Exact solutions of the nonlocal (2+1)-dimensional complex modified Korteweg-de Vries Equation

This article centres on the construction of the nonlocal (2+1)-dimensional complex modified Korteweg-de Vries (cmKdV) equation, as well as the exploration of its exact solutions. By employing Darboux transformations, specific expressions for the first-order and n-order solutions of the nonlocal equation are derived. By choosing a non-zero seed solution and characteristic function, and applying constraints on the relationships between key parameters that affect the solution structure, three exact solutions are obtained: Soliton solutions, Kink solutions, and Rational soliton solutions. By employing various limiting techniques for asymptotic analysis of these solutions, their dynamic behaviors are revealed. Through research, it has been found that the Soliton solutions consistently move along parabolic trajectories while preserving their shape. The Kink solutions are characterised by inelastic collisions between bright and dark solitons, with observable changes in height before and after the collisions. In contrast, the Rational soliton solutions exhibit elastic collisions, where the velocity, amplitude, and phase remain unchanged before and after the collisions. To vividly illustrate these solutions and their collision characteristics, 3D images of these solutions are presented in this paper, showing their time-evolving trajectories and the shapes at the moment of collision.