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Modulation instability and periodic solutions of the nonlinear Schrödinger equation. (English. Russian original) Zbl 0625.35015

Theor. Math. Phys. 69, 1089-1093 (1986); translation from Teor. Mat. Fiz. 69, No. 2, 189-194 (1986).
A very simple exact analytic solution of the nonlinear Schrödinger equation is found in the class of periodic solutions. It describes the time evolution of a wave with constant amplitude on which a small periodic perturbation is superimposed. Expressions are obtained for the evolution of the spectrum of this solution, and these expressions are analyzed qualitatively. It is shown that there exists a certain class of periodic solutions for which the real and imaginary parts are linearly related, and an example of a one-parameter family of such solutions is given.

MSC:

35G20 Nonlinear higher-order PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
35B10 Periodic solutions to PDEs
35B20 Perturbations in context of PDEs
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