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

2010/08/31 by Jonatan Lenells, Lenells, Jonatan
Mathematics · Physics and Astronomy · #37K15 #Advanced Mathematical Physics Problems #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Photonic Systems #Quantum Mechanics and Non-Hermitian Physics

paper · pdf · doi:10.48550/arxiv.1008.5379

openalex publication_date 2010/08/31 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

We consider initial-boundary value problems for the derivative nonlinear\nSchr "odinger (DNLS) equation on the half-line x > 0. In a previous work, we\nshowed that the solution q(x,t) can be expressed in terms of the solution of\na Riemann-Hilbert problem with jump condition specified by the initial and\nboundary values of q(x,t). However, for a well-posed problem, only part of\nthe boundary values can be prescribed; the remaining boundary data cannot be\nindependently specified, but are determined by the so-called global relation.\nIn general, an effective solution of the problem therefore requires solving the\nglobal relation. Here, we present the solution of the global relation in terms\nof the solution of a system of nonlinear integral equations. This also provides\na construction of the Dirichlet-to-Neumann map for the DNLS equation on the\nhalf-line.\n

Related

• The derivative nonlinear Schrödinger equation on the half-line
• Long-time asymptotics for the derivative nonlinear Schrödinger equation on the half-line
• Long-time asymptotics for the derivative nonlinear Schrödinger equation on the half-line
• The derivative nonlinear Schrödinger equation on the half line
• The derivative nonlinear Schrodinger equation on the interval