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Instability of standing waves of the Schrödinger equation with inhomogeneous nonlinearity


Authors: Yue Liu, Xiao-Ping Wang and Ke Wang
Journal: Trans. Amer. Math. Soc. 358 (2006), 2105-2122
MSC (2000): Primary 35B35, 35B60, 35Q35, 35Q40, 35Q55, 76B25, 76E25, 76E30, 78A15
Published electronically: May 9, 2005
MathSciNet review: 2197450
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with the inhomogeneous nonlinear Shrödinger equation (INLS-equation)

\begin{displaymath}i u_t + \Delta u + V(\epsilon x) \vert u\vert^p u = 0, \; x \in {\mathbf R}^N. \end{displaymath}

In the critical and supercritical cases $ p \ge 4/N, $ with $ N \ge 2, $ it is shown here that standing-wave solutions of (INLS-equation) on $ H^1({\mathbf R}^N) $ perturbation are nonlinearly unstable or unstable by blow-up under certain conditions on the potential term V with a small $ \epsilon > 0.$


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  • 1. Henri Berestycki and Thierry Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), no. 9, 489–492 (French, with English summary). MR 646873
  • 2. Cazenave, T., An introduction to nonlinear Schrödinger equation, Textos de Métodos Matemaáticos 26 (1989), Instituto de Matemática, UFRJ, Rio de-Janeiro.
  • 3. T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), no. 4, 549–561. MR 677997
  • 4. Gadi Fibich and Xiao-Ping Wang, Stability of solitary waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities, Phys. D 175 (2003), no. 1-2, 96–108. MR 1957907, 10.1016/S0167-2789(02)00626-7
  • 5. Gill, T. S., Optical guiding of laser beam in nonuniform plasma, Pramana Journal of Physics, 55 (2000), 845-852.
  • 6. J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal. 32 (1979), no. 1, 1–32. MR 533218, 10.1016/0022-1236(79)90076-4
    J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. II. Scattering theory, general case, J. Funct. Anal. 32 (1979), no. 1, 33–71. MR 533219, 10.1016/0022-1236(79)90077-6
  • 7. J. M. Gonçalves Ribeiro, Instability of symmetric stationary states for some nonlinear Schrödinger equations with an external magnetic field, Ann. Inst. H. Poincaré Phys. Théor. 54 (1991), no. 4, 403–433 (English, with French summary). MR 1128864
  • 8. Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), no. 1, 160–197. MR 901236, 10.1016/0022-1236(87)90044-9
  • 9. Kato, T., On the blowing-up of solutions, Ann. Inst. Henri Poincaré, Physique Théorique, 49 (1987), 113-129.
  • 10. Yue Liu and Xiao-Ping Wang, Nonlinear stability of solitary waves of a generalized Kadomtsev-Petviashvili equation, Comm. Math. Phys. 183 (1997), no. 2, 253–266. MR 1461958, 10.1007/BF02506406
  • 11. Franck Merle, Nonexistence of minimal blow-up solutions of equations 𝑖𝑢_{𝑡}=-Δ𝑢-𝑘(𝑥)\vert𝑢\vert^{4/𝑁}𝑢 in 𝑅^{𝑁}, Ann. Inst. H. Poincaré Phys. Théor. 64 (1996), no. 1, 33–85 (English, with English and French summaries). MR 1378233
  • 12. Jalal Shatah and Walter Strauss, Instability of nonlinear bound states, Comm. Math. Phys. 100 (1985), no. 2, 173–190. MR 804458
  • 13. Michael I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), no. 3, 472–491. MR 783974, 10.1137/0516034
  • 14. Michael I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1982/83), no. 4, 567–576. MR 691044
  • 15. Wang, K., On the strong instability of standing wave solutions of inhomogeneous NLS equation, Thesis, Department of Mathematics, The Hong Kong University of Science and Technology, 2001.
  • 16. Xuefeng Wang and Bin Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal. 28 (1997), no. 3, 633–655. MR 1443612, 10.1137/S0036141095290240

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Additional Information

Yue Liu
Affiliation: Department of Mathematics, University of Texas, Arlington, Texas 76019
Email: yliu@uta.edu

Xiao-Ping Wang
Affiliation: Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
Email: mawang@ust.hk

Ke Wang
Affiliation: California Institute of Technology, MC 217-50, 1200 E. California Boulevard, Pasadena, California 91125
Email: wang@acm.caltech.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03763-3
Keywords: Nonlinear Schr\"odinger equation, inhomogeneous nonlinearities, blow-up, standing waves, ground state, stability theory
Received by editor(s): April 16, 2003
Received by editor(s) in revised form: April 29, 2004
Published electronically: May 9, 2005
Article copyright: © Copyright 2005 American Mathematical Society