Coupled system description of perturbed KdV equation

TL;DR

This paper develops a coupled system framework for the perturbed KdV equation, separating the integrable normal form from obstacle contributions, and analyzes the solution up to third order.

Contribution

It introduces a method to transform the perturbed KdV equation into a coupled system, explicitly accounting for obstacles to asymptotic integrability.

Findings

The auxiliary equation's solution is a conserved quantity up to third order.

The approach clarifies the role of obstacles in perturbed KdV dynamics.

The method extends the analysis of perturbed soliton equations beyond first order.

Abstract

In the multiple-soliton case, the freedom in the expansion of the solution of the perturbed KdV equation is exploited so as to transform the equation into a system of two equations: The (inte-grable) Normal Form for KdV-type solitons, which obey the usual infinity of KdV-conservation laws, and an auxiliary equation that describes the contribution of obstacles to asymptotic inte-grability, which arise from the second order onwards. The analysis has been carried through the third order in the expansion. Within that order, the solution of the auxiliary equation is a con-served quantity.