## A generalized $$\left( \frac{G'}G\right)$$-expansion method and its applications in nonlinear mathematical physics equations.(English)Zbl 1224.35007

Summary: A generalized $$\left( \frac{G'}G\right)$$-expansion method is proposed by studying the $$\left( \frac{G'}G\right)$$-expansion method proposed by M. L. Wang, Y. Zhou and Z. Li [Phys. Lett., A 216, No. 1–5, 67–75 (1996; Zbl 1125.35401)] and the first order nonlinear ordinary differential equation with a sixth-order nonlinear term. The method is applied to $$(2+1)$$-dimensional dispersive long wave equations and double sine-Gordon equation. As a result, some new exact travelling wave solutions are obtained which include solitary wave solutions, triangular periodic wave solutions, hyperbolic solutions, rational solutions and Jacobi elliptic doubly periodic wave solutions. This method can also be applied to other nonlinear evolution equations in mathematical physics.

### MSC:

 35A24 Methods of ordinary differential equations applied to PDEs 35A35 Theoretical approximation in context of PDEs 35Q53 KdV equations (Korteweg-de Vries equations)

Zbl 1125.35401