Initial‐Boundary Value Problems for the Coupled Nonlinear Schrödinger Equation on the Half‐Line

Abstract

Initial‐boundary value problems for the coupled nonlinear Schrödinger equation on the half‐line are investigated via the Fokas method. It is shown that the solution can be expressed in terms of the unique solution of a matrix Riemann–Hilbert problem formulated in the complex k‐plane, whose jump matrix is defined in terms of the matrix spectral functions and that depend on the initial data and all boundary values, respectively. If there exist spectral functions satisfying the global relation, it can be proved that the function defined by the above Riemann–Hilbert problem solves the coupled nonlinear Schrödinger equation and agrees with the prescribed initial and boundary values. The most challenging problem in the implementation of this method is to characterize the unknown boundary values that appear in the spectral function . For a particular class of boundary conditions so‐called linearizable boundary conditions, it is possible to compute the spectral function in terms of and given boundary conditions by using the algebraic manipulation of the global relation. For the general case of boundary conditions, an effective characterization of the unknown boundary values can be obtained by employing perturbation expansion.