312 (1979).

77. R. Hirota, Dincrete Analogue of a Generalized Toda Equation, J. Phys. Soc. Jpn.

50, 3785 (1981).

78. R Huota and J. Satsuma, Nonlinear Evolution Equations Generated from the

Biddund Tkdormation for the To& Lattice, J. Phys. Soc. Jpn. 40. 891

(1976).

79. B. Grammaticod, A. Ramani lrnd V. G: Papgagemgiou, Do Integrable Mappirgs

have the PdnhvC Prom?, Phys. Rev. Lett. 67, 1825 (1991).

80. A. Rsmani, B. G r d c r m andJ. Hktsr i ta , Discrde Versions of the Painlev4

squrtbnr, Phys. Rav. Lett. 67, 1829 (1991).

81. V. G. P-, F. W. N i j M , B. Grunmrtiox and A. Ramani, Isomon-

odromic Deformation Problem for D i e t e Andogues of Painlev6 Equations,

Phys. Lett. 164A, 57 (1992).

82. B. Grsmmsticos, A. Ramani and I . C. Moreira, Delay-Difierential Equations and

the Painlev6 Tramcmdents, Physica lMA, 574 (1993).

83. B. Grammati- and A. Ramani, D i e t e Painlwd equations : Derivation and

Properties, NATO AS1 C413, 299 (1993).

84. A. Ramani and B. Grammaticos, Miura 'Ikansforms for Discrete Painlev6 Equa-

tions, J. Phys. A : Math. Gen. 25, L633 (1992).

85. A. Ramani, B. Grammaticos and G. Karra, Liearizable Mappings, Physica 181A,

115 (1992).

86. B. Grammaticos and A. Ramani, Integrability - and How to Detect It, in Trite-

gability of Nonlinear Systems", Lect. Notes. in Physics, Proc. of Cimpa Inter-

national Winter School on "Nonlinear Systemsn, Pondicherry Eds. Y. Kosmann-

Schwanbach, B. Grammaticos and K. M. Tamizhmani, Springer Verlag, Berlin,

30 (1996).

87. K.Kajiwara, Y. Ohta, J. Satsuma, B. Grammaticos and A. Ramani, Casorati

Determinant Solutions for the Discrete PainlevGI1 Equation, J. Phys. A : Math.

Gen. 27, 915 (1994).

88. B. Grammaticos and A. Ramani, Investigating The Integrability of Discrete Sys-

tems. Int. J. Mod. Phys. 7B, 3551 (1993).

89. B. Gramrnaticos. V. Papageorgiou and A. Ramani. Discrete Dressing Transforma-

tions and Painlw6 Quations, Phys. Lett. !USA, 475 (1997).

90. A. Ramani, B. Grammsticos and J. Satsurna, Bilinear Discrete Painlev6 Equations,

J. Php. A : Math. Gen. 38, 4655 (1995).

91. Y. Ohta, A. Rsmani, B. Grammatiox and K. M. T ' ' ' , F b m D k r e t e t o

Continuous Painleve Equations : A Bilinear Approach, Phys. Lett. 216A, 255

(leas) 92. A. Itammi, B. G r d c o 8 , K. M. 'Igmizhmsni and S. Lafortune, Again, Lin-

adzable Mappings, Phynica 251A, 138 (1998).

83. A. 5. Rho, B. Grammaticai and A. Fbmni, R.om Continuous to D i t e Painlev6

Equetiom, J. Math. Anal. Appl. 180, 342 (1993).

94. K. M. 'Ihmiehmani, B. Grammaticm a d A.Ramani, Schlesiqer aansfom for

the Diecrete Painlev6 Equation, Lett. Math. Phys. 28,49 (1993).

95. B. Grammaticm, Y. Ohta, A. Ramani, J. Sateuma and K. M. Tamizhmani, A

Miura of the Painlev4 I &ustion and Its D i Analogs, Lett. Math. phys. 39,

179 (1997).

96. J. SBtsuma, K. Kajiwara, B. Grammaticos, J. Hietarinta and A. Ramani, Bilinear

Discrete PainlevbII and Its Particular Solutions, J. Phys. A : Math. Gen. 28,

3541 (1995).

97. A. Ramani, B. Granunaticos and K. M. Tamizhmani, An Integrability test for

Di5'erential-Difference Systems, J. Phys. A : Math. Gen. 25, L883 (1992).

98. A. Ramani, B. Grammatiem and K. M. Tamizhmani, Painlev4 Analysis and Sin-

gularity Confinement : The Ultimate Conjecture, J. Phys. A : Math. Gen. 26,

LA3 (1993).

99. K. M. Tamizhmani, A. Ramani and B. Grammatiem, S i t y Confinement

Analysis of I n ~ ~ e r e n t i a l Equations of Benjamin-Ono type, J. Phys. A :

Math. Gen. 30, 1017 (1997)

100. S. Maeda, Canonical Structure and Symmetries for Discrete Systems, Math. Jap.

26, 8.5 (1980).

101. S. Maeda, Extension of Discrete Noether Theorem, Math. Jap. 26, 85 (1981).

102. S. Maeda, The Similarity Method for Difference Equations, IMA J. Appl. Math.

S8, 129 (1987).

103. D. LRvi and P. Wintemjtz, Continuous Symmetries of Discrete Equations, Phys.

Lett. 152A, 335 (1991).

104. D. Lmi and P. Witernitz, Symmetrb and Conditional Symmetries of Differential-

Diaeoce equations, J. Math. Phys. S4, 3713 (1993).

105. D. Levi and P. Wintenaitz, Symmetries of Diecrete Dynamical Syetems, J. Math.

Php. 37, 5551 (1996).

106. D. M, L. Vinet and P. Wintenaitz, Lie Group Formdism for Difference Equ&

ticma, J. Phys. A : Math. Gen. SO, 633 (1997).

107. G. R W. Quispel and H. W. Capel and R. !%adevan, Continuous Symmetries of

-a equations : the Kmvan Moerbeke Equation and Painkvk

ReductJon, Phys. Lstt. A. 170 379 (1992)

108. G. R W. Quispel and H. W. Capel and R Sahadevan, Continuoue Symmetries

end Painled Reduction of the Kac-van Mmbeke Equation in Proc. NATO Work-

shop on Applications of Analytic and Geometric Methods to Nonlinear Differential

Equations, Ed. Clarkeon, Kluwer (1993).

109. G. R. W. Quispel and R Sahadevsn, Lie symmetries and the Integration of Dif-

ference Equation, Phys. Lett. 184A, 64 (1993).

110. G. Gaeta, Lie Point Symmetries of Discrete Versus Continuous Dynamicat Sys-

tems, Phys. Lett. 178A, 376 (1993).

111. W.Oeve1, Poisson Brackets for Integrable Lattice Systems, in Algebraic Aspects of

integrable Systems : In Memory of Irne D o h , eds. A.S.Fokas and I.M.Gelfand,

Birkhauser, Boston, 261 (1996).

112. A. P. Veselw and A. B. Shabat, Dreasing Chains and the Spectral Theory of the

Schriidinger Operator, Funct. Anal. Appl. 27, 81 (1993).

113. J. Weiss. Biicklund Ttansformations and the Painleve Property in Partially Inte-

gable equations Eds. R. Conte and N. Boccera, NATO AS1 Series C : Mathemat-

ical and Physical Sciences, 310, Kluwer, Dordrecht 375, (1990)

114. S. Kmaga vel and K. M. Tamizhmani, Lax pairs, Symmetries and Conservation

Laws of a Differential-Difference Equation-Sato's Approach. C h . Solitons &

Ractals. 8. 917 (1997).

115. R. Hirota. Reduction of Soliton Equations in Bilinear Form, Physica 18D. 161

(1986).

116. J.Sidorenko and W.Strampp, Symmetry Constraints of the KP Hierarchy, Inv.

Prob. 7, L37 (1991).

117. B.Konopelchenko and W.Strampp, The AKNS Hierarchy as Symmetry Constraint

of the KP Hierarchy, Inv. Prob. 7, L17 (1991).

118. A. N h u r a , A Direct Method of Calculating periodic-wave Solutions to Nonlin-

ear Evolution Equations.1. Exact two-periodic wave Solutions, J. Phys. Soc. Jpn.

47, 1701 (1979).

119. N. Yajima, Applicstion of Hirota's Method to a Perturbed System, J. Phys. Soc.

Jpn. 51, 1298 (1982).

120. H. Airurlt, Rationd Solutions of Painlev6 Equations, Stud. Appl. Math. 61, 31

(isn).

121. H. Airsult, H. P. McKesn and J. Mceer, Rations and Elliptic Solutions of the

KdV Equation and Related Many-Body Problems, Commn. Pure Appl. Math.,

SO, 95 (1977).

122. M. J. Ablowitz and J. Sateuma, Solitons and Rational Solutions of Nonlinear

Evolution Equations, J. Math. Phys. 19, 2180 (1978).

123. J. J. C. Nimmo and N. C. Reeman, Rational Solutions of the Korteweg-de Vries

Equation in Wronskian Form, Phys. Lett. M A , 443 (1983).

124. Y. Matsuno, A New Proof of the Rational N-Soliton Solution for the Kadomtsev-

Petviashvili Equation, J. Phys. Soc. Jpn. 68, 67 (1989).

125. M. Adler and J. Moser, On a Class of Polynomials Connected with the Korteweg-

de Vries Equation, Commn. Math. Phys. 61, 1 (1978).

126. J . Satsuma and Y. Ishimori, Periodic Wavee and Rational Soliton Solutions of the

Benjamin-Ono Equation, J. Phys. Soc. Jpn., 46, 681 (1979).

127. N. A. Kudryashw and V. A. Nikitin, Painlwk Analysis, Rational and Special

Solutions of a variable Coefficient Korteweg-de Vries Equations, J. Phys. A :

Math. Gen. 27, L101 (1994).

128. V. M. Galkin, D. E. Pelinmky, Y. A. Stepanpyants, The Structure of the Rational

Solutions to the Boussinesq Equation, Physica 80D. 246 (1995).

129. D. Pelinwsky, Rational Solutions of the Kadomtsev-Petviashvili Hierarchy and

the Dynamics of Their Poles. I. New Form of a General Rational Solution, J.

Math. Phys. 35, 5820 (1994).

130. X. B. Hu, Rationd Solutions of Integrable Equations via Nonlinear Superpoeition

Formulae : Preprint

131. V. E. Adler, On the Rational Solutions of the Shabet Equation, in Nonlinear

Physics Theory and Expaiment eds E. Alfinito, M. Boiti, L. Martina and F.

Pcrmpinelli, World Scientific, Singapore, 3, (1996).

132. V. B. Matvow and M. A. Sak, Darboux ~ ~ i o r t s and Solitons, Springez

Valag, Barlin (1991).

133. R Hirota, Solutions of the Clsasical Bowheoq Equation and the Spherical Boussi-

mmq EQU.tion : The W r o W Tachniquc, J. Phys. Soc. Jpn. 55, 2137 (1986).

1%. R Hirota, M. Ik, and F. K&o, Two-dimmhnd Toda Lsttica Equations, Rog.

T h . Phyl. Suppl. 9442 (lS88).

1%. Y. Ohta aud R Hirota, A Dbcrete KdV -MI d Its Cssorati D e t .

Solution, J. Phys. Soc. Jpn. 60,2095 (1991).

136. Y. Oh& R. Hirota, S. Tsujimoto and T. Iami, Caeorati and Discrete Gram Type

DetQminant Representations of Solutions to the Discrete KP Hierarchy, J. Phys.

Soc. Jpn. 62,1872 (1983).

137. Y. Ohta, K. Kajiwara, J. Mat-sukidaira and J. Satsuma, Cawrati Determinant

Solution for the Relativistic Toda Lattice Equation, J. Math. Phys. 34, 5190

(1993).

138. K. Kajiwara and Y. Ohta, Determinant Structure of the Rational Solutions for

the Painlev4 IV Equation, J. Phys. A : Math. Gen. 31, 2431 (1998).

139. Thamizharasi Tarnizhmani, S. Kanaga vel and K. M. Tamizhmani, Wronskian

and Rational Solutions of Dierentid-Difference KP Equation, J. Phys. A : Math.

Gen. Sl, 7627 (1998).

140. E. L. Ince, Ordinary Dierential Equations, Dover Publ. New York (1956)

141. J. D. Gibbon and M. Tabor, On the one-and two Dimensional Toda lattices and

the Painleve Property, J. Math. Phys. 26, 1956 (1985).

142. M. Blaszak and R. Marciniak, R-Matrix Approach to Lattice Integrable Systems,

J. Math. Phys. 35, 4661 (1994).

143. D. Levi, L. Pilloni and P. M. Santini, Integrable Three-dimensional Lattices, J.

Phys. A : Math. Gea. 14, 1567 (1981).

144. G. W i a d H. W. Capel, Lattice Equations, Hierarchies and Hamiltonian

Structures, Physica 142A, 199 (1987).

145. F. W. Nijhoff and H. W. Capel, The Direct Linearisation Approach to Hierar-

chia of Lntelgable PDE's in (2+1) Dimensions : I. Lattice Equations and the

Dierentid-Diercnce H i , Inv. prob. 6, 567 (1990).

146. F. W. Nijhofl, V. G. Pspageorgiou, H. W. Capel and G. R W. Quispel, The

Lattice Gel'fanbDikii Hieaedty, Inv. Prob. 8, 597 (1992).

147. V. E. A&, N o n h C h a b and Painlev6 Equations, Physica 7SD, 335 (1994).

148. B. G ~ C O B , A. Ibmmi d K. hi. Tsmiahmani, Nonproliferation of Preim-

h t q y d e Mappings, J. Phys. A : M d b . Cen. 27, 559 (1994).

149. M. rlhhmuun d P. Kahpgm, Lie 'Thud-, Nonlinear Evolution

Epurtknr, ud P M RWXM, J. W h . Php. U, 795 (1983).

150. K. M. Tamizhmaai and P. Punithavathi, M n i t e - D i n $ Lie Algebraic Struc-

ture and the Symmetry Reduction of a Nonlinear Higher-Dimell~iod Equation,

J. Phys. Soc. Jpn. 59, 843 (1990).

151. W. Miller, Symmetry and Separation of Variables, Addison-Wesley, Readmg (1977).

152. K. M. Tamizhmani, A. Ramani and B. Gramrnaticsos, Lie Symmetries of Hirota's

Bilinear Equations, J. Math. Phys. 32, 2635 (1991).

153. P. A. Clarkson and M. D. Kruskal, New Similarity Reduction of the Boussinesq

Equation, J. Math. Phys. 30, 2201 (1989).

154. P. A. Clarkson and S. Hood, New Symmetry Reductions and Exact Solutions of

the Davey-Stewartson System. I. Reductions to Ordinary Differential Equations,

J. Math. Phys. 35, 255 (1994).

155. L. Martina, G. Soliani and P. Witernitz, Partially Invariant Solutions of a Class

of Nonlinear Schriidinger Equations, J. Phys. A : Math. Gen. 25, 4425 (1992).

156. G. W. Bluman and J. D. Cole, The Generd Sirnilairity Solution of the Heat

Equation. J. Math. Mech.. 18, 1025 (1969).

157. P. J. Olver and P. Rosenau. The Construction of Special Solutions to Partial

Dierentid Equations, Phys. Lett. 114A, 107 (1986).

158. G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer.

New York (1989).

159. D. Levi and P. Winternitz. Nonclensical Symmetry Reduction : Example of the

Bouseinesq Equation, J. Phys. A. Math. Gen., 22, 2915 (1989).

160. G. W. Bluman, G. J. Reid and S. Kumei, New Classas of Symmetries for Parital

Difiermtid Equations, J. Math. Phys. 29, 806 (1988).

161. B. Champagne, W. Hereman and P. Wmternitz, The Computer Calculation of

Lie point Symmetries of Large Systems of Dierentid Equations. Comp. Phys.

Comni 66, 319 (1991).

162. W. Hereman, CRC Handbook of Lie Group Analysis of Differential Equations Vol.

3 : New l'kenh in Theoretical De&lopments and Computational Methca, Ed. N.

H. Ibragimov, Boca Raton, Florida, CRC Prass (1996).

163. A. Herd, Lie : a MUMATH Program for the Calculation of the Lie Algebra of

Merentin1 Equations, Comp. Phys. Comm. 77, 241 (1993).

164. D. David, N. Kamran, D. Levi and P. Witernitz, Symmetry Reduction for the

Ksdomtsev-Petviaahviii Equation Using a Loop Algebra , J. Math. Phys. 27,

ins (1986).

165. M. Wadati and K. Sogo, Gauge Treneformatione in Soliton Theory, J. Phys. Soc.

Jpn. 62, 394 (1983).

166. Y. Ishimori, A Relationship Between the Ablowit~KwpNewell-Segur and Wadati-

Kon.nc+Ichidawa Schema of the eveme Scattering Method, J. Phys. Soc. Jpn.

61, 3036 (1982).

167. K. Kim, A Remark on the Commuting Flows Defined by Lax Equations, Prog.

Theor. Phys. 83, 1108 (1990).

168. W. Oevel and C. Rogers, Gauge 'Ikausfarmations and Reciprocal L i in (2+1)

Dimensions, Rev. Math. Phys. 5, 299 (1993).

169. B. Konopelchenko and W. Oevel, An r-matrix Approach to Nonstandard Clanses

of Integrable Equations, Publ. RIMS. Kyoto Univ. 29, 581 (1993).

170. K. M. Tamizhmani and S. Kanaga vel, Gauge Equivalence and I-reduction of

Ditferential-Ditference KP Equation, - To be Published in Chaos, Solitons & Rac-

tals

171. K . M. Tamizhmani, S. Kanaga Vel, B. Grammatics and A. Ramani, Singular-

ity and Algebraic Properties of the Differential-Dierence Kadomtsev-Petviashvili

Equation - (Preprint)

172, G. R W. Quispel, J. A. G. Roberta and C. J. Thompson, Integrable Mappings

and Soliton Equations.1. Phys. Lett. 126A, 419 (1988).

173, G. R W. Quispel, J. A. G. Roberts and C. J. Thompson, Integrable Mappings

and Soliton Equations.11. Physica MD, 183 (1989).