**On the reflection of certain anisotropic waves and some hint
on their connection with wave turbulence**

Roberta Bianchini

*National Research Council - CNR*

Abstract: When an internal wave hits a sloping boundary, the angle between the incident wave and the vertical must be preserved after reflection. If the angle is critical, this phenomenon generates a mechanism of energy focusing along the slope, in such a way that a biharmonic wave is radiated far from the boundary when the nonlinearity of the system comes into play. After presenting the mathematical aspects of the reflection problem (in collaboration with Anne-Laure Dalibard, Thierry Paul and Laure Saint-Raymond), I will discuss some (open) mathematical problems in connection with wave turbulence.

** Large Deviation Principle for the cubic NLS equation**

Ricardo Grande Izquierdo

*ENS, Paris*

Abstract: In this talk we will explore the weakly nonlinear cubic Schrödinger equation with random initial data as a model for the formation of large waves in deep sea. First we will prove a large deviation principle for the solution of the equation, i.e. we derive the top order asymptotics for the probability of seeing a large wave at a certain time as the height of the wave tends to infinity. Then we will study a related problem: if we do see a large wave, what is the most likely initial datum that produced it? We answer this question in the weakly nonlinear regime by giving a probabilistic characterization of the set of rogue waves. This is joint work with M. Garrido, K. Kurianski and G. Staffilani.

References:

- Paper: https://arxiv.org/pdf/2110.15748.pdf
- Chapter 2 of
*Large Deviations Techniques and Applications*by Dembo & Zeitouni

**On the wave turbulence theory for a stochastic KdV type equation**

Gigliola Staffilani

*MIT*

Abstract: This talk is a summary of a recent work completed with Binh Tran. Starting from the stochastic Zakharov-Kuznetsov (ZK) equation, a multidimensional KdV type equation on a hypercubic lattice, we provide a derivation of the 3-wave kinetic equation. We show that the two point correlation function can be asymptotically expressed as the solution of the 3-wave kinetic equation at the kinetic limit under very general assumptions: the initial condition is out of equilibrium, the dimension is d>1, the smallness of the nonlinearity is allowed to be independent of the size of the lattice, the weak noise is chosen not to compete with the weak nonlinearity and not to inject energy into the equation. Unlike the cubic nonlinear Schrödinger equation, for which such a general result is commonly expected without the noise, the kinetic description of the deterministic lattice ZK equation is unlikely to happen. One of the key reasons is that the dispersion relation of the lattice ZK equation leads to a singular manifold, on which not only 3-wave interactions but also all m-wave interactions are allowed to happen. To the best of our knowledge, the work provides the first rigorous derivation of nonlinear 3-wave kinetic equations. Also this is the first derivation for wave kinetic equations in the lattice setting and out-of-equilibrium.