The GLM representation of the global relation for the two-component nonlinear Schr\"odinger equation on the interval

TL;DR

This paper develops a Gelfand-Levitan-Marchenko approach to characterize unknown boundary values for the two-component nonlinear Schrödinger equation on an interval, providing an alternative to spectral domain analysis and establishing their equivalence.

Contribution

It introduces a new physical domain representation for the Dirichlet-to-Neumann map, complementing previous spectral domain methods for the two-component nonlinear Schrödinger equation.

Findings

Derived an explicit expression for the Dirichlet-to-Neumann map.

Showed the equivalence of physical and spectral domain representations.

Enhanced understanding of boundary value problems for integrable PDEs.

Abstract

In a previous work, we show that the solution of the initial-boundary value problem for the two-component nonlinear Schr\"odinger equation on the finite interval 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 the three matrix-value spectral functions $s(k)$, $S(k)$ and $S_L(k)$, which in turn are defined in terms of the initial values, boundary values at $x=0$ and boundary values at $x=L$, respectively. However, for a well-posed problem, only part of the boundary values can be prescribed, the remaining boundary data cannot be independently specified, but are determined by the so-called global relation. Here, we use a Gelfand-Levitan-Marchenko representation to derive an expression for the generalized Dirichlet-to-Neumann map to characterize the unknown boundary values in physical domain, which is different from the approach, in fact it analyzed the global relation in spectral domain, used in the previous work. And, we can show that these two representations are equivalent.