Analytical and numerical aspects of certain nonlinear evolution equations. I. Analytical

Abstract Nonlinear partial difference equations are obtained which have as limiting forms the nonlinear Schrodinger, Korteweg-deVries and modified Korteweg-deVries equations. These difference equations have a number of special properties. They are constructed by methods related to the inverse scattering transform. They can be used as a basis for numerical schemes to the associated nonlinear evolution equations. Experiments have shown that they compare very favorably with other known numerical methods (papers II, III). In paper II, the Ablowitz-Ladik scheme for the nonlinear Schrodinger equation is compared to other known numerical schemes, and generally proved to be faster than all utilized finite difference schemes but somewhat slower than the finite Fourier (pseudospectral) methods. In paper III, a proposed scheme for the Korteweg-de Vries equation proved to be faster than both the finite difference and finite Fourier methods already considered.