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.

58 pages, 3 figures

References in corpus (8)

• Integrable $\mathcal{P}T$-symmetric local and nonlocal vector nonlinear Schrödinger equations: a unified two-parameter model
• The three-component defocusing nonlinear Schrodinger equation with nonzero boundary conditions
• An initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a 4x4 Lax pair on the half-line
• Initial-boundary value problem for an integrable spin-1 Gross-Pitaevskii system with a 4x4 Lax pair on a finite interval
• Boundary value problems for the stationary axisymmetric Einstein equations: a disk rotating around a black hole
• The Generalized Dirichlet to Neumann map for the KdV equation on the half-line
• Initial-boundary value problem for integrable nonlinear evolution equations with $3\times 3$ Lax pairs on the interval
• The mKdV equation on a finite interval