Focusing Singularity in a Derivative Nonlinear Schr\"odinger Equation

TL;DR

This paper numerically investigates a derivative nonlinear Schrödinger equation with supercritical nonlinearity, revealing finite-time singularities and detailed blowup behavior similar to classical NLS equations.

Contribution

It provides the first detailed numerical analysis of blowup phenomena in a derivative NLS with general power nonlinearity, characterizing singularity formation and local solution structure.

Findings

Finite time singularity observed in supercritical regime

Blowup rate and profile characterized numerically

Solution behavior similar to classical supercritical NLS

Abstract

We present a numerical study of a derivative nonlinear Schr\"odinger equation with a general power nonlinearity, $|\psi|^{2\sigma}\psi_x$. In the $L^2$-supercritical regime, $\sigma>1$, 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 Schr\"odinger equation with supercritical power law nonlinearity.