arXiv subjects

James Atkinson

Publications and source records attributed to James Atkinson.

21 records · Page 2Linked to original sources

arXiv2008

Soliton solutions for Q3

We construct N-soliton solutions to the equation called Q3 in the recent Adler-Bobenko-Suris classification. An essential ingredient in the construction is the relationship of $(Q3)_{δ=0}$ to the equation proposed by Nijhoff, Quispel and Capel in 1983 (the NQC equation). This latter equation has two extra parameters, and depending on their sign choices we get a 4-to-1 relationship from NQC to $(Q3)_{δ=0}$. This leads to a four-term background solution, and then to a 1-soliton solution using a Backlund transformation. Using the 1SS as a guide allows us to get the N-soliton solution in terms of the tau-function of the Hirota-Miwa equation.

nlin.SI↗

arXiv2007

Solutions of Adler's lattice equation associated with 2-cycles of the Backlund transformation

The BT of Adler's lattice equation is inherent in the equation itself by virtue of its multidimensional consistency. We refer to a solution of the equation that is related to itself by the composition of two BTs (with different Backlund parameters) as a 2-cycle of the BT. In this article we will show that such solutions are associated with a commuting one-parameter family of rank-2 (i.e., 2-variable), 2-valued mappings. We will construct the explicit solution of the mappings within this family and hence give the solutions of Adler's equation that are 2-cycles of the BT.

nlin.SI↗

arXiv2006

Seed and soliton solutions for Adler's lattice equation

Adler's lattice equation has acquired the status of a master equation among 2D discrete integrable systems. In this paper we derive what we believe are the first explicit solutions of this equation. In particular it turns out necessary to establish a non-trivial seed solution from which soliton solutions can subsequently be constructed using the Bäcklund transformation. As a corollary we find the corresponding solutions of the Krichever-Novikov equation which is obtained from Adler's equation in a continuum limit.

nlin.SI↗