View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Archive Ouverte en Sciences de l'Information et de la Communication Experimental Evidence of a Hydrodynamic Soliton Gas Ivan Redor, Eric Barthélemy, Hervé Michallet, Miguel Onorato, Nicolas Mordant To cite this version: Ivan Redor, Eric Barthélemy, Hervé Michallet, Miguel Onorato, Nicolas Mordant. Experimental Evidence of a Hydrodynamic Soliton Gas. Physical Review Letters, American Physical Society, 2019, 122 (21), 10.1103/PhysRevLett.122.214502. hal-02144580 HAL Id: hal-02144580 https://hal.archives-ouvertes.fr/hal-02144580 Submitted on 22 Aug 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Experimental evidence of a hydrodynamic soliton gas Ivan Redor,1 Eric Barth´elemy,1 Herv´eMichallet,1 Miguel Onorato,2 and Nicolas Mordant1, ∗ 1Laboratoire des Ecoulements Geophysiques et Industriels, Universite Grenoble Alpes, CNRS, Grenoble-INP, F-38000 Grenoble, France 2Dipartimento di Fisica, Universit`adi Torino and INFN, 10125 Torino, Italy We report on an experimental realization of a bi-directional soliton gas in a 34 m-long wave flume in shallow water regime. We take advantage of the fission of a sinusoidal wave to inject continuously solitons that propagate along the tank, back and forth. Despite the unavoidable damping, solitons retain adiabatically their profile, while decaying. The outcome is the formation of a stationary state characterized by a dense soliton gas whose statistical properties are well described by a pure integrable dynamics. The basic ingredient in the gas, i.e. the two-soliton interaction, is studied in details and compared favourably with the analytical solutions of the Kaup-Boussinesq integrable equation. High resolution space-time measurements of the surface elevation in the wave flume provide a unique tool for studying experimentally the whole spectrum of excitations. In 1965 Zabusky and Kruskal coined the word \soli- e.g. [7, 16, 22, 23]. Soliton gas have been extensively ob- ton" to characterize two pulses that \shortly after the served in optics [e.g. 24, 25] while experimental evidence interaction, they reappear virtually unaffected in size or in a hydrodynamic context is scarce. In [26] it has been shape" [1]. This property, that makes solitons fascinat- claimed that the low frequency component of sea surface ing objects, is a common feature of solutions of integrable elevations measured in the Currituck Sound (NC, USA) equations, such as for example the celebrated Korteweg- behave as a dense soliton gas, displaying a power law de Vries (KdV) equation that describes long waves in energy spectrum with exponent equal to -1; another ap- dispersive media, or the Nonlinear Shr¨odingerequation, proach is described in [27] where the soliton content in suitable for describing cubic nonlinear, narrow-band pro- laboratory shallow water wind waves is estimated. cesses. Those equations find applications in many fields In this Letter we describe a unique experiment that is of physics such as nonlinear optics, water waves, plasma designed to build and monitor a hydrodynamic soliton waves, condensed matter, etc [2]. In analogy to a gas gas in a laboratory. We focus on shallow water gravity of interacting particles described mesoscopically by the waves where the dynamics is described to the leading or- classical Boltzmann equation, in the presence of a large der in nonlinearity and dispersion by the KdV equation. number of interacting solitons, Zakharov in 1971 derived In planning the experiment, many issues had to be faced. a kinetic equation for the velocity distribution function The main one is that dissipation is present in any ex- of solitons [3], see also [4{6]. Some of the theoretical perimental set-up; thus, the integrable equations cannot predictions have been confirmed via numerical simula- describe experiments over long time scales. In order to tions of the KdV equation in [7]. The wave-counterpart reach a stationary regime, energy must be injected by a of the particle-like interpretation of solitons is known as forcing device. These two features, dissipation and forc- \integrable wave turbulence"; such a concept was intro- ing, are clearly at odds with the integrable turbulence duced more recently by Zakharov [8]. The major ques- framework. Therefore, is it possible to produce in the tion in this field is again the understanding of the statis- laboratory a soliton gas described in statistical terms by tical properties of an interacting ensemble of nonlinear integrable turbulence? Answering positively this ques- waves, described by integrable equations, in the presence tion in laboratory experiments would strongly support or not of randomness; the latter may arise from initial the application of the integrable turbulence framework conditions which evolve under the coaction of linear and to in-situ data, as reported in [26, 28]. nonlinear effects, [9{17]. In contrast to many noninte- Experiments are performed in a 34 m-long wave flume, grable closed wave systems that reach a thermalized state 55 cm-wide with a water depth at rest equal to h = 12 cm. characterized by the equipartition of energy among the Waves are generated by a piston-type wavemaker (see degrees of freedom (Fourier modes) [18, 19], integrable a more detailed description in [29, 30]). The free sur- equations are characterized by an infinite number of con- face vertical displacement is measured along the flume served quantities and their dynamics is confined on spe- by imaging through the lateral glass walls. We use cial surfaces in the phase space. This prevents the phe- seven synchronized monochrome cameras with resolution nomenon of classical thermalization and it opens up the 1920 × 1080 pixels; each camera records the image of the fundamental quest on what is the large time state of in- contact line of water over 2 m. Thus, the field of view tegrable systems for a given class of initial conditions. consists of the 14 m-long central part of the flume. The So far the question has no answer and, apart from re- spatial resolution is close to 1 mm/pix. A grid is used cent theoretical approaches [20, 21], most of the results to calibrate the images and correct their geometrical dis- on the KdV problem relies on numerical simulations, see tortions. The contact line is extracted by detecting the 2 strongest grayscale gradient in a vertical line of pixels. 5 (a) Subpixel resolution is obtained by polynomial interpo- 6 lation in the vicinity of the pixel of steepest gradient. 4 (s) 0 2 We estimate that the resulting error on the surface ele- t 0 vation is close to 0:1 mm. The cameras are operated at 20 frames/s. −5 η (cm) 10 12 14 16 18 20 22 The second important issue to deal with during the 2 (b) design of a soliton gas experiment is that wave flumes 1.5 are usually not long enough to observe many soliton col- (a) lisions in the KdV regime which account only for wave (cm) 1 η propagating in one direction. To cope with such limita- 0.5 tion, the end of the flume, opposite to the piston, consists 0 of a vertical wall that reflects the waves. Thus the waves 10 12 14 16 18 20 22 propagate in both directions and are also reflected on the (c) x (m) piston. We take advantage of the reflections to observe 2 the propagation over time scales larger than the dura- (cm) tion restricted to a one-way trip along the flume. An η 1 integrable system of equations that deals with bidirec- tional wave propagation in shallow water is the Kaup- 0 Boussinesq (KB) system: 10 12 14 16 18 20 22 x (m) h3 @ η + @ [(h + η)u] = − @ u t x 3 xxx FIG. 1. Two-soliton interaction from experiment: (a) head-on interaction of large amplitude solitons in a space-time repre- @tu + u@xu + g@xη = 0; (1) sentation. The wavemaker is located at x = 0 m. Small dis- where η = η(x; t) is the free surface elevation, u = u(x; t) persive waves radiated from the collision can be seen in the is the fluid velocity, g is the acceleration of gravity and background, a sign of small departure from integrability. (b) Snapshots of head-on interaction of small amplitude solitons h is the water depth. Single soliton, as well as multi- (a=h = 0:1) traveling in opposite directions. Red: 2 s before solitons solutions, are known for this system and reported the collision, blue: at the collision, green: 5 s after. Black in [31]. The multi-solitons solutions of KB include the dashed lines: solution of the KB equations (see text for de- collision of two solitons: either overtaking collisions (two tails). (c) Collision of solitons traveling in the same direction solitons of distinct amplitude propagating in the same di- (overtaking interaction), same colour code as in (b). rection) or head-on collisions. Such collisions are elastic, i.e. the amplitudes of the solitons are not altered after the interaction. The effect of the collision is to induce induced by the collision. The value a=h=0.4 is below a phase shift. For head-on collisions such a phase shift the critical threshold, (a=h)c= 0.6, for which the head- is much smaller than for overtaking solitons For the lat- on collision leads to the formation of a residual jet, see ter, the larger solitons \jump" forward, while the smaller [33].