# Séminaire du LPTMS : Pierre Suret (PhLAM – Lille)

## Integrable turbulence and soliton gas: experiments and theoretical approaches

### Pierre Suret (PhLAM – Lille)

Online seminar — Zoom Meeting ID: 996 1840 3246 — Passcode: Ask L. Mazza or D. Petrov —

Exactly integrable partial differential equations (PDEs) such as the Korteweg-de-Vries (KdV) or the one-dimensional nonlinear Schrödinger equation (1DNLSE) can be studied in the framework of the Inverse Scattering Transform (IST). Integrable PDEs exhibit an infinite hierarchy of invariants that prevent the development of “standard” Wave Turbulence and energy cascade. Despite the existence of the IST technique, there is no general theory describing of the propagation of random waves in integrable systems such as 1DNLSE. For this reason, Integrable Turbulence, which deals with random fields, has been recently introduced as a completely “new chapter of turbulence theory” by V.E. Zakharov, one of the creators both of the wave turbulence theory and of the IST [1].

Soliton gas (SG) is one example of integrable turbulence. The concept of SG as a large ensemble of solitons randomly distributed on an infinite line and elastically interacting with each other originates from the work of Zakharov [2], who introduced the kinetic equation for a nonequilibrium diluted gas of weakly interacting solitons of the KdV equation. Zakharov’s kinetic equation has been generalized to the case of a dense SG in Ref. [3].

Optical fibers and 1D water tanks are very favorable experimental platforms for the investigation of integrable turbulence and soliton gas (described by the focusing 1DNLSE). In this talk, I will present our recent experimental results obtained both in optical fibers and water tanks [4-6]. In the second part of my talk, by using the famous example of the modulation instability, I will show that SG is a promising model to describe the statistical properties of integrable turbulence. The spontaneous modulation instability (MI) also named “noise-induced MI” arises when a plane wave is perturbed by noise in 1DNLSE. We will show that the long-term evolution of MI can be described by a carefully designed SG [7].

[1] V. E. Zakharov, Stud. Appl. Math. 122, 219 (2009)

[2] V. E. Zakharov, Sov. Phys. JETP 33, 538 (1971)

[3] G. El, Phys. Lett. A 311, 374 (2003)

[4] A. Tikan, S. Bielawski, C. Szwaj, S. Randoux, and P. Suret, Nature Photonics 12, 228 (2018)

[5] A. E. Kraych, D. Agafontsev, S. Randoux, and P. Suret, Phys. Rev. Lett. 123, 093902 (2019).

[6] P Suret et al. Phys. Rev. Lett. 125, 264101 (2020)

[7] A Gelash, D Agafontsev, V Zakharov, G El, S Randoux, P Suret, Phys. Rev. Lett 123, 234102 (2019)

**Date/Time : ** 02/02/2021 - *11:00 - 12:00*

**Location : ** LPTMS (100% online seminar)

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