A Numerical Energy Conserving Method for the DNLS Equation

An implicit, numerical energy conserving method is developed for the derivative nonlinear Schrödinger (DNLS) equation for periodic boundary conditions. We find no numerical high frequency modulational instabilities in addition to the modulational instability from a linear analysis around a nonlinear state for the DNLS equation if the modulation is small and ( {k ₀ -a ²}/{2}) ² t < π (k ₀ is the wavenumber and a the amplitude). The numerical scheme is used to follow the nonlinear behavior of the DNLS modulational instability. The numerical code is also tested by the evolution for one soliton initial data. These tests show that if the modulation is not small compared to the background wave amplitude, new nonlinear numerical instabilities are introduced.