Available in electronic format
Available in print format		 Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Instability of standing waves of the Schrödinger equation with inhomogeneous nonlinearity

Author(s): Yue Liu; Xiao-Ping Wang; Ke Wang
Journal: Trans. Amer. Math. Soc. 358 (2006), 2105-2122.
MSC (2000): Primary 35B35, 35B60, 35Q35, 35Q40, 35Q55, 76B25, 76E25, 76E30, 78A15
Posted: May 9, 2005
Retrieve article in: PDF DVI PostScript

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.$

References:

1.
Berestycki, H and Cazenave, T., Instabilityé des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad Sc. Paris, 293 (1981), 489-492. MR 0646873 (84f:35120)

2.
Cazenave, T., An introduction to nonlinear Schrödinger equation, Textos de Métodos Matemaáticos 26 (1989), Instituto de Matemática, UFRJ, Rio de-Janeiro.

3.
Cazenave, T. and Lions, P. L., Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.MR 0677997 (84i:81015)

4.
Fibich, G. and Wang, X. P., Stability of solitary waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities, Physica D ; 175, (2003) P96-108.MR 1957907 (2003m:35225)

5.
Gill, T. S., Optical guiding of laser beam in nonuniform plasma, Pramana Journal of Physics, 55 (2000), 845-852.

6.
Ginibre J. and Velo, G., On a class of nonlinear Schrödinger equations I, II. The Cauchy problem, general case, J. Func. Anal., 32 (1979), 1-71. MR 0533218 (82c:35057); MR 0533219 (82c:35058)

7.
Goncalves Rebeiro, J. M., Instability of symmetric stationary states for some nonlinear Schrödinger equations with an external magnetic field, Ann. Inst. Henri Poincaré Physique Théorique, 54 (1991), 403-433.MR 1128864 (92i:35113)

8.
Grillakis, M., Shatah, J. and Strauss, W., Stability theory of solitary waves in the presence of symmetry, I, J. Funct. Anal. 74 (1987), 160-197.MR 0901236 (88g:35169)

9.
Kato, T., On the blowing-up of solutions, Ann. Inst. Henri Poincaré, Physique Théorique, 49 (1987), 113-129.

10.
Liu, Yue and Wang, X. P., Nonlinear stability of solitary waves of a generalized Kadomtsev-Petvishvili equation, Comm. Math. Phys. 183 (1997), 253-266. MR 1461958 (99b:35184)

11.
Merle, F., Nonexistence of minimal blow-up solutions of equations $ iu_t = \Delta u - k(x) \vert u\vert^{4/N} u $ in $ R^N, $ Ann. Inst. Henri Poincaré Physique Théorique, 64 (1996), 33-85. MR 1378233 (97g:35073)

12.
Shatah, J. and Strauss, W., Instability of nonlinear bound states, Comm. Math. Phys., 100 (1985), 173-190. MR 0804458 (87b:35159)

13.
Weinstein, M. I., Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.MR 0783974 (86i:35130)

14.
Weinstein, M. I., Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576. MR 0691044 (84d:35140)

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.
Wang, X. F. and Zeng, B., On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal. 28 (1997), 633-655. MR 1443612 (98e:81032)

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35B35, 35B60, 35Q35, 35Q40, 35Q55, 76B25, 76E25, 76E30, 78A15

Retrieve articles in all Journals with MSC (2000): 35B35, 35B60, 35Q35, 35Q40, 35Q55, 76B25, 76E25, 76E30, 78A15

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: 10.1090/S0002-9947-05-03763-3
PII: S 0002-9947(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
Posted: May 9, 2005
Copyright of article: Copyright 2005, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google