Solitons, the Korteweg-de Vries equation with step boundary values and pseudo-embedded eigenvalues

TL;DR

This paper investigates soliton behavior in the Korteweg-de Vries equation with step boundary conditions, revealing how solitons can propagate or become trapped depending on their size, and introduces the concept of pseudo-embedded eigenvalues affecting soliton trapping.

Contribution

It introduces the concept of pseudo-embedded eigenvalues in the KdV equation with step boundaries and analyzes their role in soliton trapping using inverse scattering methods.

Findings

Large solitons propagate with phase shifts calculated via inverse scattering.

Small solitons become trapped near pseudo-embedded eigenvalues.

Continuous spectrum near pseudo-embedded eigenvalues can mimic discrete spectra.

Abstract

The Korteweg-deVries (KdV) equation with step boundary conditions is considered, with an emphasis on soliton dynamics. When one or more initial solitons are of sufficient size they can propagate through the step; in this case the phase shift is calculated via the inverse scattering transform. On the other hand, when the amplitude is too small they become trapped. In the trapped case the transmission coefficient of the associated associated linear Schr\"odinger equation can become large at a point exponentially close to the continuous spectrum. This point is referred to as a {\it pseudo-embedded eigenvalue}. Employing the inverse problem it is shown that the continuous spectrum associated with a branch cut in the neighborhood of the pseudo-embedded eigenvalue plays the role of discrete spectra, which in turn leads to a trapped soliton in the KdV equation.