ABSTRACT
This chapter aims to present some mathematical methods of integration of the ordinary differential equation (o.d.e) and explains intrinsic links between the nonlinear dispersive and dissipative wave propagation problem and the general reduction problem of the classical o.d.e theory. It focuses on mathematical problems, and a reader oriented to nonlinear elastic problems and experiments in solitary wave’s propagation may omit, however, in that case, he or she will be asked to accept the mathematical results. The governing equation usually involves a linear wave operator supplied with nonlinear, dispersive and dissipative terms, all of which may contain variable coefficients, depending on space and time variables. A wave propagation problem may be described by means of a nonlinear hyperbolic equation or its corresponding reduction to a nonlinear evolution equation that will extract the unidirectional wave. Many travelling wave propagation problems of mathematical physics are governed by a short version of the nonlinear Lie equation that allows the employment of further reduction.
