Nonlinear analysis of instability produced by linear mode coupling

We consider a system of two first-order complex partial differential equations with cubic nonlinear terms, which is a generic asymptotic model for wave envelopes in the situation when a linear instability is produced by mode coalescence (intersection of the dispersion curves) in the presence of a weak linear coupling between the two modes. Analytical considerations produce a simple condition for onset of collapse in this system, in the form of an inequality on the coefficients of the nonlinear terms. Exactly the same inequality turns out to determine the modulational stability of the continuous wave solutions with the wave number which corresponds to the center of the linear instability band. When collapse is absent, the continuous wave solutions are subject only to a long-wave modulational instability which can be eliminated by appropriate periodic boundary conditions, but in the opposite case modulational instability extends to all wave numbers. Numerical simulations demonstrate that there are two basic scenarios of principal interest. Either collapse occurs (exactly when the collapse condition is satisfied), or otherwise an initial periodic wave breaks down due to successive sideband instabilities into an apparently chaotic state. The predicted behaviors may be expected as a result of an interface instability in a number of hydrodynamic systems.