General soliton solutions for the complex reverse space-time nonlocal mKdV equation on a finite background

Three kinds of Darboux transformations are constructed by means of the loop group method for the complex reverse space-time (RST) nonlocal modified Korteweg–de Vries equation, which are different from that for the PT symmetric (reverse space) and reverse time nonlocal models. The N-periodic, the N-soliton, and the N-breather-like solutions, which are, respectively, associated with real, pure imaginary, and general complex eigenvalues on a finite background are presented in compact determinant forms. Some typical localized wave patterns such as the doubly periodic lattice-like wave, the asymmetric double-peak breather-like wave, and the solitons on singly or doubly periodic waves are graphically shown. The essential differences and links between the complex RST nonlocal equations and their local or PT symmetric nonlocal counterparts are revealed through these explicit solutions and the solving process.