Riemann–Hilbert approach to coupled nonlinear Schrödinger equations on a half-line

Abstract

We use the Fokas method to investigate coupled derivative nonlinear Schrödinger equations on a half-line. The solutions are represented in terms of solutions of two matrix Riemann–Hilbert problems (RHPs) formulated in the complex plane of the spectral parameter. The elements of jump matrices are composed of spectral functions and are derived from the initial and boundary values. The spectral functions are not independent of each other, but satisfy a compatibility condition, the so-called global condition. Therefore, if the initial boundary and values and the defined spectral functions satisfy the global condition, the RHP is solvable and hence the derivative nonlinear Schrödinger equations on a half-line are solvable.