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Inverse scattering transform for the focusing nonlinear Schrödinger equation with counterpropagating flows
Author(s) -
Biondini Gino,
Lottes Jonathan,
Mantzavinos Dionyssios
Publication year - 2021
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.12347
Subject(s) - inverse scattering transform , inverse scattering problem , eigenfunction , mathematical analysis , mathematics , scattering , eigenvalues and eigenvectors , classification of discontinuities , inverse problem , nonlinear system , quantum inverse scattering method , complex plane , inverse , scattering theory , matrix (chemical analysis) , homogeneous space , jump , riemann surface , physics , quantum mechanics , geometry , materials science , composite material
The inverse scattering transform for the focusing nonlinear Schrödinger equation is presented for a general class of initial conditions whose asymptotic behavior at infinity consists of counterpropagating waves. The formulation takes into account the branched nature of the two asymptotic eigenvalues of the associated scattering problem. The Jost eigenfunctions and scattering coefficients are defined explicitly as single‐valued functions on the complex plane with jump discontinuities along certain branch cuts. The analyticity properties, symmetries, discrete spectrum, asymptotics, and behavior at the branch points are discussed explicitly. The inverse problem is formulated as a matrix Riemann‐Hilbert problem with poles. Reductions to all cases previously discussed in the literature are explicitly discussed. The scattering data associated to a few special cases consisting of physically relevant Riemann problems are explicitly computed.