Special features of the KdV-Sawada-Kotera equation

TL;DR

This paper investigates the KdV-Sawada-Kotera equation, revealing its asymptotic integrability and inelastic soliton collision processes through normal form analysis and exact solutions.

Contribution

It develops a third-order asymptotic expansion showing the equation's asymptotic integrability and characterizes inelastic soliton interactions, including exact two-soliton solutions.

Findings

The equation is asymptotically integrable up to third order.

Higher-order corrections describe inelastic soliton collisions.

Exact two-soliton solutions exhibit purely inelastic scattering.

Abstract

The KdV-Sawada-Kotera equation has single-, two- and three-soliton solutions. However, it is not known yet whether it has N-soliton solutions for any N. Viewing it as a perturbed KdV equation, the asymptotic expansion of the solution is developed through third order within the framework of a Normal Form analysis. It is shown that the equation is asymptotically integrable through the order considered. Focusing on the soliton sector, it is shown that the higher-order corrections in the Normal Form expansion represent purely inelastic KdV-soliton-collision processes, and vanish identically in the single-soliton limit. These characteristics are satisfied by the exact two-soliton solution of the KdV-Sawada-Kotera equation: The deviation of this solution from its KdV-type two-soliton approximation describes a purely inelastic scattering process: The incoming state is the faster KdV soliton. It propagates until it hits a localized perturbation, which causes its transformation into the outgoing state, the slower soliton. In addition, the effect of the perturbation on the exact two-soliton solution vanishes identically in the single-soliton limit (equal wave numbers for the two solitons).