Stability of KdV solitons on the half-line

TL;DR

This paper investigates the stability of KdV solitons on half-lines, showing strong stability on the right and stability for negative times on the left, using almost conserved quantities and boundary condition adaptations.

Contribution

It provides the first stability analysis of KdV solitons on half-lines, including new results for boundary conditions and stability over negative times.

Findings

Strong stability of solitons on the right half-line with homogeneous boundary conditions

Stability for all negative times on the left half-line

Use of almost conserved quantities tailored to half-line evolution

Abstract

In this paper we study the stability problem for KdV solitons on the left and right half-line. Unlike standard KdV, these are not exact solutions to the equations posed in the half-line. However, we are able to show that solitons placed initially far away from the origin are strongly stable for the problem posed on the right half-line, assuming homogeneous boundary conditions. For the problem posed on the left half-line, the positive infinitetime stability problem makes no sense for the case of KdV solitons, but in this setting we prove a result of stability for all negative times. The proof involves the use of almost conserved quantities adapted to the evolution of the KdV soliton on the particular case of the half-line. Adaptations to other boundary conditions or star graphs are also discussed.