Evolution equation for bidirectional surface waves in a convecting fluid

Surface waves in a heated viscous fluid exhibit a long-wave oscillatory instability. The non-linear evolution of unidirectional waves is known to be described by a modified Korteweg-deVries-Kuramoto-Sivashinsky equation. In the present work, we eliminate the restriction of unidirectional waves and find that the evolution of the wave is governed by a modified Boussinesq system. A perturbed Boussinesq equation of the form ytt−yxx−ϵ2[yxxtt+(y2)xx]+ϵ3[yxxt+yxxxxt+(y2)xxt]=0, which includes instability and dissipation, can be derived from this system.