Some exact wave solutions of nonlinear partial differential equations by means of comparison with certain standard ordinary differential equations

We consider applications of a comparison method for obtaining some exact solutions of nonlinear partial differential equations (PDEs) through their traveling wave reductions. The method, proposed by N. A. Kudryashov, 39(18), 5733-5742, Applied Mathematical Modelling, is extended to encompass not only solutions expressible in terms of the logistic function but also to the tanh$$ \tanh $$-class of functions in a unified manner. The standard set of second-order ordinary differential equations (ODEs) admitting the logistic and tanh$$ \tanh $$ functions as solutions is derived and extensions to the third-order case are enunciated. A number of examples are included to illustrate the method.

Automated summary

This article discusses the use of a comparison method to obtain exact solutions for nonlinear partial differential equations (PDEs) through their traveling wave reductions. The method, proposed by N. A. Kudryashov, is extended to include solutions expressed in terms of both the logistic function and the tanh$$ \tanh $$-class of functions. The article derives the standard set of second-order ordinary differential equations (ODEs) that have the logistic and tanh$$ \tanh $$ functions as solutions and also extends the analysis to third-order cases.