Integrable discretization of 'time' and its application on the Fourier pseudospectral method to the Korteweg-de Vries equation

This paper presents a new integrable discretization of the Korteweg-de Vries (KdV) equation. Different from other discrete analogues, we discretize the variable 'time' and obtain an integrable differential-difference system. This system has the original KdV equation as a standard limit when the step size tends to zero. The main idea is based on Hirota's bilinear method and Bäcklund transformation and can be applied to other integrable systems. By applying the Fourier pseudospectral method to the space variable, we derive a new numerical scheme for the KdV equation. Numerical results are found to agree with the exact solution and the first five conservation quantities are preserved quite well.