Exact Solutions of a Nonclassical Nonlinear Equation of the Fourth Order

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. There are books and papers about exact solutions of partial equations, but they are devoted mainly to classical equations, where the first or second order derivative with respect to time or the derivative with respect to time of the first order derivative of the unknown function with respect to the space variable is equal to a stationary expression. Therefore it is interesting to study exact solutions of Sobolev-type equations. In the present paper, a fourth order nonlinear partial equation is studied. Three classes of its exact solutions are built. They are expressed in terms of special functions (solutions of some ordinary differential equations). For two of these classes subsets that can be expressed in elementary functions are built, for the third one subsets that can be described in elementary functions and an implicit function (without a quadrature) are built.