Bäcklund transformation of partial differential equations from the Painlevé-Gambier classification. I. Kaup-Kupershmidt equation

Journal of Mathematical Physics

Volumn 39, Issue 10, 1998, Pages 5617-5630

  Musette, Micheline ^a   Conte, Robert ^b  

^a VRIJE UNIVERSITEIT BRUSSEL   (Belgium)

^b SERVICE DE PHYSIQUE DE L ETAT CONDENSÉ   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032357771     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.532554     Document Type: Article

References (44)

1. A. V. Bäcklund, "Om ytor med konstant negativ krökning," Lunds Universitets Arsskrift Avd. 2, 19 (1883).
2. C. Rogers and W. F. Shadwick, Bäcklund Transformations and their Applications (Academic, New York, 1982).
3. G. Darboux, Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal (Gauthier-Villars, Paris, 1894), Vol. III; Théorie générale des surfaces (Chelsea, New York, 1972), reprint; (Gabay, Paris, 1993), reprint, see original title.
4. G. Darboux, Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal (Gauthier-Villars, Paris, 1894), Vol. III; Théorie générale des surfaces (Chelsea, New York, 1972), reprint; (Gabay, Paris, 1993), reprint, see original title.
5. V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons (Springer, Berlin, 1991).
6. J. Weiss, M. Tabor, and G. Carnevale, "The Painlevé property for partial differential equations," J. Math. Phys. 24, 522-526 (1983).
7. J. Weiss, "The Painlevé property for partial differential equations. II. Bäcklund transformation, Lax pairs, and the Schwarzian derivative," J. Math. Phys. 24, 1405-1413 (1983).
8. M. Musette and R. Conte, "Algorithmic method for deriving Lax pairs from the invariant Painlevé analysis of nonlinear partial differential equations," J. Math. Phys. 32, 1450-1457 (1991).
9. M. Musette and R. Conte, "The two-singular manifold method. I. Modified KdV and sine-Gordon equations," J. Phys. A 27, 3895-3913 (1994).
10. R. Conte, M. Musette, and A. Pickering, "The two-singular manifold method. II. Classical Boussinesq system," J. Phys. A 28, 179-187 (1995).
11. R. Conte and M. Musette, "Beyond the two-singular manifold method, in Nonlinear Physics: Theory and Experiment, edited by E. Alfinito, M. Boiti, L. Martina and F. Pempinelli (World Scientific, Singapore, 1996), pp. 67-74.
12. A. Pickering, "The singular manifold method revisited," J. Math. Phys. 37, 1894-1927 (1996).
13. x + 6Rψ=λ," Stud. Appl. Math. 62, 189-216 (1980).
14. A. P. Fordy and J. Gibbons, "Some remarkable nonlinear transformations," Phys. Lett. 75A, 325-325 (1980).
15. M. Musette, "Painlevé analysis for nonlinear partial differential equations," in The Painlevé Property, One Century Later, edited by R. Conte (Springer, Berlin, 1998), 54 pages.
16. J. Weiss, "On classes of integrable systems and the Painlevé property," J. Math. Phys. 25, 13-24 (1984).
17. C. Rogers and S. Carillo, "On reciprocal properties of the Caudrey-Dodd-Gibbon and Kaup-Kupershmidt hierarchies," Phys. Scr. 36, 865-869 (1987).
18. G. Darboux, "Sur une proposition relative aux équations linéaires," C. R. Acad. Sci. Paris 94, 1456-1459 (1882).
19. R. Anderson, J. Hamad, and P. Winternitz, "Systems of ordinary differential equations with nonlinear superposition principles," Physica D 4, 164-182 (1982).
20. M. Musette and R. Conte, "Solitary waves and Lax pairs from polynomial expansions of nonlinear differential equations," in Nonlinear Evolution Equations and Dynamical Systems, edited by M. Boiti, L. Martina, and F. Pempinelli (World Scientific, Singapore, 1992), pp. 161-170.
21. H. H. Chen, "General derivation of Bäcklund transformations from inverse scattering problems," Phys. Rev. Lett. 33, 925-928 (1974).
22. V. E. Zakharov and A. B. Shabat, "A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I." Funktsional'nyi Analiz i Ego Prilozheniya 8, 43-53 (1974) [English: Funct. Anal. Appl. 8, 226-235 (1974)].
23. V. E. Zakharov and A. B. Shabat, "A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I." Funktsional'nyi Analiz i Ego Prilozheniya 8, 43-53 (1974) [English: Funct. Anal. Appl. 8, 226-235 (1974)].
24. J. Weiss, "The Painlevé property and Bäcklund transformations for the sequence of Boussinesq equations," J. Math. Phys. 26, 258-269 (1985).
25. R. Conte, "Invariant Painlevé analysis of partial differential equations," Phys. Lett. A 140, 383-390 (1989).
26. V. E. Zakharov, "On stochastization of one-dimensional chains of nonlinear oscillators," Zh. Eksp. Teor. Fiz. 65, 219-225 (1973) [English: Sov. Phys. JETP 38, 108-110 (1974)].
27. V. E. Zakharov, "On stochastization of one-dimensional chains of nonlinear oscillators," Zh. Eksp. Teor. Fiz. 65, 219-225 (1973) [English: Sov. Phys. JETP 38, 108-110 (1974)].
28. H. C. Morris, "Prolongation structures and a generalized inverse scattering problem," J. Math. Phys. 17, 1867-1869 (1976).
29. R. Hirota and J. Satsuma, "Nonlinear evolution equations generated from the Bäcklund transformation for the Boussinesq equation," Prog. Theor. Phys. 57, 797-807 (1977).
30. R. Hirota and J. Satsuma, "A simple structure of superposition formula of the Bäcklund transformation," J. Phys. Soc. Jpn. 45, 1741-1750 (1978).
31. B. Gambier, "Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est à points critiques fixes," thèse, Paris, 1909; Acta Math. 33, 1-55 (1910).
32. B. Gambier, "Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est à points critiques fixes," thèse, Paris, 1909; Acta Math. 33, 1-55 (1910).
33. P. Painlevé, "Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme," Acta Math. 25, 1-85 (1902).
34. 3u," Phys. Lett. 79A, 264-268 (1980).
35. D. Levi and O. Ragnisco, "Non-isospectral deformations and Darboux transformations for the third-order spectral problem," Inverse Probl. 4, 815-828 (1988).
36. R. Willox (private communication).
37. R. Hirota and J. Satsuma, "N-soliton solutions of model equations for shallow water waves," J. Phys. Soc. Jpn. 40, 611-612 (1976).
38. R. Conte, "Painlevé singular manifold equation and integrability," in Inverse Methods in Action, edited by P. C. Sabatier (Springer, Berlin, 1990), pp. 497-504.
39. J. Chazy, "Sur les équations différentielles du troisième ordre et d'ordre supérieur dont l'intégrale générale a ses points critiques fixes," thèse, Paris. 1910; Acta Math. 34, 317-385 (1911). Table des matières commentée avec index, R. Conte, 6 pages (1991).
40. J. Chazy, "Sur les équations différentielles du troisième ordre et d'ordre supérieur dont l'intégrale générale a ses points critiques fixes," thèse, Paris. 1910; Acta Math. 34, 317-385 (1911). Table des matières commentée avec index, R. Conte, 6 pages (1991).
41. R. Conte, "Towards the equivalence between integrability and Painlevé test for partial differential equations," in Nonlinear and Chaotic Phenomena in Plasmas, Solids and Fluids, edited by W. Rozmus and J. A. Tuszyński (World Scientific, Singapore, 1991), pp. 94-101.
42. J. Satsuma and D. J. Kaup, "A Bäckiund transformation for a higher order Korteweg-de Vries equation," J. Phys. Soc. Jpn. 43, 692-697 (1977).
43. R. Conte and M. Musette, "Linearity inside nonlinearity: Exact solutions to the complex Ginzburg-Landau equation," Physica D 69, 1-17 (1993).
44. M. Jimbo and T. Miwa, "Solitons and infinite dimensional Lie algebras," Publ. RIMS, Kyoto 19, 943-1001 (1983).