Rogue waves, modulation instability of the (2+1)-dimensional complex modified Korteweg-de Vries equation on the periodic background

Abstract

In this paper, we construct rogue periodic wave solutions (rogue wave solutions under the Jacobian elliptic dn- and cn-periodic waves background) and the modulation instability of a (2+1)-dimensional complex modified Korteweg–de Vries (CMKdV) equation. Under the linear stability analysis, we reveal that modulation instability is the condition for the existence of rogue waves. Next, we derive the expression of eigenvalues and corresponding squared periodic eigenfunctions by the method of nonlinearization of the Lax pair. Then, we construct the periodic and non-periodic solution of the Lax pair. At last, we present the rogue periodic wave solutions of the (2+1)-dimensional CMKdV equation by the Darboux transformation (DT) method, and analyze the non-linear dynamics in some figures.