# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1704.08534