Rational solutions of the (2+1)-dimensional cmKdV equations

The order-n periodic solutions for the (2+1)-D complex modified Korteweg-de Vries (cmKdV) equations are investigated with the aid of Darboux transformation (DT) method. By using Taylor expansion considering the limits lambda(2k-1) -> lambda(0) = -alpha/2 + ci(k = 1, ... , n), order-n rational solutions are obtained, among which the order-1 and order-2 solutions are analyzed in detail. By varying different parameter l1, two kinds of rational solutions are deduced, namely, the line rogue wave solutions and the lump solutions. Dynamical properties of these solitons, including speed, amplitude, and extreme values, are investigated. It is shown that the line rogue wave solutions appear and disappear, while the lump solutions are localized traveling wave solutions.