Novel nonlocal three-component mKdV equations and classification of solutions

Abstract

A kind of nonlocal reduction for the unreduced modified Korteweg–de Vries (mKdV) system is presented, which yields the reverse space–time nonlocal complex three-component mKdV (NCTC-mKdV) equation. This equation can be regarded as a new member of the Ablowitz–Kaup–Newell–Segur (AKNS) integrable hierarchy. We develop the Cauchy matrix approach to investigate the solution structure of the nonlocal system systematically, where the Sylvester equation is pivotal in constructing explicit solutions. In fact, the analytical expressions of the solutions can be classified according to the eigenvalue structure of the coefficient matrix $K$ in the Sylvester equation. Specially, various explicit solutions of the NCTC-mKdV equation are derived, including soliton solution, Jordan solution and diagonal-Jordan-block mixed solution. Notably, the conditions for generating one-soliton solution, two-soliton solution, mixed solution, periodic solution, double-periodic solution, quasi-periodic solution and dark soliton solution are presented and their dynamic behaviors are analyzed. The results reveal the structural features of solutions to the three-component mKdV equation under nonlocal reduction.