A note on the integrable discretization of the nonlinear Schrödinger equation

Abstract

We revisit integrable discretizations for the nonlinear Schrödinger equation due to Ablowitz and Ladik. We demonstrate how their main drawback, the non-locality, can be overcome. Namely, we factorize the non-local difference scheme into the product of local ones. This must improve the performance of the scheme in the numerical computations dramatically. Using the equivalence of the Ablowitz--Ladik and the relativistic Toda hierarchies, we find the interpolating Hamiltonians for the local schemes and show how to solve them in terms of matrix factorizations.

24 pages, LaTeX, revised and extended version

References in corpus (3)

• A discrete time relativistic Toda lattice
• Discrete time Bogoyavlensky lattices
• A discrete time peakons lattice

Cited by in corpus (13)

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• Functional representation of the Ablowitz-Ladik hierarchy. II
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• Modified melting crystal model and Ablowitz-Ladik hierarchy
• A systematic method for constructing time discretizations of integrable lattice systems: local equations of motion
• Orbifold melting crystal models and reductions of Toda hierarchy
• From Quantum Bäcklund Transforms to Topological Quantum Field Theory
• On an integrable discretisation of the Ablowitz-Ladik hierarchy
• Generalized Ablowitz-Ladik hierarchy in topological string theory
• Modified melting crystal model and Ablowitz-Ladik hierarchy
• A q-difference Baxter's operator for the Ablowitz-Ladik chain
• Discretizations of the generalized AKNS scheme