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We investigate the initial-boundary value problem for the generalthree-component nonlinear Schrodinger (gtc-NLS) equation with a 4x4 Lax pair ona finite interval by extending the Fokas unified approach. The solutions of thegtc-NLS equation can be expressed in terms of the solutions of a 4x4 matrixRiemann-Hilbert (RH) problem formulated in the complex k-plane. Moreover, therelevant jump matrices of the RH problem can be explicitly found via the threespectral functions arising from the initial data, the Dirichlet-Neumannboundary data. The global relation is also established to deduce two distinctbut equivalent types of representations (i.e., one by using the large k ofasymptotics of the eigenfunctions and another one in terms of theGelfand-Levitan-Marchenko (GLM) method) for the Dirichlet and Neumann boundaryvalue problems. Moreover, the relevant formulae for boundary value problems onthe finite interval can reduce to ones on the half-line as the length of theinterval approaches to infinity. Finally, we also give the linearizableboundary conditions for the GLM representation.

An initial-boundary value problem of the general three-component nonlinear Schrodinger equation with a 4x4 Lax pair on a finite interval