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A note on the integrable discretization of the nonlinear Schr

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 factorise the non-local difference scheme into the product of two 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 two local schemes and show how to solve them in terms of the matrix factorisations.