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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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