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Local structure of singular profiles for a Derivative Nonlinear Schrödinger Equation

The Derivative Nonlinear Schrödinger equation is an $L^2$-critical nonlinear dispersive equation model for Alfvén waves in space plasmas. Recent numerical studies on an $L^2$-supercritical extension of this equation provide evidence of finite time singularities. Near the singular point, the solution is described by a universal profile that solves a nonlinear elliptic eigenvalue problem depending only on the strength of the nonlinearity. In the present work, we describe the deformation of the profile and its parameters near criticality, combining asymptotic analysis and numerical simulations.