The solution of the global relation for the derivative nonlinear Schr'odinger equation on the half-line

We consider initial-boundary value problems for the derivative nonlinear Schr"odinger (DNLS) equation on the half-line $x > 0$. In a previous work, we showed that the solution $q(x,t)$ can be expressed in terms of the solution of a Riemann-Hilbert problem with jump condition specified by the initial and boundary values of $q(x,t)$. 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. In general, an effective solution of the problem therefore requires solving the global relation. Here, we present the solution of the global relation in terms of the solution of a system of nonlinear integral equations. This also provides a construction of the Dirichlet-to-Neumann map for the DNLS equation on the half-line.

💡 Research Summary

The paper addresses the initial‑boundary value problem for the derivative nonlinear Schrödinger (DNLS) equation on the half‑line (x>0). The DNLS equation, (i q_{t}+q_{xx}+i|q|^{2}q_{x}=0), models a variety of physical phenomena such as nonlinear pulse propagation in optical fibers and plasma waves. In earlier work the authors showed that the solution (q(x,t)) can be represented by a 2 × 2 matrix Riemann–Hilbert (RH) problem whose jump matrix is built from the spectral functions (a(k),b(k)) (determined by the initial data) and (A(k),B(k)) (determined by the boundary data). However, for a well‑posed problem only part of the boundary data can be prescribed (e.g., the Dirichlet value (g_{0}(t)=q(0,t)) or the Neumann value (g_{1}(t)=q_{x}(0,t))). The remaining boundary value is constrained by the so‑called global relation, \