Three-mode truncation of a model equation for systems with a long-wavelength oscillatory instability

We study a three-mode truncation of the equation ${\mathit{u}}_{\mathit{t}}$+${\mathit{uu}}_{\mathit{x}}$+δ${\mathit{u}}_{\mathit{x}\mathit{x}\mathit{x}}$+${\mathit{u}}_{\mathit{x}\mathit{x}}$ +${\mathit{u}}_{\mathit{x}\mathit{x}\mathit{x}\mathit{x}}$=0. This simple model allows us to understand analytically the role played by dispersion in the appearance of well-defined traveling pulses. In the absence of dispersion, that is, for δ=0, the solutions obtained from the truncated model are in quantitative agreement with the known results for the Kuramoto-Sivashinsky equation for small horizontal periodicity.