Nonlinear Wave Research Articles

Non-linear torsional Alfv\'en waves evolving in stratified viscous plasmas: Coronal hole plumes

We model solar atmospheric structures characterised by parallel structuring. We focus on Alfv\'en waves in the weakly non-linear regime to highlight the efficiency of non-linear wave steepening when dissipative effects are prominent. We also consider the local and equilibrium conditions involved in shock formation and the shock's contributions to coronal seismology. Coronal plumes were modelled analytically by implementing the magnetohydrodynamic (MHD) theory in cylindrical geometry. Here, the stratification and viscosity are present internal to the plume, whilst effects of the external medium, together with equilibrium conditions, are implied where the magnetic fields are parallel to the plume axis. We implemented a second-order thin flux tube approximation to obtain a wave equation that points to effects tied to non-linear, dissipative, and stratification terms, as well as terms representing atmospheric conditions. The impact of shear viscosity on non-linear Alfv\'en waves extracted by the Cohen-Kulsrud-Burgers-type equation proves more efficient when propagated to higher altitudes. The dissipative effects linked to the dimensionless viscosity indicate that the dissipative effects are not linear. Meanwhile, the delay in shock formation enables energy conversions at higher altitudes, thereby maintaining coronal heating at higher levels. The efficiency of parallel structuring and viscous damping is enhanced by such transverse structuring, as it is directly proportional to the external plasma-beta . It is observed that Alfv\'en pulses may undergo a backward shock, either in the lower levels of coronal plasma or as they propagate toward higher regions, implying a conversion of energy occurring at various altitudes. A peak was observed, indicating that the interplay reverses at heights around $1.5$ solar radii. Such effects are shown to play a key role in the context of coronal seismology.

Analytical and numerical solutions to the generalized Schrödinger equation with fourth-order dispersion and nonlinearity

This study aims to solve the generalized Schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity (4-Order-GNLSE) using advanced analytical and numerical methods. The 4-Order-GNLSE is critical in understanding complex wave propagation phenomena in various physical contexts, including optical fibers and fluid dynamics, where higher-order dispersion and nonlinear effects are significant. We employ the Khater II (Khat II) and modified Rational (MRat) methods to construct analytical solutions. To validate these solutions, we utilize the trigonometric-quantic-B-spline (TQBS) method as a numerical scheme, demonstrating a close match between analytical and numerical results. Our findings indicate that the proposed methods effectively capture the intricate dynamics governed by the 4-Order-GNLSE. The results highlight the relevance of these solutions in practical applications, offering new insights into the wave propagation behaviors influenced by higher-order dispersion and nonlinearities. This research provides a significant contribution to the field of nonlinear wave equations, presenting novel solutions and validating methodologies that enhance our understanding and potential applications of these complex systems.

Variable dust charge generates multi solitons, breather soliton structures in Saturn’s ring

In this study, we have investigated nonlinear wave structures that are localized along with the exact stationary wave solutions with oscillating, i.e. the breather structures in an unmagnetized dusty plasma with variable dust charge concentration with two temperatures of ions. We have derived GE from the normalized governing equations employing the reductive perturbative technique (RPT). The GE is the extended Korteweg–de Vries (KdV) equation that includes the united effect of quadratic and cubic nonlinearities. Employing the Hirota bilinear method (HBM), multi-soliton solutions and the breather soliton solutions have been obtained. Breathers (oscillating wave packets) play a significant role in hydrodynamics and optics, and their interaction can change the dynamics of the wave fields. It is also important to propagate finite-amplitude waves in the astrophysical atmosphere, plasma dynamics, ocean, optic fibers, signal processing, etc. In the circumstances of Saturn’s ring, it is noticed that the breather soliton structures are modified significantly with variations in dust charge concentration, ion temperature ratio, and other physical parameters.

Diving into plasma physics: dynamical behaviour of nonlinear waves in (3 + 1)-D extended quantum Zakharov–Kuznetsov equation

Diving into plasma physics: dynamical behaviour of nonlinear waves in (3 + 1)-D extended quantum Zakharov–Kuznetsov equation

Exploration of soliton structures in the Hirota–Maccari system with stability analysis

In this research, the modified extended tanh-function (METF) and the extended Jacobi elliptic function expansion (EJEFE) techniques are used to investigate the generation and detection of soliton structures in the Hirota–Maccari (HM) model. Consequently, we obtain soliton solutions with advanced structures, including singular bright soliton, dark soliton, periodic waves, breather waves, periodic breather waves, and multiple bright and dark breather waves. In addition, a lump-type breather wave is also included in the presented solutions. Stability analysis of the obtained solutions is addressed by employing the Hamiltonian technique. 3D surfaces and 2D visuals of the outcomes are represented with the help of a computer application. These findings contribute to understanding nonlinear wave phenomena with potential applications in optics, fluid dynamics, and plasma physics.

Directional effects on the nonlinear response of vehicle-bridge system under correlated wind and waves

Long-span railway sea-crossing bridges are subjected to strong winds and high waves. Due to variations in structural features, including stiffness and size across diverse orientations, the dynamic response of vehicles and bridges can exhibit significant differences under distinct wind and wave directions. Consequently, a novel approach for analyzing bridge and vehicle dynamic performance called the directional wind-wave-vehicle-bridge (DWWVB) model has been developed. To acquire the statistical characterization of the correlation between wind and wave conditions, a three-dimensional joint probability distribution of wind and wave characteristics derived from collected data at the bridge site is established. The wind loads are derived from wind tunnel tests and computational fluid dynamics simulations. The nonlinear wave loads are solved utilizing the hydrodynamic solver ANSYS-AQWA. Based on the finite element bridge model and a 23-degree-of-freedom mass-spring-damper vehicle model, the aerodynamic nonlinearity and initial geometric nonlinearity are considered during the iteration process. The Newmark-β method is employed to solve the dynamic response of bridges and vehicles. The calculation results reveal that the most unfavorable response occurs when the wind direction is 45° and the wave direction is 90°. The directional analysis provides valuable insights for selecting optimal bridge locations and reasonable design environmental parameters.

On nonlinear wave structures, stability analysis and modulation instability of the time fractional perturbed dynamical model in ultrafast fibers

On nonlinear wave structures, stability analysis and modulation instability of the time fractional perturbed dynamical model in ultrafast fibers

Rescaling method for blow‐up solutions of nonlinear wave equations

Rescaling method for blow‐up solutions of nonlinear wave equations

Soliton solutions and sensitive analysis to nonlinear wave model arising in optics

In this study, we use analytical algorithms, specifically the auxiliary equation (AE) approach, the improved F-expansion method, and the modified Sardar sub-equation (MSSE) method to investigate complex wave structures for plentiful solutions associated with the fractional perturbed Gerdjikov-Ivanov (PGI) model with the M-fractional operator. The investigated model is a well-established mathematical model used to represent a variety of physical events in nonlinear dynamics and mathematical physics. By using the aforementioned techniques, we scrutinize some new optical wave solutions in the framework of dark, bright, periodic, combo, W-shaped, M-shape, V-shape, kink type, singular rational, exponential, trigonometric, and hyperbolic solutions. The acquired solutions address a wide range of optical solutions in the form of 3D plots, contour plots, and 2D plots, declaring the free parameters of such optical soliton solutions and comprehending their dynamic behavior. Also, the sensitive analysis of the selected model is analyzed. The main contribution of this study is to extract diverse solitary wave solutions of the adopted model. Some of the solutions are similar and some diverge from the previous solutions which justifies the novelty of the study. Finally, we discovered that the current technique provides a reliable instrument for investigating the analytic solutions of fractional differential equations. The proposed PGI model can be used to transmit ultra-fast pulses across optical fibers. This research goes beyond to the advancement of mathematical techniques for solving fractional differential equations and broadens their application to a wide range of real-world scientific and engineering problems.

New Solitary Waves for Thin-Film Ferroelectric Material Equation Arising in Dielectric Materials

Abstract In this paper, the thin-film ferroelectric material equation (TFFME), which enables the propagation of solitary polarisation in thin-film ferroelectric materials is investigated, will be expressed through the non-linear evolution models. Ferroelectrics are dielectric materials that explain wave propagation non-linear demeanors. The non-linear wave propagation form is administrated by TFFME. To investigate the characterisations of new waves and solitonic properties of the TFFME, the modified exponential Jacobi technique and rational exp(−ϕ(η))-expansion technique are used. Plenty of alternative responses may be achieved by employing individual formulas; each of these solutions is offered by some plain graphs. The validity of such schemes and solutions may be exhibited by assessing how well the relevant schemes and solutions match up. The effect of the free variables on the manner of acting of reached plots to a few solutions in the exact forms was also explored depending upon the nature of non-linearities. The descriptive characteristics of the reached results are presented and analysed by some density, two- and three-dimensional figures. We believe that our results would pave the way for future research generating optical memories based on non-linear solitons.

Trident edge coupler on thin-film lithium niobate for optimized coupling of octave-separated wavelengths for nonlinear applications

We introduce a trident edge coupler design optimized for the simultaneous coupling of two widely separated wavelengths (2 µm and 1 µm) between a lensed fiber and a 600-nm-thick X-cut lithium-niobate-on-insulator (LNOI) waveguide. These wavelengths are commonly encountered in nonlinear wave mixing applications, representing either the fundamental and second harmonics in second harmonic generation (SHG) processes or the leading and trailing edges of an octave-spanning spectrum generated through broadband nonlinear processes such as frequency comb or supercontinuum generation. Achieving efficient coupling between fibers and strongly confined waveguides in integrated platforms, such as LNOI, can be challenging due to the significant difference in spot sizes between the two wavelengths. Our trident edge coupler offers coupling losses below 1.4 dB for the 2 µm and 1 µm spots simultaneously, showcasing an average transmission enhancement of around 10% compared to the baseline of a single linear taper. Furthermore, it enables a reduction of transmission at 1.5 µm, a typical pump wavelength, with an attenuation of transmission over 10 dB compared to those at the 2 µm and 1 µm wavelengths.

Feasibility study of nonlinear ultrasonic imaging using a contact pulse-echo method for fatigue damage evaluation

ABSTRACT A contact pulse-echo nonlinear imaging method is proposed for the detection and localisation of fatigue damage in solid materials. A dual element transducer is designed based on nonlinear wave simulations and manufactured to improve the signal-to-noise ratio of harmonic waves. The transducer is clamped on a fixture device for scanning. The proposed method is introduced for scanning and imaging stainless steel specimens with different fatigue damage states. Experimental results for the specimen in the initial state show that both the fundamental and second harmonic waves are well measured and the imaging of nonlinear parameters is very stable over the scanning area. The imaging results for the specimen with an initial fatigue crack show that the fundamental waves remain the same but the second harmonic waves vary significantly in the potential growth area of the fatigue crack. Furthermore, the imaging results for the seriously fatigued specimen show that the amplitudes of fundamental waves decrease near the fatigue crack, while the amplitudes of second harmonic waves increase. These results indicate that the proposed pulse-echo method is effective for nonlinear ultrasonic imaging and early damage evaluation. The factors affecting the nonlinear waves in the lab experiments and possible problems with practical applications are also discussed for the proposed method.

ANN-assisted prediction of wave run-up around a tension leg platform under irregular wave conditions

This study proposes a novel prediction method for nonlinear wave run-ups around a TLP by combining the linear diffraction method with deep learning techniques. Initially, the linear diffraction method evaluates relative wave motion around the platform based on measured incident waves and frequency domain analysis. Subsequently, an artificial neural network (ANN) predicts the peak values of nonlinear wave run-ups, utilizing incident wave data and linear calculation results as inputs. Two different ANN models were proposed, each tailored to specific output variables. The first, the ζ-based ANN model, predicts the peak value of wave run-ups, while the second, the α-based ANN model, predicts the nonlinear amplification factor. The incorporation of a neural network significantly enhances the prediction accuracy for the peak values of wave run-ups. Moreover, the α-based ANN model demonstrates superior accuracy in predicting high run-up events and better generalization ability across various wave conditions compared to the ζ-based ANN model. The study also investigates the effect of various preprocessing methods on prediction performance. The over-sampling method improves accuracy for high run-up events but degrades predictions for low run-up regions, while the polynomial feature transformation reduces prediction errors for high run-up in the ζ-based ANN model.

Optical solitons for the concatenation model with power–law of self–phase modulation by lie symmetry

This paper investigates the concatenation model under the influence of power-law self-phase modulation through the Lie symmetry. We employ two integration schemes, namely the extended tanh approach and the F-expansion algorithm, to rigorously integrate the reduced ordinary differential equations governing the system. Through this methodological framework, we uncover a diverse array of soliton solutions and systematically classify them, shedding light on their intricate dynamics and characteristics. Our research unveils previously undiscovered soliton solutions, enriching the existing understanding of concatenation models. We introduce a comprehensive classification scheme for these solitons, providing valuable insights into their behavior and interactions. Numerical simulations validate the stability and persistence of the identified soliton solutions across various parameter regimes. Our findings contribute to the theoretical framework of nonlinear wave dynamics and hold potential for innovative applications in fields such as nonlinear optics and information processing.

An Analytical Study of the Mikhailov–Novikov–Wang Equation with Stability and Modulation Instability Analysis in Industrial Engineering via Multiple Methods

Solitary waves, inherent in nonlinear wave equations, manifest across various physical systems like water waves, optical fibers, and plasma waves. In this study, we present this type of wave solution within the integrable Mikhailov–Novikov–Wang (MNW) equation, an integrable system known for representing localized disturbances that persist without dispersing, retaining their form and coherence over extended distances, thereby playing a pivotal role in understanding nonlinear dynamics and wave phenomena. Beyond this innovative work, we examine the stability and modulation instability of its gained solutions. These new solitary wave solutions have potential applications in telecommunications, spectroscopy, imaging, signal processing, and pulse modeling, as well as in economic systems and markets. To derive these solitary wave solutions, we employ two effective methods: the improved Sardar subequation method and the (℧′/℧, 1/℧) method. Through these methods, we develop a diverse array of waveforms, including hyperbolic, trigonometric, and rational functions. We thoroughly validated our results using Mathematica software to ensure their accuracy. Vigorous graphical representations showcase a variety of soliton patterns, including dark, singular, kink, anti-kink, and hyperbolic-shaped patterns. These findings highlight the effectiveness of these methods in showing novel solutions. The utilization of these methods significantly contributes to the derivation of novel soliton solutions for the MNW equation, holding promise for diverse applications throughout different scientific domains.

Optical soliton solution of the perturbed Biswas-Milovic equation having cubic-quintic-septic law nonlinearity in the presence of spatio-temporal and chromatic dispersion

In this manuscript, we investigate the analytical and soliton solutions of the cubic-quintic-septic law for the perturbed Biswas-Milovic equation, considering spatio-temporal and chromatic dispersions. The perturbed Biswas-Milovic equation with the spatio-temporal and chromatic dispersion terms provides a comprehensive study for describing nonlinear optical wave propagation in optical fiber. We use the wave transformation to reduce the main equation to a nonlinear ordinary differential equation. The transformation of the original equation into a more simplified form aims to attain a more profound comprehension of the fundamental dynamics of the system. We retrieve the analytical solutions of the presented model by implementing the new Kudryashov technique and a subversion of the new extended auxiliary equation approach. Besides, bright, singular, and V-shape soliton structures are represented. By employing powerful analytical techniques, we systematically derive a wide range of soliton solutions. This approach successfully captures diverse soliton types highlighting the novelty of applying the new Kudryashov technique and a subversion of the new extended auxiliary equation method to this complex model. Moreover, we analyze the soliton behavior influenced by various parameters. The analysis of the parameter influences reveals the complicated relationship governing the dynamics of the perturbed Biswas-Milovic model. Furthermore, this manuscript includes the modulation instability analysis for the presented model. Conducting modulation instability analysis for the presented equation enhances our understanding of the system’s stability and dynamics.

Nonlinear wave propagation in homogenized strain gradient 1D and 2D lattice materials: Applications to hexagonal and triangular networks

AbstractThis work aims to analyze the propagation of fully nonlinear waves, encompassing shear, extension, and bending deformation modes, within homogenized periodic nonlinear hexagonal and triangular networks, successively considering 1D and 2D situations. The wave analysis is conducted from the expression of the effective strain energy density of periodic hexagonal and triangular lattices in the nonlinear regime by a continualization of the discrete lattice equations, considering all forms of energy. We incorporate strain gradient effects into the continuous model to account for the wave‐dispersive nature. The resulting second‐gradient nonlinear continuum exhibits subsonic and supersonic propagation modes. We first examine in a 1D situation the dynamical response of the hexagonal and triangular lattices, considering varying levels of nonlinearity quantified by a single scalar valued parameter. We further evaluate the impact of a fully nonlinear analysis compared to an analysis solely based on the shear energy, regarding both supersonic and subsonic modes. The nonlinear wave propagation analysis is then extended to a 2D situation for the same two lattices. It is shown that the longitudinal mode exhibits a higher frequency at a low degree of nonlinearity; however, as the degree of nonlinearity increases, the shear mode surpasses the longitudinal mode in terms of frequency. As the wavenumber increases, the nonlinearity has a lesser impact on the frequency behavior, and the phase velocity is more influenced by other factors, such as the second gradient contributions of the effective constitutive law. Such a behavior indicates a transition from a highly nonlinear behavior at lower wave numbers to a more linear behavior at higher wave numbers.

Anti-symmetric and positivity preserving formulation of a spectral method for Vlasov-Poisson equations

We analyze the anti-symmetric properties of a spectral discretization for the one-dimensional Vlasov-Poisson equations. The discretization is based on a spectral expansion in velocity with the symmetrically weighted Hermite basis functions, central finite differencing in space, and an implicit Runge Kutta integrator in time. The proposed discretization preserves the anti-symmetric structure of the advection operator in the Vlasov equation, resulting in a stable numerical method. We apply such discretization to two formulations: the canonical Vlasov-Poisson equations and their continuously transformed square-root representation. The latter preserves the positivity of the particle distribution function. We derive analytically the conservation properties of both formulations, including particle number, momentum, and energy, which are verified numerically on the following benchmark problems: manufactured solution, linear and nonlinear Landau damping, two-stream instability, bump-on-tail instability, and ion-acoustic wave.

Exact Solutions for the Sharma–Tasso–Olver Equation via the Sardar Subequation Method with a Comparison between Atangana Space–Time Beta-Derivatives and Classical Derivatives

The Sharma–Tasso–Olver (STO) equation is a nonlinear, double-dispersive, partial differential equation that is physically important because it provides insights into the behavior of nonlinear waves and solitons in various physical areas, including fluid dynamics, optical fibers, and plasma physics. In this paper, the STO equation is generalized to a fractional equation by using Atangana (or Atangana–Baleanu) fractional space and time beta-derivatives since they have been found to be useful as a model for a variety of traveling-wave phenomena. Exact solutions are obtained for the integer-order and fractional-order equations by using the Sardar subequation method and an appropriate traveling-wave transformation. The exact solutions are obtained in terms of generalized trigonometric and hyperbolic functions. The exact solutions are derived for the integer-order STO and for a range of values of fractional orders. Numerical solutions are also obtained for a range of parameter values for both the fractional and integer orders to show some of the types of solutions that can occur. As examples, the solutions are obtained showing the physical behavior, such as the solitary wave solutions of the singular kink-type and periodic wave solutions. The results show that the Sardar subequation method provides a straightforward and efficient method for deriving new exact solutions for fractional nonlinear partial differential equations of the STO type.

Nonlinear spatial evolution of degenerate quartets of water waves

In this manuscript we investigate the Benjamin–Feir (or modulation) instability for the spatial evolution of water waves from the perspective of the discrete, spatial Zakharov equation, which captures cubically nonlinear and resonant wave interactions in deep water without restrictions on spectral bandwidth. Spatial evolution, with measurements at discrete locations, is pertinent for laboratory hydrodynamic experiments, such as in wave flumes, which rely on time-series measurements at fixed gauges installed along the facility. This setting is likewise appropriate for experiments in electromagnetic and plasma waves. Through a reformulation of the problem for a degenerate quartet, we bring to bear techniques of phase-plane analysis which elucidate the full dynamics without recourse to linear stability analysis. In particular we find hitherto unexplored breather solutions and discuss the optimal transfer of energy from carrier to sidebands. We show that the maximal energy transfer consistently occurs for smaller side-band separation than the fastest linear growth rate. Finally, we discuss the observability of such discrete solutions in light of numerical simulations.