# Numerical solution of a locally Lipschitz differential equation. (Gross-Pitaevskii)

A (very simplified) version of the GP-equation, can be written as

$$\frac{d\alpha}{dt}=-Ui\alpha^*\alpha^2=-Ui|\alpha|^2\alpha$$,

Where $$i$$ is the imaginary unit. In contrast to what I first thought, this problem seems not one-sided Lipschitz, so that an Implicit Euler method is generally not sufficient to tame numerical instability...; it is locally Lipschitz however.

Are there other implicit or explicit methods that are guaranteed to be numerically stable requiring only local Lipschitzness?