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Dressing the boundary: On soliton solutions of the nonlinear Schrödinger equation on the half‐line
Author(s) -
Zhang Cheng
Publication year - 2019
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12248
Subject(s) - integrable system , soliton , lax pair , inverse scattering problem , inverse scattering transform , bounded function , mathematical analysis , nonlinear system , nonlinear schrödinger equation , boundary value problem , mathematics , boundary (topology) , line (geometry) , mathematical physics , space (punctuation) , scattering , physics , inverse problem , schrödinger equation , quantum mechanics , geometry , linguistics , philosophy
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 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.