1:2 resonance mediated second harmonic generation in a 1-D nonlinear discrete periodic medium

We derive traveling wave solutions in a nonlinear diatomic particle chain near the 1:2 resonance (κ*, ω*), where ω* = D(κ*), 2ω* = D(2κ*) and ω = D(κ) is the linear dispersion relation. To leading order, the waves have form ±εsin(κn - ωt) + δsin(2κn - 2ωt), where the near-resonant acoustic frequency ω and the amplitude ε of the first harmonic are given to first order in terms of the wavenumber difference κ - κ* and the amplitude δ 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.

Duke Scholars

Author Stephanos Venakides Mathematics