Abstract
A brief review of the Painlevé singularity structure analysis of some autonomous and nonautonomous nonlinear partial differential equations is discussed. We point out how the Painlevé analysis of solutions of these equations systematically provides the integrability properties of the equation. The Lax pair, Bäcklund transformation and bilinear forms are constructed from the analysis.
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References
S Kovalevskaya,Acta Math. 12, 177, (1889);14, 81 (1889)
P Painlevé,Oeuvres, Vol. I, CNRS, Paris (1973)
E L Ince,Ordinary differential equations (Dover, New York, 1956)
T T Wu, B M McCoy, C A Tracy and E Barouch,Phys. Rev. B13, 316 (1976)
M Tabor and J Weiss,Phys. Rev. A24, 2157 (1981)
Y F Chang, M Tabor and J Weiss,J. Math. Phys. 23, 531 (1982)
T Bountis, H Segur and F Vivaldi,Phys. Rev. A25, 1257 (1982)
A Ramani, B Dorizzi and B Grammaticos,Phys. Rev. Lett. 49, 1539 (1982)
M J Alowitz, D J Kaup, A C Newell and H Segur,Stud. Appl. Math. Phys. 53, 249 (1974)
R Hirota, inSolitons edited by R K Bullough and P J Caudrey (Springer, Berlin, 1980)
M J Alowitz, A Ramani and H Segur,J. Math. Phys. 27, 1006 (1980)
J Weiss, M Tabor and G Carnevale,J. Math. Phys. 24, 532 (1983)
M D Kruskal, Private Communication (1982)
J Weiss,J. Math. Phys. 24, (1983), 1405, 532;25, (1984), 13,2226;26, (1985), 258, 2174 (1986)
A C Newell, M Tabor and Y B Zeng,Physica D29, 1 (1987)
F Lund and T Regge,Phys. Rev. D14, 1524 (1976)
K Pohlymeyer,Com. Math. Phys. 46, 207 (1977)
B S Getmanov, JINR Preprint E2-11093, Dubna 1977
K Porsezian,Chaos, solitons and fractals (Submitted)
P Goldstein and E Infeld,Phys. Lett. A103, 8 (1984)
K Porsezian and K Nakkeeran,J. Mod. Opt. 42, 1953 (1995)
K Porsezian and K Nakkeeran,Phys. Rev. Lett. 76, 3955 (1996)
S Liu and W Z Wang,Phys. Rev. E49, 5726 (1994)
K Porsezian,Pure. App. Opt. 5, L345 (1996)
F Cariello and M Tabor,Physica D39, 77 (1989)
M Lakshmanan and P Kaliappan,J. Math. Phys. 24, 795 (1983)
J D Gibbon and M Tabor,J. Math. Phys. 26, 1956 (1985)
R Conte, A P Fordy and A Pickering,Physica D69, 33 (1993)
R Conte,Phys. Lett. A140, 383 (1989)
A Pickering,J. Math. Phys. 37, 1894 (1996)
F Calogero and A Degasperis,Lett. Nuovo Cimento 22, 139, 271 421 (1978)
R Balakrishnan,Phys. Rev. A32, 1144 (1985)
H H Chen and C S Liu,Phys. Fluids 21, 377 (1978)
S P Burtsev, V E Zakharov and A V Mikhailov,Teor. Math. Fiz. 70, 323 (1987)
M Daniel, K Porsezian and M Lakshmanan,J. Math. Phys. 35, 6498 (1994)
K Porsezian and K Nakkeeran,Phys. Lett. A206, 183 (1995)
K Porsezian, M Daniel and R Bharathikannan,Phys. Lett. A156, 206 (1991)
K Porsezian and M Lakshmanan,J. Math. Phys. 32, 2923 (1992)
K Porsezian,Phys. Lett. A191, 229 (1994)
K Porsezian,Phys. Lett. A221, 163 (1996)
W H Steeb and J A Louw,Phys. Lett. A113, 61 (1985)
S Puri,Phys. Lett. A107, 359 (1985)
N Joshi,Phys. Lett. A125, 456 (1987)
L Halavaty,J. Phys. Soc. Japan 55, 1405 (1995);Phys. Lett. A128, 335 (1988)
W H Steeb, M Kloke, B M Spieker and W Oevel,J. Phys. A16, L447 (1983)
N Nirmala, M J Vedan and B V Baby,J. Math. Phys. 27, 2640, 2644 (1986)
P A Clarkson,Proc. R. Soc. Edinb. A109, 109 (1988)
R Radha and M Lakshmanan,J. Phys. A28, 6977 (1995)
K Porsezian and A Uthyakumar,Phys. Lett. A183, 371 (1993)
W Oevel and W H Steeb,Phys. Lett. A103, 239 (1984)
S Ganesan and M Lakshmanan,J. Phys. A20, L1147 (1988)
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Porsezian, K. Painlevé analysis and integrability aspects of some nonlinear partial differential equations. Pramana - J Phys 48, 143–161 (1997). https://doi.org/10.1007/BF02845628
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DOI: https://doi.org/10.1007/BF02845628