Spontaneous stochasticity and anomalous dissipation in collapsing wave turbulence

Abstract

We study a focusing Majda--McLaughlin--Tabak type equation undergoing finite-time wave collapse. This singularity terminates the classical smooth solution and opens a post-blowup regime where infinitely many solutions may exist. To probe this nonunique regime, we regularize the dynamics either by viscous diffusion or by nonlinear saturation and study the corresponding vanishing-regularization limits. Both regularizations prevent blowup at fixed parameter and recover the same inviscid collapse as the parameter vanishes. Before collapse, they converge to the same smooth inviscid solution. After collapse, however, their limits differ. The viscous approximation undergoes anomalous mass dissipation whereas the saturating approximation remains conservative. Moreover, neither regularization selects a unique post-blowup solution. Vanishing perturbations of the regularization parameter or of the initial condition survive the singular limit and generate finite post-blowup uncertainty. This places collapsing wave turbulence in the setting of spontaneous stochasticity, where the inviscid limit is better described in terms probability law on inviscid solutions rather than deterministically. Scale-by-scale fluctuation budgets identify collapse events as localized sources of uncertainty production. While spontaneous stochasticity is usually associated with fluid turbulence, these results provide numerical evidence that it can be applied to a broader class of systems, including dispersive media in which experiments could be conducted.