Adiabatic midpoint rule for the dispersion-managed nonlinear Schrödinger equation

The dispersion-managed nonlinear Schrodinger equation contains a small parameter $$\varepsilon $$ , a rapidly changing piecewise constant coefficient function, and a cubic nonlinearity. Typical solutions are highly oscillatory and have a discontinuous time-derivative, and hence solving this equation numerically is a challenging task. We present and analyze a tailor-made time integrator which attains the desired accuracy with a significantly larger step-size than traditional methods. The construction of this method is based on a favorable transformation to an equivalent problem and the explicit computation of certain integrals over highly oscillatory phases. The error analysis requires the thorough investigation of various cancellation effects which result in improved accuracy for special step-sizes.