Discontinuous depinning/yielding transition of elastic manifolds with tailored internal elasticity

We consider elastic manifolds evolving on disordered energy potentials under the action of an external uniform driving. This scenario includes the cases of {\em depinning} and {\em yielding}, which provide paradigmatic examples of out of equilibrium phase transitions. In both cases, velocity of the manifold is zero at low driving force, and increases smoothly when a critical driving is exceeded,defining a continuous flow-curve for these systems. We show that when more general forms of the manifold elasticity are considered, the flow curve may become reentrant, and the transition hysteretic, or discontinuous. This constitutes a novel scenario for a discontinuous transition out of equilibrium.