Generalized Solitary Waves in a Finite-Difference Korteweg-de Vries Equation

TL;DR

This paper investigates generalized solitary wave solutions in higher-order KdV equations and their finite-difference lattice discretizations, revealing conditions for their existence and connecting continuous and discrete models.

Contribution

It introduces conditions for generalized solitary waves in seventh-order and higher KdV equations and demonstrates their existence in a finite-difference lattice KdV model.

Findings

Generalized solitary waves exist in higher-order KdV equations.

Conditions for solitary wave solutions are identified.

Lattice KdV discretization supports solitary wave solutions.

Abstract

Generalized solitary waves with exponentially small non-decaying far field oscillations have been studied in a range of singularly-perturbed differential equations, including higher-order Korteweg-de Vries (KdV) equations. Many of these studies used exponential asymptotics to compute the behaviour of the oscillations, revealing that they appear in the solution as special curves known as Stokes lines are crossed. Recent studies have identified similar behaviour in solutions to difference equations. Motivated by these studies, the seventh-order KdV and a hierarchy of higher-order KdV equations are investigated, identifying conditions which produce generalized solitary wave solutions. These results form a foundation for the study of infinite-order differential equations, which are used as a model for studying lattice equations. Finally, a lattice KdV equation is generated using finite-difference discretization, in which a lattice generalized solitary wave solution is found.