The order-n breather and degenerate breather solutions of the (2+1)-dimensional cmKdV equations

Abstract

Starting with a plane wave seed, the order-[Formula: see text] breather for the (2+1)-D complex modified Korteweg-de Vries (cmKdV) equations is obtained by the use of Darboux transformation. The dynamic evolution of order-2 and order-3 breather solutions is shown in the form of pictures. Afterward, we obtain the order-[Formula: see text] degenerate breather solution by using the Taylor expansion concerning the limits [Formula: see text] and focus on the order-2 degenerate breather solution. We show the dynamic evolution with time and discuss the degradation process from a breather solution through getting [Formula: see text] closer and closer to [Formula: see text]. Furthermore, the approximate trajectories of the order-2, order-3, order-4 degenerate breather solutions are depicted by explicit expressions, respectively.