Model equation of surface waves of viscous fluid down an inclined plane (Q1323262)

The purpose of this paper is to study the periodic solutions of the following partial differential equation (we call it Korteweg-de Vries-Kuramoto-Sivashinsky (KdV-KS) equation): \[ u_t+ u_{xxxx}+ u_{xxx}+ Ru_{xx}+ uu_x= 0,\tag{1} \] where \(R\) is the parameter corresponding to the Reynolds number. It is an approximate equation for surface waves of a two-dimensional incompressible viscous fluid down an inclined plane under the assumption of small amplitude and long wave. We investigate the form of time periodic solutions of (1) bifurcating from the zero solution. We show that (1) has a Hopf bifurcation from the zero solution, however KS equation does not. Moreover, we consider the travelling wave solution of KdV-KS equation \(u(z)= u(x- ct)\), and prove that it has periodic solutions bifurcating from the zero solution. Additionally, we prove that the time periodic solution bifurcating from the zero solution corresponds to some travelling wave solution.