The Painleve Analysis of an Ordinary Differential Equation Which Describes a Traveling-Wave Solution of the Oskolkov–Benjamin–Bona–Mahony–Burgers Equation

Abstract

Since the second half of the twentieth century, wide studies of Sobolev-type equations are undertaken. These equations contain items that are derivatives with respect to time of the second order derivatives of the unknown function with respect to space variables. They can describe nonstationary processes in semiconductors, in plasm, phenomena in hydrodinamics and other ones. Notice that wide studies of qualitative properties of solutions of Sobolev-type equations exist. Namely, results about existence and uniqueness of solutions, their asymptotics and blow-up are known. But there are few results about exact solutions of Sobolev-type equations. In the present paper, we study the ordinary differential equation that describes a traveling-wave solution of a Sobolev-type equation. Namely, we apply the Painleve test to it and find all possible combinations of parameters that allow us to build the general solution as Laurent series of a special form.