Lyapunov stability of ground states of nonlinear dispersive evolution equations
1986; Wiley; Volume: 39; Issue: 1 Linguagem: Inglês
10.1002/cpa.3160390103
ISSN1097-0312
Autores Tópico(s)Nonlinear Waves and Solitons
ResumoCommunications on Pure and Applied MathematicsVolume 39, Issue 1 p. 51-67 Article Lyapunov stability of ground states of nonlinear dispersive evolution equations Michael I. Weinstein, Michael I. Weinstein Princeton UniversitySearch for more papers by this author Michael I. Weinstein, Michael I. Weinstein Princeton UniversitySearch for more papers by this author First published: January 1986 https://doi.org/10.1002/cpa.3160390103Citations: 483AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Bibliography 1 Akhamanov, S. A., Sukhorukov, A. P., and Khokhlov, R. V., Self-focusing and self-trapping of intense light beams in a nonlinear medium, Sov. Phys. JETP 23, 1966, pp. 1025– 1033. 2 Arnold, V. I., On an a priori estimate in the theory of hydrodynamical stability, Am. Math. Soc. Transl. 79, 1969, pp. 267– 269. 3 Benjamin, T. B., The stability of solitary waves, Proc. R. Soc. Lond. A 328, 1972, pp. 153– 183. 4 Berestycki, H., and Lions, P. L., Nonlinear scalar field equations I— Existence of a ground state, Arch. Rat. Mech. Anal. 82, 1983, pp. 313– 345. 5 Berestycki, H., and Cazenave, T., Instabilité des états stationnaires dans les équations de Schrödinger et des Klein-Gordon nonlinéaires, C. R. Acad. Sc. 293, 1981, pp. 489– 492. 6 Bona, J., On the stability of solitary waves, Proc. R. Soc. Lond. A 344, 1975, pp. 363– 374. 7 Cazenave, T., Stable solutions of the logarithmic Schrödinger equation, Nonlinear Analysis, Theory, Methods & Applications 7, 1983, pp. 1127– 1140. 8 Cazenave, T., and Lions, P. L., Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85, 1982, pp. 549– 561. 9 Coffman, C. V., Uniqueness of the ground state solution for μu - u + u3 = 0 and a variational characterization of other solutions, Arch. Rat. Mech. Anal. 46, 1972, pp. 81– 95. 10 Dushane, T. E., Generalization of the Korteweg-de Vries equation, Proc. Symp. Pure Math. 23, and Ann. Math. Soc., 1973, pp. 393– 307. 11 Friedman, A., Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York, 1969. 12 Ginibre, J., and Velo, G., On a class of nonlinear Schrödinger equations I. The Cauchy problem, general case, J. Func. Anal. 32, 1979, pp. 1– 32. 13 Henry, D. B., Perez, J. F., and Wreszinski, W. F., Stability theory for solitary-wave solutions of scalar field equations, Comm. Math. Phys. 85, 1982, pp. 351– 361. 14 Holm, D. D., Marsden, J. E., Ratiu, T., and Weinstein, A., Nonlinear stability conditions and a priori estimates for barotropic hydrodynamics, Phys. Lett. 98A, 1983, pp. 15– 21. 15 Kato, T., On the Cauchy problem for the (generalized) Korteweg-de Vries equation, in Studies in Appl. Math. Advanced in Mathematics Supplementary Studies 8, Academic Press, New York, 1983. 16 Laedke, E. W., and Spatschek, K. H., Stable three-dimensional envelope solitons, Phys. Rev. Let 52, 1984, pp. 279– 282. 17 Laedke, E. W., and Spatschek, K. H., Stability theorem for KdV-type equations, J. Plasma Physics, to appear. 18 McLeod, K., and Serrin, J., Uniqueness of solutions of semilinear Poisson equations, Proc. Nat. Aca. Sci. USA 78, 1981, pp. 6592– 6595. 19 Strauss, W. A., Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55, 1977, pp. 149– 162. 20 Strauss, W. A., Dispersion of low-energy waves for two conservative equations, Arch. Rat. Mech. Anal. 55, 1974, pp. 86– 92. 21 Shatah, J., Stable standing waves of nonlinear Klein-Gordon equations, Commun. Math. Phys. 91, 1983, pp. 313– 327. 22 Shatah, J., and Strauss, W., Instability of nonlinear bound states, preprint. 23 Weinstein, M. I., Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87, 1983, pp. 567– 576. 24 Weinstein, M. I., Modulational stability of ground states of nonlinear Schrödinger equations, Siam J. Math. Anal. 16, 1985, pp. 567– 576. 25 Weinstein, M. I., Self-focusing and modulational analysis of nonlinear Schrödinger equations, Ph.D. thesis, NYU, New York, 1982. 26 Weinstein, M. I., On the structure and formation of singularities in solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, in press. 27 Whitham, G. B., Linear and Nonlinear Waves, John Wiley and Sons, New York, 1974. 28 Zakharov, V. E., Collapse of Langmuir waves, Sov. Phys. JETP 35, 1972, pp. 908– 922. 29 Zakharov, V. E., and Rubenchik, Instability of waveguides and solutions in nonlinear media, Sov. Phys. JETP 38, 1974. Citing Literature Volume39, Issue1January 1986Pages 51-67 ReferencesRelatedInformation
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