Numerical Simulations and Parallel Implementation of Some Nonlinear Schrödinger Systems

Abstract

As it is well known, the nonlinear Schrödinger equation (NLS)

$$ \left\{ i \right.{W_t} + {W_{xx}} + a{\left| W \right|^2}W = 0,\quad W\left( {x,0} \right) = {W_0}\left( x \right),$$

(1)

where a =const, and W(x, t) being a complex function, can be exactly solved by using the Inverse Scattering Theory. At the same time several numerical schemes have been proposed to solve this equation^1–7. In general, the perturbations of the NLS equation as driving forces, dissipations, stochastic potentials, coupled NLS systems are not tractable analytically. The same happens for the NLS systems in two and three space dimensions. As a consequence, large and massive computations are needed and a possible way to make them is by the parallel computing. In this context, we started a project to implement suitable algorithms for the complex nonlinear Schrödinger systems on a distributed array of transputers. Our first result was to propose an alternative finite difference scheme for the unperturbed NLS equation in one space dimension⁸. This algorithm has been tested in several relevant physical simulations, that is described in Sections 2 and 3. In Sections 4 to 7 we describe some features of the parallelization of the scheme as well as its parallel implementation on an array of transputers.