Akhmediev, N. N.; Korneev, V. I. 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. Cited in 127 Documents 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 Keywords:exact analytic solution; nonlinear Schrödinger equation; periodic solutions; time evolution; wave with constant amplitude; small periodic perturbation; spectrum PDFBibTeX XMLCite \textit{N. N. Akhmediev} and \textit{V. I. Korneev}, Theor. Math. Phys. 69, 1089--1093 (1986; Zbl 0625.35015); translation from Teor. Mat. Fiz. 69, No. 2, 189--194 (1986) Full Text: DOI References: [1] S. P. Novikov, in: R. K. Bullough and P. J. Caudrey (eds), Solitons, Springer, Berlin (1980). (The reference is to pp. 348-362 of the Russian translation published by Mir, Moscow (1983). [2] A. R. Its and V. P. Kotlyarov, Dokl. Akad. Nauk Uzb. SSR, Ser. A, No. 11, 965 (1976). [3] M. V. Babich, A. I. Bobenko, and V. B. Mateev Izv. Akad. Nauk SSSR, Ser. Mat.,49, 511 (1985). [4] Y. C. Ma and M. J. Ablowitz, Stud. Appl. Math.,65, 113 (1981). [5] E. R. Tracy, H. H. Chen, and Y. C. Lee, Phys. Rev. Lett.,53, 218 (1984). · doi:10.1103/PhysRevLett.53.218 [6] A. Hasegawa, Opt. Lett.,9, 288 (1984). · doi:10.1364/OL.9.000288 [7] D. Anderson and M. Lisak, Opt. Lett.,9, 468 (1984). · doi:10.1364/OL.9.000468 [8] V. I. Bespalov and V. I. Talanov, Pis’ma Zh. Eksp. Teor. Fiz.,3, 471 (1966). [9] L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media [in Russian], Nauka, Mir (1982). (English translation of earlier edition published by Pergamon Press, Oxford (1960).) · Zbl 0122.45002 [10] H. C. Yuen and B. Lake, in: Solitons in Action (Proc. of a Workshop, Redstone Arsenal, 1977, eds K. Longren and A. Scott), United States Army Research Office, Durham, Math. Div., New York (1978). [11] H. C. Yuen and W. E. Ferguson, Phys. Fluids,21, 1275 (1978). · doi:10.1063/1.862394 [12] N. N. Akhmediev, V. I. Korneev, and Yu. V. Kuz’menko, Zh. Eksp. Teor. Fiz.,88, 107 (1985). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.