Testing efficiency of the generalised $$\left( {{{G}'}/G} \right) $$ G ′ / G -expansion method for solving nonlinear evolution equations

Abstract

In this investigation, we employ the generalised $$(G'{/}G)$$ -expansion method to test its efficiency in extracting travelling wave solutions of nonlinear evolution equations (NLEEs). As test cases, the modified Kuramoto–Sivashinsky (mK-S) and the modified Burgers–Korteweg–de Vries (mB-KdV) equations are considered because of their importance in soliton theory. The general solutions are obtained in hyperbolic, trigonometric and rational function forms for both the equations. Taking specific parametric values in the corresponding general solutions, some new exact travelling waves in trigonometric and hyperbolic forms and only in hyperbolic form are obtained for the mK-S and mB-KdV equations, respectively. The obtained results are checked to see whether the criticism made by Parkes (Comput. Fluids 42, 108 (2011)), that the so-called ‘new’ solutions derived by the $$(G'{/}G)$$ -expansion method are often erroneous and are merely disguised versions of previously known solutions, is justified also for the generalised $$(G'{/}G)$$ -expansion method. The solutions were checked with Maple by putting them back into their corresponding equations. With specific values of parameters, some of our obtained solutions satisfied directly and some solutions never satisfied the considered NLEEs. Among the satisfactory solutions, some are found to be in disguised versions of some results obtained in this study.