Purely nonlinear instability of standing waves with minimal energy

We consider Hamiltonian systems with U(1) symmetry. We prove that in the generic situation the standing wave that has the minimal energy among all other standing waves is unstable, in spite of the 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 the nonlinear Schrodinger equation. (C) 2003 Wiley Periodicals, Inc.