# Simons Collaboration on Wave Turbulence Annual Meeting 2021

Date & Time

Meeting Goals:
Wave turbulence theory describes a statistical theory for describing the nonlinear interaction of waves. The theory has broad applications including describing turbulent behavior in the ocean, in magnetized fluids, condensed matter and nonlinear optics. The predictions of the theory are critical in the day-to-day forecasting of surface waves in the oceans. The Simons Collaboration on Wave Turbulence brings together diverse experts from Physics, Numerics and Mathematical Analysis in order to further develop the foundations of theory. The Annual Simons Collaboration on Wave Turbulence Conference aims at providing an update on some of the research conducted under the umbrella of the collaboration, as well as highlighting external research from related fields.

Speakers:
Eric Falcon
Zaher Hani
Jani Lukkarinen
Andrea Nahmod
Miguel Onorato
Paola Malanotte Rizzoli
Eric Vanden-Eijnden
HT Yau

Organizer:
Jalal Shatah, New York University

• Agenda

#### Thursday, December 2

 9:30 AM Paola Malanotte Rizzoli | Resilience of Venice: Past, Present and Future 11:00 AM Eric Falcon | Wave turbulence experiments on the surface of a fluid 1:00 PM Zaher Hani | The mathematical theory of wave turbulence 2:30 PM Jani Lukkarinen | Cumulant hierarchy and wave turbulence of the discrete nonlinear Schrodinger equation 4:00 PM Andrea Nahmod | Propagation of randomness, Gibbs measures and random tensors for NLS

#### Friday, December 3

 9:30 AM Miguel Onorato | Anomalous conduction in one-dimensional chains: A wave turbulence approach 11:00 AM Eric Vanden-Eijnden | Extreme events, tail statistics and large deviation theory in fluid flows 1:00 PM HT Yau | Quantum diffusion of random band matrices in high dimensions
• Abstracts

Eric Falcon
Centre national de la recherche scientifique/University of Paris

Wave turbulence experiments on the surface of a fluid

Falcon will discuss experimental advances in wave turbulence systems such as gravity waves, gravity-capillary waves or capillary waves. Except for capillary wave turbulence, which is rather well described by weak turbulence theory, he will show that dissipation, finite-system size effects and finite nonlinearity effects usually affect the experimental energy cascade towards small scales. Falcon will also present a recent experiment on the properties of large scales (i.e., larger than the forcing scale) in gravity wave turbulence showing an inverse cascade (towards large scales) that was theoretically predicted in the 1980s.

Zaher Hani
University of Michigan

The mathematical theory of wave turbulence

The kinetic theory of waves, ‘wave turbulence theory,’ has been formulated in various fields of physics to describe the statistical behavior of interacting wave systems. This started early in the past century with the pioneering works of Peierls, Hasselman, Zakharov and others, and developed into the highly successful and informative paradigm widely employed nowadays, both in physical theory and practice. However, for the longest time, the mathematical foundation of the theory has not been established, with all its derivations based on formal manipulations and unproven postulates. The central objects here are the ‘wave kinetic equation,’ which describes the effective dynamics of an interacting wave system in the thermodynamic limit, and the ‘propagation of chaos’ hypothesis, which is a fundamental postulate in the field that lacks mathematical justification.

This problem has attracted considerable interest in the mathematical community over the past decade or so. The culmination of this effort came recently in a series of joint works with Yu Deng (University of Southern California), in which Hani will provide the first rigorous derivation of the wave kinetic equation and justify the propagation of chaos hypothesis in the same setting. The proof features a nice interplay of analysis, probability theory, combinatorics, and analytic number theory.

Jani Lukkarinen
University of Helsinki

Cumulant hierarchy and wave turbulence of the discrete nonlinear Schrodinger equation

Wick polynomials of random variables provide a convenient regularization scheme of the corresponding monomials. In this talk, Lukkarinen will apply the Wick polynomial regularization to the discrete nonlinear Schrodinger evolution (DNLS) with suitable random initial data and show how they can be employed to simplify the cumulant hierarchy of the system. In particular, Lukkarinen will use the simplified hierarchy to study evolution of second moments of the field. This allows correctly identifying the kinetic collision term of the Boltzmann–Peierls equation for the DNLS and, hence, studying its wave turbulence. The talk is based on joint works with Matteo Marcozzi, Alessia Nota, Herbert Spohn and Aleksis Vuoksenmaa.

Andrea Nahmod
University of Massachusetts, Amherst

Propagation of randomness, Gibbs measures and random tensors for NLS

In groundbreaking work, Jean Bourgain put forward a random data theory to study the existence of strong solutions on the statistical ensemble of Gibbs measures associated to dispersive equations. Despite numerous follow up works to Bourgain’s, fundamental questions remain, such as: How does a given initial random data get transported by the nonlinear flow? If it is Gaussian initially, how does this Gaussianity propagate? What’s the description of the solution beyond the linear evolution?

In recent work — joint with Yu Deng and Haitian Yue — Nahmod developed the theory of random tensors a powerful, new framework that allows us to unravel the propagation of randomness under the nonlinear flow beyond the linear evolution of random data and answer these questions in an optimal range relative to what Nahmod will define as the probabilistic scaling. In particular, Nahmod will establish the invariance of Gibbs measures for 2D NLS and 3D Hartree NLS equations using the method of random averaging operators, a first order approximation to the full random tensor theory.

Miguel Onorato
University of Torino

Anomalous conduction in one-dimensional chains: A wave turbulence approach

Heat conduction in 3D macroscopic solids is in general well described by Fourier’s law. However, low-dimensional systems, for example nanotubes, may be characterized by a conductivity that is size-dependent. This phenomenon, known as anomalous conduction, has been widely studied in one dimensional chains like FPUT, mostly using deterministic simulations of the microscopic model. Here, Onorato will present a mesoscopic approach based on the wave turbulence theory and give the evidence, through extensive numerical simulations and theoretical arguments, that the anomalous conduction is the result of the presence of long waves that rapidly propagate from one thermostat to the other without interacting with other modes. Onorato will also show that the scaling of the conductivity with the length of the chain obtained from the mesoscopic approach is consistent with the one obtained from microscopic simulations.

Paola Malanotte Rizzoli
Massachusetts Institute of Technology

Resilience of Venice: Past, Present and Future

During its more than millennial history Venice has demonstrated exceptional examples of resilience , from the invasions of the barbarians in the 500s, to the wars with the major Europeans powers in the 1500s, which determined remarkable engineering feats for the preservation of the lagoon.The present challenge for its resilience are the floods which submerge the city and the lagoon islands. Their number is increasing exponentially due to the challenge of global climate change.The most important reason for these floods are the storm surges of the Adriatic sea, produced by special meteorological conditions whose frequency and strength have also increased exponentially, and are now called ” megacanes”, i.e. Mediterranean hurricanes. Simple analytical hydrodynamic models will be presented to explain and predict the storm surges. Presently very complex numerical models are used for their prediction . The storm surges are often followed by multiple seiches of the sea, resonant modes of oscillation for which analytical models can also be formulated, and numerically implemented. A byproduct of the storm surges is the very intense wind wave field, with waves up to 4 m. of height which can destroy the islands protecting the lagoon. Wind wave models are also used for the prediction of these extreme wave conditions. Finally, examples will be presented of the real hurricanes producing the floods and the extreme sea levels with multiple events submerging the city.

Eric Vanden-Eijnden
Courant Institute of Mathematical Sciences – New York University

Extreme events, tail statistics and large deviation theory in fluid flows

In recent years, several analytical and numerical methods have been introduced to characterize the pathway, rate and likelihood of rare but important events. These methods build on large deviation theory, which indicates that the way such events occur is often predictable and offers ways to compute them via solution of an optimization problem for their most likely path. In this talk, Vanden-Eijnden will discuss the applicability of these techniques to fluid flows, in particular in the context of regime transition in atmospheric flows and rogue waves in deep sea.

H.T. Yau
Harvard University

Quantum diffusion of random band matrices in high dimensions

Yau will consider Hermitian random band matrices $$H=(h_{xy})$$ on the $$d$$-dimensional lattice $$(\mathbb Z/L\mathbb Z)^d$$. The entries $$h_{xy}$$ are independent (up to Hermitian conditions) centered complex Gaussian random variables with variances $$s_{xy}=\mathbb E|h_{xy}|^2$$. The variance matrix $$S=(s_{xy})$$ has a banded structure so that $$s_{xy}$$ is negligible if $$|x-y|$$ exceeds the band width $$W$$. In dimensions $$d\ge 8$$, Yau will prove that, as long as $$W\ge L^\varepsilon$$ for a small constant $$\varepsilon>0$$, with high probability most bulk eigenvectors of $$H$$ are delocalized in the sense that their localization lengths are comparable to $$L$$. Denote by $$G(z)=(H-z)^{-1}$$ the Green’s function of $$H$$. For $$\text{Im } z\gg W^2/L^2$$, Yau will also prove a widely used criterion in physics for quantum diffusion of this model, namely, the leading term in the Fourier transform of $$\mathbb E|G_{xy}(z)|^2$$ with respect to $$x-y$$ is of the form $$(\text{Im } z + a(p))^{-1}$$ for some $$a(p)$$ quadratic in $$p$$, where $$p$$ is the Fourier variable.

• Simons Foundation Lecture: December 1, 2021

Gigliola Staffilani
Massachusetts Institute of Technology

The Study of Wave Interactions: Where Beautiful Mathematical Ideas Come Together

Phenomena involving wave interactions happen at different scales and media. For example, gravitational waves ripple through space-time, ocean waves move through water and quantum particles behave like wave packets. These phenomena are difficult to study in a rigorous mathematical manner, but maybe because of this challenge, mathematicians have developed interdisciplinary approaches that are powerful and beautiful.

In this lecture, Gigliola Staffilani will describe some of these approaches. She will show, for example, how the need to understand certain multilinear and periodic interactions provided the tools necessary to prove a famous conjecture in number theory and how classical tools in probability provided the framework for viable theories behind certain deterministic counterexamples.

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