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

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 $3\times 3$ Riemann-Hilbert problem. The relevant jump matrices are explicitly given in terms of three matrix-value spectral functions $s(\hat{I}»)$, $S(\hat{I}»)$ and $S_L(\hat{I}»)$, 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.

arXiv admin note: substantial text overlap with arXiv:1509.02617, arXiv:1304.4586; text overlap with arXiv:1108.2875 by other authors