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

The modulational instability of two pairs of plane counter-propagating waves in a two-component system is considered within the framework of two coupled Sine-Gordon equations. The emphasis is on the generic case when the system is not integrable, and the group velocities of each pair of waves are arbitrary and usually different from each other. We consider the dynamics of weakly nonlinear wave packets, and using an asymptotic multiple-scales expansion we obtain several sets of evolution equations describing the two components of the system. Depending on the wave packet length scale vis-a-vis the wave amplitude scale, these evolution equations are either four non-dispersive and nonlinearly coupled envelope equations, or four non-locally coupled nonlinear Schrِdinger equations. We also consider a set of fully coupled nonlinear Schrِdinger equations, even though in general this system contains small dispersive terms which are strictly beyond the leading order of the asymptotic multiple-scales expansion procedure. Although each set of amplitude equations can be used to describe a range of dynamical phenomena, we focus here on stability of plane-wave solutions, and show that they may be modulationally unstable. dy 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.