A. Comech and D. Pelinovsky

Purely nonlinear instability of standing waves with minimal energy

Comm. Pure Appl. Math. 56, 1565-1607 (2003)

Abstract:
We consider Hamiltonian systems with U(1) symmetry. We prove that in the generic situation the standing wave e-i ω t φω that has the minimal energy among all other standing waves is unstable, in spite of absence of linear instability. Essentially, the instability is caused by higher algebraic degeneracy of the zero eigenvalue in the spectrum of the linearized system. We apply our theory to nonlinear Schrodinger equation.

Keywords:
NONLINEAR SCHRODINGER EQUATION, HAMILTONIAN SYSTEMS, INSTABILITY OF NONLINEAR WAVES, MULTIPLE EIGENVALUES, NORMAL FORMS