MODULATION OF MULTIPHASE WAVES IN THE PRESENCE OF RESONANCE

The phenomenon of spatio-temporal phase modulation made possible by re sonance is investigated in detail through the analysis of an example p roblem. A simple family of exact solutions to the Ablowitz-Ladik equat ions is found to be modulationally stable in some regimes. This family of solutions is determined by fixing antiperiod 2 boundary conditions , which determines two wavenumbers. Within the family of solutions, th e frequencies do not depend on amplitude; this feature ensures that th e antiperiod 2 boundary conditions will be enforced under modulation. The family of solutions is described by four parameters, two being act ions that foliate the phase space, and two being macroscopically obser vable functions of the phase constants. The modulation of the actions is described by a closed hyperbolic system of first order equations, w hich is consistent with the full set of four genus 1 modulation equati ons. The modulation of the phase information, easily observed due to t he presence of two resonances, is described by two more equations that are driven by the actions. The results are confirmed by numerical exp eriments.