## Purely nonlinear instability of standing waves with minimal energy.(English)Zbl 1072.35165

Based on a general theory by M. G. Grillakis, J. Shatah and W. Strauss about the non-linear stability (with respect to the flow) of solitary waves in abstract evolutionary partial differential equations in presence of symmetries [J. Funct. Anal. 74, 160–197 (1987; Zbl 0656.35122); J. Funct. Anal. 94, 308–348 (1990; Zbl 0711.58013)], in the paper under review it is proved the generic nonlinear stability of the standing waves with minimal energy for Hamiltonian evolutionary partial differential equations admitting a $$U(1)$$ symmetry. The authors apply their results to the nonlinear Schrödinger equation with $$U(1)$$ symmetry.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems

### Keywords:

stability; nonlinear evolutionary equations; solitons

### Citations:

Zbl 0656.35122; Zbl 0711.58013
Full Text:

### References:

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