Stability of KdV solitons on the half-line

Abstract

In this paper we study the stability problem for KdV solitons on the left and right half-lines. Unlike standard KdV, these are not exact solutions to the equations posed on the half-line, and, contrary to NLS, no exact soliton solution seems to exist. However, we are able to show that solitons posed 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 infinite-time 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 finding and using two almost conserved quantities adapted to the evolution of the KdV soliton in the particular case of the half-line. Adaptations to other boundary conditions or star graphs are also discussed.