Elliptic \(N\)-soliton solutions of ABS lattice equations. (English) Zbl 1217.37068

Nijhoff, Frank; Atkinson, James

Int. Math. Res. Not. 2010, No. 20, 3837-3895 (2010).

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.].

Reviewer: Thomas J. Bartsch (Gießen)

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)

Keywords:

quadrilateral lattice equations; partial differential equations; Bäcklund transformations