Whitham theory for perturbed Korteweg-de Vries equation

TL;DR

This paper extends Whitham's modulation theory to perturbed Korteweg-de Vries equations, deriving generalized equations for non-uniform and modified velocity cases, with applications to steady-state solutions under dissipation.

Contribution

It provides a general framework for deriving Whitham modulation equations for perturbed KdV systems, including non-uniform and velocity-modified cases, with illustrative examples.

Findings

Derived general form of perturbed Whitham equations for two cases

Illustrated differences using generalized KdV equation example

Analyzed steady-state solutions under dissipative perturbations

Abstract

Original Whitham's method of derivation of modulation equations is applied to systems whose dynamics is described by a perturbed Korteweg-de Vries equation. Two situations are distinguished: (i) the perturbation leads to appearance of right-hand sides in the modulation equations so that they become non-uniform; (ii) the perturbation leads to modification of the matrix of Whitham velocities. General form of Whitham modulation equations is obtained for each case. The essential difference between them is illustrated by an example of so-called `generalized Korteweg-de Vries equation'. Method of finding steady-state solutions of perturbed Whitham equations in the case of dissipative perturbations is considered.