Quasi-integrability of deformations of the KdV equation

TL;DR

This paper explores the quasi-integrability of deformed KdV equations, demonstrating asymptotic conservation of charges during soliton scattering through analytical and numerical methods, and providing new soliton solutions via Hirota's method.

Contribution

It introduces a framework for analyzing quasi-integrability in deformed KdV equations, including analytical solutions and numerical evidence of asymptotic charge conservation.

Findings

Charges are asymptotically conserved during soliton scattering.

Analytical one-soliton solutions are obtained for specific deformation parameters.

Numerical simulations confirm charge conservation and reveal soliton-radiation interactions.

Abstract

We investigate the quasi-integrability properties of various deformations of the Korteweg-de Vries (KdV) equation, depending on two parameters $\varepsilon_1$ and $\varepsilon_2$, which include among them the regularized long-wave (RLW) and modified regularized long-wave (mRLW) equations. We show, using analytical and numerical methods, that the charges, constructed from a deformation of the zero curvature equation for the KdV equation, are asymptotically conserved for various values of the deformation parameters. By that we mean that, despite the fact that the charges do vary in time during the scattering of solitons, they return after the scattering to the same values they had before it. That property was tested numerically for the scattering of two and three solitons, and analytically for the scattering of two solitons in the mRLW theory ($\varepsilon_2=\varepsilon_1=1$). We also show that the Hirota method leads to analytical one-soliton solutions of our deformed equation for $\varepsilon_1 = 1$, and any value of $\varepsilon_2$. We also mention some properties of soliton-radiation interactions seen in some of our simulations.