A study of periodic solutions and periodic background solutions for the reverse-space–time modified nonlinear Schrödinger equation

The modified nonlinear Schrödinger (MNLS) equation is used to describe the effect of femtosecond pulses and nonlinear dispersion in long single-mode fibers and the propagation of modulated Alfvén waves along a magnetic field in a cold plasma. Under investigation in this paper is its nonlocal reverse-space–time type, extending it to the more general case, which has much physical significance in the field of nonlocal nonlinear dynamical systems. Through constructing nonlocal type Darboux transformations (DTs), we will derive a series of new solutions as follows: (1)Bright and dark solitons on plane wave backgrounds, single and double periodic solutions; (2)Breather, rogue wave solutions and their coexistence mechanism on single periodic backgrounds via odd-fold DTs; (3) Bright and dark breather-like solutions on single periodic backgrounds, breather, rogue wave solutions and their degenerate and coexistence forms on double periodic backgrounds via even-fold DTs. In addition, we will discuss dynamic behaviors of those solutions through graphic simulations.