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The $\left(\frac{{G}^{\text{'}}}{G}\right)$-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. (English) Zbl 1217.76023
Summary: The ($\frac{{G}^{\text{'}}}{G}$)-expansion method is firstly proposed, where $G=G\left(\xi \right)$ satisfies a second order linear ordinary differential equation (LODE for short), by which the travelling wave solutions involving parameters of the KdV equation, the mKdV equation, the variant Boussinesq equations and the Hirota-Satsuma equations are obtained. When the parameters are taken as special values the solitary waves are also derived from the travelling waves. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. The proposed method is direct, concise, elementary and effective, and can be used for many other nonlinear evolution equations.

##### MSC:
 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76B25 Solitary waves (inviscid fluids) 35Q53 KdV-like (Korteweg-de Vries) equations 35Q51 Soliton-like equations