Experimental Evidence of a Hydrodynamic Soliton Gas

Abstract

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 "soliton" to characterize two pulses that "shortly after the interaction, they reappear virtually unaffected in size or shape" [1]. This property, that makes solitons fascinating objects, is a common feature of solutions of integrable equations, such as for example the celebrated Kortewegde Vries (KdV) equation that describes long waves in dispersive media, or the Nonlinear Shrödinger equation, suitable for describing cubic nonlinear, narrow-band processes. Those equations find applications in many fields of physics such as nonlinear optics, water waves, plasma waves, condensed matter, etc [2]. In analogy to a gas of interacting particles described mesoscopically by the classical Boltzmann equation, in the presence of a large number of interacting solitons, Zakharov in 1971 derived a kinetic equation for the velocity distribution function of solitons [3], see also [4][5][6]. Some of the theoretical predictions have been confirmed via numerical simulations of the KdV equation in [7]. The wave-counterpart of the particle-like interpretation of solitons is known as "integrable wave turbulence"; such a concept was introduced more recently by Zakharov [8]. The major question in this field is again the understanding of the statistical properties of an interacting ensemble of nonlinear waves, described by integrable equations, in the presence or not of randomness; the latter may arise from initial conditions which evolve under the coaction of linear and nonlinear effects, [9][10][11][12][13][14][15][16][17]. In contrast to many nonintegrable closed wave systems that reach a thermalized state characterized by the equipartition of energy among the degrees of freedom (Fourier modes) [18,19], integrable equations are characterized by an infinite number of conserved quantities and their dynamics is confined on special surfaces in the phase space. This prevents the phenomenon of classical thermalization and it opens up the fundamental quest on what is the large time state of integrable systems for a given class of initial conditions. So far the question has no answer and, apart from recent theoretical approaches [20,21], most o...

Quasi-integrable KdV models, towers of infinite number of anomalous charges and soliton collisions

“…Finally, it would be an interesting issue to analyze, in the context of the quasi-integrable KdV models, the behavior of the so-called statistical moments defined by the integrals of the type (4.1), which would be relevant to the study of certain structures in (quasi-)integrable systems, such as soliton turbulence, soliton gas dynamics and rogue waves [18,19].…”

“…We believe that those types of charges will play an important role in the study of soliton gases and formation of certain structures in (quasi-)integrable systems, such as soliton turbulence, soliton gas dynamics and rogue waves [18,19]. It has recently been achieved the experimental realization of a hydrodynamic soliton gas in a wave flume in a shallow water regime [19]. In this scenario a pure integrable dynamics and the two-soliton interaction are the basic ingredients in the formation of the soliton gas.…”

“…the socalled statistical moments of the KdV model, have been argued to play a fundamental role in the undertanding of the phenomena of soliton gas and soliton turbulence, see e.g. [18,19] and references therein. Moreover, we perform the deformation of the KdV model in the framework of the Riccati-type pseudo-potential approach.…”

Quantized Vortex Rings and Loop Solitons

“…For high T values the solitons will pass through each other with no reconnection taking place before continuing to move away from each other with their shapes unchanged. It is well known that hump solitons tend to pass through each other without changing shape [21][22][23] (although there can be a phase shift). However, the interaction between loop solitons generally leads to the solitons reconnecting with each other with the emission of one or two vortex rings, with some Kelvin wave deformations left behind on the line.…”

Solutions to the Kaup–Broer system and its (2＋1) dimensional integrable generalization via the dressing method

“…al. that a soliton gas consisting of many solitons propagating in both directions can be formed in a flume [22]. Moreover, they demonstrated that the counter propagating Kaup-Broer (Kaup-Boussinesq) 2soliton soliton collision compares favorably to laboratory data provided that we ignore non-integrable effects which are of small amplitude relative to the solitons in small amplitude regime [22].…”