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The Korteweg-deVries (KdV) equation with step boundary conditions isconsidered, with an emphasis on soliton dynamics. When one or more initialsolitons are of sufficient size they can propagate through the step; in thiscase the phase shift is calculated via the inverse scattering transform. On theother hand, when the amplitude is too small they become trapped. In the trappedcase the transmission coefficient of the associated associated linearSchr\"odinger equation can become large at a point exponentially close to thecontinuous spectrum. This point is referred to as a {\it pseudo-embeddedeigenvalue}. Employing the inverse problem it is shown that the continuousspectrum associated with a branch cut in the neighborhood of thepseudo-embedded eigenvalue plays the role of discrete spectra, which in turnleads to a trapped soliton in the KdV equation.

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