Abstract

Soliton perturbation theory is used to obtain analytical solutions describing
solitary wave tails or shelves, due to numerical discretization error, for
soliton solutions of the nonlinear Schr¨odinger equation. Two important implicit
numerical schemes for the nonlinear Schr¨odinger equation, with secondorder
temporal and spatial discretization errors, are considered. These are the Crank–
Nicolson scheme and a scheme, due to Taha [1], based on the inverse scattering
transform. The firstorder correction for the solitary wave tail, or shelf, is in
integral form and an explicit expression is found for large time. The shelf decays
slowly, at a rate of t−12
, which is characteristic of the nonlinear Schr¨odinger
equation. Singularity theory, usually used for combustion problems, is applied
to the explicit largetime expression for the solitary wave tail. Analytical results
are then obtained, such as the parameter regions in which qualitatively different
types of solitary wave tails occur, the location of zeros and the location and
amplitude of peaks. It is found that three different types of tail occur for the
Crank–Nicolson and Taha schemes and that the Taha scheme exhibits some
unusual symmetry properties, as the tails for left and right moving solitary
waves are different. Optimal choices of the discretization parameters for the
numerical schemes are also found, whichminimize the amplitude of the solitary
wave tail. The analytical solutions are compared with numerical simulations,
and an excellent comparison is found.