Inverse scattering transform numerical schemes for nonlinear evolution equations and the method of lines (Q1917452)

The author presents a systematic method for deriving differential-difference equations based on the inverse scattering transform (IST) that have as limiting forms the Korteweg-de Vries (KdV), the nonlinear Schrödinger (NLS), the modified KdV, the complex modified KdV, and the higher NLS nonlinear evolution equations. These equations are used as numerical schemes for their associated nonlinear evolution equations. These IST numerical schemes are continuous in time and discrete in space. These can be solved by the method of lines. A numerical example is included to show that the IST numerical scheme is more accurate than the combination IST scheme. It appears that methods related to IST can play an important role in the proper discretization of nonlinear terms in nonlinear differential equations.