Inverse scattering for an AKNS problem with rational reflection coefficients
Abbreviated Journal Title
NONLINEAR SCHRODINGER-EQUATION; SHABAT EIGENVALUE PROBLEM; BOUNDARY-VALUE PROBLEM; EVOLUTION-EQUATIONS; CANONICAL SYSTEMS; MATRIX; FUNCTIONS; HALF-SOLITONS; TRANSFORM; OPERATORS; FORMULAS; Mathematics, Applied; Physics, Mathematical
We study the AKNS (Ablowitz-Kaup-Newell-Segur) inverse scattering problem for rational reflection coefficients on a semi-infinite interval. We demonstrate that the Marchenko integral equations for the AKNS version on such an interval can be solved in a direct and straightforward way by algebraic methods for any set of rational reflection coefficients, vanishing at infinity. The general AKNS scattering problem for this case, as well as the usual symmetry reductions, are discussed. The connection to an alternative procedure by Rourke and Morris (1992 Phys. Rev. A 46 3631) is pointed out. Our procedure is built around a constant matrix, M, which can be constructed from the poles and residues of the rational reflection coefficients. Under certain conditions, which we define as minimal symmetry and which then suitably constrains the distribution of eigenvalues, we show that it is always possible to represent the resulting potentials as truncated N-soliton potentials. This procedure is of interest for solving initial-boundary value problems of integrable hyperbolic systems by the inverse scattering transform (IST) applied to a semi-infinite or finite interval.
"Inverse scattering for an AKNS problem with rational reflection coefficients" (2008). Faculty Bibliography 2000s. 1013.