Curvature Driven Complexity in the Defocusing Parametric Nonlinear Schrödinger System

The parametric nonlinear Schrödinger equation models a variety of parametrically forced and damped dispersive waves. For the defocusing regime, we derive a normal velocity for the evolution of curved dark-soliton fronts that represent a \(\pi \) $\pi$ -phase shift across a thin interface. We establish a simple mechanism through which the parametric term transitions the normal velocity evolution from a curvature-driven flow to motion against curvature regularized by surface diffusion of curvature. In the former case interfacial length shrinks, while in the latter case interface length generically grows until self-intersection followed by a transition to complex motion.