An Inverse Scattering Transform for the Lattice Potential KdV 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.

Cited by in corpus (10)

• On non-multiaffine consistent-around-the-cube lattice equations
• Darboux and binary Darboux transformations for discrete integrable systems 1. Discrete potential KdV equation
• Multidimensional Inverse Scattering of Integrable Lattice Equations
• Darboux and Binary Darboux Transformations for Discrete Integrable Systems. II. Discrete Potential mKdV Equation
• Twisted reductions of integrable lattice equations, and their Lax representations
• Discrete Crum's Theorems and Integrable Lattice Equations
• Solutions to the ABS lattice equations via generalized Cauchy matrix approach
• The Sylvester equation and the elliptic Korteweg-de Vries system
• Spectrum transformation and conservation laws of the lattice potential KdV equation
• Eigenfunction equations of lattice KdV equations and connections to ABS lattice equations with a $\delta$ term