Symplectic Integration and the Nonlinear Schrödinger
Equation
Abstract
The nonlinear Schrödinger equation is the basis for many simulations of
nanoscale semiconductor devices where quantum mechanical effects
prevail (e.g., quantum dot devices). In order to ensure the long-term
stability of the numerical solution, advanced numerical methods have to be
used. One approach is to try to conserve certain geometric quantities,
which leads to symplectic integrators, whose theory is outlined. A
generalization of symplectic integrators, namely Poisson integrators,
are explained and applied to the nonlinear Schrödinger equation with
arbitrary potential. Several numerical experiments are performed and
illustrate the properties of the numerialc integrators and the solutions
of the Schrödinger equation.
