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Reflectionless potentials and soliton series of the nonlinear Schrödinger equation. (English) Zbl 1194.35421
Summary: Potentials of the Dirac system, slowly decreasing at infinity, generate an infinite discrete spectrum converging to zero. The inverse scattering problem in the class of such potentials is solved in a constructive way similarly to the classical soliton theory. An infinite-dimensional dressing chain generated by Bäcklund transformations over soliton solutions plays the role of determinant representation of the potential. The asymptotics at infinity is derived by use of the Poisson summation formula. An application to the long-time asymptotics of the solution of the NLS equation is considered.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
81U40 Inverse scattering problems in quantum theory
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