All traveling wave exact solutions of two kinds of nonlinear evolution equations

Abstract

In this article, we employ the complex method to obtain all meromorphic solutions of complex Korteweg–de Vries (KdV) equation and the modified Benjamin–Bona–Mahony (mBBM) equation at first, then find out all traveling wave exact solutions of the Eqs. (KdV) and (mBBM). The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all rational and simply periodic solutions of the Eqs. (KdV) and (mBBM) are solitary wave solutions, the complex method is simpler than other methods, and there exist some rational solutions w2r,2(z)w2r,2(z) and simply periodic solutions w2s,1(z)w2s,1(z) which are not only new but also not degenerated successively by the elliptic function solutions. We give some computer simulations to illustrate our main results.