Finite speed of disturbance for the nonlinear Schrödinger equation

Simão Correia

doi:10.1017/prm.2018.69

Published online by Cambridge University Press: 27 December 2018

Simão Correia

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Simão Correia

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CMAF-CIO and FCUL, Campo Grande, Edifício C6, Piso 2, 1749-016 Lisboa, Portugal (sfcorreia@fc.ul.pt)

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sfcorreia@fc.ul.pt

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.

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.CrossRef Google Scholar

4Cazenave, T.. Semilinear Schrödinger equations (New York: New York University, Courant Institute of Mathematical Sciences; Providence, RI: American Mathematical Society, 2003).CrossRef 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.CrossRef 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.CrossRef 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

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Finite speed of disturbance for the nonlinear Schrödinger equation

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Finite speed of disturbance for the nonlinear Schrödinger equation

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