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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?

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