Focusing Singularity in a Derivative Nonlinear Schrödinger Equation

We present a numerical study of a derivative nonlinear Schrödinger equation with a general power nonlinearity, $|ψ|^{2σ}ψ_x$. In the $L^2$-supercritical regime, $σ>1$, our simulations indicate that there is a finite time singularity. We obtain a precise description of the local structure of the solution in terms of blowup rate and asymptotic profile, in a form similar to that of the nonlinear Schrödinger equation with supercritical power law nonlinearity.

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