The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations

Abstract

The tanh (or hyperbolic tangent) method is a powerful technique to search for travelling waves coming out from one-dimensional nonlinear wave and evolution equations. In particular, in those problems where dispersive effects, reaction, diffusion and/or convection play an important role. To show the strength of the method, an overview is given to find out which kind of problems are solved with this technique and how in some nontrivial cases this method, adapted to the problem at hand, still can be applied. Single as well as coupled equations, arising from wave phenomena which appear in different scientific domains such as physics, chemical kinetics, geochemistry and mathematical biology, will be treated. Next, attention is focussed towards approximate solutions. As a result, solitary- and shock-wave profiles are derived together with the associated width and velocity. The same method can be easily extended so that difference-differential equations can be similarly solved. Finally, some extensions of the method are discussed.