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Finite gap integration of the derivative nonlinear Schrödinger equation: a Riemann-Hilbert method. (English) Zbl 1453.35161

Summary: In this paper we retrieve finite gap solutions of the Gerdjikov-Ivanov type derivative nonlinear Schrödinger equation by using the algebro-geometric method and the Riemann-Hilbert method. We show that the Baker-Akhiezer function of the derivative nonlinear Schrödinger equation can be described in terms of two solvable matrix Riemann-Hilbert problems on \(\mathbb C\) with \(\sigma_2\)- and \(\sigma_3\)-symmetry conditions based on the technique developed in the study of long-time asymptotics. Our main tools include matrix Baker-Akhiezer function, asymptotic analysis, algebraic curve and Riemann theta function, matrix Riemann-Hilbert problem and associated deformation procedures.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions

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