Integrable theory of the perturbation equations

Chaos, Solitons and Fractals

Volumn 7, Issue 8, 1996, Pages 1227-1250

  Ma, W X ^a   Fuchssteiner, B ^b  

^a FUDAN UNIVERSITY   (China)

^b UNIVERSITY OF PADERBORN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000913370     PISSN: 09600779     EISSN: None     Source Type: Journal    
DOI: 10.1016/0960-0779(95)00104-2     Document Type: Article

References (34)

1. V. I. Karpman, Soliton evolution in the presence of perturbation, Phys. Scr. 20, 462-478 (1979).
2. J. P. Keener and D. W. McLaughlin, Solitons under perturbations, Phys. Rev. A 16, 777-790 (1977).
3. Y. Kodama and M. J. Ablowitz, Perturbations of solitons and solitary waves, Stud. Appl. Math. 64, 225-245 (1981).
4. P. J. Olver, Hamiltonian perturbation theory and water waves, Contemporary Mathematics 28, 231-249 (1984).
5. R. L. Herman, A direct approach to studying soliton perturbations, J. Phys. A, Math. Gen. 23, 2327-2362 (1990).
6. M. Tanaka, Perturbations on the K-dV equation - An approach based on the multiple time scale expansion, J. Phys. Soc. Jpn 49, 807-812 (1980).
7. R. Grimshaw and H. Mitsudera, Slowly varying solitary wave solutions of the perturbed Korteweg-de Vries equation revisited, Stud. Appl. Math. 90, 75-86 (1993).
8. J. N. Elgin, T. Brabec and S. M. J. Kelly, A perturbative theory of soliton propagation in the presence of the third order dispersion. Opt. Commun. 114, 321-328 (1995).
9. G. J. Ballantyne, P. T. Gough and D. P. Taylor, Deriving average soliton equations with a perturbation method, Phys. Rev. E 5, 825-828 (1995).
10. C. Currò, A. Donate and A. Ya. Povzner, Perturbation method for a generalized Burger's equation, Int. J. Non-Linear Mech. 27, 149-155 (1992).
11. Y. Matsuno, Multisoliton perturbation theory for the Benjamin-Ono equation and its application to real physical systems, Phys. Rev. E 51, 1471-1483 (1995).
12. Y. S. Kivshar and B. A. Malomed, Dynamics of solitons in nearly integrable systems, Rev. Mod. Phys. 61, 763-915 (1989).
13. K. M. Tamizhmani and M. Lakshmanan, Complete integrability of the Korteweg-de Vries equation under perturbation around its solution: Lie-Bäcklund symmetry approach, J. Phys. A, Math. Gen. 16, 3773-3782 (1983).
14. W. X. Ma and B. Fuchssteiner, The bi-Hamiltonian structure of the perturbation equations of KdV hierarchy, Phys. Lett. A 213, 49 (1995).
15. R. A. Kraenkel, M. A. Manna and J. G. Pereira, The Korteweg-de Vries hierarchy and long water-waves, J. Math. Phys. 36, 307-320 (1995).
16. R. A. Kraenkel, M. A. Manna, J. C. Montero and J. G. Pereira, Boussinesq solitary-wave as a multiple-time solution of the Korteweg-de Vries hierarchy J. Math. Phys. 36, 6822-6828 (1995).
17. B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Backlund transformations and hereditary symmetries, Physica D 4, 47-66 (1981).
18. B. Fuchssteiner, Coupling of completely integrable systems: the perturbation bundle, In Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, edited by P. A. Clarkson, pp. 125-138 Kluwer, Dordrecht (1993).
19. W. Oevel, Dirac constraints in the field theory: lifts of Hamiltonian systems to the cotangent bundle, J. Math. Phys. 29, 210-219 (1988).
20. W. X. Ma, Poisson manifolds and classical Hamiltonian operators, Northeastern Mathematical Journal 6, 346-356 (1990).
21. B. Fuchssteiner, Application of hereditary symmetries to nonlinear evolution equations, Nonlinear Anal. Theory Methods Appl. 3, 849-862 (1979).
22. I. M. Gel'fand and I. Y. Dorfman, Hamiltonian operators and algebraic structures related to them, Funct. Anal. Appl. 13, 248-262 (1979).
23. F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19, 1156-1162 (1978).
24. H. H. Chen and J. E. Lin, On the integrability of multidimensional nonlinear evolution equations, J. Math. Phys. 28, 347-350 (1987).
25. W. X. Ma, Symmetry constraint of MKdV equations by binary nonlinearization. Physica A 219, 467-481 (1995).
26. M. Wadati, The modified Korteweg-de Vries equation, J. Phys. Soc. Jpn 34, 1289-1296 (1973).
27. W. X. Ma, K-symmetries and τ-symmetries of evolution equations and their Lie algebras, J. Phys. A, Math. Gen. 23, 2707-2716 (1990).
28. W. X. Ma, The algebraic structures of isospectral Lax operators and applications to integrable equations, J. Phys. A, Math. Gen. 25, 5329-5343 (1992).
29. V. E. Zakharov and B. G. Konopelchenko, On the theory of recursion operator, Commun. Math. Phys. 94, 483-509 (1984).
30. I. Ya. Dorfman and A. S. Fokas, Hamiltonian theory over noncommutative rings and integrability in multidimensions, J. Math. Phys. 33, 2504-2514 (1992).
31. F. Schwarz, Symmetries of the two-dimensional Korteweg-de Vries equation, J. Phys. Soc. Jpn 51, 2387-2388 (1982).
32. D. David, N. Kamran, D. Levi and P. Winternitz, Symmetry reduction for the Kadomtsev-Petviashvili equation using a loop algebra, J. Math. Phys. 27, 1225-1237 (1986).
33. W. X. Ma, The generators of vector fields and the time dependent symmetries of evolution equations, Sci. China A 34, 769-782 (1991).
34. B. Fuchssteiner, Master symmetries, higher order time-dependent symmetries and conserved densities of nonlinear evolution equations. Prog. Theor. Phys. 70, 1508-1522 (1983).