Chaotic Wave Field Research Articles

Integrable turbulence and statistical characteristics of chaotic wave field in the Kundu-Eckhaus equation.

Integrable turbulence, as an irregular behavior in dynamic systems, has attracted a lot of attention in integrable and Hamiltonian systems. This article focuses on the studies of integrable turbulence phenomena of the Kundu-Eckhaus (KE) equationas well as the generation of rogue waves from the numerical and statistical viewpoints. First, via the Fourier collocation method, we obtain the spectral portraits of different analytical solutions. Second, we perform the numerical simulation on the KE equationunder the initial condition of a plane wave with random noise to simulate the chaotic wave fields. Then, we analyze the influences of standard deviation and correlation length on the integrable turbulence and amplitude of wave field. It's found that both of the two parameters have positive effects on the generation probability of rogue wave caused by the interactions. But only the variation of standard deviation can lead to the transition from the breather turbulence to soliton turbulence. Furthermore, by analyzing the effects of additional higher-order nonlinear terms on the chaotic wave field, we find that those two higher-order nonlinear effects in the KE equationcan lead to a larger amplitude of the chaotic wave field and a higher probability of generating rouge waves compared with the NLS equation.

Pilot-wave dynamics: Using dynamic mode decomposition to characterize bifurcations, routes to chaos, and emergent statistics.

We develop a data-driven characterization of the pilot-wave hydrodynamic system in which a bouncing droplet self-propels along the surface of a vibrating bath. We consider drop motion in a confined one-dimensional geometry and apply the dynamic mode decomposition (DMD) in order to characterize the evolution of the wave field as the bath's vibrational acceleration is increased progressively. Dynamic mode decomposition provides a regression framework for adaptively learning a best-fit linear dynamics model over snapshots of spatiotemporal data. Thus, DMD reduces the complex nonlinear interactions between pilot waves and droplet to a low-dimensional linear superposition of DMD modes characterizing the wave field. In particular, it provides a low-dimensional characterization of the bifurcation structure of the pilot-wave physics, wherein the excitation and recruitment of additional modes in the linear superposition models the bifurcation sequence. This DMD characterization yields a fresh perspective on the bouncing-droplet problem that forges valuable new links with the mathematical machinery of quantum mechanics. Specifically, the analysis shows that as the vibrational acceleration is increased, the pilot-wave field undergoes a series of Hopf bifurcations that ultimately lead to a chaotic wave field. The established relation between the mean pilot-wave field and the droplet statistics allows us to characterize the evolution of the emergent statistics with increased vibrational forcing from the evolution of the pilot-wave field. We thus develop a numerical framework with the same basic structure as quantum mechanics, specifically a wave theory that predicts particle statistics.

Diffuse field cross-correlation in a programmable-metasurface-stirred reverberation chamber

Programmable metasurfaces can endow complex scattering environments with reconfigurability. Here, we make use of these configurational degrees of freedom to retrieve the impulse response between two passive antennas via cross-correlation techniques. An ensemble of stirred chaotic wave fields in a reverberation chamber (RC) can play the role of thermal noise in passive Green's function retrieval. Instead of using a conventional mechanical mode stirrer, we generate the ensemble of RC configurations with random configurations of a programmable metasurface. We adapt the data processing of the diffuse field cross-correlation technique to this stirring mechanism, which is, given the size of the RC, nominally inefficient, and we investigate the convergence of the cross-correlated signals toward the impulse response. Finally, we apply our scheme to phaseless imaging in complex propagation environments, with potential applications in indoor context-awareness.

Robust position sensing with wave fingerprints in dynamic complex propagation environments

Irregular propagation environments with complex scattering effects challenge traditional ray-tracing-based localization. However, the environment's complexity enables solutions based on wave fingerprints (WFPs). Yet, since WFPs rely on the extreme sensitivity of the chaotic wave field to geometrical details, it is not clear how viable WFP techniques may be in a realistic dynamically evolving environment. Here, we reveal that environmental perturbations reduce both the diversity of the WFP dictionary and the effective signal-to-noise ratio (SNR), such that the amount of information that can be obtained per measurement is reduced. This unfavorable effect can, however, be fully compensated by taking more measurements. We show in simulations and experiments with a low-cost software-defined radio that WFP localization of non-cooperative objects is possible even when the scattering strength of the environmental perturbation significantly exceeds that of the object to be localized. Our results underline that diversity is only one important ingredient to achieve high sensing accuracy in compressed sensing, the other two being SNR and the choice of decoding method. We find that sacrificing diversity for SNR may be worthwhile and observe that artificial neural networks outperform traditional decoding methods in terms of the achieved sensing accuracy, especially at low SNR.

Self-localized solitons of a q-deformed quantum system

Beyond a pure mathematical interest, q-deformation is promising for the modeling and interpretation of various physical phenomena. In this paper, we numerically investigate the existence and properties of the self-localized soliton solutions of the nonlinear Schrödinger equation (NLSE) with a q-deformed Rosen–Morse potential. By implementing a Petviashvili method (PM), we obtain the self-localized one and two soliton solutions of the NLSE with a q-deformed Rosen–Morse potential. In order to investigate the temporal behavior and stabilities of these solitons, we implement a Fourier spectral method with a 4th order Runge–Kutta time integrator. We observe that the self-localized one and two solitons are stable and remain bounded with a pulsating behavior and minor changes in the sidelobes of the soliton waveform. Additionally, we investigate the stability and robustness of these solitons under noisy perturbations. A sinusoidal monochromatic wave field modeled within the frame of the NLSE with a q-deformed Rosen–Morse potential turns into a chaotic wavefield and exhibits rogue oscillations due to modulation instability triggered by noise, however, the self-localized solitons of the NLSE with a q-deformed Rosen–Morse potential are stable and robust under the effect of noise. We also show that soliton profiles can be reconstructed after a denoising process performed using a Savitzky–Golay filter.

Chaotic breathers and breather fission/fusion for a vector nonlinear Schrödinger equation in a birefringent optical fiber or wavelength division multiplexed system

Investigation is made on a vector nonlinear Schrödinger equation in the anomalous dispersion regime describing the optical pulses in a birefringent optical fiber or in a wavelength division multiplexed system. We derive the analytical breather solutions. Fission and fusion of the breathers are investigated via the relation between the phase velocity and group velocity of the plane wave. Stability of the numerical breather is studied via the pseudospectral method: The breather with the white noise propagates stably. Breathers in the chaotic wave fields are derived via the split-step Fourier method: The breathers in the positive x axis are observed more distinctly than in the negative, where x is the spatial coordinate. Complex eigenvalue is gotten to study those phenomena. Effects of the modulation instability on the breathers in the chaotic wave fields are investigated.

Magnetic breathers and chaotic wave fields for a higher-order [formula omitted]-dimensional nonlinear Schrödinger-type equation in a Heisenberg ferromagnetic spin chain

Ferromagnetism, whose physical properties include the Giant-Magneto resistance, spin-transfer torque, spin-Hall effect and so on, attracts researchers’ interest in the search for the spin polarizers in spin electronics. In this paper, we investigate a higher-order (2+1)-dimensional nonlinear Schrödinger-type equation for a Heisenberg ferromagnetic spin chain with the bilinear and biquadratic interaction. Magnetic breathers are derived. Conditions of the transformation from the magnetic breather to the magnetic soliton are obtained. We get the magnetic chaotic wave fields, in which the magnetic breather-like solitons and rogue waves appear. Effects of γ on the magnetic chaotic wave fields are studied: As γ increases, mean amplitude of the magnetic waves in the chaotic wave field increases, and amplitude dispersion degree decreases, where γ represents the lattice parameter.

Inelastic interactions, numerical breathers and chaotic wave fields for a focusing Kundu–Eckhaus equation in a nonlinear optical fiber

Investigation is made on a focusing Kundu–Eckhaus equation in nonlinear/quantum optics and fluid mechanics, for describing the optical properties of the femtosecond lasers and femtochemistry objects, and for examining the stability of the Stokes waves in weakly nonlinear dispersive media. We derive the one and two optical breather solutions via the Darboux transformation. Inelastic interactions between two single-hump optical breathers and between a single-hump optical breather and a double-hump optical breather are depicted. Numerical optical breathers are obtained via the split-step Fourier method: β affects the speeds of the numerical optical breathers and space energy distribution, while β has no effects on the total energy, where β is a real constant representing the quintic nonlinearity. Optical chaotic wave fields via three initial conditions, including the plane wave, one soliton and one breather, are derived. Among them, the optical chaotic wave field via the one breather is the most orderless via the probability density function. Total energy of the optical chaotic wave field via the one breather is the largest.

Optical breathers and multiple rogue waves via the modulation instability for a complex Ginzburg–Landau equation in the mode-locked laser operation

Investigation is made on a complex Ginzburg–Landau equation for the mode-locked laser operation. Optical breathers are numerically derived via the split-step Fourier method. Effects of the modulation instability on the dynamic behaviors of the optical breathers are investigated: when the growth rate of the modulation instability increases, the optical breather splits into two optical breathers earlier, and the two optical breathers separate faster. Via the energy distributions of the optical breathers, we obtain that when the fission phenomenon occurs, the energy decreases more than that of the fusion phenomenon. Multiple rogue waves in the chaotic wave field are obtained via the modulation instability. It is observed that multiple rogue waves occur earlier as the growth rate of the modulation instability becomes larger. Occurrence of the rogue wave can be predicted via the spectrum of the optical chaotic wave field.

Optical breathers and rogue waves via the modulation instability for a higher-order generalized nonlinear Schrödinger equation in an optical fiber transmission system

For describing the propagation of ultrashort pulses in a high-speed, long-distance optical fiber transmission system with the fourth-order dispersion, cubic–quintic nonlinearity, self-steepening and self-frequency shift, a higher-order generalized nonlinear Schrodinger equation is investigated. We get the rogue-wave solutions. Effects of the modulation instability on the optical rogue waves are studied: Increasing the growth rate of the modulation instability makes the existence time of the optical rogue wave shorter. We numerically derive the optical breathers in the chaotic wave fields via the modulation instability. Spectrum of the optical chaotic wave field can be used to indicate the appearance of the optical breather in the chaotic wave field. Optical rogue waves in the chaotic wave fields are also gotten via the modulation instability.

Breather turbulence versus soliton turbulence: Rogue waves, probability density functions, and spectral features

Turbulence in integrable systems exhibits a noticeable scientific advantage: it can be expressed in terms of the nonlinear modes of these systems. Whether the majority of the excitations in the system are breathers or solitons defines the properties of the turbulent state. In the two extreme cases we can call such states "breather turbulence" or "soliton turbulence." The number of rogue waves, the probability density functions of the chaotic wave fields, and their physical spectra are all specific for each of these two situations. Understanding these extreme cases also helps in studies of mixed turbulent states when the wave field contains both solitons and breathers, thus revealing intermediate characteristics.

Rogue wave spectra of the Kundu-Eckhaus equation

In this paper we analyze the rogue wave spectra of the Kundu-Eckhaus equation (KEE). We compare our findings with their nonlinear Schrödinger equation (NLSE) analogs and show that the spectra of the individual rogue waves significantly differ from their NLSE analogs. A remarkable difference is the one-sided development of the triangular spectrum before the rogue wave becomes evident in time. Also we show that increasing the skewness of the rogue wave results in increased asymmetry in the triangular Fourier spectra. Additionally, the triangular spectra of the rogue waves of the KEE begin to develop at earlier stages of their development compared to their NLSE analogs, especially for larger skew angles. This feature may be used to enhance the early warning times of the rogue waves. However, we show that in a chaotic wave field with many spectral components the triangular spectra remain as the main attribute as a universal feature of the typical wave fields produced through modulation instability and characteristic features of the KEE's analytical rogue wave spectra may be suppressed in a realistic chaotic wave field.

Rogue waves of the Kundu-Eckhaus equation in a chaotic wave field.

In this paper we study the properties of the chaotic wave fields generated in the frame of the Kundu-Eckhaus equation (KEE). Modulation instability results in a chaotic wave field which exhibits small-scale filaments with a free propagation constant, k. The average velocity of the filaments is approximately given by the average group velocity calculated from the dispersion relation for the plane-wave solution; however, direction of propagation is controlled by the β parameter, the constant in front of the Raman-effect term. We have also calculated the probabilities of the rogue wave occurrence for various values of propagation constant k and showed that the probability of rogue wave occurrence depends on k. Additionally, we have showed that the probability of rogue wave occurrence significantly depends on the quintic and the Raman-effect nonlinear terms of the KEE. Statistical comparisons between the KEE and the cubic nonlinear Schrödinger equation have also been presented.

Early detection of rogue waves by the wavelet transforms

We discuss the possible advantages of using the wavelet transform over the Fourier transform for the early detection of rogue waves. We show that the triangular wavelet spectra of the rogue waves can be detected at early stages of the development of rogue waves in a chaotic wave field. Compared to the Fourier spectra, the wavelet spectra are capable of detecting not only the emergence of a rogue wave but also its possible spatial (or temporal) location. Due to this fact, wavelet transform is also capable of predicting the characteristic distances between successive rogue waves. Therefore multiple simultaneous breaking of the successive rogue waves on ships or on the offshore structures can be predicted and avoided by smart designs and operations.

Rogue wave spectra of the Sasa–Satsuma equation

We analyze the rogue wave spectra of the Sasa–Satsuma equation and their appearance in the spectra of chaotic wave fields produced through modulation instability. Chaotic wave fields occasionally produce high peaks that result in a wide triangular spectrum, which could be used for rogue wave detection.

Rogue waves of the Sasa-Satsuma equation in a chaotic wave field.

We study the properties of the chaotic wave fields generated in the frame of the Sasa-Satsuma equation (SSE). Modulation instability results in a chaotic pattern of small-scale filaments with a free parameter-the propagation constant k. The average velocity of the filaments is approximately given by the group velocity calculated from the dispersion relation for the plane-wave solution. Remarkably, our results reveal the reason for the skewed profile of the exact SSE rogue-wave solutions, which was one of their distinctive unexplained features. We have also calculated the probability density functions for various values of the propagation constant k, showing that probability of appearance of rogue waves depends on k.

Experiments on wind-perturbed rogue wave hydrodynamics using the Peregrine breather model

Being considered as a prototype for description of oceanic rogue waves, the Peregrine breather solution of the nonlinear Schrödinger equation has been recently observed and intensely investigated experimentally in particular within the context of water waves. Here, we report the experimental results showing the evolution of the Peregrine solution in the presence of wind forcing in the direction of wave propagation. The results show the persistence of the breather evolution dynamics even in the presence of strong wind and chaotic wave field generated by it. Furthermore, we have shown that characteristic spectrum of the Peregrine breather persists even at the highest values of the generated wind velocities thus making it a viable characteristic for prediction of rogue waves.

RADIATIVE HEATING OF THE SOLAR CORONA

We investigate the effect of solar visible and infrared radiation on electrons in the Sun's atmosphere using a Monte Carlo simulation of the wave-particle interaction and conclude that sunlight provides at least 40% and possibly all of the power required to heat the corona, with the exception of dense magnetic flux loops. The simulation uses a radiation waveform comprising 100 frequency components spanning the solar blackbody spectrum. Coronal electrons are heated in a stochastic manner by low solar electromagnetic radiation. The wave coherence time and coherence for each component is determined from optical theory. The low of solar radiation allows moving electrons to gain energy from the chaotic wave field which imparts multiple random velocity kicks to these particles causing their velocity distribution to broaden or heat. Monte Carlo simulations of broadband solar radiative heating on ensembles of 1000 electrons show heating at per particle levels of 4.0 × 10–21 to 4.0 × 10–20 W, as compared with non-loop radiative loss rates of ≈1 × 10–20 W per electron. Since radiative losses comprise nearly all of the power losses in the corona, sunlight alone can explain the elevated temperatures in this region. The volume electron heating rate is proportional to density, and protons are assumed to be heated either by plasma waves or through collisions with electrons.

Early detection of rogue waves in a chaotic wave field

The ability to unveil growing rogue waves in the ocean is essential for safe marine travel in stormy conditions. This vital problem has not been adequately addressed so far. We show that the specific triangular spectra of rogue waves can be detected at early stages of their development in a chaotic wave field. Continuously measuring the spectra of various parts of the wave field allows us to find a rogue wave before the dangerous peak appears. This possibility of early detection is a necessary part of a rogue wave early-warning system.

Ray chaos in optical cavities based upon standard laser mirrors

We present a composite optical cavity made of standard laser mirrors; the cavity consists of a suitable combination of stable and unstable cavities. In spite of its very open nature the composite cavity shows ray chaos, which may be either soft or hard, depending on the cavity configuration. This opens a convenient route for experimental studies of the quantum aspects of a chaotic wave field.