Stability of bound states of Hamiltonian PDEs in the degenerate cases. (English) Zbl 1244.35008

Summary: We consider a Hamiltonian system which is invariant under a one-parameter unitary group and give a criterion for the stability and instability of bound states for the degenerate case. We apply our theorem to the single power nonlinear Klein-Gordon equation and the double power nonlinear Schrödinger equation.


35B35 Stability in context of PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q40 PDEs in connection with quantum mechanics
Full Text: DOI arXiv


[1] Comech, Andrew; Pelinovsky, Dmitry, Purely nonlinear instability of standing waves with minimal energy, Comm. pure appl. math., 56, 11, 1565-1607, (2003), MR 1995870 (2005h:37176) · Zbl 1072.35165
[2] Ginibre, J.; Velo, G., The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z., 189, 4, 487-505, (1985), MR 786279 (86f:35149) · Zbl 0549.35108
[3] Grillakis, Manoussos; Shatah, Jalal; Strauss, Walter, Stability theory of solitary waves in the presence of symmetry. I, J. funct. anal., 74, 1, 160-197, (1987), MR MR901236 (88g:35169) · Zbl 0656.35122
[4] Maeda, Masaya, Stability and instability of standing waves for 1-dimensional nonlinear Schrödinger equation with multiple-power nonlinearity, Kodai math. J., 31, 2, 263-271, (2008), MR 2435895 (2009k:35304) · Zbl 1180.35483
[5] Maeda, Masaya, Instability of bound states of nonlinear Schrödinger equations with Morse index equal to two, Nonlinear anal., 72, 3-4, 2100-2113, (2010), MR 2577607 (2010k:35466) · Zbl 1180.35484
[6] Ohta, Masahito, Instability of bound states for abstract nonlinear Schrödinger equations, J. funct. anal., 261, 1, 90-110, (2011) · Zbl 1228.34092
[7] Ohta, Masahito; Todorova, Grozdena, Strong instability of standing waves for the nonlinear Klein-Gordon equation and the Klein-Gordon-Zakharov system, SIAM J. math. anal., 38, 6, 1912-1931, (2007), (electronic). MR 2299435 (2008a:35198) · Zbl 1128.35074
[8] Shatah, Jalal, Stable standing waves of nonlinear Klein-Gordon equations, Comm. math. phys., 91, 3, 313-327, (1983), MR 723756 (84m:35111) · Zbl 0539.35067
[9] Shatah, Jalal; Strauss, Walter, Instability of nonlinear bound states, Comm. math. phys., 100, 2, 173-190, (1985), MR MR804458 (87b:35159) · Zbl 0603.35007
[10] Tao, Terence, Nonlinear dispersive equations. local and global analysis, CBMS reg. conf. ser. math., vol. 106, (2006), published for the Conference Board of the Mathematical Sciences, Washington, DC. MR 2233925 (2008i:35211) · Zbl 1106.35001
[11] Weinstein, Michael I., Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. math. anal., 16, 3, 472-491, (1985), MR MR783974 (86i:35130) · Zbl 0583.35028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.