The derivative nonlinear Schrodinger equation on the interval

TL;DR

This paper applies the Fokas method to analyze the derivative nonlinear Schrödinger equation on a finite interval, representing solutions via a Riemann-Hilbert problem linked to initial and boundary data.

Contribution

It formulates a Riemann-Hilbert problem for the DNLS equation on an interval, explicitly relating spectral functions to initial and boundary conditions.

Findings

Representation of solutions via a matrix Riemann-Hilbert problem

Explicit jump matrices in terms of spectral functions

Derivation of a global relation linking spectral functions

Abstract

We use the Fokas method to analyze the derivative nonlinear Schr\"odinger (DNLS) equation $iq_t(x,t)=-q_{xx}(x,t)+(r q^2)_x$ on the interval $[0,L]$. Assuming that the solution $q(x,t)$ exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter $\x$. This problem has explicit $(x,t)$ dependence, and it has jumps across $\{\x \in \C|\im{\x^4}=0 \}$. The relevant jump matrices are explicitly given in terms of the spectral functions $\{a(\x),b(\x)\},\{A(\x),B(\x)\}$, and $\{\ca({\x}),\cb(\x)\}$, which in turn are defined in terms of the initial data $q_0(x)=q(x,0)$, the boundary data $g_0(t)=q(0,t),g_1(t)=q_x(0,t)$, and another boundary values $f_0(t)=q(L,t),f_1(t)=q_x(L,t)$. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.