Initial-boundary value problem for the two-component Gerdjikov-Ivanov equation on the interval

TL;DR

This paper applies the Fokas unified method to solve initial-boundary value problems for the two-component Gerdjikov-Ivanov equation on a finite interval, expressing solutions via a Riemann-Hilbert problem and analyzing boundary maps.

Contribution

It extends the Fokas method to the two-component Gerdjikov-Ivanov equation on finite intervals, providing explicit spectral functions and boundary value analysis.

Findings

Solution expressed via a 3x3 Riemann-Hilbert problem

Explicit jump matrices in terms of spectral functions

Analysis of Dirichlet to Neumann map via global relation

Abstract

In this paper, we apply Fokas unified method to study initial-boundary value problems for the two-component Gerdjikov-Ivanov equation formulated on the finite interval with $3 \times 3$ Lax pairs. The solution can be expressed in terms of the solution of a $3\times3$ Riemann-Hilbert problem. The relevant jump matrices are explicitly given in terms of three matrix-value spectral functions $s(\lambda)$, $S(\lambda)$ and $S_L(\lambda)$, which arising from the initial values at $t=0$, boundary values at $x=0$ and boundary values at $x=L$, respectively. Moreover, The associated Dirichlet to Neumann map is analyzed via the global relation. 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 tends to infinity.