The application of a novel computational approach to physically important nonlinear evolution equations of higher order

Abstract

A new algebraic procedure is introduced to find new classes of solutions of (1+1)-nonlinear partial differential equations (nPDEs) of physical relevance. The proposed new computational algebraic approach leads to traveling-wave solutions and moreover a new class of explicit one-soliton-solutions can be obtained. The crucial step of the method is the basic assumption that unknown solutions satisfy an ordinary differential equation (ODE) of first order that can be integrated easily.

A further important aspect of this paper, however, is the fact that one is able to calculate a distinctive class of solutions which cannot be found in the current literature. By increasing the number of arbitrary constants the approach allows one to handle problems occurring in solving nonlinear algebraic systems closely related with nPDEs. It is worth mentioning that the benefit of the given approach is surprisingly the easiness in deriving different classes of solutions; so it can therefore be applied in the physical sciences, biology, chemistry and related domains alike. On the contrary it is worth stressing the necessity of such sophisticated methods since a general theory of nPDEs does not exist.