An initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a 4x4 Lax pair on the half-line

TL;DR

This paper formulates and analyzes an initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations on the half-line, using a 4x4 Lax pair and Riemann-Hilbert problem approach.

Contribution

It introduces a method to solve the spin-1 GP equations with boundary conditions via a 4x4 Lax pair and spectral analysis, establishing the global relation between spectral functions.

Findings

Explicit jump matrices for the RH problem derived from spectral functions

Global relation connecting initial and boundary spectral data established

Analysis of boundary value mappings in the spectral domain

Abstract

We investigate the initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii (GP) equations with a 4x4 Lax pair on the half-line. The solution of this system can be obtained in terms of the solution of a 4x4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. The relevant jump matrices of the RH problem can be explicitly found using the two spectral functions s(k) and S(k), which can be defined by the initial data, the Dirichlet-Neumann boundary data at x=0. The global relation is established between the two dependent spectral functions. The general mappings between Dirichlet and Neumann boundary values are analyzed in terms of the global relation.