×
Yu, Fajun; Feng, Shuo; Zhao, Yanyu

A complex integrable hierarchy and its Hamiltonian structure for integrable couplings of WKI soliton hierarchy. (English) Zbl 1470.37089

Abstr. Appl. Anal. 2014, Article ID 146537, 7 p. (2014).
Summary: We generate complex integrable couplings from zero curvature equations associated with matrix spectral problems in this paper. A direct application to the WKI spectral problem leads to a novel soliton equation hierarchy of integrable coupling system; then we consider the Hamiltonian structure of the integrable coupling system. We select the \(\overline{U}\), \(\overline{V}\) and generate the nonlinear composite parts, which generate new extended WKI integrable couplings. It is also indicated that the method of block matrix is an efficient and straightforward way to construct the integrable coupling system.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q51 Soliton equations
PDF BibTeX XML Cite
\textit{F. Yu} et al., Abstr. Appl. Anal. 2014, Article ID 146537, 7 p. (2014; Zbl 1470.37089)
Full Text: DOI

References:

[1] Pickering, A., A new truncation in Painlevé analysis, Journal of Physics A: Mathematical and General, 26, 17, 4395-4405 (1993) · Zbl 0803.35131
[2] Zhang, J.-F., Multiple soliton solutions of the dispersive long-wave equations, Chinese Physics Letters, 16, 1, 4-5 (1999)
[3] Fan, E.; Zhang, H., New exact solutions to a solutions to a system of coupled KdV equations, Physics Letters A, 245, 5, 389-392 (1998) · Zbl 0947.35126
[4] Lou, S. Y., Obtaining higher-dimensional integrable models by means of Miura type noninvertible transformations, Acta Physica Sinica, 49, 9, 1657-1662 (2000) · Zbl 1202.37100
[5] Ma, W. X.; Fuchssteiner, B., Integrable theory of the perturbation equations, Chaos, Solitons and Fractals, 7, 8, 1227-1250 (1996) · Zbl 1080.37578
[6] Ma, W. X.; Ma, W. X.; Kaup, D., Integrable couplings and matrix loop algebra, Nonlinear and Modern Mathematical Physics. Nonlinear and Modern Mathematical Physics, AIP Conference Proceedings, 1562, 105-122 (2013), Melville, NY, USA: American Institute of Physics, Melville, NY, USA
[7] Ma, W.-X., Enlarging spectral problems to construct integrable couplings of soliton equations, Physics Letters A, 316, 1-2, 72-76 (2003) · Zbl 1042.37057
[8] Ma, W.-X., Integrable couplings of vector AKNS soliton equations, Journal of Mathematical Physics, 46, 3 (2005) · Zbl 1067.37096
[9] Zhang, Y.; Zhang, H., A direct method for integrable couplings of TD hierarchy, Journal of Mathematical Physics, 43, 1, 466-472 (2002) · Zbl 1052.37055
[10] Zhang, Y.; Tam, H.-W., The multi-component KdV hierarchy and its multi-component integrable coupling system, Chaos, Solitons and Fractals, 23, 2, 651-655 (2005) · Zbl 1071.37050
[11] Ma, W. X., On the two sorts of Hamiltonian operators, Journal of Mathematical Physics, 43, 1408-1421 (2002) · Zbl 1059.37052
[12] Frappat, L.; Sciarrino, A.; Sorba, P., Dictionary on Lie Algebras and Superalgebras (2000), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0965.17001
[13] Sakovich, S. Yu., On integrability of a \((2 + 1)\)-dimensional perturbed KdV equation, Journal of Nonlinear Mathematical Physics, 5, 3, 230-233 (1998) · Zbl 0946.35092
[14] Sakovich, S. Yu., Coupled KdV equations of Hirota-Satsuma type, Journal of Nonlinear Mathematical Physics, 6, 3, 255-262 (1999) · Zbl 0947.35142
[15] Ma, W.-X., A spectral problem based on \(\operatorname{ so }(3, R)\) and its associated commuting soliton equations, Journal of Mathematical Physics, 54, 10 (2013) · Zbl 1304.37047
[16] Fan, E., Integrable evolution systems based on Gerdjikov-Ivanov equations, bi-Hamiltonian structure, finite-dimensional integrable systems and \(N\)-fold Darboux transformation, Journal of Mathematical Physics, 41, 11, 7769-7782 (2000) · Zbl 0986.37059
[17] Ma, W.-X.; Xu, X.-X.; Zhang, Y., Semidirect sums of Lie algebras and discrete integrable couplings, Journal of Mathematical Physics, 47, 5 (2006) · Zbl 1111.37059
[18] Tsuchida, T.; Wadati, M., New integrable systems of derivative nonlinear Schrödinger equations with multiple components, Physics Letters A, 257, 1-2, 53-64 (1999) · Zbl 0936.37043
[19] Tsuchida, T.; Wadati, M., Multi-field integrable systems related to WKI-type eigenvalue problems, Journal of the Physical Society of Japan, 68, 7, 2241-2245 (1999) · Zbl 0943.82009
[20] Fuchssteiner, B., Coupling of Completely Integrable Systems (1993), Dordrecht, The Netherlands: Kluwer, Dordrecht, The Netherlands · Zbl 0786.35123
[21] Gu, C. H.; Hu, H. S.; Zhou, Z. X., Soliton Theory and Its Application (1990), Zhejiang Publishing House of Science and Technology
[22] Hu, X. B., A powerful approach to generate new integrable systems, Journal of Physics A, 27, 7, 2497-2514 (1994) · Zbl 0838.58018
[23] Zhang, Y.; Guo, X.; Tam, H., A new Lie algebra, a corresponding multi-component integrable hierarchy and an integrable coupling, Chaos, Solitons & Fractals, 29, 1, 114-124 (2006) · Zbl 1138.17312
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.