Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium

Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation

In this paper, using the Darboux transformation, we demonstrate the generation of first-order breather and higher-order rogue waves from a generalized nonlinear Schrödinger equation with several higher-order nonlinear effects representing femtosecond pulse propagation through nonlinear silica fiber. The same nonlinear evolution equation can also describe the soliton-type nonlinear excitations in classical Heisenberg spin chain. Such solutions have a parameter γ(1), denoting the strength of the higher-order effects. From the numerical plots of the rational solutions, the compression effects of the breather and rogue waves produced by γ(1) are discussed in detail.

Emergence of scaling associated with complex branched wave structures in optical medium

Branched wave structures, an unconventional wave propagation pattern, can arise in random media. Experimental evidence has accumulated, revealing the occurrence of these waves in systems ranging from microwave and optical systems to solid-state devices. Experiments have also established the universal feature that the wave-intensity statistics deviate from Gaussian and typically possess a long-tail distribution, implying the existence of spatially localized regions with extraordinarily high intensity concentration ("hot" spots). Despite previous efforts, the origin of branched wave pattern is currently an issue of debate. Recently, we proposed a "minimal" model of wave propagation and scattering in optical media, taking into account the essential physics for generating robust branched flows: (1) a finite-size medium for linear wave propagation and (2) random scatterers whose refractive indices deviate continuously from that of the background medium. Here we provide extensive numerical evidence and a comprehensive analytic treatment of the scaling behavior to establish that branched wave patterns can emerge as a general phenomenon in wide parameter regime in between the weak-scattering limit and Anderson localization. The basic physical mechanisms to form branched waves are breakup of waves by a single scatterer and constructive interference of broken waves from multiple scatterers. Despite simplicity of our model, analysis of the scattering field naturally yields an algebraic (power-law) statistic in the high wave-intensity distribution, indicating that our model is able to capture the generic physical origin of these special wave patterns. The insights so obtained can be used to better understand the origin of complex extreme wave patterns, whose occurrences can have significant impact on the performance of the underlying physical systems or devices.

Rogue events in the group velocity horizon

The concept of rogue waves arises from a mysterious and potentially calamitous phenomenon of oceanic surfaces. There is mounting evidence that they are actually commonplace in a variety of different physical settings. A set of defining criteria has been advanced; this set is of great generality and therefore applicable to a wide class of systems. The question arises naturally whether there are generic mechanisms responsible for extreme events in different systems. Here we argue that under suitable circumstances nonlinear interaction between weak and strong waves results in intermittent giant waves with all the signatures of rogue waves. To obtain these circumstances only a few basic conditions must be met. Then reflection of waves at the so-called group-velocity horizon occurs. The connection between rogue waves and event horizons, seemingly unrelated physical phenomena, is identified as a feature common in many different physical systems.

Observation of multifractal intermittent dust-acoustic-wave turbulence

Intermittent dust acoustic wave turbulence self-excited by downward ion flow in dissipative dusty plasma is experimentally observed and investigated. The power spectra of the temporal dust density fluctuation show distinct bumps in the low-frequency regime and power-law scaling in the high-frequency regime. The structure-function analysis demonstrates the multifractal dynamics of the wave turbulence. Decreasing dissipation by decreasing neutral pressure leads to a more turbulent state with a less distinct low-frequency bump in the power spectrum, more stretched non-Gaussian tails in the histogram of the wave-height increment at a small time interval τ, and a higher degree of multifractality. The loss of long time memory with increasing τ for a more turbulent state causes a change from the distribution with stretched non-Gaussian tails to Gaussian with increasing τ.

Matter rogue waves in an F=1 spinor Bose-Einstein condensate

We report new types of matter rogue waves of a spinor (three-component) model of the Bose-Einstein condensate governed by a system of three nonlinearly coupled Gross-Pitaevskii equations. The exact first-order rational solutions containing one free parameter are obtained by means of a Darboux transformation for the integrable system where the mean-field interaction is attractive and the spin-exchange interaction is ferromagnetic. For different choices of the parameter, there exists a variety of different shaped solutions including two peaks in bright rogue waves and four dips in dark rogue waves. Furthermore, by utilizing the relation between the three-component and the one-component versions of the nonlinear Schrödinger equation, we can devise higher-order rational solutions, in which three components have different shapes. In addition, it is noteworthy that dark rogue wave features disappear in the third-order rational solution.

Amplitude modulation of hydromagnetic waves and associated rogue waves in magnetoplasmas

It is shown that the dynamics of amplitude-modulated compressional dispersive Alfvénic (CDA) waves in a collisional megnetoplasma is governed by a complex Ginzburg-Landau (CGL) equation. The nonlinear dispersion relation for the modulational instability of the CDA waves is derived and investigated numerically. It is found that the growth rate of the modulational instability decreases (increases) with the increase of the normalized electron-ion collision frequency α (the plasma β). The modulational instability criterion for the CGL equation is defined precisely and investigated numerically. The region of the modulational instability becomes narrower with the increase of α and β, indicating that the system dissipates the wave energy by collisions, and a stable CDA wave envelope packet in the form of a hole will be a dominant localized pulse. For a collisionless plasma, i.e., α=0, the CGL equation reduces to the standard nonlinear Schrödinger (NLS) equation. The latter is used to investigate the modulational (in)stability region for the CDA waves in a collisionless magnetoplasma. It is shown that, within unstable regions, a random set of nonlinearly interacting CDA perturbations leads to the formation of CDA rogue waves. In order to demonstrate that the characteristics of the CDA rogue waves are influenced by the plasma β, the relevant numerical analysis of the appropriate nonlinear solution of the NLS equation is presented. The application of our investigation to space and laboratory magnetoplasmas is discussed.

Peregrine rogue waves induced by the interaction between a continuous wave and a soliton

Based on the soliton solution on a continuous wave background for an integrable Hirota equation, the reduction mechanism and the characteristics of the Peregrine rogue wave in the propagation of femtosecond pulses of optical fiber are discussed. The results show that there exist two processes of the formation of the Peregrine rogue wave: one is the localized process of the continuous wave background, and the other is the reduction process of the periodization of the bright soliton. The characteristics of the Peregrine rogue wave are exhibited by strong temporal and spatial localization. Also, various initial excitations of the Peregrine rogue wave are performed and the results show that the Peregrine rogue wave can be excited by a small localized (single peak) perturbation pulse of the continuous wave background, even for the nonintegrable case. The numerical simulations show that the Peregrine rogue wave is unstable. Finally, through a realistic example, the influence of the self-frequency shift to the dynamics of the Peregrine rogue wave is discussed. The results show that in the absence of the self-frequency shift, the Peregrine rogue wave can split into several subpulses; however, when the self-frequency shift is considered, the Peregrine rogue wave no longer splits and exhibits mainly a peak changing and an increasing evolution property of the field amplitude.

Dust-acoustic rogue waves in a nonextensive plasma

We present an investigation for the generation of a dust-acoustic rogue wave in a dusty plasma composed of negatively charged dust grains, as well as nonextensive electrons and ions. For this purpose, the reductive perturbation technique is used to obtain a nonlinear Schrödinger equation. The critical wave-number threshold k(c), which indicates where the modulational instability sets in, has been determined precisely for various regimes. Two different behaviors of k(c) against the nonextensive parameter q are found. For small k(c), it is found that increasing q would lead to an increase of k(c) until q approaches a certain value q(c), then further increase of q beyond q(c) decreases the value of k(c). For large k(c), the critical wave-number threshold k(c) is always increasing with q. Within the modulational instability region, a random perturbation of the amplitude grows and thus creates dust-acoustic rogue waves. In order to show that the characteristics of the rogue waves are influenced by the plasma parameters, the relevant numerical analysis of the appropriate nonlinear solution is presented. The nonlinear structure, as reported here, could be useful for controlling and maximizing highly energetic pulses in dusty plasmas.

Deterministic optical rogue waves

Experimental observations of rare giant pulses or rogue waves were done in the output intensity of an optically injected semiconductor laser. The long-tailed probability distribution function of the pulse amplitude displays clear non-Gaussian features that confirm the rogue wave character of the intensity pulsations. Simulations of a simple rate equation model show good qualitative agreement with the experiments and provide a framework for understanding the observed extreme amplitude events as the result of a deterministic nonlinear process.

Granularity and inhomogeneity are the joint generators of optical rogue waves

In the presence of many waves, giant events can occur with a probability higher than expected for random dynamics. By studying linear light propagation in a glass fiber, we show that optical rogue waves originate from two key ingredients: granularity, or a minimal size of the light speckles at the fiber exit, and inhomogeneity, that is, speckles clustering into separate domains with different average intensities. These two features characterize also rogue waves in nonlinear systems; thus, nonlinearity just plays the role of bringing forth the two ingredients of granularity and inhomogeneity.

Capillary rogue waves

We report the first observation of extreme wave events (rogue waves) in parametrically driven capillary waves. Rogue waves are observed above a certain threshold in forcing. Above this threshold, frequency spectra broaden and develop exponential tails. For the first time we present evidence of strong four-wave coupling in nonlinear waves (high tricoherence), which points to modulation instability as the main mechanism in rogue waves. The generation of rogue waves is identified as the onset of a distinct tail in the probability density function of the wave heights. Their probability is higher than expected from the measured wave background.

Non-Gaussian statistics and extreme waves in a nonlinear optical cavity

A unidirectional optical oscillator is built by using a liquid crystal light valve that couples a pump beam with the modes of a nearly spherical cavity. For sufficiently high pump intensity, the cavity field presents complex spatiotemporal dynamics, accompanied by the emission of extreme waves and large deviations from the Gaussian statistics. We identify a mechanism of spatial symmetry breaking, due to a hypercycle-type amplification through the nonlocal coupling of the cavity field.

Statistical properties of strongly nonlinear waves within a resonator

An experimental investigation of nonlinear waves is reported for a system of one-dimensional second sound waves in superfluid helium within a cylindrical resonator of high Q quality factor. The strong nonlinear dependence of the wave velocity on amplitude distorts the wave shape and leads to the formation of multiple harmonics. The restricted geometry of the resonator results in a discrete energy spectrum, where the energy is transmitted from the driving frequency to the high-frequency edge of the spectrum, where dissipation occurs-a Kolmogorov-like energy distribution. It is found that the main resonance occurs at the driving frequency, and that the next few harmonics are approximately sinusoidal, coherent with the driving force, but that higher harmonics appear to be chaotic and are no longer phase coherent with the drive. For developed turbulence, the probability density function of the high-frequency harmonics is well approximated by a Gaussian distribution. Thus, the nonlinear acoustic waves exhibit the statistical properties distinctive of weak turbulence, confirming that they can properly be treated in terms of a statistical description.