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Anna Georgieva : Resonances in Nonlinear Discrete Periodic Medium

We derive traveling wave solutions in a nonlinear diatomic particle chain near the 1:2 resonance (k*, omega*), where omega*=D(k*), 2omega*=D(2k*) and omega=D(k) is the linear dispersion relation. To leading order, the waves have form +/- epsilon sin(k n-omega t) + delta sin(2 k n-2 omega t), where the near-resonant acoustic frequency omega and the amplitude epsilon of the first harmonic are given to first order in terms of the wavenumber difference k-k* and the amplitude delta of the second harmonic. These traveling wave solutions are unique within a certain set of symmetries.

We find that there is a continuous line in parameter space, that transfers energy from the first to the second harmonic, even in cases where initially almost all energy is in the first harmonic, connecting these waves to pure optical waves that have no first harmonic content. The analysis is extended to higher resonances.

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