Symmetry and non-lie reduction of the nonlinear Schrödinger equation
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The nonlinear Schrödinger-type equations invariant with respect to the extended Galilean group are described. We study the conditional symmetry of such equations, realize the reduction procedure, and construct the classes of exact solutions.
KeywordsExact Solution Reduction Procedure Galilean Group Conditional Symmetry
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- 1.V. I. Fushchich, V. M. Shtelen', andN. I. Serov, Symmetry Analysis and Exact Solutions of Nonlinear Equations in Mathematical Physics [in Russian], Naukova Dumka, Kiev (1989).Google Scholar
- 2.V. I. Chopik, “Symmetry and reduction of multi-dimensional Schrödinger equation with the logarithmic nonlinearity,” in:Symmetry Analysis of Equations of Mathematical Physics [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1992), pp. 54–62.Google Scholar
- 3.I. Bialynicki-Birula andJ. Mycielski, “Nonlinear wave mechanics,”Ann. Phys.,100, No. 1, 62–93 (1976).Google Scholar
- 4.D. Schuch, K.-M. Chung, andH. Hartman, “Nonlinear Schrödinger-type field equation for the description of dissipative systems. I. Derivation of the nonlinear field equation and one-dimensional examples,”J. Math. Phys.,24, No. 6, 1652–1660 (1983).Google Scholar
- 5.L. Brüll andH. Lange, “The Schrödinger-Langevin equation: special solutions and nonexistence of solitary waves,”J. Math. Phys.,25, No. 4, 786–790 (1984).Google Scholar
- 6.V. I. Fushchich andV. I. Chopik, “Conditional invariance of the nonlinear Schrödinger equation,”Dokl. Akad. Nauk Ukr. SSR, Ser. A., No. 4, 30–33 (1990).Google Scholar
- 7.V. I. Fushchich., L. F. Barannik, andA. F. Barannik, The Subgroup Analysis of the Galilei and Poincaré Groups and the Reduction of Nonlinear Equations [in Russian], Naukova Dumka, Kiev (1991).Google Scholar
- 8.G. Burdet, J. Parera, M. Perrin, and P. Wintemitz, “The optical group and its subgroups,”J. Math. Phys.,19, No. 8, 1758–1786 (1978).Google Scholar
- 9.P. Clarkson, “Dimensional reductions and exact solutions of a generalized nonlinear Schrödinger equation,”Nonlinearity (UK),5, 453–472 (1992).Google Scholar
- 10.V. I. Chopik, “Non-Lie reduction of the nonlinear Schrödinger equation,”Ukr. Mat. Zh.,43, No. 11, 1504–1509 (1991).Google Scholar
- 11.P. Myronyuk andV. Chopyk, “Conditional Galilei-invariance of multi-dimensional nonlinear heat equation,” in:Symmetry Analysis of Equations of Mathematical Physics [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1992), pp. 66–68.Google Scholar