2012/01/31 by Samuel Butler · 1 citation
Physics and Astronomy · #nlin.SI
paper · pdf · doi:10.1088/0951-7715/25/6/1613
published as Nonlinearity 25 (2012) 1613-1634 · 18 pages
arxiv created 2012/03/21 · arxiv updated 2015/06/03
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 affine-linear lattice equations which possess the multidimensional consistency property. Due to this property it is natural to consider these equations living in an N-dimensional lattice, where the solutions depend on N distinct independent variables and associated parameters. The direct scattering procedure, which is one-dimensional, 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 inverse scattering solutions of some previously known lattice equations, such as the lattice KdV equation.