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

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 Schrödinger 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.