Nonlinear wave and Schrödinger equations. I: Instability of periodic and quasiperiodic solutions. (English) Zbl 0780.35106

Stability of periodic and quasiperiodic solutions of linear wave and Schrödinger equations under nonlinear perturbations is investigated. Using the Mourre estimate and the Fermi Golden rule for non-elliptic and nonlinear equations the author shows that for the wave equation such solutions are unstable for generic perturbations and gives spectral conditions which guarantee instability. In this case of the Schrödinger equation periodic solutions are stable while quasiperiodic ones are not.
These results are extended to periodic solutions of nonlinear equations. Some illustrative examples are considered.


35Q55 NLS equations (nonlinear Schrödinger equations)
35K55 Nonlinear parabolic equations
35B35 Stability in context of PDEs
35B10 Periodic solutions to PDEs
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