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Elliptic \(N\)-soliton solutions of ABS lattice equations. (English) Zbl 1217.37068
V. E. Adler, A. I. Bobenko and Yu. B. Suris [Commun. Math. Phys. 233, No. 3, 513–543 (2003; Zbl 1075.37022]) classified discrete integrable systems on quad-graphs, i.e., surface cell decompositions with quadrilateral faces. In the present paper, the authors construct \(N\)-soliton-type solutions for all equations in the classification but the \(Q4\) equation. They generalize the Cauchy matrix approach of an earlier paper joint with J. Hietarinta [J. Phys. A, Math. Theor. 42, No. 40, Article ID 404005, 34 p. (2009; Zbl 1184.35281)].
The main construction consists of an application of \(N\) consecutive Bäcklund transformations to a so-called seed solution. It is performed for the \(Q3\) equation; the corresponding \(N\)-solition solutions for the degenerate subcases of \(Q3\) are then obtained by passing to limits of certain parameters as in [loc. cit.].

MSC:
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K60 Lattice dynamics; integrable lattice equations
39A12 Discrete version of topics in analysis
35Q53 KdV equations (Korteweg-de Vries equations)
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