Quasi-periodic Wave Research Articles

Chaos analysis and traveling wave solutions for fractional (3+1)-dimensional Wazwaz Kaur Boussinesq equation with beta derivative

Wazwaz Kaur Boussinesq (WKB) equation can effectively simulate the behavior of water waves in shallow water, including the nonlinear effect and dispersion phenomenon of waves, which is of great significance for understanding the dynamic process of ocean, river and other water bodies. To enrich the wave equation theory, the (3+1)-dimensional integer order derivative of WKB equation is changed to the fractional one with beta derivative. The current work deals with the fractional (3+1)-dimensional WKB equation for discussing its chaotic behavior and establishing some new analytic solutions. The chaotic properties of the equation are verified by the trend of evolution along with time, Lyapunov exponents and initial sensitivity analysis. And then complete discrimination system for polynomial method is applied to derive some trigonometric, hyperbolic, Jacobi elliptic and other solutions. The graphical demonstrations are provided for part of these solutions. From these visualized graphs, the solitary, periodic and quasi-periodic wave are shown and the effect of fractional derivatives on the equation can be seen intuitively.

Conversion mechanisms and transformed waves for the (3 + 1)-dimensional nonlinear equation

In this paper, we focus on investigating the (3 + 1)-dimensional nonlinear equation which is used to describe the propagation of waves in the shallow water. The study begins with the application of the Hirota bilinear method to obtain N-soliton solution. Building on this foundation, the research delves into the construction of first-order breather wave by imposing complex conjugate constraints on the parameters of two solitons. Further analysis of the characteristic lines of breathers leads to the derivation of conversion conditions. Under this specific condition, a series of nonlinear transformed waves are presented, including quasi-kink solitons, W-shaped kink solitons, oscillation W-shaped kink solitons, multipeaks solitons, quasi-periodic waves, and line rogue waves. Each of these transformed waves exhibits unique structural and dynamic properties, enriching the understanding of wave behavior in higher-dimensional nonlinear systems. The study also explores the nonlinear superposition mechanism between solitary wave and periodic wave. This mechanism elucidates the formation process of nonlinear waves, explaining how their locality and oscillatory characteristics emerge from the superposition of different wave components. Moreover, the geometric properties of the two characteristic lines of the waves are analyzed to understand the time-varying nature of the transformed waves. This temporal analysis is crucial for predicting the evolution and interaction of these waves over time. Finally, the research extends to the higher-order breather wave and explores the interactions among various waves. These interactions reveal the complex dynamics that may arise in the (3 + 1)-dimensional nonlinear systems and provide deeper insights into the interactions among different wave structures.

Investigating high-frequency quasi-periodic oscillations in the 6374 Å red coronal line observed during the total solar eclipse on August 21, 2017

Context. We present the results of the search for high-frequency quasi-periodic oscillations (HFQPOs) in the 6374-Å red line emission from the solar corona during the 2017 total solar eclipse (TSE). Aims. The existence of HFQPOs of coronal intensity remains unclear. Motivated by previous observations, this paper focuses on searching for the HFQPOs (> 0.1 Hz) during the 2017 TSE. Methods. Wavelet analysis tools were used to analyze the radiation intensity oscillations in different height layers, local regions of interest (such as the east coronal loop, west cavity, etc.), and coronal feature points as well as changes in oscillations along coronal feature structures and the kinematic wobble of the coronal magnetic features. Results. We have observed oscillatory signals at multiple spatial locations, which reveal the existence of quasi-periodic slow magnetoacoustic waves. These signals display periods ranging from approximately 3 to 9 seconds, with a notable focus on periods around 4 seconds. Conclusions. High-frequency quasi-periodic slow magnetoacoustic oscillations (with periods less than 9.63 seconds) have been identified in the inner corona (at heights less than 1.4 R⊙) in the 6374 Å red coronal line radiation.

On the Origin of a Broad Quasiperiodic Fast-propagating Wave Train: Unwinding Jet as the Driver

Large-scale extreme-ultraviolet waves commonly exhibit as single wave front and are believed to be caused by coronal mass ejections. Utilizing high spatiotemporal resolution imaging observations from the Solar Dynamics Observatory, we present two sequentially generated wave trains originating from the same active region: a narrow quasiperiodic fast-propagating (QFP) wave train that propagates along the coronal loop system above the jet and a broad QFP wave train that travels along the solar surface beneath the jet. The measurements indicate that the narrow QFP wave train and the accompanying flare’s quasiperiodic pulsations (QPPs) have nearly identical onsets and periods. This result suggests that the accompanying flare process excites the observed narrow QFP wave train. However, the broad QFP wave train starts approximately 2 minutes before the QPPs of the flare, but it is consistent with the interaction between the unwinding jet and the solar surface. Moreover, we find that the period of the broad QFP wave train, approximately 130 s, closely matches that of the unwinding jet. This period is significantly longer than the 30 s period of the accompanying flare’s QPPs. Based on these findings, we propose that the intermittent energy release of the accompanying flare excited the narrow QFP wave train confined propagating in the coronal loop system. The unwinding jet, rather than the intermittent energy release in the accompanying flare, triggered the broad QFP wave train propagating along the solar surface.

[formula omitted]-periodic wave solutions of the [formula omitted] supersymmetric KdV equation

In this paper, the N-periodic wave solutions of the N=2 supersymmetric KdV equation are studied by combining the super Hirota bilinear form with the super Riemann-theta function, which can be used to describe new phenomena on super quasi-periodic waves with the fermionic field. With the aid of the Gauss–Newton method, the three-periodic and four-periodic wave solutions are obtained. In particular, these quasi-periodic waves can produce parallel, crossed and degenerated patterns. The analytical method related to the characteristic lines is used to analyze the dynamic characteristics of the three-periodic and four-periodic waves. In addition, it has been indicated that N-periodic waves can exist in the supersymmetric integrable systems.

Reducibility of Linear Quasi-periodic Hamiltonian Derivative Wave Equations and Half-Wave Equations Under the Brjuno Conditions

Reducibility of Linear Quasi-periodic Hamiltonian Derivative Wave Equations and Half-Wave Equations Under the Brjuno Conditions

Automatic Detection of Quasi-Periodic Emissions from Satellite Observations by Using DETR Method

The ionospheric quasi-periodic wave is a type of typical and common electromagnetic wave phenomenon occurring in extremely low-frequency (ELF) and very low-frequency ranges (VLF). These emissions propagate in a distinct whistler-wave mode, with varying periodic modulations of the wave intensity over time scales from several seconds to a few minutes. We developed an automatic detection model for the QP waves in the ELF band recorded by the China Seismo-Electromagnetic Satellite. Based on the 827 QP wave events, which were collected through visual screening from the electromagnetic field observations, an automatic detection model based on the Transformer architecture was built. This model, comprising 34.27 million parameters, was trained and evaluated. It achieved mean average precision of 92.3% on the validation dataset, operating at a frame rate of 39.3 frames per second. Notably, after incorporating the proton cyclotron frequency constraint, the model displayed promising performance. Its lightweight design facilitates easy deployment on satellite equipment, significantly enhancing the feasibility of on-board detection.

Nonlinear waves and transitions mechanisms for (2+1)-dimensional Korteweg–de Vries-Sawada-Kotera-Ramani equation

In this paper, state transition waves are investigated in a (2+1)-dimensional Korteweg–de Vries-Sawada-Kotera-Ramani equation by analyzing characteristic lines. Firstly, the N-soliton solutions are given by using the Hirota bilinear method. The breather and lump waves are constructed by applying complex conjugation limits and the long-wave limit method to the parameters. In addition, the transition condition of breather and lump wave are obtained by using characteristic line analysis. The state transition waves consist of quasi-anti-dark soliton, M-shaped soliton, oscillation M-shaped soliton, multi-peak soliton, W-shaped soliton, and quasi-periodic wave soliton. Through analysis, when solitary wave and periodic wave components undergo nonlinear superposition, it leads to the formation of breather waves and transformed wave structures. It can be used to explain the deformable collisions of transformation waves after collision. Furthermore, the time-varying property of transformed waves are studied using characteristic line analysis. Based on the high-order breather solutions, the interactions involving breathers, state transition waves, and solitons are exhibited. Finally, the dynamics of these hybrid solutions are analyzed through symbolic computations and graphical representations.

Broad and Bidirectional Narrow Quasiperiodic Fast-propagating Wave Trains Associated with a Filament-driven Halo Coronal Mass Ejection on 2023 April 21

This paper presents three distinct wave trains that occurred on 2023 April 21: a broad quasiperiodic fast-propagating (QFP) wave train and bidirectional narrow QFP wave trains. The broad QFP wave train expands outward in a circular wave front, while bidirectional narrow QFP wave trains propagate in the northward and southward directions, respectively. The concurrent presence of the wave trains offers a remarkable opportunity to investigate their respective triggering mechanisms. Measurement shows that the speed of the broad QFP wave train is in the range of 300–1100 km s−1 in different propagating directions. There is a significant difference in the speed of the bidirectional narrow QFP wave trains: the southward propagation achieves 1400 km s−1, while the northward propagation only reaches about 550 km s−1 accompanied by a deceleration of about 1–2 km s−2. Using the wavelet analysis, we find that the periodicity of the propagating wave trains in the southward and northward directions closely matches the quasiperiodic pulsations exhibited by the flares. Based on these results, the narrow QFP wave trains were most likely excited by the intermittent energy release in the accompanying flare. In contrast, the broad QFP wave train had a tight relationship with the erupting filament, probably attributed to the unwinding motion of the erupting filament, or the leakage of the fast sausage wave train inside the filament body.

Bifurcation analysis and new waveforms to the fractional KFG equation

This study presents novel waveforms and bifurcation analysis for the fractional Klein–Fock–Gordon (KFG) structure, which is widely used in particle and condensed matter physics. To examine the bifurcation, chaos, and sensitivity of the model, we derive the dynamical system from the Galilean transformation. We present and explore a diverse range of waveforms, including periodic waves, quasi-periodic waves, bright and dark solitons, kink waves, and anti-kink waves. Graphic diagrams from simulations illustrate the diverse features and existence of these solutions. Nonlinear models can benefit from the potency, brevity, and effectiveness of integration procedures used in contemporary scientific and engineering contexts.

Wronskian solutions, bilinear Bäcklund transformation, quasi-periodic waves and asymptotic behaviors for a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation

In this paper, we investigate the integrability of a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation, which is widely used in fluid mechanics and theoretical physics. The N-soliton solution is obtained via the Hirota’s bilinear method. The Wronskian solution is derived by using the Wronskian technique for the bilinear form. Through the exchange formula, we deduce the bilinear Bäcklund transformation consisting of four equations and six parameters. In order to consider the quasi-periodic wave having complex structure, one-, two- and three-periodic waves are investigated systemically by combining the Hirota’s bilinear method with Riemann theta function. Furthermore, the corresponding graphs of periodic wave are presented by considering the geometric properties between the characteristic lines. The propagation characteristics of periodic waves are investigated by virtue of the characteristic lines. Finally, the asymptotic relationships between quasi-periodic wave solutions and soliton solutions are established theoretically under a condition of the small amplitude limit. The analytical method used in this paper can be applied in other integrable systems.

Solitons, nonlinear wave transitions and characteristics of quasi-periodic waves for a (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation in fluid mechanics and plasma physics

A (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation describing many nonlinear phenomena in fluid dynamics and plasma physics is considered. The N-solitons and breathers are obtained by basing on its Hirota’s bilinear form and taking the complex conjugate condition on parameters of N-solitons. What is more, breathers can be transformed into a series of nonlinear localized waves by the mechanism of breather transformation. Then through the multi-dimensional Riemann-theta function and the bilinear method, the high-dimensional complex three-periodic wave solutions are constructed systematically, which are the generalization of one-periodic wave and two-periodic wave solutions. By a limiting procedure, the asymptotic relations between the quasi-periodic waves and solitons are strictly established. Additionally, a novel analytical method of characteristic line is introduced to analyze statistically the dynamical characteristics of the quasi-periodic waves. The analytical method employed in this paper can be further extended to investigate the other complex high-dimensional nonlinear integrable equations.

Consecutive narrow and broad quasi-periodic fast-propagating wave trains associated with a flare

Consecutive narrow and broad quasi-periodic fast-propagating wave trains associated with a flare

Effect of Ion Temperature Anisotropy on Damped and Undamped Ion-Acoustic KdV Solitons in Plasmas and Quasi-periodic Wave Structures with Kappa-Distributed Electrons

Effect of Ion Temperature Anisotropy on Damped and Undamped Ion-Acoustic KdV Solitons in Plasmas and Quasi-periodic Wave Structures with Kappa-Distributed Electrons

Quasi-Periodic Traveling Waves on an Infinitely Deep Perfect Fluid Under Gravity

We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and we establish the existence and the linear stability of small amplitude, quasi-periodic in time, traveling waves. This provides the first existence result of quasi-periodic water waves solutions bifurcating from a completely resonant elliptic fixed point. The proof is based on a Nash–Moser scheme, Birkhoff normal form methods and pseudo differential calculus techniques. We deal with the combined problems of small divisors and the fully-nonlinear nature of the equations. The lack of parameters, like the capillarity or the depth of the ocean, demands a refined nonlinear bifurcation analysis involving several nontrivial resonant wave interactions, as the well-known “Benjamin-Feir resonances”. We develop a novel normal form approach to deal with that. Moreover, by making full use of the Hamiltonian structure, we are able to provide the existence of a wide class of solutions which are free from restrictions of parity in the time and space variables.

Quasi-periodic waves to the defocusing nonlinear Schrödinger equation

A direct approach for the quasi-periodic wave solutions to the defocusing nonlinear Schrödinger equation is proposed based on the theta functions and Hirota’s bilinear method. We transform the problem into a system of algebraic equations, which can be formulated into a least squares problem and then solved by using numerical iterative methods. A rigorous asymptotic analysis demonstrates that these solutions can be classified into two categories: quasi-periodic oscillatory waves and quasi-periodic dark solitons. Singular behaviors may arise in the former case. The numerical results obtained for both the (1+1) -dimensional and (2+1) -dimensional equations are consistent with the theoretical results. Additionaly, the system of algebraic equations can be further extended to address the Riemann–Schottky problem for hyperelliptic curves with 2 infinite points.

Excitation of Quasiperiodic Fast-propagating Waves in the Early Stage of the Solar Eruption

We propose a mechanism for the excitation of large-scale quasiperiodic fast-propagating magnetoacoustic (QFP) waves observed on both sides of the coronal mass ejection. Through a series of numerical experiments, we successfully simulated the quasi-static evolution of the equilibrium locations of the magnetic flux rope in response to the change of the background magnetic field, as well as the consequent loss of the equilibrium that eventually gives rise to the eruption. During the eruption, we identified QFP waves propagating radially outward of the flux rope, and tracing their origin reveals that they result from the disturbance within the flux rope. Acting as an imperfect waveguide, the flux rope allows the internal disturbance to escape to the outside successively via its surface, invoking the observed QFP waves. Furthermore, we synthesized the images of QFP waves on the basis of the data given by our simulations and found consistency with observations. This indicates that the leakage of the disturbance outside the flux rope could be a reasonable mechanism for QFP waves.

Evidence for Electron Precipitation Diffused by Rising‐Tone Quasiperiodic Whistler Waves in Extremely Low L−Shells Observed by CSES

AbstractBased on the observations by the sun‐synchronous circular orbit China Seismo‐Electromagnetic Satellite(CSES), a typical case of the rising‐tone quasiperiodic (QP) emissions with a large period of around 2 min was reported to appear on the dayside on 23 October 2021. The frequency of QP waves ranges from 2,000 ∼ 3,000 Hz and the wave spectral structure consists of four elements with right‐handed polarization. Three elements of them are located at extremely low L‐shells from 2 ∼ 3.5. The periodic precipitating fluxes of 100 ∼ 400 keV electrons were found in the conjugate region also observed by CSES, which appeared in the same magnetic field lines with the elements of QP waves, respectively. By numerical simulations of quasi‐linear diffusion theory, we confirm that QP emissions can effectively precipitate energetic electrons near the loss cone into the atmosphere in the inner radiation belt. To our best knowledge, this is the first evidence that the rising‐tone QP whistler waves scatter electrons in the inner radiation belt. This new finding will help to deepen our understanding of the electron precipitation dynamics in radiation belt physics.

Stabilization for the Klein–Gordon–Zakharov system

This paper deals with global stability dynamics for the Klein–Gordon–Zakharov system in R 2 . We first establish that this system admits a family of linear mode unstable explicit quasi-periodic wave solutions. Next, we prove that the Kelvin–Voigt damping can help to stabilize those linear mode unstable explicit quasi-periodic wave solutions for the Klein–Gordon–Zakharov system in the Sobolev space H s + 1 ( R 2 ) × H s + 1 ( R 2 ) × H s + 1 ( R 2 ) for any s ⩾ 1. Moreover, the Kelvin–Voigt damped Klein–Gordon–Zakharov system admits a unique Sobolev regular solution exponentially convergent to some special solutions (including quasi-periodic wave solutions) of it. Our result can be extended to the n-dimension dissipative Klein–Gordon–Zakharov system for any n ⩾ 1.

Characteristics and longitudinal extent of VLF quasi-periodic emissions using multi-point ground-based observations

Quasi-periodic (QP) emissions are a type of magnetospheric ELF/VLF waves characterized by a periodic intensity modulation ranging from tens of seconds to several minutes. Here, we present 63 QP events observed between January 2017 and December 2018. Initially detected at the VLF receiver in Kannuslehto, Finland (KAN, MLAT = 67.7°N, L = 5.5), we proceeded to check whether these events were simultaneously observed at other subauroral receivers. To do so we used the following PWING stations: Athabasca (ATH, MLAT = 61.2°N, L = 4.3, Canada), Gakona (GAK, MLAT = 63.6°N, L = 4.9, Alaska), Husafell (HUS, MLAT = 64.9°N, L = 5.6, Iceland), Istok (IST, MLAT = 60.6°N, L = 6.0, Russia), Kapuskasing (KAP, MLAT = 58.7°N, L = 3.8, Canada), Maimaga (MAM, MLAT = 58.0°N, L = 3.6, Russia), and Nain (NAI, MLAT = 65.8°N, L = 5.0, Canada). We found that: (1) QP emissions detected at KAN had a relatively longer observation time (1–10 h) than other stations, (2) 11.3% of the emissions at KAN were observed showing one-to-one correspondence at IST, and (3) no station other than IST simultaneously observed the same QP emission as KAN. Since KAN and IST are longitudinally separated by 60.6°, we estimate that the maximum meridional spread of conjugated QP emissions should be close to 60° or 4 MLT. Comparison with geomagnetic data shows half of the events are categorized as type II, while the rest are mixed (type I and II). This study is the first to clarify the longitudinal spread of QP waves observed on the ground by analyzing simultaneous observations over 2 years using multiple ground stations.Graphical