Generalized Darboux transformation and higher-order rogue wave solutions to the Manakov system

In this paper, we propose a recursive Darboux transformation in a generalized form of a focusing vector Nonlinear Schrödinger Equation (NLSE) known as the Manakov System. We apply this generalized recursive Darboux transformation to the Lax-pairs of this system in view of generating the Nth-order vector generalization rogue wave solutions with a rule of iteration. We discuss from first- to three-order vector generalizations of rogue wave solutions while illustrating these features with some 3D, 2D graphical depictions. We illustrate a clear connection between higher-order rogue wave solutions and their free parameters for better understanding the physical phenomena described by the Manakov system

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

The order-[Formula: see text] 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 [Formula: see text], order-n rational solutions are obtained, among which the order-1 and order-2 solutions are analyzed in detail. By varying different parameter [Formula: see text], 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.