Negative order MKdV hierarchy and a new integrable Neumann-like system

Physica A: Statistical Mechanics and its Applications

Volumn 313, Issue 3-4, 2002, Pages 365-380

  Qiao, Zhijun ^a,b   Strampp, Walter ^c  

^a LOS ALAMOS NATIONAL LABORATORY   (United States)

^b FUDAN UNIVERSITY   (China)

^c UNIVERSITY OF KASSEL   (Germany)

Author keywords

Lax representation; MKdV hierarchy; Negative order; Neumann like system; Parametric solution; Positive order

Indexed keywords

CONSTRAINT THEORY; HIERARCHICAL SYSTEMS; INTEGRATION; INVERSE PROBLEMS; MATHEMATICAL OPERATORS; NONLINEAR EQUATIONS; SOLITONS;

NEUMANN-LIKE SYSTEMS;

HAMILTONIANS;

EID: 0037107993     PISSN: 03784371     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0378-4371(02)00995-0     Document Type: Article

References (36)

1. Zakharov V.E., Manakov S.V., Novikov S.P., Pitaevsky L.P. Soliton Theory: The Method of Inverse Problem. 1980;Nauka, Moscow.
2. Ablowitz M.J., Segur H. Solitons and Inverse Scattering Transform. 1981;SIAM, Philadelphia.
3. O.I. Bogoyavlensky, S.P. Novikov, The relationship between Hamiltonian formalisms of stationary and nonstationary problems (English. Russian original) Funct. Anal. Appl. 10 (1976) 8-11; (translation from Funkt. Anal. Priloz. 10 (1978) 9-13).
4. Antonowicz M., Fordy A.P., Wojciechowski S. Integrable stationary flows. Mirua maps and bi-Hamiltonian structures Phys. Lett. A. 124:1987;143-150.
5. C.W. Cao, Nonlinearization of Lax system for the AKNS hierarchy, Sci. China A 32 (1989) 701-707 (in Chinese) (also see English Edition, Nonlinearization of Lax system for the AKNS hierarchy, Sci. Sinica A 33 (1990) 528-536).
6. C.W. Cao, Nonlinearization of Lax system for the AKNS hierarchy, Sci. China A 32 (1989) 701-707 (in Chinese) (also see English Edition, Nonlinearization of Lax system for the AKNS hierarchy, Sci. Sinica A 33 (1990) 528-536).
7. C.W. Cao, X.G. Geng, Classical integrable systems generated through nonlinearization of eigenvalue problems in: Chaohao Gu, Yishen Li, Guizhang Tu (Eds.), Reports in Physics, Nonlinear Physics, Springer, Berlin, 1990, pp. 68-78.
8. Fordy A.P., Holm D.D. A tri-Hamiltonian formulation of the self-induced transparency equations. Phys. Lett. A. 160:1991;143-148.
9. Antonowicz M., Wojciechowski S.R. Constrained flows of integrable PDEs and bi-Hamiltonian structure of the Garnier system. Phys. Lett. A. 147:1990;455-462.
10. Camassa R., Holm D.D. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71:1993;1661-1664.
11. Alber M., Camassa R., Fedorov Y.N., Holm D.D., Marsden J.E. On billiard solutions of nonlinear PDE's. Phys. Lett. A. 264:2000;171-178.
12. Wadati M. The exact solution of the modified Korteweg-de Vries equation. J. Phys. Soc. Japan. 32:1972;1681-1682.
13. Wadati M. The modified Korteweg-de Vries equation. J. Phys. Soc. Japan. 34:1973;1289-1296.
14. Chern S.S., Peng C.K. Lie groups and KdV equations. Manuscr. Math. 28:1979;207-217.
15. Olver P.J. Applications of Lie Groups to Differential Equation. 1986;Springer, Berlin.
16. Case K.M., Roos A.M. Sine-Gordon and modified Korteweg-de Vries charges. J. Math. Phys. 23:1982;392-395.
17. Verosky J.M. Negative powers of Olver recursion operators. J. Math. Phys. 32:1991;1733-1736.
18. Andree V.A., Shmakova M.V. Hierarchy of lower Korteweg-de Vries equations and supersymmetry structure of Miura transformation. J. Math. Phys. 34:1993;3491-3506.
19. H.W. Capel, F.W. Nijhoff, Integrable lattice equation, in: A.S. Fokas, V.E. Zakharov (Eds.), Important Developments in Soliton Theory, Springer Lecture Notes in Nonlinear Dynamica, Springer, Berlin, 1993, pp. 38-57.
20. Qiao Z.J. Two new hierarchies containing the sine-Gordon and sinh-Gordon equations, and their Lax representations. Physica A. 243:1997;141-151.
21. Z.J. Qiao, Generation of the soliton hierarchies and the general structure of the commutator representations, preprint 1992, Acta Appl. Math. Sinica 18 (1995) 287-301.
22. Qiao Z.J. Commutator representations of three isospectral equation hierarchies. preprint 1991 Chinese J. Contemporary Math. 14:1993;41-51.
23. Zhou R. Hierarchy of negative order equation and its Lax pair. J. Math. Phys. 36:1995;4220-4225.
24. Zhou R. Dynamical. r -matrices for the constrained Harry-Dym flows Phys. Lett. A. 220:1996;320-330.
25. Cao C.W. Commutator representation of isospectral equation. Chin. Sci. Bull. 34:1989;723-724.
26. Qiao Z.J. Commutator representation for the D-AKNS hierarchy of evolution equation. preprint 1989 Math. Appl. 4(4):1991;64-70.
27. Z.J. Qiao, Lax representations of the Levi hierarchy, preprint 1989, Chinese Sci. Bull. 35 (1990) 1353-1354.
28. Arnol'd V.I. Mathematical Methods of Classical Mechanics. 1978;Springer, Berlin.
29. Zakharov V.E., Shabat A.B. Exact theory of two dimensional self focusing and one dimensional self modulation of waves in nonlinear media. Sov. Phys. JETP. 34:1972;62-69.
30. G.Z. Tu, D.Z. Meng, Trace identity - a powerful tool to Hamiltonian structure of integrable system (II), Acta Math. Appl. Sinica (English Series) 5 (1989) 89-97.
31. C.H. Gu et al., Soliton Theory and its Applications, Zhejiang Publishing House of Science and Technology, Hangzhou, PR China, 1990 (also see the English Edition: Springer, Berlin, 1995 (Chapters 3, 4)).
32. Alber M., Camassa R., Holm D.D., Marsden J.E. The geometry of peaked solitons and billiards solutions of a class of integrable PDEs. Lett. Math. Phys. 32:1994;137-151.
33. Qiao Z.J. Involutive system and integrable C. Neumann system associated with the MKdV hierarchy. J. Math. Phys. 35:1994;2978-2983.
34. J. Moser, Integrable Hamiltonian systems and spectral theory, in: Proceedings of 1983 Beijing Symposium on Differential Geometry and Differential Equations, Science Press, Beijing, 1986, 157-229.
35. Knoerrer H. Geodesics on quadrics and a mechanical problem of C. Neumann. J. Rein. Ang. Math. 334:1982;69-78.
36. Z.J. Qiao, C.W. Cao, W. Strampp, Category of nonlinear evolution equations, algebraic structure, and r-matrix, preprint 2000, 45p., submitted for publication.