We consider the dynamics of weakly nonlinear multi-phase wavetrains within the framework of two pairs of counter-propagating waves in a system 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. Using an asymptotic multiple-scales expansion we obtain a hierarchy of asymptotically exact coupled evolution equations describing the amplitudes of the waves. 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. We study these instabilities in the context of solutions exhibiting an energy exchange between the two physical components of the system and 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.
References: [1] S.D. Griffiths, R.H.J. Grimshaw, K.R. Khusnutdinova, The influence of modulational instability on energy exchange in coupled Sine-Gordon equations, Theor. Math. Phys. 137 (2003) 1448-1458. [2] S.D. Griffiths, R.H.J. Grimshaw, K.R. Khusnutdinova, Modulational instability of two pairs of counter-propagating waves and energy exchange in a two-component system, Phys. D 214 (2006) 1-24.