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Painlevé analysis and integrability aspects of some nonlinear partial differential equations

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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

  1. S Kovalevskaya,Acta Math. 12, 177, (1889);14, 81 (1889)

    Article  MathSciNet  Google Scholar 

  2. P Painlevé,Oeuvres, Vol. I, CNRS, Paris (1973)

    MATH  Google Scholar 

  3. E L Ince,Ordinary differential equations (Dover, New York, 1956)

    Google Scholar 

  4. T T Wu, B M McCoy, C A Tracy and E Barouch,Phys. Rev. B13, 316 (1976)

    ADS  Google Scholar 

  5. M Tabor and J Weiss,Phys. Rev. A24, 2157 (1981)

    ADS  Google Scholar 

  6. Y F Chang, M Tabor and J Weiss,J. Math. Phys. 23, 531 (1982)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. T Bountis, H Segur and F Vivaldi,Phys. Rev. A25, 1257 (1982)

    ADS  MathSciNet  Google Scholar 

  8. A Ramani, B Dorizzi and B Grammaticos,Phys. Rev. Lett. 49, 1539 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  9. M J Alowitz, D J Kaup, A C Newell and H Segur,Stud. Appl. Math. Phys. 53, 249 (1974)

    Google Scholar 

  10. R Hirota, inSolitons edited by R K Bullough and P J Caudrey (Springer, Berlin, 1980)

    Google Scholar 

  11. M J Alowitz, A Ramani and H Segur,J. Math. Phys. 27, 1006 (1980)

    Article  ADS  Google Scholar 

  12. J Weiss, M Tabor and G Carnevale,J. Math. Phys. 24, 532 (1983)

    ADS  MathSciNet  Google Scholar 

  13. M D Kruskal, Private Communication (1982)

  14. J Weiss,J. Math. Phys. 24, (1983), 1405, 532;25, (1984), 13,2226;26, (1985), 258, 2174 (1986)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. A C Newell, M Tabor and Y B Zeng,Physica D29, 1 (1987)

    ADS  MathSciNet  Google Scholar 

  16. F Lund and T Regge,Phys. Rev. D14, 1524 (1976)

    ADS  MathSciNet  Google Scholar 

  17. K Pohlymeyer,Com. Math. Phys. 46, 207 (1977)

    Article  ADS  Google Scholar 

  18. B S Getmanov, JINR Preprint E2-11093, Dubna 1977

  19. K Porsezian,Chaos, solitons and fractals (Submitted)

  20. P Goldstein and E Infeld,Phys. Lett. A103, 8 (1984)

    ADS  MathSciNet  Google Scholar 

  21. K Porsezian and K Nakkeeran,J. Mod. Opt. 42, 1953 (1995)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  22. K Porsezian and K Nakkeeran,Phys. Rev. Lett. 76, 3955 (1996)

    Article  ADS  Google Scholar 

  23. S Liu and W Z Wang,Phys. Rev. E49, 5726 (1994)

    ADS  Google Scholar 

  24. K Porsezian,Pure. App. Opt. 5, L345 (1996)

  25. F Cariello and M Tabor,Physica D39, 77 (1989)

    ADS  MathSciNet  Google Scholar 

  26. M Lakshmanan and P Kaliappan,J. Math. Phys. 24, 795 (1983)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  27. J D Gibbon and M Tabor,J. Math. Phys. 26, 1956 (1985)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  28. R Conte, A P Fordy and A Pickering,Physica D69, 33 (1993)

    ADS  MathSciNet  Google Scholar 

  29. R Conte,Phys. Lett. A140, 383 (1989)

    ADS  MathSciNet  Google Scholar 

  30. A Pickering,J. Math. Phys. 37, 1894 (1996)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  31. F Calogero and A Degasperis,Lett. Nuovo Cimento 22, 139, 271 421 (1978)

    Google Scholar 

  32. R Balakrishnan,Phys. Rev. A32, 1144 (1985)

    ADS  Google Scholar 

  33. H H Chen and C S Liu,Phys. Fluids 21, 377 (1978)

    Article  MATH  ADS  Google Scholar 

  34. S P Burtsev, V E Zakharov and A V Mikhailov,Teor. Math. Fiz. 70, 323 (1987)

    MathSciNet  Google Scholar 

  35. M Daniel, K Porsezian and M Lakshmanan,J. Math. Phys. 35, 6498 (1994)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  36. K Porsezian and K Nakkeeran,Phys. Lett. A206, 183 (1995)

    ADS  Google Scholar 

  37. K Porsezian, M Daniel and R Bharathikannan,Phys. Lett. A156, 206 (1991)

    ADS  MathSciNet  Google Scholar 

  38. K Porsezian and M Lakshmanan,J. Math. Phys. 32, 2923 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  39. K Porsezian,Phys. Lett. A191, 229 (1994)

    ADS  MathSciNet  Google Scholar 

  40. K Porsezian,Phys. Lett. A221, 163 (1996)

    ADS  MathSciNet  Google Scholar 

  41. W H Steeb and J A Louw,Phys. Lett. A113, 61 (1985)

    ADS  MathSciNet  Google Scholar 

  42. S Puri,Phys. Lett. A107, 359 (1985)

    ADS  MathSciNet  Google Scholar 

  43. N Joshi,Phys. Lett. A125, 456 (1987)

    ADS  Google Scholar 

  44. L Halavaty,J. Phys. Soc. Japan 55, 1405 (1995);Phys. Lett. A128, 335 (1988)

    ADS  Google Scholar 

  45. W H Steeb, M Kloke, B M Spieker and W Oevel,J. Phys. A16, L447 (1983)

  46. N Nirmala, M J Vedan and B V Baby,J. Math. Phys. 27, 2640, 2644 (1986)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  47. P A Clarkson,Proc. R. Soc. Edinb. A109, 109 (1988)

    MathSciNet  Google Scholar 

  48. R Radha and M Lakshmanan,J. Phys. A28, 6977 (1995)

    ADS  MathSciNet  Google Scholar 

  49. K Porsezian and A Uthyakumar,Phys. Lett. A183, 371 (1993)

    ADS  Google Scholar 

  50. W Oevel and W H Steeb,Phys. Lett. A103, 239 (1984)

    ADS  MathSciNet  Google Scholar 

  51. S Ganesan and M Lakshmanan,J. Phys. A20, L1147 (1988)

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Authors and Affiliations

  1. Institut für Theoretische Physik, Universität Hannover, 30167, Hannover, Germany

    K Porsezian

  2. Centre for Laser Technology, Department of Physics, Anna University, 600 025, Madras, India

    K Porsezian

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  1. K Porsezian
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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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