The nonlinear Schrödinger equation with forcing involving products of eigenfunctions

Abstract

We elaborate on a new methodology, which starting with an integrable evolution equation in one spatial dimension, constructs an integrable forced version of this equation. The forcing consists of terms involving quadratic products of certain eigenfunctions of the associated Lax pair. Remarkably, some of these forced equations arise in the modelling of important physical phenomena. The initial value problem of these equations can be formulated as a Riemann-Hilbert problem, where the "jump matrix" has explicit x and t dependence and can be computed in terms of the initial data. Thus, these equations can be solved as efficiently as the nonlinear integrable equations from which they are generated. Details are given for the forced versions of the nonlinear Schrodinger.

arXiv comment20 pages, no figure, this paper is published in the journal Open Communications in Nonlinear Mathematical Physics

References in corpus (11)

• The Nonlinear Schrödinger Equation on the Interval
• Negaton and Positon solutions of the soliton equation with self-consistent sources
• The Solutions of the NLS Equations with Self-Consistent Sources
• On the Toda Lattice Equation with Self-Consistent Sources
• Self-Consistent Sources for Integrable Equations via Deformations of Binary Darboux Transformations
• Integrable Rosochatius deformations of higher-order constrained flows and the soliton hierarchy with self-consistent sources
• Darboux Transformation and Exact Solutions of the integrable Heisenberg ferromagnetic equation with self-consistent potentials
• Beyond nonlinear saturation of backward Raman amplifiers
• Bäcklund transformations for the constrained dispersionless hierarchies and dispersionless hierarchies with self-consistent sources
• The quasiclassical limit of the symmetry constraint of the KP hierarchy and the dispersionless KP hierarchy with self-consistent sources
• The Degasperis-Procesi equation with self-consistent sources