Discrete Darboux Transformation for Ablowitz-Ladik Systems Derived from Numerical Discretization of Zakharov-Shabat Scattering Problem
For researchers in integrable systems and numerical inverse scattering, this work shows that the discrete Darboux transformation yields only first-order accuracy, which is a limitation for high-precision applications.
The paper derives discrete Ablowitz-Ladik systems from numerical discretization of the Zakharov-Shabat scattering problem and studies their Darboux transformation. The resulting numerical scheme for computing multisoliton potentials is first-order accurate regardless of the underlying discretization order.
The numerical discretization of the Zakharov-Shabat Scattering problem using integrators based on the implicit Euler method, trapezoidal rule and the split-Magnus method yield discrete systems that qualify as Ablowitz-Ladik systems. These discrete systems are important on account of their layer-peeling property which facilitates the differential approach of inverse scattering. In this paper, we study the Darboux transformation at the discrete level by following a recipe that closely resembles the Darboux transformation in the continuous case. The viability of this transformation for the computation of multisoliton potentials is investigated and it is found that irrespective of the order of convergence of the underlying discrete framework, the numerical scheme thus obtained is of first order with respect to the step size.