Soliton gas

Pierre Suret (Prof.), Stéphane Randoux (Prof.) et Francois Copie (lecturer)

PhD : Loic Fache, Elias Charnay, Adrien Escoubet

Postdoc : Thibault Bonnemain

In nonlinear waves systems described by integrable equations such as the Korteweg-de Vries (KdV) or the one-dimensional nonlinear Schrödinger (NLS) equations, non-interacting solitons are wave packets propagating without any change of their shape and of their velocity. Remarkably, the solitons collide elastically, i.e. they recover their shape and velocity at large propagation distance after their mutual (nonlinear) interaction. Inspired by this particle-like behavior, V. Zakharov has proposed in 1971 the concept of soliton gas defined as an infinite collection of weakly interacting solitons in the framework of KdV equation. In this theoretical construction of a diluted (rarefied) soliton gas, solitons with random amplitude and phase parameters are rarely overlapping. More recently, the concept has been extended to dense gases in which solitons strongly and continuously interact.

The notion of soliton gas is inherently associated with integrability and the tool of the inverse scattering transform (often seen as a nonlinear version of the Fourier transform). In this framework, solitons are associated with pair of discrete eigenvalues of some linear operator and can be considered as nonlinear modes of propagation.

Moreover, integrable equations exhibit an infinite number of constant of motions. As a consequence, the theory of soliton gas can be seen as the statistical thermodynamics of waves in the presence of an infinite hierarchy of temperature. This approach has recently led to the emergence of a new field of research named “Generalyzed Hydrodynamics” in which the generalized Gibbs ensemble replaces the standard Gibbs ensemble of statistical Physics.

Our group has proposed pioneering experiments in the field of soliton gas and is now internationally recognized for its original and interdisciplinary approach including theory, numerical simulations and experiments in optical fibers and hydrodynamics. We work in strong synergy with several theoreticians experts of integrability (Andrey Gelash EPFL, Lausanne Switzerland, Dmitry Agafontsev, Gennady El, Thibault Congy, Northumbria UK, Alexander Tovbis, Univ of Florida USA) and experts in the field of water waves turbulence (Eric Falcon, MSC Paris, Guillaume Ducrozet and Félicien Bonnefoy Ecole Centrale de Nantes, France).

Among our recent theoretical significant results, we have shown that the soliton gas dynamics underlies some fundamental nonlinear wave phenomena such as spontaneous modulation instability and the formation of rogue waves (A Gelash et al., PRL 123, 234102 (2019)). From the experimental point of view, we have developed several optical fiber platforms enabling the study of soliton gas dynamic and statistics, including ultrafast measurement of phase and amplitude (A Tikan et al., Nat. Photon. 12, 228-234 (2018)) or the first observation in single-shot of the spatio-temporal dynamics of the modulation instability (AE Kraych et al., Phys. Rev. Lett. 122 , 054101 (2019) and AE Kraych et al., PRL 123, 093902 (2019)).

The evolution of physical systems described at leading order by integrable nonlinear equations such as the one-dimensional nonlinear Schrodinger equation (1D-NLSE) can be investigated using the inverse scattering transform (IST) method. Although the discovery of this elaborate mathematical framework is considered one of the major achievements of mathematical physics in the 20th century, its practical applications and benefits have only been discovered relatively recently. Our research group has developed expertise in the numerical implementation of IST tools over the last five years, with the aim of providing new insights into laboratory and field experiments.

For example, nonlinear spectral analysis of Peregrine breathers observed in famous previous experiments with optical fibers and water tanks revealed that they exhibit spectral portraits associated with finite-gap NLSE solutions that are more general than thought using conventional (fitting) data analysis (Randoux et al., PRE 98, 022219 (2018)). Another example is a giant nonlinear wavepacket observed on the surface of the ocean during a storm that we analyzed using numerical IST tools. We demonstrated that this giant wave recorded by a buoy in natural environment is nonlinear with some solitonic content (Onorato et al., Sci. Rep. 11, 23606 (2021)). We have also successfully applied numerical IST tools in optics, in the context of the so-called noise-induced modulation instability (Lebel et al., Opt. Lett. 46, 298 (2021)).

In addition to the tools of direct scattering transform devoted to the analysis of experimental data, we have developed numerical tools to perform the nonlinear spectral synthesis of waves. This has required the specific development of new algorithms based on the recursive Darboux transform, which must be implemented in arbitrary numerical precision. Among the most important achievements, this approach has enabled us to carry out the nonlinear spectral synthesis of soliton gases in deep-water surface gravity waves (Suret et al., PRL 125, 264101 (2020)) and also to synthesize new nonlinear objects, such as breather gases (Roberti et al., PRE 103, 042205 (2021)).

In 2021, the group has been awarded an ANR project involving a consortium of four French experimental groups in the fields of hydrodynamics and optics united under a common approach aimed at developing unified scientific knowledge in experiments dealing with the subject of soliton gases. In preliminary hydrodynamic experiments funded in 2019 by the Labex CEMPI, we studied the topic nonlinear diffraction in a 140-m long water tank available at Ecole Centrale de Nantes.

Using our numerical tools of IST analysis, we demonstrated that the evolution of soliton bound-states is governed by an underlying dynamics that is integrable at the leading order (PRF 5, 034802 (2020)). This work has been later extended to the prediction and manipulation of hydrodynamic rogue waves via nonlinear spectral engineering (Tikan et al., PRF 7, 054401 (2022)).

The most important achievement in our water experiments is represented by the synthesis of a soliton gas in a 140-m long flume where the propagation of waves is governed by the integrable focusing 1D-NLSE. Using an approach fully based on the IST formalism, the soliton gas has been generated and detected using tools of nonlinear spectral analysis. In particular, the density of states (i.e. the probability density function of the discrete eigenvalues parameterizing the soliton gas in the framework of the IST) has been measured for the first time, thus bridging the first connection between experiments and the kinetic theory of soliton gas (Suret et al., PRL 125, 264101 (2020)).

One experimental strategy that the team has developed to investigate the nonlinear dynamics of optical waves is the technique of fiber loop recirculation. The origin of recirculating fiber loop systems traces back to the 1980s when they were first conceptualized to explore the feasibility of ultra-long-distance telecommunications based on the propagation of solitons in optical fibers.

In the recent years, we have revived these systems by exploiting recent technological advancements in fast signal generation and detection to create unique recirculating fiber loop setups. Particularly, the systems developed by the team allow (i) on-demand creation of fast, complex-shaped optical wavepackets, (ii) their propagation over very long distance, and (iii) their stroboscopic detection in single-shot after each roundtrip. For the first time, we have investigated the real-time spatiotemporal dynamics of some fundamental nonlinear processes and have reported, among others, the long-awaited observation of the dynamics of modulation instability in multiple configurations (AE Kraych et al., Phys. Rev. Lett. 122 , 054101 (2019), AE Kraych et al., PRL 123, 093902 (2019) and Copie et al., Opt. Lett. 47, 3560 (2022)).

The team’s current research revolves around the concept of soliton gas (i.e. large random ensembles of solitons) and its practical implementation, notably in the field of optics. The fiber optics systems developed by the team are exceptional testbeds for such investigations and they have already enabled unprecedented experimental studies on the interaction between solitons and soliton gases that confirm predictions of the inverse scattering transform (IST) theory [Suret et al., PRR 5, L042002 (2023), Copie et al., Opt. Commun. 545, 129647 (2023)].

In 2021, the group obtained an ANR JCJC grant (StormWave) to develop an innovative experimental platform that builds on the concept of recirculating fiber loop with advanced capabilities. The originality of the new system, with respect to existing fiber optics platforms, is that it enables precise manipulation of the light dynamics during its propagation by application of a highly-reconfigurable spatio-temporal potential. The construction of this original system has been made possible thanks to the combined skills of François Copie (MCF recruited in sept. 2018), Stéphane Randoux and Pierre Suret on the development of state-of-the-art fiber optics setups, fast electronics and complex signal processing. First results have been presented in international conferences (Copie et al., CLEO US FTu4B.6 (2023), Copie et al., EQEC/Europe EF_7.6 (2023))