That’s a fascinating question. From my point of view, one promising direction is to see these extreme events (rogue waves, soliton explosions, etc.) not just as random anomalies, but as manifestations of deeper universal structures in nonlinear dynamics.

We know that modulational instability, for example, appears in fluids, optics, plasmas, and even quantum systems. This suggests that there may be universal statistical or analytic laws governing their onset—something like the way random matrix theory or Riemann-type functions capture global regularities that emerge locally in number theory.

In that sense, prediction (and perhaps partial control) may be possible if we can bridge two levels:

• Global analytic frameworks (integrable models, spectral theory, even holographic or theta-function formalisms)
• Local nonlinear dynamics (the specific instabilities and bifurcations leading to extreme events)

I think the key is to develop discrete/continuous analogues that let us translate between those two. If successful, this wouldn’t just help us forecast rogue waves at sea or instabilities in optics, but also point toward universal organizing principles across physics and mathematics.

Dear Arturo Ortiz Tapia

Thank you very much for your insightful response. I fully agree that treating these extreme events as manifestations of deeper universal structures in nonlinear dynamics is a promising path. The analogy with random matrix theory is particularly striking. Bridging global analytic frameworks (integrable models, spectral approaches) with local nonlinear instabilities could indeed uncover unifying principles. I also wonder whether machine learning–assisted spectral analysis might help to detect precursors of such events and strengthen this global–local connection.

Best regards,

Bensafi Toufik

Yes — positive. Machine-learning applied to spectral (and spectral-like) representations has already been used to detect precursors and to predict extreme events (rogue waves, bursts in Kerr cavities, flares, etc.). Two directly relevant papers that can be mentioned:

Häfner et al., Real-world rogue wave probabilities (Scientific Reports, 2021) — uses large observational datasets and interpretable machine-learning/data-mining on wave records (including spectral/state features) to relate sea-state spectral properties to rogue-wave occurrence.

Nature

Pokhrel et al., Random Forest Classifier Based Prediction of Rogue waves on Deep Oceans (arXiv, 2020) — shows that spectral features are significant predictors and reports good classification performance using spectral inputs (random forest). This paper explicitly argues the spectral features’ usefulness for early warning.

arXiv

If you want a narrower example in optics / Kerr resonators, see Coulibaly et al. (precursor-driven ML prediction of chaotic extreme pulses) and Mohaghegh et al. (rapid phase-resolved ML prediction using spectral information).

ScienceDirect

arXiv