A new hierarchy of (1 + 1)-dimensional soliton equations and its quasi-periodic solutions. (English) Zbl 1142.37046
Summary: A new spectral problem is proposed, from which a hierarchy of (1 + 1)-dimensional soliton equations is derived. With the help of nonlinearization approach, the soliton systems in the hierarchy are decomposed into two new compatible Hamiltonian systems of ordinary differential equations. The generating function flow method is used to prove the involutivity and the functional independence of the conserved integrals. The Abel-Jacobi coordinates are introduced to straighten out the associated flows. Using the Riemann-Jacobi inversion technique, the explicit quasi-periodic solutions for the (1 + 1)-dimensional soliton equations are obtained.
MSC:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 35B15 | Almost and pseudo-almost periodic solutions to PDEs |
| 35Q51 | Soliton equations |
| 37K15 | Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems |
Keywords:
integrable systems; spectral problem; nonlinearization approach; soliton systems; Hamiltonian systems; generating function flow method; Abel-Jacobi coordinatesReferences:
| [1] | Cao, C. W.; Geng, X. G., Classical integrable systems generated through nonlinearization of eigenvalue problems, (Proceedings of the conference on nonlinear physics (Shanghai 1989), Research Reports in Physics (1990), Springer: Springer Berlin), 66-78 · Zbl 0728.70012 |
| [2] | Cao, C. W., Nonlinearization of the Lax system for AKNS hierarchy, Sci China A, 33, 528-536 (1990) · Zbl 0714.58026 |
| [3] | Cao, W.; Geng, X. G., C. Neumann and Bargmann systems associated with the coupled KdV soliton hierarchy, J Phys A, 23, 4117-4125 (1990) · Zbl 0719.35082 |
| [4] | Belokolos, E. D.; Bobenko, A. I.; Enolskii, V. Z.; Its, A. R.; Matveev, V. B., Algebro-geometric approach to nonlinear integrable equations (1994), Springer: Springer Berlin · Zbl 0809.35001 |
| [5] | Gesztesy, F.; Ratnaseelan, R., An alternative approach to algebro-geometric solutions of the AKNS hierarchy, Rev Math Phys, 10, 345-391 (1998) · Zbl 0974.35107 |
| [6] | Cao, C. W.; Wu, Y. T.; Geng, X. G., Relation between the Kadometsev-Petviashvili equation and the confocal involutive system, J Math Phys, 40, 3948-3970 (1999) · Zbl 0947.35138 |
| [7] | Geng, X. G.; Wu, Y. T.; Cao, C. W., Quasi-periodic solutions of the modified Kadomtsev-Petviashvili equation, J Phys A, 32, 3733-3742 (1999) · Zbl 0941.35090 |
| [8] | Cao, C. W.; Geng, X. G.; Wu, Y. T., From the special 2+1 Toda lattice to the Kadomtsev-Petviashvili equation, J Phys A, 32, 8059-8078 (1999) · Zbl 0977.37036 |
| [9] | Geng, X. G.; Cao, C. W., Quasi-periodic solutions of the (2+1)-dimensional modified Korteweg-de Vries equation, Phys Lett A, 261, 289-296 (1999) · Zbl 0937.35155 |
| [10] | Geng, X. G.; Cao, C. W., Decomposition of the (2+1)-dimensional Gardner equation and its quasi-periodic solutions, Nonlinearity, 14, 1433 (2001) · Zbl 1160.37405 |
| [11] | Hao, Y. H.; Du, D. L., Two (2+1)-dimensional soliton equations and their quasi-periodic solutions, Chaos, Solitons & Fractals, 26, 979-996 (2005) · Zbl 1080.35110 |
| [12] | Li, X. M.; Geng, X. G., Lax matrix and a generalized coupled KdV hierarchy, Phys A, 327, 357-370 (2003) · Zbl 1098.35562 |
| [13] | Li, X. M.; Zhang, J. S., A generalized TD hierarchy through the Lax matrix and their explicit solutions, Phys Lett A, 321, 14-23 (2004) · Zbl 1118.81393 |
| [14] | Arnold, V. I., Mathematical methods of classical mechanics (1978), Springer: Springer Berlin · Zbl 0386.70001 |
| [15] | Griffiths, P.; Harris, J., Principles of algebraic geometry (1999), Wiley: Wiley New York |
| [16] | Mumford, D., Tata lectures on theta (1984), Birkhäuser: Birkhäuser Boston · Zbl 0744.14033 |
| [17] | Zhou, R. G., The finite-band solution of the Jaulent-Miodek equation, J Math Phys, 38, 2535-2546 (1997) · Zbl 0878.58039 |
| [18] | Tu, G. Z.; Meng, D. Z., The trace identity, a powerful tool for constructing the hamiltonian. Structure of integrable system, Acta Math Appl Sin, 5, 89-96 (1989) · Zbl 0698.70013 |
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