EVOLUTION AND FORMATION OF DISPERSIVE-DISSIPATIVE PATTERNS

A variety of interfacial phenomena, including non-Boussinesq and Maran goni effects are described by a dispersive-dissipative model u(tau)+al pha uu(xi)+beta u(xi xi xi)[(1+2u)u(xi)]xi+yu xi xi xi xi=0. A critica l surface alpha=alpha(c)(beta,gamma) is found such that for alpha<alph a(c)(beta,gamma) the amplitude becomes unbounded within a finite time and the model breaks down. For alpha>alpha(c)(beta,gamma), if the init ial perturbation is not too large, bounded patterns emerge. The intera ction between dispersion and advection dislocates the critical surface (favorably when dispersion and convection cooperate) and suppresses t he temporally irregular nature of the resulting patterns. In the first of the two regularized variants of the model considered, the amplitud e runaway is mitigated and a formation of cusps is observed. In the se cond variant with a quadratic dispersion, the emerging solutions are b ounded save for a strip in a parameter space, where both the amplitude and the gradients were found to grow at competing rates.