Solitons, boundary value problems and a nonlinear method of images

TL;DR

This paper develops a nonlinear method of images for the nonlinear Schrödinger equation on a half line, characterizing soliton boundary interactions and discrete spectrum properties using symmetry and extension techniques.

Contribution

It introduces a nonlinear analogue of the method of images, revealing the structure of soliton reflections and eigenvalues for boundary value problems.

Findings

Discrete eigenvalues form quartets, not pairs.

Reflection of solitons is due to a 'mirror' soliton beyond the boundary.

Explicit relations for norming constants are derived.

Abstract

We characterize the soliton solutions of the nonlinear Schroedinger equation on the half line with linearizable boundary conditions. Using an extension of the solution to the whole line and the corresponding symmetries of the scattering data, we identify the properties of the discrete spectrum of the scattering problem. We show that discrete eigenvalues appear in quartets (as opposed to pairs in the initial value problem), and we obtain explicit relations for the norming constants associated to symmetric eigenvalues. The apparent reflection of each soliton at the boundary of the spatial domain is due to the presence of a "mirror" soliton, with equal amplitude and opposite velocity, located beyond the boundary. We then calculate the position shift of the physical solitons as a result of the nonlinear reflection. These results provide a nonlinear analogue of the method of images that is used to solve boundary value problems in electrostatics.