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Grillakis, Manoussos; Shatah, Jalal; Strauss, Walter

Stability theory of solitary waves in the presence of symmetry. I. (English) Zbl 0656.35122

J. Funct. Anal. 74, 160-197 (1987).
Consider an abstract Hamiltonian system which is invariant under a one- parameter unitary group of operators. By a “solitary wave” we mean a solution the time development of which is given exactly by the one- parameter group. We find sharp conditions for the stability and instability of solitary waves. Applications are given to bound states and traveling waves of nonlinear PDEs such Klein-Gordon and Schrödinger equations.

Cited in 39 Reviews
Cited in 561 Documents

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35G20 Nonlinear higher-order PDEs
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35A30 Geometric theory, characteristics, transformations in context of PDEs

Keywords:

Hamiltonian system; one-parameter unitary group; solitary wave; stability; instability; bound states; traveling waves; Klein-Gordon; Schrödinger
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\textit{M. Grillakis} et al., J. Funct. Anal. 74, 160--197 (1987; Zbl 0656.35122)
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