A Riemann–Hilbert approach for a new type coupled nonlinear Schrödinger equations

Abstract

We develop a Riemann–Hilbert approach to the Cauchy problem on the line for a new type of coupled nonlinear Schrödinger (CNLS) equationsiq1,t+q1,xx+2(|q1|2−2|q2|2)q1−2q22q1⁎=0,iq2,t+q2,xx+2(2|q1|2−|q2|2)q2+2q12q2⁎=0. This approach allows us to give a representation of the solution to the Cauchy problem of the CNLS equations in terms of the solution of a 4×4 Riemann–Hilbert problem formulated in the complex k-plane. Due to the energy conservation law of above system is ∫−∞+∞(|q1|2−|q2|2)dx, it is difficult to obtain a solution for this system by using the energy estimate method of PDE's. Therefore, this approach efficiently provides a new way in studying the nonlinear problems that PDE's theory can't solve. Furthermore, this representation is then used for retrieving the soliton solutions.