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.
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M. C. Depassier (2006) studied this question.
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