Nonlinear steady-state traveling solutions of the Kuramoto-Sivashinsky equation coupled with the linear dissipative equation

A generalization of the known active-dissipative Kuramoto-Sivashinsky equation coupled with a linear dissipative equation is considered. In such a model the region of zero solution instability is shown to depend in a complex way on the specific values of the problem parameters. The families of periodic nonlinear steady-state traveling solutions bifurcating from the zero solution in the vicinity of neutral wave numbers are constructed numerically. The investigation of the stability of these solutions enables obtaining new families that appear as a result of subsequent bifurcations. Among these families the ones that extend into the region of small wave numbers and turn into soliton solutions in the limit by the wave numbers are found among them. Various two-hump solitons are constructed. The research results on the stability of a soliton solution are presented.