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Zhang, Cheng

Dressing the boundary: on soliton solutions of the nonlinear Schrödinger equation on the half-line. (English) Zbl 1420.35388

Stud. Appl. Math. 142, No. 2, 190-212 (2019).
Summary: 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 Schrödinger 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, which 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.

Cited in 8 Documents

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
35C08 Soliton solutions

Keywords:

dressing transformations; integrable boundary conditions; inverse scattering transform; soliton solutions; NLS equation on the half-line
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