AI 中文总结
研究聚焦Majda-McLaughlin-Tabak型方程的有限时间波坍塌，通过粘性扩散或非线性饱和正则化动力学，发现坍塌前后两种正则化极限不同，产生自发随机性，为其可应用于更广泛系统提供数值证据。
AI 中文摘要
我们研究了一个经历有限时间波坍塌的聚焦Majda-McLaughlin-Tabak型方程。这种奇点终止了经典的光滑解，并开启了一个爆破后状态，其中可能存在无限多个解。为了探究这个非唯一状态，我们通过粘性扩散或非线性饱和对动力学进行正则化，并研究相应的消失正则化极限。两种正则化都能在固定参数下防止爆破，并在参数消失时恢复相同的无粘坍塌。坍塌前，它们收敛到相同的光滑无粘解。坍塌后，它们的极限不同。粘性近似经历反常质量耗散，而饱和近似保持守恒。此外，两种正则化都没有选择唯一的爆破后解。正则化参数或初始条件的消失扰动在奇异极限中幸存，并产生有限的爆破后不确定性。这将坍塌波湍流置于自发随机性的背景下，其中无粘极限用无粘解上的概率定律而非确定性来更好地描述。逐尺度波动预算将坍塌事件识别为不确定性产生的局部源。虽然自发随机性通常与流体湍流相关，但这些结果提供了数值证据，表明它可以应用于更广泛的系统类别，包括可以进行实验的色散介质。
英文摘要
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.
