A step-like approximation and a new numerical schema for the Korteweg-de Vries equation in the soliton region

We discuss a numerical schema for solving the initial value problem for the Korteweg-de Vries equation in the soliton region which is based on a new method of evaluation of bound state data. Using a step-like approximation of the initial profile and a fragmentation principle for the scattering data, we obtain an explicit procedure for computing the bound state data. Our method demonstrates an improved accuracy on discontinuous initial data. We also discuss some generalizations of this algorithm and how it might be improved by using Haar and other wavelets.