Solutions to ABS lattice equations via generalized Cauchy matrix approach. (English) Zbl 1338.37113
Summary: The usual Cauchy matrix approach starts from a known plain wave factor vector \(\mathbf{r}\) and known dressed Cauchy matrix \(\mathbf{M}\). In this paper, we start from a determining matrix equation set with undetermined \(\mathbf{r}\) and \(\mathbf{M}\). From the determining equation set we can build shift relations for some defined scalar functions and then derive lattice equations. The determining equation set admits more choices for \(\mathbf{r}\) and \(\mathbf{M}\) and in the paper we give explicit formulae for all possible \(\mathbf{r}\) and \(\mathbf{M}\). As applications, we get more solutions than usual multisoliton solutions for many lattice equations including the lattice potential KdV equation, the lattice potential modified KdV equation, the lattice Schwarzian KdV equation, NQC equation, and some lattice equations in ABS list.
MSC:
| 37K60 | Lattice dynamics; integrable lattice equations |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 39A12 | Discrete version of topics in analysis |
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