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Non
Linear
Physics
Group -
Eric Falcon
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Solitons, Solitary waves and Nonlinear waves (papers by our group)
- Depression KdV solitary wave
160 years after the first
observation of a solitary wave hump (positive) on the surface of
water and modelled by Korteweg and De Vries (KdV) equation, we
observed experimentally for the first time a KdV solitary wave
of depression type (negative). It was nevertheless
known since 1895 that dispersion could change sign, and thus the
shape of the wave (positive or negative) if the effect of the
surface tension is important. These results were widely
commented in the scientific press. MORE and papers
Axisymmetric solitons
Solitary waves or solitons are localized nonlinear waves that propagate almost without deformation due to the balance between the
nonlinearity and the dispersion. Solitons are ubiquitous in
hydrodynamics, optics and condensed matter. However, most of
them propagates within a quasi-one-dimensional plane system.
Observation of axisymmetric solitary waves is much more
scarce. We have designed an experiment to observe axisymmetric
solitary waves on the surface of a cylindrical magnetic fluid
layer. Generally, in a usual fluid, a cylindrical fluid layer
is unstable and droplets appear (Rayleigh-Plateau
instability). By using a ferrofluid (a colloidal suspension of
magnetic nanoparticles), it is possible to stabilize this
cylindrical fluid layer. To wit, a metallic tube carrying an
electrical current creates a magnetic field which stabilizes
the cylindrical layer of ferrofluid around the tube due to the
magnetic centripetal force. Both the shape and the speed of
the solitary waves are modified by the strength of the
magnetic field. Such a system allows us to observe for the
first time the axisymmetric magnetic solitary waves predicted
theoretically in the 80s. The study of collisions between
these new type of solitary waves should be of particular
interest. MORE and papers
Rogue waves
We have achieved to generate an inverse cascade regime of surface-gravity wave turbulence in a large-scale basin (50 m x 30 m and 5 m in depth) [see PRL2020]. Such wave fields are valuable for the study of rogue waves since the random waves involved in their dynamics are generated by the nonlinear wave interactions rather than directly forced by a wave maker. We have then reported statistics of thousands of rogue waves in a large-scale basin and demonstrated that some of their properties crucially depend on four-wave resonant interactions (large crests being for instance more likely than predicted by second-order models) [see JFM2022].
Solitons and integrable turbulence
We have performed several campaigns within a 150 m long wave channel at ECN, Nantes, France, to experimentally test various aspects related to integrable turbulence, i.e., non-equilibrium, stationary theoretical states arising from the dynamics of a random initial condition within an integrable equation. We have obtained the first controlled synthesis of a dense gas of solitons in deep-water random surface gravity waves, using the tools of nonlinear spectral theory (inverse scattering transform - IST) for the one-dimensional focusing nonlinear Schrödinger equation (NLS). The soliton gas is experimentally generated in a 150-m long water tank instrumented with twenty wave elevation gauges. The obtained results represent the first crucial step for the experimental validation of the kinetic theory of soliton gases, and pave the way for further studies in nonlinear optics, superfluid or oceanography [see PRL2020b].
We also reported the first observation of the emergence of Peregrine solitons within an ensemble of 1D hydrodynamic random waves and provided their statistics [see PRF2018b] and [PRF2020b]. To do this, we sent random waves into the 150-m long water tank for five hours. We are now able to control the point of appearance of the Peregrine soliton, in space and time, by employing inverse scattering transform (IST) for the synthesis of the initial data [see PRF2022].
The nonlinear dispersion relation of unidirectional nonlinear random gravity waves was also measured experimentally and provided signatures of the deviation from integrable turbulence for the 1D NLS equation [see SciRep2022].
We also observed the theoretically predicted dispersive shock wave regime arising from the destabilization of a nonlinear wave packet [see PRF2020a].
Nonlinear waves and solitons along a torus of fluid
By means of an original technique, we achieved experimentally to form a stable torus of liquid and to report the resonance frequencies of a torus of liquid [see PRL2019]. By improving the technique, we have reported the full dispersion relation of azimuthal waves along a torus of fluid [see PRL2021]. We have also reported the observation of nonlinear three-wave resonant interactions between two different branches of the dispersion relation, namely the gravity-capillary and sloshing modes [see PRE2023].
For a stronger and impulsion forcing, we reported the observation of new Korteweg–deVries type solitons, due to the periodicity of the torus, with significant differences compared to the classical rectilinear geometry (see EPL2022). In particular, we highlight the observation of unreported subsonic elevation solitons, and a nonlinear dependence of the soliton velocity on its amplitude.
Sommerfeld precursors
One feature of the propagation
of linear waves in dispersive medium is the existence of
precursors (or forerunners). This terminology traces back to the
fact that they generally arrive earlier than the main signal.
This transient response is due to the propagation of the fastest
high frequency components of the spectrum of the initial
excitation. Although predicted since 1914 by Sommerfeld and
Brillouin, the experimental observations remain rare and
qualitative, and relate to mainly the electromagnetic waves in a
dielectric medium.
We observed
two types of Sommerfeld precursors on the surface of a layer
of mercury. They are interpreted within the framework of the
analysis first introduced by Sommerfeld and Brillouin. This
study also makes it possible to connect the precursor
concept of electromagnetic waves to the well-known transient
phenomena of hydrodynamic surface waves, and their
applications to the submarine eruptions. MORE and papers
Solitary waves in a chain
of beads
Dynamical behaviors linked to the contact between beads (Hertz
contact): Collective processes of collisions (see EPJB1998) and solitary waves propagation (see PRE1997) in granular media.
- PUBLICATIONS on solitons, solitary waves and nonlinear waves:
For wave turbulence papers (click here)
Surface waves
on a fluid:
27. G. Ricard and E. Falcon 2024
in press in Physical Review Letters (2024)
Soliton Dynamics over a Disordered Topography
26. G. Ricard, F. Novkoski, and E. Falcon 2024
Nature Communications 15, 5726 (2024)
Anderson localization of nonlinear surface gravity waves
25. L. Fache, F. Bonnefoy, G. Ducrozet, F. Copie, F. Novkoski, G. Ricard, G. Roberti, E. Falcon, P. Suret, G. El, and S. Randoux 2024 
Physical Review E 109, 03427 (2024)
Interaction of soliton gases in deep-water surface gravity waves
24. F. Novkoski, E. Falcon, and C.T. Pham 2023
The European Physical Journal Plus 138, 1146 (2023)
A numerical direct scattering method for the periodic sine-Gordon equation
23. G. Ricard and E. Falcon 2023
Physical Review E 108, 045106 (2023)
Experimental evidence of random shock-wave intermittency