Integrable discretizations of derivative nonlinear Schroedinger equations

Abstract

We propose integrable discretizations of derivative nonlinear Schroedinger (DNLS) equations such as the Kaup-Newell equation, the Chen-Lee-Liu equation and the Gerdjikov-Ivanov equation by constructing Lax pairs. The discrete DNLS systems admit the reduction of complex conjugation between two dependent variables and possess bi-Hamiltonian structure. Through transformations of variables and reductions, we obtain novel integrable discretizations of the nonlinear Schroedinger (NLS), modified KdV (mKdV), mixed NLS, matrix NLS, matrix KdV, matrix mKdV, coupled NLS, coupled Hirota, coupled Sasa-Satsuma and Burgers equations. We also discuss integrable discretizations of the sine-Gordon equation, the massive Thirring model and their generalizations.

24 pages, LaTeX2e (IOP style), final version

References in corpus (5)

• Integrable semi-discretization of the coupled nonlinear Schrödinger equations
• New integrable systems of derivative nonlinear Schrödinger equations with multiple components
• Classification of integrable polynomial vector evolution equations
• Complete integrability of derivative nonlinear Schrödinger-type equations
• Symmetrically coupled higher-order nonlinear Schroedinger equations: singularity analysis and integrability

Cited by in corpus (23)

• Darboux transformations and Recursion operators for differential--difference equations
• Noncommutative Burgers Equation
• Darboux transformation and soliton solutions of the semi-discrete massive Thirring model
• Rational recursion operators for integrable differential-difference equations
• Systematic method of generating new integrable systems via inverse Miura maps
• On vector analogs of the modified Volterra lattice
• New reductions of integrable matrix PDEs: $Sp(m)$-invariant systems
• From AKNS to derivative NLS hierarchies via deformations of associative products
• Symmetries, conservation laws and Noether's theorem for differential-difference equations
• On the recursion operators for integrable equations
• On a (2+1)-dimensional generalization of the Ablowitz-Ladik lattice and a discrete Davey-Stewartson system
• Integrable semi-discretization of the massive Thirring system in laboratory coordinates
• Integrable 7-point discrete equations and evolution lattice equations of order 2
• Hamiltonian structures for integrable nonabelian difference equations
• Derivative non-linear Schrödinger equation: Singular manifold method and Lie symmetries
• An unusual series of autonomous discrete integrable equations on the square lattice
• On a new integrable discretization of the derivative nonlinear Schrödinger (Chen-Lee-Liu) equation
• A refined and unified version of the inverse scattering method for the Ablowitz-Ladik lattice and derivative NLS lattices
• Unifying scheme for generating discrete integrable systems including inhomogeneous and hybrid models
• Integrable discretization of the vector/matrix nonlinear Schrödinger equation and the associated Yang-Baxter map
• Direct and inverse scattering problems for the first-order discrete system associated with the derivative NLS system
• Soliton solutions and their dynamics in reverse-space and reverse-space-time nonlocal discrete derivative nonlinear Schrödinger equations
• Affine Lie group approach to a derivative nonlinear Schrödinger equatoin and its similarity reduction