Analytical and numerical aspects of certain nonlinear evolution equations. IV: Numerical, modified Korteweg-de Vries equation (Q1104063)

In part III [ibid. 55, 231-253 (1984; Zbl 0541.65083)] we derived nonlinear partial difference equations which have as limiting forms the Korteweg-de Vries (KdV) and the MKdV equations. These difference equations have a number of special properties and are constructed by methods related to the inverse scattering transform (IST). We have also implemented similar schemes for the nonlinear Schrödinger (NLS) and the KdV equations and compared them with known numerical schemes. Experiments have shown that the IST schemes for the NLS and KdV equations compare very favorably with the other known numerical methods. This work aims to implement and compare the proposed schemes which were developed in part III with certain other known numerical methods for the MKdV equation \(u_ t+6u^ 2u_ x+u_{xxx}=0.\)