Conservation laws, periodic and rational solutions for an extended modified Korteweg–de Vries equation

Abstract

We study an integrable extended modified Korteweg–de Vries equation, which contains the fifth-order dispersion and relevant higher-order nonlinear terms. The infinitely many conservation laws are constructed based on the Lax pair. A general Nth-order periodic solution is obtained by means of the N-fold Darboux transformation (DT), and a simple representation of the Nth-order rational solution is derived from the generalized DT by using the limit approach. As an application, the explicit periodic and rational solutions from first to second order are given, and some typical nonlinear wave patterns such as the doubly periodic lattice-like and doubly localized high-peak waves are shown. It is interestingly found that, the doubly localized high-peak wave can be converted into a W-shaped soliton in the second-order rational solution due to the existence of higher-order terms.