Periodic travelling waves and rogue waves for the higher-order modified Korteweg-de Vries equation

Abstract

In this paper, we construct the rogue wave solutions of the higher-order modified Korteweg-de Vries equation on the background of the general periodic travelling wave solutions. By the sub-equation method, we derive the general periodic travelling wave solutions expressed in terms of the Jacobi elliptic functions. By using the Darboux transformation method, we derive one-, two-, and three-fold rogue wave solutions on the background of elliptic functions. We provide some examples of two families of rogue wave solutions for illustration. The method provided in this paper can also be applicable to the construction of rogue wave solutions of other higher-order nonlinear integrable models on the periodic wave background.