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Finite speed of disturbance for the nonlinear Schrödinger equation

Part of: Equations of mathematical physics and other areas of application Partial differential equations

Published online by Cambridge University Press:  27 December 2018

Simão Correia
Simão Correia*
Affiliation:
CMAF-CIO and FCUL, Campo Grande, Edifício C6, Piso 2, 1749-016 Lisboa, Portugal ([email protected])
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Abstract

We consider the Cauchy problem for the nonlinear Schrödinger equation on the whole space. After introducing a weaker concept of finite speed of propagation, we show that the concatenation of initial data gives rise to solutions whose time of existence increases as one translates one of the initial data. Moreover, we show that, given global decaying solutions with initial data u0, v0, if |y| is large, then the concatenated initial data u0 + v0(· − y) gives rise to globally decaying solutions.

Keywords

nonlinear Schrödinger equationglobal well-posednessfinite speed of disturbance

MSC classification

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger)
Secondary: 35A01: Existence problems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 6 , December 2019 , pp. 1405 - 1419
Copyright
Copyright © Royal Society of Edinburgh 2018 

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References

1Berestycki, H. and Cazenave, T.. Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires. C. R. Acad. Sci. Paris 56 (1981), 489–492.Google Scholar
2Brézis, H.. Functional analysis, Sobolev spaces and partial differential equations (New York: Springer-Verlag Inc., 2011).Google Scholar
3Carles, R.. Critical nonlinear Schrödinger equations with and without harmonic potential. Math. Models Methods Appl. Sci. 12 (2002), 1513–1523.Google Scholar
4Cazenave, T.. Semilinear Schrödinger equations (New York: New York University, Courant Institute of Mathematical Sciences; Providence, RI: American Mathematical Society, 2003).Google Scholar
5Ginibre, J. and Velo, G.. Scattering theory in the energy space for a class of nonlinear Schrödinger equations. J. Math. Pures Appl. (9) 64 (1985), 363–401.Google Scholar
6Glassey, R. T.. On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations. J. Math. PhEs. 18 (1977), 7794–7797.Google Scholar
7Kato, T.. On nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Phys. Théor. 46 (1987), 113–129.Google Scholar
8Kato, T.. On nonlinear Schrodinger equations. II. H s-solutions and unconditional well-posedness. J. Anal. Math. 67 (1995), 281–306.Google Scholar
9Kenig, C., Ponce, G. and Vega, L.. On the unique continuation for nonlinear Schrödinger equations. Commun. Pure. Appl. Math. LV (2002), 1247–1262.Google Scholar
10Martel, Y. and Merle, F.. Multi solitary waves for nonlinear Schrödinger equations. Ann. I. H. Poincaré 23 (2006), 849–864.Google Scholar
11Tao, T.. A pseudoconformal compactification of the nonlinear Schrödinger equation and applications. New York J. Math. 15 (2009), 265–282, electronic only.Google Scholar