Geodesic
Modulation instability and periodic solutions of the nonlinear Schr\"odinger equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 69 (1986) no. 2, pp. 189-194

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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. 
@article{TMF_1986_69_2_a2,
     author = {N. N. Akhmediev and V. I. Korneev},
     title = {Modulation instability and periodic solutions of the nonlinear {Schr\"odinger} equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {189--194},
     publisher = {mathdoc},
     volume = {69},
     number = {2},
     year = {1986},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1986_69_2_a2/}
}
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N. N. Akhmediev; V. I. Korneev. Modulation instability and periodic solutions of the nonlinear Schr\"odinger equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 69 (1986) no. 2, pp. 189-194. http://geodesic.mathdoc.fr/item/TMF_1986_69_2_a2/