New solitary solutions and a conservative numerical method for the Rosenau–Kawahara equation with power law nonlinearity

Abstract

Recently, Biswas et al. (Phys Wave Phenom 19:24–29, 2011) derived exact bright and dark solitary solutions for the Rosenau–Kawahara equation with power law nonlinearity, and Hu et al. (Adv Math Phys 11, Article ID 217393, 2014) proposed two conservative finite difference schemes for the usual Rosenau–Kawahara equation. In this paper, we obtain another set of exact solitary solutions for the Rosenau–Kawahara equation with power law nonlinearity. More importantly, a conservative finite difference method is presented. The fundamental energy conservation law is preserved by the current numerical scheme. And the existence and uniqueness of the numerical solution are proved. The convergence and stability of the numerical solution are also shown. The method is second-order convergent both in time and in space. Finally, numerical results confirm well with the theoretical results and show that the current method can be well used to study the solitary wave at long time.