Dressing the boundary: on soliton solutions of the nonlinear Schr\"odinger equation on the half-line

TL;DR

This paper develops a direct dressing method based on integrable boundary conditions to explicitly compute soliton solutions of the focusing nonlinear Schrödinger equation on the half-line, including boundary-bound solitons.

Contribution

It introduces a novel dressing approach that preserves boundary constraints, enabling explicit construction of half-line solitons and boundary-bound solutions for the NLS equation.

Findings

Explicit half-line soliton solutions derived

Boundary-bound solitons constructed and analyzed

Inverse scattering interpretation provided

Abstract

Based on the theory of integrable boundary conditions (BCs) developed by Sklyanin, we provide a direct method for computing soliton solutions of the focusing nonlinear Schr\"odinger (NLS) equation on the half-line. The integrable BCs at the origin are represented by constraints of the Lax pair, and our method lies on dressing the Lax pair by preserving those constraints in the Darboux-dressing process. The method is applied to two classes of solutions: solitons vanishing at infinity and self-modulated solitons on a constant background. Half-line solitons in both cases are explicitly computed. In particular, the boundary-bound solitons, that are static solitons bounded at the origin, are also constructed. We give a natural inverse scattering transform interpretation of the method as evolution of the scattering data determined by the integrable BCs in space.