A method for practical realization of the inverse scattering transform
method for the Korteweg-de Vries equation is proposed. It is based on
analytical representations for Jost solutions and for integral kernels
of transformation operators obtained recently by the authors. The
representations have the form of functional series in which the first
coefficient plays a crucial role both in solving the direct scattering
and the inverse scattering problems. The direct scattering problem
reduces to computation of a number of the coefficients following a
simple recurrent integration procedure with a posterior calculation of
scattering data by well known formulas. The inverse scattering problem
reduces to a system of linear algebraic equations from which the first
component of the solution vector leads to the recovery of the potential.
We prove the applicability of the finite section method to the system of
linear algebraic equations and discuss numerical aspects of the proposed
method. Numerical examples are given, which reveal the accuracy and
speed of the method.