Variational identities and applications to Hamiltonian structures of soliton equations

Nonlinear Analysis, Theory, Methods and Applications

Volumn 71, Issue 12, 2009, Pages

  Ma, Wen Xiu ^a  

^a ZHEJIANG NORMAL UNIVERSITY   (China)

Author keywords

Hamiltonian structures; Variational identities; Zero curvature equations

Indexed keywords

AKNS HIERARCHY; CONSERVATION LAW; HAMILTONIAN STRUCTURES; ILLUSTRATIVE EXAMPLES; INTEGRABLE COUPLING; LIE ALGEBRA; SOLITON EQUATION; SOLITON HIERARCHY; SPECTRAL PROBLEM; TWO-HIERARCHY; VOLTERRA LATTICES; ZERO-CURVATURE EQUATION;

ALGEBRA; SOLITONS;

HAMILTONIANS;

EID: 72149125297     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2009.02.045     Document Type: Article

References (52)

1. Ablowitz M.J., and Clarkson P.A. Solitons Nonlinear Evolution Equations and Inverse Scattering (1991), Cambridge University Press, Cambridge
2. Calogero F., and Degasperis A. Spectral Transform and Solitons, Vol. I. Studies in Mathematics and its Applications vol. 13 (1982), North-Holland Publishing Co., Amsterdam, New York
3. Faddeev L.D., and Takhtajan L.A. Hamiltonian Methods in the Theory of Solitons (1987), Springer-Verlag, Berlin
4. Ma W.X., Xu X.X., and Zhang Y.F. Semi-direct sums of Lie algebras and continuous integrable couplings. Phys. Lett. A 351 (2006) 125-130
5. Ma W.X., Xu X.X., and Zhang Y.F. Semidirect sums of Lie algebras and discrete integrable couplings. J Math. Phys. 47 (2006) 053501 16 pp
6. Jacobson N. Lie Algebras (1962), Interscience Publishers, New York, London
7. Ma W.X., and Fuchssteiner B. Integrable theory of the perturbation equations. Chaos Solitons Fractals 7 (1996) 1227-1250
8. Ma W.X. Integrable couplings of soliton equations by perturbations I - A general theory and application to the KdV hierarchy. Methods Appl. Anal. 7 (2000) 21-55
9. Ma W.X. Enlarging spectral problems to construct integrable couplings of soliton equations. Phys. Lett. A 316 (2003) 72-76
10. Guo F.K., and Zhang Y.F. A unified expressing model of the AKNS hierarchy and the KN hierarchy as well as its integrable coupling system. Chaos Solitons Fractals 19 (2004) 1207-1216
11. Xia T.C., Yu F.J., and Zhang Y. The multi-component coupled burgers hierarchy of soliton equations and its multi-component integrable couplings system with two arbitrary functions. Phys. A 343 (2004) 238-246
12. Ma W.X. Integrable couplings of vector AKNS soliton equations. J. Math. Phys. 46 (2005) 033507 19 pp
13. Ding H.Y., Sun Y.P., and Xu X.X. A hierarchy of nonlinear lattice soliton equations, its integrable coupling systems and infinitely many conservation laws. Chaos Solitons Fractals 30 (2006) 227-234
14. 6 and their applications. Internat. J. Modern Phys. B 21 (2007) 3809-3824
15. Olshanetsky M.A., and Perelomov A.M. Classical integrable finite-dimensional systems related to lie algebra. Phys. Rep. C 71 (1981) 313-400
16. Calogero F. Classical Many-Body Problems Amenable to Exact Treatments. Lecture Notes in Physics, New Series m: Monographs vol. 66 (2001), Springer-Verlag, Berlin
17. Clarkson P.A., and Winternitz P. Symmetry reduction and exact solutions of nonlinear partial differential equations, in: The Painlevé Property. CRM Ser. Math. Phys. (1999), Springer, New York 591-660
18. Basarab-Horwath P., Lahno V., and Zhdanov R. The structure of Lie algebras and the classification problem for partial differential equations. Acta Appl. Math. 69 (2001) 43-94
19. Hurtubise J.C., and Markman E. Calogero-Moser systems and Hitchin systems. Commun. Math. Phys. 223 (2001) 533-552
20. Tu G.Z. On Liouville integrability of zero-curvature equations and the Yang hierarchy. J. Phys. A: Math. Gen. 22 (1989) 2375-2392
21. Ma W.X. A new family of Liouville integrable generalized Hamilton equations and its reduction. Chinese Ann. Math. Ser. A 13 (1992) 115-123
22. Ma W.X. Isospectral Lax operator algebras and integrable systems. Proceedings of CAST First Academic Annual Meeting of Youths, Science Volume (1992), Chinese Science and Technology Press 7-14
23. Ma W.X. The algebraic structure of zero curvature representations and application to coupled KdV systems. J. Phys. A: Math. Gen. 26 (1993) 2573-2582
24. Ma W.X. An approach for constructing nonisospectral hierarchies of evolution equations. J. Phys. A: Math. Gen. 25 (1992) L719-L726
25. Ma W.X. Lax representations and Lax operator algebras of isospectral and nonisospectral hierarchies of evolution equations. J. Math. Phys. 33 (1992) 2464-2476
26. Qiao Z.J. New hierarchies of isospectral and non-isospectral integrable NLEEs derived from the Harry-Dym spectral problem. Phys. A 252 (1998) 377-387
27. Tu G.Z. A trace identity and its applications to the theory of discrete integrable systems. J. Phys. A: Math. Gen. 23 (1990) 3903-3922
28. Ma W.X., and Fuchssteiner B. Algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations. J. Math. Phys. 40 (1999) 2400-2418
29. Ma W.X. A simple scheme for generating nonisospectral flows from zero curvature representation. Phys. Lett. A 179 (1993) 179-185
30. Blaszak M. Multi-Hamiltonian Theory of Dynamical Systems, Texts and Monographs in Physics (1998), Springer-Verlag, Berlin
31. Ma W.X. A discrete variational identity on semi-direct sums of Lie algebras. J. Phys. A: Math. Theor. 40 (2007) 15055-15069
32. Ma W.X., and Zhou R.G. Adjoint symmetry constraints of multicomponent AKNS equations. Chin. Ann. Math. Ser. B 23 (2002) 373-384
33. Gel'fand I.M., and Dorfman I.Ja. Hamiltonian operators and algebraic structures associated with them. Funktsional. Anal. i Prilozhen 13 (1979) 13-30 96
34. Dickey L.A. Soliton Equations and Hamiltonian System. second ed. Advanced Series in Mathematical Physics vol. 26 (2003), World Scientific Publishing Co., Inc., River Edge, NJ
35. Luo L., Ma W.X., and Fan E.G. The algebraic structure of zero curvature representations associated with integrable couplings. Internat. J. Modern Phys. A 23 (2008) 1309-1325
36. Luo L., and Fan E.G. The algebraic structure of discrete zero curvature equations associated with integrable couplings and application to enlarged Volterra systems. Sci. China Ser. A 52 (2009) 147-159
37. Tu G.Z. Constrained formal variational calculus and its applications to soliton equations. Sci. Sin. Ser. A 29 (1986) 138-148
38. Tu G.Z. The trace identity a powerful tool for constructing the Hamiltonian structure of integrable systems. J. Math. Phys. 30 (1989) 330-338
39. Ma W.X., and Chen M. Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras. J. Phys. A: Math. Gen. 39 (2006) 10787-10801
40. Hu X.B. A powerful approach to generate new integrable systems. J. Phys. A: Math. Gen. 27 (1994) 2497-2514
41. Zhang D.J., and Chen D.Y. Hamiltonian structure of discrete soliton systems. J. Phys. A: Math. Gen. 35 (2002) 7225-7241
42. Guo F.K., and Zhang Y.F. The quadratic-form identity for constructing the Hamiltonian structure of integrable systems. J. Phys. A: Math. Gen. 38 (2005) 8537-8548
43. Li Z., Zhang Y.J., and Dong H.H. Integrable couplings of the TC hierarchy and its Hamiltonian structure. Modern Phys. Lett. B 21 (2007) 595-602
44. Liu B., Dong H.H., and Li Z. The Hamiltonian structure of two integrable expanding models. J. Partial Differential Equations 20 (2007) 337-348
45. Ablowitz M.J., Kaup D.J., Newell A.C., and Segur H. The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math. 53 (1974) 249-315
46. Fuchssteiner B. Application of hereditary symmetries to nonlinear evolution equations. Nonlinear Anal. 3 (1979) 849-862
47. Fuchssteiner B., and Fokas A.S. Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys. D 4 (1981-82) 47-66
48. Ma W.X., and Zhou R.G. Nonlinearization of spectral problems for the perturbation KdV systems. Phys. A 296 (2001) 60-74
49. W.X. Ma, L. Gao, Coupling integrable couplings, Modern Phys. Lett. B, 2008 (in press)
50. Ma W.X., He J.S., and Qin Z.Y. A supertrace identity and its applications to superintegrable systems. J Math. Phys. 49 (2008) 033511 13 pp
51. Frappat L., Sciarrino A., and Sorba P. Dictionary on Lie Algebras and Superalgebras (2000), Academic Press Inc., San Diego CA
52. Nersessian A. Elements of (super-)Hamiltonian formalism. Supersymmetric Mechanics Vol.1. Lecture Notes in Phys. vol. 698 (2006), Springer, Berlin 139-188