Multidimensional inverse scattering of integrable lattice equations
Abstract
We present a discrete inverse scattering transform for all ABS equations excluding Q4. The nonlinear partial difference equations presented in the ABS hierarchy represent a comprehensive class of scalar affinelinear lattice equations which possess the multidimensional consistency property. Due to this property it is natural to consider these equations living in an Ndimensional lattice, where the solutions depend on N distinct independent variables and associated parameters. The direct scattering procedure, which is onedimensional, is carried out along a staircase within this multidimensional lattice. The solutions obtained are dependent on all N lattice variables and parameters. We further show that the soliton solutions derived from the Cauchy matrix approach are exactly the solutions obtained from reflectionless potentials, and we give a short discussion on solutions of some previously known lattice equations, such as the lattice KdV equation.
 Publication:

Nonlinearity
 Pub Date:
 June 2012
 DOI:
 10.1088/09517715/25/6/1613
 arXiv:
 arXiv:1201.4626
 Bibcode:
 2012Nonli..25.1613B
 Keywords:

 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 18 pages