Title: An initial-boundary value problem of the general three-component nonlinear Schrodinger equation with a 4x4 Lax pair on a finite interval,
Abstract: We investigate the initial-boundary value problem for the general
three-component nonlinear Schrodinger (gtc-NLS) equation with a 4x4 Lax pair on
a finite interval by extending the Fokas unified approach. The solutions of the
gtc-NLS equation can be expressed in terms of the solutions of a 4x4 matrix
Riemann-Hilbert (RH) problem formulated in the complex k-plane. Moreover, the
relevant jump matrices of the RH problem can be explicitly found via the three
spectral functions arising from the initial data, the Dirichlet-Neumann
boundary data. The global relation is also established to deduce two distinct
but equivalent types of representations (i.e., one by using the large k of
asymptotics of the eigenfunctions and another one in terms of the
Gelfand-Levitan-Marchenko (GLM) method) for the Dirichlet and Neumann boundary
value problems. Moreover, the relevant formulae for boundary value problems on
the finite interval can reduce to ones on the half-line as the length of the
interval approaches to infinity. Finally, we also give the linearizable
boundary conditions for the GLM representation.