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New hierarchies of integrable positive and negative lattice models and Darboux transformation. (English) Zbl 1081.37040

Starting from a new discrete isospectral problem, two hierarchies of nonlinear integrable differential-difference equations are derived. It is shown that every equation in the resulting models is integrable in Liouville’s sense. It is also shown that these two hierarchies correspond to positive and negative power expansions with respect to spectral parameter, and they are of rational and polynomial-type equations about potentials, respectively. As an application, exact solutions are given. To this end, the authors’ Darboux transformation is established.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35A30 Geometric theory, characteristics, transformations in context of PDEs
35Q51 Soliton equations
37K60 Lattice dynamics; integrable lattice equations
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References:

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