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

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
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[1] Akhmediev, N.N, Novel class of nonlinear surfaces waves: asymmetric modes in a symmetric layered structure, Soviet phys. JETP, 56, 299-303, (1982)
[2] Berestycki, H; Cazenave, T, Instabilité des états stationnaires dans LES équations de Schrödinger et de Klein-Gordon non-linéaires, C. R. acad. sci., 293, 489-492, (1981) · Zbl 0492.35010
[3] Bona, J; Souganidis, P; Strauss, W, Stability and instability of solitary waves of KdV type, (), to appear · Zbl 0648.76005
[4] Cazenave, T; Lions, P.L, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. math. phys., 85, 549-561, (1982) · Zbl 0513.35007
[5] {\scA. Floer and A. Weinstein}, Non-spreading wave packets for the cubic Schrödinger equation with a bounded potential, preprint. · Zbl 0613.35076
[6] {\scC. K. R. T. Jones}, Instability of standing waves for nonlinear Schrödinger type equations, preprint.
[7] {\scC. K. R. T. Jones and J. V. Moloney}, Instability of standing waves in nonlinear optical waveguides, Phy. Lett. A, to appear.
[8] Maddocks, J.H, Restricted quadratic forms and constrained variational principles, SIAM J. math. anal., 16, 47-68, (1985) · Zbl 0581.47049
[9] {\scJ. H. Maddocks}, Stability and folds, preprint.
[10] Reed, M; Simon, B, ()
[11] Shatah, J, Stable standing waves of nonlinear Klein-Gordon equations, Comm. math. phys., 91, 313-327, (1983) · Zbl 0539.35067
[12] Shatah, J, Unstable ground state of nonlinear Klein-Gordon equations, Trans. amer. math. soc., 290, 701-710, (1985) · Zbl 0617.35072
[13] Shatah, J; Strauss, W, Instability of nonlinear bound states, Comm. math. phys., 100, 173-190, (1985) · Zbl 0603.35007
[14] Weinstein, M, Liapunov stability of ground states of nonlinear dispersive evolution equations, Comm. pure appl. math., 39, (1986)
[15] {\scM. Weinstein}, Stability analysis of ground states of nonlinear Schrödinger equations, preprint.
[16] Weinstein, M, On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations, Comm. partial differential equations, 11, 545-565, (1986) · Zbl 0596.35022
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