Modulational instability of two pairs of counter-propagating waves and energy exchange in a two-component system

We study these instabilities in the context of solutions exhibiting an energy exchange between the two physical components of the system. We also perform numerical simulations of the original unapproximated coupled Sine-Gordon equations to assess the accuracy of the instability predictions derived from the various asymptotic models, and consider the range of validity of each asymptotic model. We show that the instabilities can lead to the formation of localized structures, and to a modification of the linear energy exchange, which then continues for some time into the nonlinear regime as an energy exchange between these localized structures. When the system is close to being integrable, the time evolution is distinguished by a remarkable almost periodic sequence of energy exchange scenarios, with spatial patterns alternating between approximately uniform wavetrains and localized structures, which invites further theoretical study of this special case.