PERIODIC AND QUASI-PERIODIC SOLUTIONS OF DEGENERATE MODULATION EQUATIONS

In some circumstances (degenerations) it is essential to add higher-order nonlinear coefficients to a Ginzburg-Landau type modulation equation (which only has one cubic nonlinearity). In this paper we study these degenerate modulation equations. We consider the important situation in which the equation has real coefficients and the case of coefficients with small imaginary parts. First we determine the stability of periodic solutions. The stationary problem is, like in the non-degenerate case, integrable: there exist families of quasi-periodic and homoclinic solutions. This system is perturbed by considering modulation equations with coefficients with small imaginary parts. We establish that there exists an unbounded domain in parameter space in which the modulation equation has quasi-periodic solutions. Moreover, we show that there is a manifold of codimension 1 (in parameter space) on which the homoclinic solutions survive the perturbation.