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

The dynamics of two pairs of counter-propagating waves in two-component media is considered within the framework of two generally nonintegrable coupled Sine-Gordon equations. We consider the dynamics of weakly nonlinear wave packets, and using an asymptotic multiple-scales expansion we obtain a suite of evolution equations to describe energy exchange between 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 Schrödinger equations. We also consider a set of fully coupled nonlinear Schrödinger equations, even though this system contains small dispersive terms which are strictly beyond the leading order of the asymptotic multiple-scales expansion method. Using both the theoretical predictions following from these asymptotic models and numerical simulations of the original unapproximated equations, we investigate the stability of plane-wave solutions, and show that they may be modulationally unstable. These instabilities can then lead to the formation of localized structures, and to a modification of the energy exchange between the components. 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. PACS numbers: 02.30 Jr, 02.30 Mv, 46.40 Cd, 47.20 Ky MSC numbers: 35 Q 55, 41 A 60, 74 J 30, 76 E 30