Positive and negative integrable hierarchies, associated conservation laws and Darboux transformation

Two hierarchies of integrable positive and negative lattice equations in connection with a new discrete isospectral problem are derived. It is shown that they correspond to positive and negative power expansions respectively of Lax operators with respect to the spectral parameter, and each equation...

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Bibliographic Details
Published in:Journal of computational and applied mathematics 2009-12, Vol.233 (4), p.1096-1107
Main Authors: Li, Xin-Yue, Zhang, Yuan-Qing, Zhao, Qiu-Lan
Format: Article
Language:English
Subjects:
Combinatorics. Ordered structures
Conservation laws
Darboux transformation
Discrete integrable system
Discrete zero curvature equation
Exact sciences and technology
Hamiltonian structure
Logic and foundations
Mathematical analysis
Mathematical logic, foundations, set theory
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Order, lattices, ordered algebraic structures
Partial differential equations
Recursion theory
Sciences and techniques of general use
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Summary:Two hierarchies of integrable positive and negative lattice equations in connection with a new discrete isospectral problem are derived. It is shown that they correspond to positive and negative power expansions respectively of Lax operators with respect to the spectral parameter, and each equation in the resulting hierarchies is Liouville integrable. Moreover, infinitely many conservation laws of corresponding positive lattice equations are obtained in a direct way. Finally, a Darboux transformation is established with the help of gauge transformations of Lax pairs for the typical lattice soliton equations, by means of which the exact solutions are given.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2009.09.009