An initial-boundary value problem for the general three-component nonlinear Schrödinger equations on a finite interval

The general three-component nonlinear Schrödinger (gtc-NLS) equations are completely integrable and contain the self-focusing, defocusing and mixed cases, which are applied in many physical fields. In this paper, we would like to use the Fokas method to explore the initial-boundary value (IBV) problem for the gtc-NLS equations with a $4\times 4$ matrix Lax pair on a finite interval based on the inverse scattering transform. The solutions of the gtc-NLS equations can be expressed using the solution of a $4\times 4$ matrix Riemann–Hilbert (RH) problem constructed in the complex $k$ -plane. The jump matrices of the RH problem can be explicitly found in terms of three spectral functions related to the initial data, and the Dirichlet–Neumann boundary data, respectively. The global relation between the distinct spectral functions is also proposed to derive two distinct but equivalent types of representations of the Dirichlet–Neumann boundary value problems. Particularly, the relevant formulae for the boundary value problems on the finite interval can generate ones on the half-line as the length of the interval closes to infinity. Finally, we also analyse the linearizable boundary conditions for the Gel'fand–Levitan–Marchenko representation. These results will be useful to further study the solution properties of the IBV problem of the gtc-NLS system by using the Deift–Zhou's nonlinear steepest descent method and some numerical methods.