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Modulation of multiphase waves in the presence of resonance. (English) Zbl 0887.34060
Summary: The phenomenon of spatio-temporal phase modulation made possible by resonance is investigated in detail through the analysis of an example problem. A simple family of exact solutions to the Ablowitz-Ladik equations is found to be modulationally stable in some regimes. This family of solutions is determined by fixing antiperiod 2 boundary conditions, which determines two wave numbers. Within the family of solutions, the frequencies do not depend on amplitude; this feature ensures that the antiperiod 2 boundary conditions will be enforced under modulation. The family of solutions is described by four parameters, two being actions that foliate the phase space, and two being macroscopically observable functions of the phase constants. The modulation of the actions is described by a closed hyperbolic system of first order equations, which is consistent with the full set of four genus 1 modulation equations. The modulation of the phase information, easily observed due to the presence of two resonances, is described by two more equations that are driven by the actions. The results are confirmed by numerical experiments.

MSC:
34G20 Nonlinear differential equations in abstract spaces
35Q35 PDEs in connection with fluid mechanics
34D20 Stability of solutions to ordinary differential equations
78A10 Physical optics
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