Nonlinear Evolution Research Articles

High-Precision computational solutions for nonlinear evolution models in graphene sheets

This study investigates the analytical solutions of a nonlinear evolution model governing the dynamics of graphene sheets, a material renowned for its exceptional electronic properties and versatile applications in nanotechnology. Three advanced analytical approaches-the Khater II (Khat II) method, the Khater III (Khat III) method, and the Generalized Rational (GRat) approach-are employed to derive exact solutions for this model with high precision. The accuracy and reliability of these solutions are validated by comparing them to numerical results obtained via He’s Variational Iteration (HVI) method, which serves as a benchmark for numerical verification. The analysis reveals a remarkable agreement between the analytical and numerical solutions, highlighting the robustness and effectiveness of the proposed methodologies. Furthermore, this study provides new insights into the nonlinear dynamics and physical properties of graphene sheets, while also identifying connections to other prominent nonlinear evolution equations. The innovative use of these analytical techniques offers practical frameworks for addressing complex nonlinear models in mathematical physics, thus advancing solution methodologies for such equations. This research contributes significantly to applied mathematics, material science, and nanotechnology by delivering accurate solutions and enhancing our understanding of graphene’s nonlinear behavior. Finally, the findings have far-reaching implications, offering potential applications in designing advanced materials with tailored properties to support technological advancements, thereby pushing the boundaries of nanotechnology and materials engineering.

Butchery activities associated with member 5 at Sterkfontein, South Africa

The origin of animal tissue consumption within the hominin lineage remains a central question in palaeoanthropology and taphonomy. This question is mostly addressed through the study of bone surface modifications (e.g., butchery marks) observed on fossils from East African sites. Albeit somewhat overlooked compared to East Africa, South Africa provides an additional body of evidence regarding the evolution of hominin behaviours. Here, we provide a comprehensive description and analysis of a butchered bone assemblage from the Sterkfontein Name Chamber and Member 5 East Oldowan infill in South Africa, dated conservatively to between 1.4 and 2.18 Ma. Based on the anatomical location and morphology of the bone surface modifications, we demonstrate that hominins using Oldowan tools were capable of performing a complete butchery sequence that included skinning, disarticulation, defleshing and marrow extraction. Furthermore, comparison with the butchered bones from the neighbouring sites of Cooper’s D and Swartkrans shows a continuity, or the repeated emergence, of similar butchery patterns through the Early Pleistocene. The identification of distinct butchery patterns, the range of exploited animals, as well as the presence of bone tools in many sites highlight the diversity of hominin subsistence behaviours during the Early Pleistocene, which we interpret as a reflection of the likely non-linear evolution of such behaviours. Finally, we argue that the research focus of taphonomic analyses should address how hominins processed carcasses in addition to how and when these were acquired. Such analyses would help identifying the development of complex butchery practices in the archaeological record.

In-depth analysis and exploration of rogue wave, lump wave, and different solitonic patterns to the Painlevé-integrable (3+1)D nonlinear evolution equations using a new extended methodology

This paper proposes a novel extended methodology, establishing the new extended Generalized Exponential Rational Function Method (GERFM), which is known as a “new extended GERFM” for solving the Painleve-integrable [Formula: see text]D generalized nonlinear evolution equation. We enhance GERFM by incorporating additional terms that combine functions and derivatives, making it a more generalized and efficient approach for solving nonlinear partial differential equations. Moreover, the unified method is applied to thoroughly delve deeper into the equation’s properties to understand its mathematical characteristics and potential solutions. This approach aims to systematically investigate and elucidate the equation’s inherent features, providing a more complete insight into its behavior and the nature of its solutions. Utilizing these methods, we have derived new exact solutions that involve exponential, trigonometric, hyperbolic, polynomial, and rational functions with additional parameters to the model. To emphasize the physical importance of these solutions, we present three-dimensional (3D) and contour plots, investigating different parameter choices. The graphs illustrate various wave patterns such as irregular periodic solitons, rogue waves, lumps, and the interaction of solitary and lump waves based on different parameter values. This visualization method enriches our comprehension of the solutions and facilitates a thorough exploration of how they could be applied to the generation of prolonged waves with small magnitudes in plasma physics, fluid dynamics, and other media with weak dispersion.

Case report: Non-linear evolution of oxytocin informs YBOCS changes post-DBS of the bed nucleus of the stria terminalis for treatment resistant OCD

IntroductionObsessive-compulsive disorder (OCD) is a challenging neuropsychiatric condition with a subset of patients remaining refractory to conventional treatments. Deep brain stimulation (DBS) of the bed nucleus of the stria terminalis (BNST) has shown promise for severe, treatment-resistant OCD. This case report examines the relationship between plasma oxytocin levels and OCD symptom severity following BNST-DBS.MethodsA 36-year-old patient with long-standing, treatment-resistant OCD underwent stereotactic implantation of DBS electrodes at the BNST. Postoperative assessments included OCD symptom severity using the Yale-Brown Obsessive Compulsive Scale (YBOCS) and plasma oxytocin levels, measured at 12 time points over three years. Longitudinal and correlational analyses were performed using linear and polynomial regression models.ResultsNon-linear trends in oxytocin levels were identified, with polynomial regression revealing a significant quadratic term, suggesting a parabolic trend. Strong positive correlations were found between changes in oxytocin levels and YBOCS total, obsession, and compulsion scores.ConclusionThe findings suggest a significant non-linear evolution of oxytocin levels and a positive correlation with OCD symptom changes following BNST-DBS. Oxytocin levels could serve as a biomarker for DBS efficacy if this finding is replicated in larger studies.

Exploring soliton and soliton-type solutions to the modified Camassa-Holm and Schrödinger-Hirota equations: An analytical approach

Abstract In this work, we investigate the solitary wave solutions of two nonlinear evolution equations (NLEEs): the modified Camassa-Holm (MCH) equation and the Schrödinger-Hirota (SH) equation. Both are widely used, especially in fluid dynamics, nonlinear optics, and quantum mechanics. We analytically extract solitary wave solutions to the considered equations by applying the (G'⁄((G'+G+A)))-expansion method which enhances the understanding of wave dynamics in various fields of mathematical physics and engineering. The graphical representations of the obtained results of these two equations for certain values are shown in 3D, 2D, contour, and density plot diagrams, all of which are of the soliton types with one compacton, one anti-flat kink-shaped, one anti-bell-shaped, and one parabolic-shaped soliton. Moreover, we analyze the role of non-algebraic functions, specifically exponential and trigonometric functions, in characterizing the intricate waveforms and their stability properties. We systematically explore the applicability of the chosen expansion method, demonstrating its effectiveness in generating explicit solutions and examining the conditions under which these solitary wave phenomena arise. The novel findings contribute new analytical insights into these classical equations and pave the way for future research into the complex behaviors exhibited by nonlinear wave interactions.

Geometry of the fiber cavity as a factor for stabilizing pulse generation in a fiber laser mode-locked by nonlinear polarization evolution

Optimal fiber cavity geometry for a fiber pulse oscillator based on nonlinear polarization evolution mode-locking is proposed for the stabilization of pulse generation against changes in ambient temperature. The method of arranging the optical fiber in an orthogonal cavity, in which the bend-induced birefringence compensates, was investigated. Experimental testing of different cavity geometries confirmed the advantage of the orthogonal distribution fiber scheme compared to conventional flat fiber winding.

Effect of plasma beta on the nonlinear evolution of m/n = 2/1 double tearing mode in high Lundquist number regime

Abstract The existing results indicate that in the large Lundquist number regime, plasmoids play an important role in the growth and saturation of the double tearing mode (DTM). In this paper, the effect of plasma beta on the nonlinear evolution of the m/n=2/1 DTM in large Lundquist number regime is numerically investigated in a cylinder geometry. The results demonstrate that the impact of plasma beta on plasmoid dynamics varies significantly with the separation distance between rational surfaces. In the small separation regime（Δr=0.1), no plasmoids are observed, regardless of plasma beta and resistivity. However, when the separation is relatively large (Δr=0.2 or Δr=0.3), plasmoids exhibit highly complex behavior under different plasma beta and resistivity conditions. For the medium separation Δr=0.2, the resistivity threshold for the emergence of plasmoids is approximately 2.5×10^(-6) and is not affected by plasma beta. Conversely, when the separation increases to Δr=0.3, the resistivity threshold is significantly influenced by plasma beta. Under certain conditions, this threshold can reach as high as 1.0×10^(-5), which is much higher than the typical value of around 10^(-7). Furthermore, a preliminary case study shows that the tokamak plasma rotation shear has a significant impact on plasmoid dynamic.

Exact solutions and their local structures of an inverse scattering integrable system with variable coefficients

An important manifestation of the integrability of nonlinear mathematical and physical models is their solvability through inverse scattering transform (IST). The problem to be investigated in this paper is a system of fifth-order nonlinear evolution equations (NLEEs) with variable coefficients related to time-varying spectral problems. The expected result is to derive the system of fifth-order NLEEs, obtain its exact solutions, verify its integrability in the sense IST solvability, and reveal some novel local structures of the obtained solutions. First, the system of fifth-order NLEEs is derived. Then, by combining the IST with a time-varying spectral parameter, the associated scattering data are determined for the reconstruction of potentials. Using the determined scattering data, exact solutions are ultimately obtained. Meanwhile, some local structures with new features are analyzed for two pairs of special solutions, including kink solitons, wide/narrow top bell solitons, and small-scale peaks in the time variable direction, as well as sine/cosine-like fluctuations in the spatial variable direction. This paper not only demonstrates that equipping the Ablowitz–Kaup–Newell–Segur (AKNS) problems with appropriate time-varying spectra can be used to derive some other inverse scattering integrable systems of NLEEs with variable coefficients, but also graphically illustrates the regulatory effect of time-varying spectral parameter on local solution structures.

Tunable all-fiber all-normal-dispersion mode-locked laser of cylindrical vector beams covering the range of 69 nm.

We propose and demonstrate an ultra-wide tunable mode-locked all-fiber laser based on nonlinear amplifying loop mirror (NALM) with the output of cylindrical vector beams (CVBs). The tuning range covers from 1029 nm to 1098 nm through the intracavity nonlinear polarization evolution (NPE) filter effect. The switchable CVBs between radially and azimuthally polarized beams with mode purity above 90% are generated by incorporating a broadband few-mode long-period fiber grating (LPFG). It is the first time to realize mode-locked CVBs near 1100 nm and the widest spectral tuning range in all-fiber laser is achieved to our knowledge. The pulsed CVBs at 1098 nm have 3 dB bandwidth of 0.31 nm with a pulse duration of 358ps.The narrow-bandwidth pulse of less than 1 nm is obtained among the whole tuning process which is of high flexibility and high tuning precision by introducing what we believe to be novel tuning mechanisms of NPE into the NALM cavity. The wide-range tunable CVBs all-fiber mode-locked laser has potential applications in high-capacity optical communication, laser imaging, and fiber optic sensing fields.

Resonant capture of stars by black hole binaries: extreme eccentricity excitation

ABSTRACT Massive black hole (MBH) binaries in galactic nuclei are one of the leading sources of ${\sim}$mHz gravitational waves (GWs) for future missions such as Laser Interferometer Space Antenna (LISA). However, the poor sky localization of GW interferometers will make it challenging to identify the host galaxy of MBH mergers absent an electromagnetic counterpart. One such counterpart is the tidal disruption of a star that has been captured into mean motion resonance with the inspiralling binary. Here we investigate the production of tidal disruption events (TDEs) through capture into, and subsequent evolution in, orbital resonance. We examine the full non-linear evolution of planar autoresonance for stars that lock into autoresonance with a shrinking MBH binary. Capture into the 2:1 resonance is guaranteed for any realistic astrophysical parameters (given a relatively small MBH binary mass ratio), and the captured star eventually attains an eccentricity $e\approx 1$, leading to a TDE. Stellar discs can be produced around MBHs following an active galactic nucleus episode, and we estimate the TDE rates from resonant capture produced when a secondary MBH begins inspiralling through such a disc. In some cases, the last resonant TDE can occur within a decade of the eventual LISA signal, helping to localize the GW event.

Nonlinear evolution of unstable charged de Sitter black holes with hyperboloidal formalism

Nonlinear evolution of unstable charged de Sitter black holes with hyperboloidal formalism

Nonlinear evolution of fluting oscillations in coronal flux tubes

Magnetic flux tubes in the solar corona support a rich variety of transverse oscillations, which are theoretically interpreted as magnetohydrodynamic (MHD) modes with a fast and/or Alfvénic character. In the standard flux tube model made of a straight cylindrical tube, these modes can be classified according to their azimuthal wavenumber, m. Sausage m = 0 modes produce periodic expansion and contraction of the tube cross section and are observed during solar flares. Kink m = 1 modes laterally displace the tube axis and are related to, for example, post-flare global transverse oscillations of coronal loops. Fluting m ≥ 2 modes produce disturbances that are mainly confined to the tube boundary, but their observation remains elusive to date. We use 3D ideal MHD numerical simulations to investigate the nonlinear evolution of fluting modes in coronal flux tubes with transversely nonuniform boundaries. The simulations show that fluting modes are short-lived as coherent, collective motions of the flux tube. Owing to the process of resonant absorption, fluting oscillations become overdamped modes in tubes with wide enough nonuniform boundaries. During the nonlinear evolution, shear flows drive the Kelvin-Helmholtz instability at the tube boundary, which further disrupts the coherent fluting oscillation. For large-enough oscillation amplitudes, baroclinic instabilities of Rayleigh-Taylor type are also present at locations in the boundary where the plasma acceleration is normal to the boundary. The evolution of the instabilities drives turbulence in the flux tube, which may inhibit the resonant damping. However, the oscillations remain strongly damped even in this case. As a result of the combination of the strong damping and the induced instabilities, it is unlikely that coronal flux tubes can support fluting modes as sufficiently enduring coherent oscillations.

Variable-time-step weighted IMEX FEMs for nonlinear evolution equations

Variable-time-step weighted IMEX FEMs for nonlinear evolution equations

Computational and numerical solutions to the Benney–Luke equation: Insights into nonlinear long wave dynamics in dispersive media

The current investigation seeks to explore the nonlinear Benney–Luke model, a governing equation pertinent to the propagation of extended waves within dispersive media, such as those observed in water waves and plasma waves, among other nonlinear wave phenomena. This model is distinguished by its incorporation of both dispersive and nonlinear effects, rendering it a valuable instrument for delving into intricate wave dynamics. Notably, it shares similarities with other prominent nonlinear evolution equations, including the Korteweg–de Vries (KdV) equation and the Boussinesq equation, both of which have been extensively examined within the domain of water waves and associated phenomena.Employing analytical methodologies, specifically the Khater (Khat III) and modified Kudryashov (MKud) methods, the research endeavors to derive exact solutions for the Benney–Luke equation. These solutions serve to elucidate various aspects of long wave behavior in dispersive media, encompassing their propagation characteristics, stability, and potential for inter-wave interactions. Furthermore, a trigonometric-quantic-B-spline (TQBS) scheme is adopted as a numerical approach to corroborate the precision of the derived analytical solutions and demonstrate their relevance across diverse physical scenarios linked to the model under examination.The research outcomes underscore the efficacy of the analytical techniques in generating dependable solutions for the Benney–Luke model. Moreover, through the utilization of the TQBS numerical scheme, the accuracy of the derived analytical solutions is affirmed, thereby ensuring their pragmatic utility across a spectrum of physical scenarios pertinent to the investigated model. This endeavor holds significance in its contribution towards advancing the comprehension of nonlinear wave phenomena within dispersive media, with ramifications spanning multiple disciplines, including fluid dynamics, plasma physics, and nonlinear optics. The novelty of this study lies in its application of the Khater (Khat III) and modified Kudryashov (MKud) methods to the Benney–Luke equation, alongside the integration of the TQBS numerical scheme for solution validation. Positioned within the domain of nonlinear partial differential equations, this research aligns with its broader applications in physics, particularly concerning the investigation of dispersive wave phenomena.

Validation of the DESI 2024 Lyα forest BAO analysis using synthetic datasets

The first year of data from the Dark Energy Spectroscopic Instrument (DESI) contains the largest set of Lyman-α (Lyα) forest spectra ever observed. This data, collected in the DESI Data Release 1 (DR1) sample, has been used to measure the Baryon Acoustic Oscillation (BAO) feature at redshift z = 2.33. In this work, we use a set of 150 synthetic realizations of DESI DR1 to validate the DESI 2024 Lyα forest BAO measurement presented in [1]. The synthetic data sets are based on Gaussian random fields using the log-normal approximation. We produce realistic synthetic DESI spectra that include all major contaminants affecting the Lyα forest. The synthetic data sets span a redshift range 1.8 < z < 3.8, and are analysed using the same framework and pipeline used for the DESI 2024 Lyα forest BAO measurement. To measure BAO, we use both the Lyα auto-correlation and its cross-correlation with quasar positions. We use the mean of correlation functions from the set of DESI DR1 realizations to show that our model is able to recover unbiased measurements of the BAO position. We also fit each mock individually and study the population of BAO fits in order to validate BAO uncertainties and test our method for estimating the covariance matrix of the Lyα forest correlation functions. Finally, we discuss the implications of our results and identify the needs for the next generation of Lyα forest synthetic data sets, with the top priority being to simulate the effect of BAO broadening due to non-linear evolution.

Nonlinear spatiotemporal and evolution characteristics of extreme precipitation during the Mei-yu period based on a state transition network

The frequent occurrence of extreme precipitation events notably affects the middle and lower reaches of the Yangtze River (MLYR) during the Mei-yu period. Based on meteorological station data from the China Meteorological Administration collected from 1970 to 2019, we employed nonlinear time series analysis and the state transition networks (STN) to identify the spatiotemporal and evolution characteristics of 10 extreme precipitation indices (EPIs). Although previous studies have largely focused on investigating the spatiotemporal characteristics, the suitability of STN for extreme precipitation has not been extensively explored. This study focused on not only the spatiotemporal characteristics of extreme precipitation but also its evolution using the STN. The results showed that the periodic oscillations in extreme precipitation ranged from 2 to 5 years and an abrupt change occurred primarily around 1990. Extreme precipitation displayed similar network parameters, yet different network structures and transition probabilities. The evolution of extreme precipitation exhibited from decreasing to increasing from 1970 to 2019, with an increasing trend after 1990. Similar network structures tended to exhibit higher Pearson correlations. Spatially, high EPIs values were mainly concentrated in the central region of the MLYR. This study could provide new methods and perspectives for studying the characteristics of extreme precipitation.

Curlometer and gradient techniques: past and future applications

We review the range of applications and use of multi spacecraft techniques, applicable to close formation arrays of spacecraft, focusing on spatial gradient based methods, and the curlometer in particular. The curlometer was originally applied to Cluster multi-spacecraft magnetic field data, but later was updated for different environments and measurement constraints such as the NASA MMS mission, small-scale formation of 4 spacecraft; the 3 spacecraft configurations of the NASA THEMIS mision, and derived 2-4 point measurements from the ESA Swarm mission. In general, spatial gradient based methods are adaptable to a range of multi-point and multi-scale arrays. We also review the range of other techniques based on the computation of magnetic field gradients and magnetic field topology in general, including: magnetic rotation analysis and various least squares approaches. We review Taylor expansion methodology (FOTE), in particular, which has also been applied to both Cluster and MMS constellations, as well as interpretation of simulations. Four-point estimates of magnetic gradients are limited by uncertainties in spacecraft separations and the magnetic field, as well as the presence of non-linear gradients and temporal evolution. Nevertheless, the techniques can be reliable in many magnetospheric regions where time stationarity is largely applicable, or when properties of the morphology can be assumed (for example, the expected orientation of underlying large-scale structure). Many magnetospheric regions have been investigated directly (illustrated here by the magnetopause, ring current and field-aligned currents at high and low altitudes), and options for variable numbers of spacecraft have been considered. The comparative use of plasma measurements and possible new methodology for arrays of spacecraft greater than four are also considered briefly.

Exact and soliton solutions of nonlinear evolution equations in mathematical physics using the generalized (G′/ G)-expansion approach

Abstract The nonlinear evolution equations play a significant role in applied mathematics, including ordinary and partial differential equations, which are frequently used in many disciplines of applied sciences. The Klein–Gordon equation, a second-order equation in both space and time, is closely related to the Schrödinger equation and describes the behavior of spinless particles in theoretical physics. The equation is utilized extensively in many areas related to modern physics, including astrophysics, cosmology, and the theory of classic mechanics. The uKdV equation is suitable for use in various fields, including fluid dynamics, oceanography, and optical solitons research. It is useful for simulating events with wave-like characteristics and soliton dynamics, which contributes to the theoretical development of mathematical physics. This article addresses the exact and soliton solutions of nonlinear evolution equations in mathematical physics and engineering using the new generalized ( G ′ / G ) -expansion approach. We examine a large number of soliton solutions, including kink-shaped, singular-kink, single soliton, singular-soliton, periodic, and others. To better understand the characteristics of the solutions, some three-dimensional plots are set up for visualization. It has been demonstrated that the suggested approaches are more attractive as well as effective for getting single solutions to nonlinear evolution equations.

Existence, comparison principle and uniqueness for doubly nonlinear anisotropic evolution equations

We prove the existence of solutions to the Cauchy-Dirichlet problem associated with a class of doubly nonlinear anisotropic evolution equations. We also demonstrate the existence of solutions to the corresponding Cauchy problem on RN×(0,T). Under some assumptions on the Caratheodory vector field we prove a comparison principle and utilize it to obtain a uniqueness result for the Cauchy-Dirichlet problem.

Nonlinear pressure evolution in wellbore during the tubing conveyed perforation process

Abstract With increasing wellbore depth, the environmental conditions become progressively more severe, characterized by elevated temperatures and pressures. Consequently, the pressure variations within the well induced by Tubing Conveyed Perforation (TCP) operations exhibit a more pronounced intensity. These nonlinear pressure fluctuations generate significant impact loads at the bottom of the perforating gun, which can cause failure of the perforating tubing. A transient nonlinear pressure field model of perforation explosion was established based on hydraulic-mechanical coupling to effectively control the risk of accidents. The model considers the effects of several hundred perforation charges and their precise placement in relation to blind holes, thus taking into account the impact of shot density. As a result, the model more accurately reflects actual field conditions. The study investigated the evolution of nonlinear pressure fields at different positions of the entire tubing under perforation explosion and analyzed the influence of different factors on nonlinear pressure fluctuations at the bottom of the perforating gun. The results showed that the peak pressure of the perforation section was much higher than that of the tubing section and the rathole section, maintaining at approximately 200 MPa. The pressure wave attenuated to the initial wellbore pressure of about 80 MPa only seven meters away from the perforation section center. The peak pressure at the bottom of the perforating gun was positively correlated with the initial wellbore pressure, perforating fluid density, shot density, and quantity of explosive. These results provided guidance for the prediction of TCP wellbore nonlinear pressure fluctuation.