  The Derivative Nonlinear Schr\"odinger equation is an $L^2$-critical
nonlinear dispersive equation model for Alfv\'en 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.
