Its, A. R.; Rybin, A. V.; Sall’, M. A. Exact integration of nonlinear Schrödinger equation. (English. Russian original) Zbl 0678.35012 Theor. Math. Phys. 74, No. 1, 20-32 (1988); translation from Teor. Mat. Fiz. 74, No. 1, 29-45 (1988). Summary: By means of degeneration of general finite-gap formulas a many-parameter set of smooth periodic and almost-periodic solutions of the nonlinear Schrödinger equation expressed in terms of elementary functions is obtained. The scheme of obtaining these solutions by the method of the Darboux transformation is presented. The soliton propagation on arbitrary background is studied. Cited in 35 Documents MSC: 35G20 Nonlinear higher-order PDEs 35Q99 Partial differential equations of mathematical physics and other areas of application 35B10 Periodic solutions to PDEs 35B15 Almost and pseudo-almost periodic solutions to PDEs 35B40 Asymptotic behavior of solutions to PDEs 35A22 Transform methods (e.g., integral transforms) applied to PDEs 35J10 Schrödinger operator, Schrödinger equation Keywords:exact integration; nonlinear Schrödinger equation; finite-gap formulas; smooth periodic; almost-periodic solutions; Darboux transformation; soliton propagation × Cite Format Result Cite Review PDF Full Text: DOI References: [1] V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, The Theory of Solitons: The Inverse Scattering Method [in Russian], Nauka, Moscow (1980). · Zbl 0598.35002 [2] N. N. Akhmedieva, V. M. Eleonskii, and N. E. Kulagin, Zh. Eksp. Teor. Fiz.,89, 1542 (1985). [3] A. R. Its, Vestn. Leningr. Univ., Ser. Mat., No. 7, 121 (1976). [4] A. R. Its and V. P. Kotlyarov, Dokl. Akad. Nauk SSSR, Ser. A, No. 11, 965 (1976). [5] A. R. Its, in: Problems of Mathematical Physics, No. 10 [in Russian], Leningrad State University (1983), p. 118. · Zbl 0516.35025 [6] B. A. Dubrovin, in: Modern Problems of Mathematics (Reviews of Science and Technology), Vol. 23 [in Russian], VINITI, Moscow (1983), p. 137. [7] I. M. Krichever, Usp. Mat. Nauk,32, 184 (1977). [8] M. A. Sall’, Teor. Mat. Fiz.,53, 227 (1982). [9] A. V. Rybin and M. A. Sall’, Teor. Mat. Fiz.,63, 333 (1985). [10] V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz.,61, 118 (1971). 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.