Solutions of matrix NLS systems and their discretisations: A unified treatment

Abstract

Using a bidifferential graded algebra approach to integrable partial differential or difference equations, a unified treatment of continuous, semi-discrete (Ablowitz-Ladik) and fully discrete matrix NLS systems is presented. These equations originate from a universal equation within this framework, by specifying a representation of the bidifferential graded algebra and imposing a reduction. By application of a general result, corresponding families of exact solutions are obtained that in particular comprise the matrix soliton solutions in the focusing NLS case. The solutions are parametrised in terms of constant matrix data subject to a Sylvester equation (which previously appeared as a rank condition in the integrable systems literature). These data exhibit a certain redundancy, which we diminish to a large extent. More precisely, we first consider more general AKNS-type systems from which two different matrix NLS systems emerge via reductions. In the continuous case, the familiar Hermitian conjugation reduction leads to a continuous matrix (including vector) NLS equation, but it is well-known that this does not work as well in the discrete cases. On the other hand there is a complex conjugation reduction, which apparently has not been studied previously. It leads to square matrix NLS systems, but works in all three cases (continuous, semi- and fully-discrete). A large part of this work is devoted to an exploration of the corresponding solutions, in particular regularity and asymptotic behaviour of matrix soliton solutions.

60 pages, 3 figures, second version: some minor amendments and additional references, to appear in Inverse Problems

References in corpus (10)

• Explicit solutions to the Korteweg-de Vries equation on the half line
• Symmetries for exact solutions to the nonlinear Schrödinger equation
• Casorati Determinant Form of Dark Soliton Solutions of the Discrete Nonlinear Schrödinger Equation
• Analytic doubly periodic wave patterns for the integrable discrete nonlinear Schroedinger (Ablowitz-Ladik) model
• Perturbation theory for bright spinor Bose--Einstein condensate solitons
• Full-time dynamics of modulational instability in spinor Bose-Einstein condensates
• From AKNS to derivative NLS hierarchies via deformations of associative products
• A Bi-Hamiltonian Structure for the Integrable, Discrete Non-Linear Schrodinger System
• Soliton Dynamics in Linearly Coupled Discrete Nonlinear Schrödinger Equations
• On the spatial asymptotics of solutions of the Ablowitz-Ladik hierarchy

Cited by in corpus (5)

• Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions
• Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations
• Pseudo-Exponential-Type Solutions of Wave Equations Depending on Several Variables
• Solitons of Some Nonlinear Sigma-Like Models
• Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy