Multi‐soliton solutions of the three‐level coupled Schrödinger Maxwell‐Bloch equations via the Riemann‐Hilbert approach

Abstract

In this paper, a study of the coupled Schrödinger Maxwell‐Bloch equations, which described the propagation of two optical pulses in an optical medium with coherent three‐level atoms, is presented via the Riemann‐Hilbert approach. First, we performed a spectral analysis on the basis of Lax pair with the negative flow, and then the Jost function, scattering matrix, and their analytic and symmetric properties are given. Second, the Riemann‐Hilbert problem is established successfully through a standard dressing procedure via the Riemann‐Hilbert approach, and then the potential function related to the solution of the Riemann‐Hilbert problem is reconstructed. By introducing the special matrix functions, we can transform the irregular Riemann‐Hilbert problem into the regular one, which can be solved by the Plemelj formula. Finally, some applications are given to solve the Riemann‐Hilbert problem without reflection for the coupled Schrödinger Maxwell‐Bloch equation, and the multi‐soliton solutions are obtained explicitly. Moreover, the dynamic behaviors of some soliton solutions are discussed graphically by choosing appropriate parameters.