A note on the integrable discretization of the nonlinear Schrödinger equation

Details A note on the integrable discretization of the nonlinear Schrödinger equation A note on the integrable discretization of the nonlinear Schrödinger equation

1997-01-14

arxiv.org/abs/solv-int/9701010v2

Inverse Problems, 1997, V. 13, p. 1121-1136.

Nonlinear Sciences

Exactly Solvable and Integrable Systems

24 pages, LaTeX, revised and extended version

Scientific paper

10.1088/0266-5611/13/4/016

We revisit integrable discretizations for the nonlinear Schr\"odinger 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.

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