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Cylindrical solitary pulses in a two-dimensional stabilized Kuramoto-Sivashinsky system. (English) Zbl 1006.76015
Summary: By linearly coupling a generalized Zakharov-Kuznetsov equation (alias the two-dimensional (2D) Benney equation) to an extra linear dissipative equation, a 2D extension of a recently proposed stabilized Kuramoto-Sivashinsky system is developed. The model applies to the description of surface waves on 2D liquid layers in various physical settings. The extra equation provides for the stability of the zero state in the system, thus paving a way to the existence of stable 2D localized solitary pulses (SPs). A perturbation theory, based on a family of cylindrical solitons existing in the conservative counterpart of the system, is developed by treating dissipation and gain in the model as small perturbations. It is shown that the system may select two steady-state solitons from the continuous family provided by the conservative counterpart, of which the one with larger amplitude is expected to be stable. Numerical simulations support the analytical predictions quite well. Additionally, it is found that a shallow quasi-one-dimensional trough is attached to the stable SP if the integration domain is not very large, and an explanation to this feature is proposed. Stable double-humped bound states of two pulses are found, too.
MSC:
76B25Solitary waves (inviscid fluids)
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q51Soliton-like equations