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On smooth solutions to the initial value problem for the mixed nonlinear Schrödinger equations*

Published online by Cambridge University Press:  14 November 2011

Guo Boling  and
Tan Shaobin
Guo Boling
Affiliation:
Centre for Nonlinear Studies, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing, China
Tan Shaobin
Affiliation:
Centre for Nonlinear Studies, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing, China
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Synopsis

Solutions to the initial value problem for the mixed nonlinear Schrödinger equation

are considered. Conditions on the constants α,β, γ, function g(·) and initial data u(x, 0) are given so that, for this problem, the unique existence of smooth solutions is proved. In addition, the decay behaviours of the smooth solutions as |x|→+∞ are discussed.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 119 , Issue 1-2 , 1991 , pp. 31 - 45
Copyright
Copyright © Royal Society of Edinburgh 1991

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References

1Rogister, A.. Phys. Fluids 14 (1971), 2733–2739.Google Scholar
2Chen, H. H., Lee, Y. C. and Lin, C. S.. Phys. Scr. 20 (1979), 490–495.CrossRefGoogle Scholar
3Ablowitz, M. J., Ramani, A., Segur, H.. A connection between nonlinear evolution equations and ordinary differential equations of P-type. II, J. Math. Phys. 21 (1980), 1006–1015.Google Scholar
4Gerdjikov, V. S. and Ivanov, I.. Bulg. J. Phys. 10 (1983), 130–134.Google Scholar
5Kundu, A.. Landau–Lifshitz ad higher-order nonlinear systems gauge generated from nonlinear Schrödinger type equations, J. Math. Phys. 25 (1984), 3433–3436.Google Scholar
6Chen, Y. M.. The initial-boundary value problem for a class of nonlinear Schrödinger equations, Acta Math. Sci. 6 (1986), 405ö418.Google Scholar
7Tsutsumi, M. and Fukuda, I.. On solutions of the derivative nonlinear Schrödinger equation, I. Funkcial. Ekvac. 23 (1980), 259–277.Google Scholar
8Tsutsumi, M. and Fukuda, I.. On solutions of the derivative nonlinear Schrödinger equation, II. Funkcial. Ekvac. 24 (1981), 85–94.Google Scholar
9Guo, B. L.. Galerkin Finite Element Method and Error Estimates for the system of Multi-dimensional Higher-order Generalized BBM-KDV Equations, Kexue Tongbao 28 (1983), 310–315.Google Scholar
10Zhou, Y. L.. Nonlinear P.D.E. in Applied Science, U.S. Japan Seminar, Tokyo, 1982.Google Scholar
11Lin, C. S.. Interpolation inequalities with weights, Comm. Partial Differential Equations 11 (1986), 1515–1538.Google Scholar
12Klainerman, S.. Global esistence for nonlinear wave equations. Comm. Pure Appl. Math. 33 (1980), 43–101.CrossRefGoogle Scholar
13Guo, B. L.. Proceedings of the 1980 Beijing symposium on Differential Geometry and Differential Equation 3 (1982), 1227–1246.Google Scholar
14Kaup, D. J. and Newell, A. C.. An exact solution for a derivative nonlinear Schrödinger equation, J. Math. Phys. 19 (1978), 798–801.CrossRefGoogle Scholar