96 research outputs found
From modulational instability to focusing dam breaks in water waves
We report water wave experiments performed in a long tank where we consider the evolution of nonlinear deep-water surface gravity waves with the envelope in the form of a large-scale rectangular barrier. Our experiments reveal that, for a range of initial parameters, the nonlinear wave packet is not disintegrated by the Benjamin-Feir instability but exhibits a specific, strongly nonlinear modulation, which propagates from the edges of the wavepacket towards the center with finite speed. Using numerical tools of nonlinear spectral analysis of experimental data we identify the observed envelope wave structures with focusing dispersive dam break flows, a peculiar type of dispersive shock waves recently described in the framework of the semi-classical limit of the 1D focusing nonlinear Schr"odinger equation (1D-NLSE). Our experimental results are shown to be in a good quantitative agreement with the predictions of the semi-classical 1D-NLSE theory. This is the first observation of the persisting dispersive shock wave dynamics in a modulationally unstable water wave system
Observation of Fermi-Pasta-Ulam-Tsingou Recurrence and Its Exact Dynamics
One of the most controversial phenomena in nonlinear dynamics is the reappearance of initial
conditions. Celebrated as the Fermi-Pasta-Ulam-Tsingou problem, the attempt to understand how these
recurrences form during the complex evolution that leads to equilibrium has deeply influenced the entire
development of nonlinear science. The enigma is rendered even more intriguing by the fact that integrable
models predict recurrence as exact solutions, but the difficulties involved in upholding integrability for a
sufficiently long dynamic has not allowed a quantitative experimental validation. In natural processes,
coupling with the environment rapidly leads to thermalization, and finding nonlinear multimodal systems
presenting multiple returns is a long-standing open challenge. Here, we report the observation of more than
three Fermi-Pasta-Ulam-Tsingou recurrences for nonlinear optical spatial waves and demonstrate the
control of the recurrent behavior through the phase and amplitude of the initial field. The recurrence period
and phase shift are found to be in remarkable agreement with the exact recurrent solution of the nonlinear
Schrödinger equation, while the recurrent behavior disappears as integrability is lost. These results identify
the origin of the recurrence in the integrability of the underlying dynamics and allow us to achieve one of
the basic aspirations of nonlinear dynamics: the reconstruction, after several return cycles, of the exact
initial condition of the system, ultimately proving that the complex evolution can be accurately predicted in
experimental conditions
The exact rogue wave recurrence in the NLS periodic setting via matched asymptotic expansions, for 1 and 2 unstable modes
The focusing Nonlinear Schr\"odinger (NLS) equation is the simplest universal
model describing the modulation instability (MI) of quasi monochromatic waves
in weakly nonlinear media, the main physical mechanism for the generation of
rogue (anomalous) waves (RWs) in Nature. In this paper we investigate the
-periodic Cauchy problem for NLS for a generic periodic initial perturbation
of the unstable constant background solution, in the case of unstable
modes. We use matched asymptotic expansion techniques to show that the solution
of this problem describes an exact deterministic alternate recurrence of linear
and nonlinear stages of MI, and that the nonlinear RW stages are described by
the N-breather solution of Akhmediev type, whose parameters, different at each
RW appearence, are always given in terms of the initial data through elementary
functions. This paper is motivated by a preceeding work of the authors in which
a different approach, the finite gap method, was used to investigate periodic
Cauchy problems giving rise to RW recurrence.Comment: 20 pages. arXiv admin note: text overlap with arXiv:1708.00762 and
substantial text overlap with arXiv:1707.0565
Numerical instability of the Akhmediev breather and a finite-gap model of it
In this paper we study the numerical instabilities of the NLS Akhmediev
breather, the simplest space periodic, one-mode perturbation of the unstable
background, limiting our considerations to the simplest case of one unstable
mode. In agreement with recent theoretical findings of the authors, in the
situation in which the round-off errors are negligible with respect to the
perturbations due to the discrete scheme used in the numerical experiments, the
split-step Fourier method (SSFM), the numerical output is well-described by a
suitable genus 2 finite-gap solution of NLS. This solution can be written in
terms of different elementary functions in different time regions and,
ultimately, it shows an exact recurrence of rogue waves described, at each
appearance, by the Akhmediev breather. We discover a remarkable empirical
formula connecting the recurrence time with the number of time steps used in
the SSFM and, via our recent theoretical findings, we establish that the SSFM
opens up a vertical unstable gap whose length can be computed with high
accuracy, and is proportional to the inverse of the square of the number of
time steps used in the SSFM. This neat picture essentially changes when the
round-off error is sufficiently large. Indeed experiments in standard double
precision show serious instabilities in both the periods and phases of the
recurrence. In contrast with it, as predicted by the theory, replacing the
exact Akhmediev Cauchy datum by its first harmonic approximation, we only
slightly modify the numerical output. Let us also remark, that the first rogue
wave appearance is completely stable in all experiments and is in perfect
agreement with the Akhmediev formula and with the theoretical prediction in
terms of the Cauchy data.Comment: 27 pages, 8 figures, Formula (30) at page 11 was corrected, arXiv
admin note: text overlap with arXiv:1707.0565
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