Evolution equation for bidirectional surface waves in a convecting fluid

Surface waves in a heated viscous fluid exhibit a long wave oscillatory instability. The nonlinear 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 $y_{tt}-y_{xx} -\epsilon^2(y_{xxtt} + (y^2)_{xx})+ \epsilon^3(y_{xxt}+y_{xxxxt} + (y^2)_{xxt}) =0$ which includes instability and dissipation is derived from this system.