Instability in supercritical nonlinear wave equations: Theoretical results and symplectic integration

Nonlinear wave evolutions involve a dynamical balance between linear dispersive spreading of the waves and nonlinear self-interaction of the waves. In sub-critical settings, the dispersive spreading is stronger and therefore solutions are expected to exist globally in time. We show that in the supercritical case, the nonlinear self-interaction of the waves is much stronger. This leads to some sort of instability of the waves. The proofs are based on the construction of high frequency approximate solutions. Preliminary numerical simulations that support these theoretical results are also reported.