Note on slightly unstable nonlinear wave systems

Abstract

The asymptotic solution for large time of the initial-value problem for weakly nonlinear wave systems is obtained by the method of matched asymptotic expansions in the case in which the linearized problem is slightly unstable. The linearized solution is valid until its small exponential growth overcomes the algebraic decay due to the dispersion of the initial energy. For larger times the nonlineax terms become important, but there are no additional dispersive or diffusive effects. For the non-diffusive case an exact solution which enables the explicit verification of the asymptotic results is found.