The Order-N Breather And Degenerate Breather Solutions Of The (2+1)-Dimensional Cmkdv Equations

Starting with a plane wave seed, the order-n 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-n degenerate breather solution by using the Taylor expansion concerning the limits lambda(2k-1) -> lambda(1)(k = 1, ... , n) 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 lambda(3) closer and closer to lambda(1). Furthermore, the approximate trajectories of the order-2, order-3, order-4 degenerate breather solutions are depicted by explicit expressions, respectively.