Orbital stability of standing wave solution for a quasilinear Schrödinger equation
Authors:
Boling Guo and Jianqing Chen
Journal:
Quart. Appl. Math. 67 (2009), 781-791
MSC (2000):
Primary 35Q55, 35A15, 35B35
DOI:
https://doi.org/10.1090/S0033-569X-09-01147-5
Published electronically:
May 27, 2009
MathSciNet review:
2588237
Full-text PDF Free Access
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Additional Information
Abstract: Via minimization arguments and the Concentration Compactness Principle, we prove the orbital stability of standing wave solutions for a class of quasilinear Schrödinger equation arising from physics.
References
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References
- J. P. Albert, Concentration compactness and the stability of solitary-wave solutions to nonlocal equations, Contemporary Mathematics 221 (1999) 1-29. MR 1647189 (99m:35199)
- A. Ambrosetti and Z.Q. Wang, Positive solutions to a class of quasilinear elliptic equations on $\mathbb {R}$, Disc. Contin. Dynam. Syst. 9 (2003), 55-68. MR 1951313 (2003m:34050)
- H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983) 486-490. MR 699419 (84e:28003)
- F.E. Browder, Variational methods for nonlinear elliptic eigenvalue problems, Bull. Amer. Math. Soc. 71 (1965) 176-183. MR 0179459 (31:3707)
- T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal. TMA 7 (1983) 1127-1140. MR 719365 (84m:35102)
- T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982) 549-561. MR 677997 (84i:81015)
- M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Functional Analysis 74 (1987), 160-197; II, J. Functional Analysis 94 (1990) 308-348. MR 901236 (88g:35169)
- S. Kurihura, Large amplitude quasi-solitons in superfluid film, J. Phys. Soc. Japan 50 (1981) 3262-3267.
- H. Lange, M. Poppenberg and H. Teismann, Nash-Moser methods for the solution of quasilinear Schrödinger equations, Comm. Partial Diff. Equations 24 (1999) 1399-1418. MR 1697492 (2000e:35212)
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- A. Nakamura, Damping and modification of exciton solitary waves, J. Phys. Soc. Japan 42 (1977) 1824-1835.
- M. Poppenberg, K. Schmitt and Z. Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Cal. Var. and PDEs 14 (2002) 329-344. MR 1899450 (2003d:35247)
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- T. Tsurumi and M. Wadati, Collapses of wavefunctions in multi-dimensional nonlinear Schrödinger equations under harmonic potential, J. Phys. Soc. Japan 66(1997) 1-8.
- J. Zhang, Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials, Z. Angew. Math. Phys. 51 (2000) 498-503. MR 1762704 (2001c:35227)
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Additional Information
Boling Guo
Affiliation:
Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, People’s Republic of China
MR Author ID:
189874
Jianqing Chen
Affiliation:
School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350007, People’s Republic of China
Email:
jqchen@fjnu.edu.cn
Keywords:
Standing wave solution,
orbital stability,
quasilinear Schrödinger equation.
Received by editor(s):
September 17, 2008
Published electronically:
May 27, 2009
Additional Notes:
The second author is supported by the National Natural Sciences Foundation of China.
Article copyright:
© Copyright 2009
Brown University