An Inverse Scattering Transform for the Lattice Potential KdV Equation

TL;DR

This paper develops a discrete inverse scattering transform method to solve the initial value problem for the lattice potential KdV equation, enabling explicit soliton solutions and a comprehensive analysis of reflectionless potentials.

Contribution

It introduces a novel discrete inverse scattering transform approach for the LKdV equation, including solving a discrete Gel'fand-Levitan equation and characterizing N-soliton solutions.

Findings

Explicit N-soliton solutions derived

Complete characterization of reflectionless potentials

Solution of a discrete Gel'fand-Levitan equation

Abstract

The lattice potential Korteweg-de Vries equation (LKdV) is a partial difference equation in two independent variables, which possesses many properties that are analogous to those of the celebrated Korteweg-de Vries equation. These include discrete soliton solutions, Backlund transformations and an associated linear problem, called a Lax pair, for which it provides the compatibility condition. In this paper, we solve the initial value problem for the LKdV equation through a discrete implementation of the inverse scattering transform method applied to the Lax pair. The initial value used for the LKdV equation is assumed to be real and decaying to zero as the absolute value of the discrete spatial variable approaches large values. An interesting feature of our approach is the solution of a discrete Gel'fand-Levitan equation. Moreover, we provide a complete characterization of reflectionless potentials and show that this leads to the Cauchy matrix form of N-soliton solutions.