Nonlinear Propagation Research Articles

Mode I fracture propagation and post-peak behavior of sandstone: Insight from AE and DIC observation

Microcracks generated at the crack tip, particularly in quasi-brittle materials such as sandstone, give rise to a non-linear region known as the fracture process zone (FPZ), which cannot be analyzed using linear elastic fracture mechanics (LEFM). To gain insights into the nonlinear fracturing behavior of rock, this study conducted three-point bending experiments on sandstone beams with prefabricated notch. Stable mode-Ⅰ fracture propagation during the post-peak stage was achieved by the monotonic increment of crack mouth opening displacement (CMOD) measured by a clip gauge. The acoustic emission (AE) and digital image correlation (DIC) techniques were utilized to visually track the detailed crack propagation process from both micro and macro perspectives. Nonlinear fracture parameters, including FPZ length, were identified by AE and DIC measurements and compared with the theoretical calculation. A nonlinear fracture criterion relating applied load, equivalent crack length, and stress intensity factor were established, from which the theoretical CMOD-P curve at the post-peak stage were obtained and validated with experimental measurement from MTS machine. The results reveal that microcracks formulating the FPZ materialize in the pre-peak stage, with the majority emerging post-peak to facilitate the propagation of the traction-free crack and FPZ in the loading direction, as evidenced by AE locations and DIC displacement gradient. Throughout the crack propagation process, over 65 % of microcracks are estimated to be tensile mode based on both AF/RA and polarity analysis, aligning with anticipated outcomes from macroscopic three-point bending fracture tests. Energy dissipation primarily occurs within the FPZ to facilitate separation of crack faces, with the dissipated energy region advancing towards the beam boundary. By integrating AE source locations and DIC displacement gradient, the FPZ length of the 3-point-bending sandstone specimens is determined to be lp = 8.1 mm, slightly exceeding the theoretical FPZ lengths calculated by the Irwin model (lp = 7.0 mm) and the Dugdale-Barenblatt model (lp = 7.6 mm). Using a non-linear fracture criterion that incorporates the FPZ and employing the in-time equivalent crack length measured by AE and DIC as an intermediate variable, the theoretical applied load-crack length curve during the post-peak stage is calculated, showing close agreement with load cell measurements with a negligible error of approximately 2 %. Furthermore, the theoretical post-peak CMOD vs. load curve aligns with experimentally obtained results, demonstrating a non-linear crack propagation speed at a constant CMOD rate, wherein crack propagation speed decreases as crack length increases.

Thermal Marangoni convection in two-phase quadratic convective flow of dusty MHD trihybrid nanofluid with non-linear heat source

The current study examines the effect of heat generation on thermal Marangoni convective boundary layer flow of dusty trihybrid nanofluid across a flat surface with thermal radiation and non-linear mixed convection. We consider, two phase dusty liquid model with non-linear heat generation. The fluctuation of surface tension gradients leads to the discovery of Marangoni convection. It can be used for growing crystals, drying silicon wafers, stabilising soap films, and wielding. The main objective of this study is to ascertain the trihybrid nanofluid thermal mobility. A trihybrid nanofluid consisting of magnesium oxide (Mgo), titanium oxide (TiO2), silver (Ag) and water as the base fluid is used. This model can improve the efficiency and dependability of thermal systems in a variety of applications by helping to optimize heat transfer processes in materials processing, electronics cooling, and the creation of cutting-edge cooling technologies in energy systems. The existing PDEs are converted into nonlinear ODEs via similarity variables. The nonlinear ODEs are then solved numerically using shooting technique (RKF-45th approach). When the Marangoni convection parameter rises, higher surface tension gradients lead to stronger induced flows and more efficient heat transfer inside the liquid. As the temperature profiles of the dust and fluid phases drop, the distribution of these characteristics in the liquid becomes more homogeneous.

Updating Nonlinear Stochastic Dynamics of an Uncertain Nozzle Model using Probabilistic Learning with Partial Observability and Incomplete dataset

Abstract This paper introduces a methodology for updating the nonlinear stochastic dynamics of a nozzle with uncertain computational model. The approach focuses on a high-dimensional nonlinear computational model constrained by a small target dataset. Challenges include the large number of degrees-of-freedom, geometric nonlinearities, material uncertainties, stochastic external loads, under-observability, and high computational costs. A detailed dynamic analysis of the nozzle is presented. An updated statistical surrogate model relating the observations of interest to the control parameters is constructed. Despite small training and target datasets, and partial observability, the study successfully applies Probabilistic Learning on Manifolds (PLoM) to address these challenges. PLoM captures geometric nonlinear effects and uncertainty propagation, improving conditional mean statistics compared to training data. The conditional confidence region demonstrates the ability of the methodology to accurately represent both observed and unobserved output variables, contributing to advancements in modeling complex systems.

Rock Layer Classification and Identification in Ground-Penetrating Radar via Machine Learning

Ground-penetrating radar (GPR) faces complex challenges in identifying underground rock formations and lithological structures. The diversity, intricate shapes, and electromagnetic properties of subsurface rock formations make their accurate detection difficult. Additionally, the heterogeneity of subsurface media, signal scattering, and non-linear propagation effects contribute to the complexity of signal interpretation. To address these challenges, this study fully considers the unique advantages of convolutional neural networks (CNNs) in accurately identifying underground rock formations and lithological structures, particularly their powerful feature extraction capabilities. Deep learning models possess the ability to automatically extract complex signal features from radar data, while also demonstrating excellent generalization performance, enabling them to handle data from various geological conditions. Moreover, deep learning can efficiently process large-scale data, thereby improving the accuracy and efficiency of identification. In our research, we utilized deep neural networks to process GPR signals, using radar images as inputs and generating structure-related information associated with rock formations and lithological structures as outputs. Through training and learning, we successfully established an effective mapping relationship between radar images and lithological label signals. The results from synthetic data indicate a rock block identification success rate exceeding 88%, with a satisfactory continuity identification of lithological structures. Transferring the network to measured data, the trained model exhibits excellent performance in predicting data collected from the field, further enhancing the geological interpretation and analysis. Therefore, through the results obtained from synthetic and measured data, we can demonstrate the effectiveness and feasibility of this research method.

Nonlinear optical diode effect in a magnetic Weyl semimetal

Diode effects are of great interest for both fundamental physics and modern technologies. Electrical diode effects (nonreciprocal transport) have been observed in Weyl systems. Optical diode effects arising from the Weyl fermions have been theoretically considered but not probed experimentally. Here, we report the observation of a nonlinear optical diode effect (NODE) in the magnetic Weyl semimetal CeAlSi, where the magnetization introduces a pronounced directionality in the nonlinear optical second-harmonic generation (SHG). We demonstrate a six-fold change of the measured SHG intensity between opposite propagation directions over a bandwidth exceeding 250 meV. Supported by density-functional theory, we establish the linearly dispersive bands emerging from Weyl nodes as the origin of this broadband effect. We further demonstrate current-induced magnetization switching and thus electrical control of the NODE. Our results advance ongoing research to identify novel nonlinear optical/transport phenomena in magnetic topological materials and further opens new pathways for the unidirectional manipulation of light.

Free propagation of resonant waves in nonlinear dissipative metamaterials

This paper deals with the free propagation problem of resonant and close-to-resonance waves in one-dimensional lattice metamaterials endowed with nonlinearly viscoelastic resonators. The resonators' constitutive and geometric nonlinearities imply a cubic coupling with the lattice. The analytical treatment of the nonlinear wave propagation equations is carried out via a perturbation approach. In particular, after a suitable reformulation of the problem in the Hamiltonian setting, the approach relies on the well-known resonant normal form techniques from Hamiltonian perturbation theory. It is shown how the constructive features of the Lie Series formalism can be exploited in the explicit computation of the approximations of the invariant manifolds. A discussion of the metamaterial dynamic stability, either in the general or in the weak dissipation case, is presented.

تخامد لانداو للأمواج السوليتونية الصوتية الغبارية في بلازما غبارية غير شاملة

تم دراسة الأمواج السوليتونية الصوتية الغبارية  تحت تأثير تخامد لانداو للبلازما الغبارية الفضائية تحوي أيونات خاضعة لتوزع سرعة غير شامل. قمنا بالتحقيق في تأثير تخامد لانداو والبارمتر غير الشامل   على البنى السوليتونية الصوتية الغبارية من خلال الحل العددي لمعادلة كورديفيك- دي فريس المعدلة بحد تخامد لانداو. باستخدام طريقة الاضطراب الاختزالية قمنا باستنتاج معادلة كورتيفيك – دي فريس مع إضافة حد تخامد لانداو.  أظهرت هذه الدراسة أنً كثافة حبيبات الغبار تلعب دوراَ أساسياً في ظهور أو عدم ظهور الأمواج السوليتونية DA في البلازما الغبارية،  والتأثير الواضح للخاصية غير الشاملة للأيونات على ظاهرة تخامد لانداو للبنى اللاخطية المتشكلة في البلازما الغبارية. تفيد نتائج هذه الدراسة في محاولة فهم الآلية الفيزيائية للانتشار الأمواج السوليتونية تحت تأثير تخامد لانداو في بيئات البلازما المغبرة غير الشاملة كالمناطق الكوكبية، والأغلفة المغناطيسية، وبيئات البلازما الفضائية.

An efficient multimode vectorial nonlinear propagation solver beyond the weak guidance approximation

An efficient multimode vectorial nonlinear propagation solver beyond the weak guidance approximation

On beat-driven and spontaneous excitations of zonal flows by drift waves

Using the slab plasma as a paradigm model, we have derived analytically equations for the nonlinear generation of zero-frequency zonal flows by electron drift waves including, on the same footing, both the beat-driven and spontaneous excitations. It is found that the beat-driven zonal flow tends to reduce the frequency mismatch between the electron drift waves and, thereby, contributes to a significant O(1) enhancement of the modulational instability drive and lowering its threshold. Implications to tokamaks plasmas as well as drift-wave soliton formation are also discussed.

New exact solutions of the (3+1)-dimensional double sine-Gordon equation by two analytical methods

The (3+1)-dimensional double sine-Gordon equation plays a crucial role in various physical phenomena, including nonlinear wave propagation, field theory, and condensed matter physics. However, obtaining exact solutions to this equation faces significant challenges. In this article, we successfully employ a modified G′G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left( \\dfrac{G'}{G^2}\\right) $$\\end{document}-expansion and improved tanϕξ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ an \\left( \\dfrac{\\phi \\left( \\xi \\right) }{2}\\right) $$\\end{document}-expansion methods to construct new analytical solutions to the double sine-Gordon equation. These solutions can be divided into four categories like trigonometric function solutions, hyperbolic function solutions, exponential solutions, and rational solutions. Our key findings include a rich spectrum of soliton solutions, encompassing bright, dark, singular, periodic, and mixed types, showcasing the (3+1)-dimensional double sine-Gordon equation ability to model diverse wave behaviors. We uncover previously unreported complex wave structures, revealing the potential for complex nonlinear interactions within the (3+1)-dimensional double sine-Gordon equation framework. We demonstrate the modified G′G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left( \\dfrac{G'}{G^2}\\right) $$\\end{document}-expansion and improved tanϕξ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ an \\left( \\dfrac{\\phi \\left( \\xi \\right) }{2}\\right) $$\\end{document}-expansion methods effectiveness in handling higher-dimensional nonlinear partial differential equations, expanding their applicability in mathematical physics. These method offers enhanced flexibility and broader solution categories compared to conventional approaches.

A study of the non-linear properties and wave generation of the multi-layer non-hydrostatic wave model SWASH

Despite the increasing use of non-hydrostatic models in the study of wave processes in coastal regions, there is still limited understanding of the non-linear properties of the governing equations and how they improve with increased vertical resolution. In this study, the governing equations of the non-hydrostatic wave model SWASH are analysed and the linear and non-linear solutions up to third-order of all dependent variables are derived, considering one to four vertical layers. The analysis concludes that the model can achieve excellent non-linear properties with respect to the Stokes theory over a large range of water depths using only a few vertical layers. Furthermore, deriving solutions for all variables enables the formulation of improved wave generation and absorption boundary conditions for non-hydrostatic models. A well-known issue of non-linear wave models is related to the generation and propagation of spurious free waves, resulting to non-homogeneous wave fields. In this study, it is proven that by imposing the derived exact mathematical solutions of the governing equations at the model’s boundaries, the target first- and second-order wave profiles can be generated with high accuracy, while the spurious waves can be entirely eliminated.

Landau damping effects on dust ion-acoustic solitary waves in a nonthermal pair-ion plasma

The Landau damping effects on the nonlinear propagation of dust ion-acoustic solitary waves (DIASWs) are studied in an electron-free unmagnetized collisionless dusty pair-ion plasma. The two species of nonthermal ions (positive and negative) are described by the kinetic Vlasov equations, whereas the massive charged dust grains form only the background plasma. Using the multi-scale reductive perturbation technique generalized for the applications to the Vlasov equation, we derive a modified Korteweg–de Vries (KdV) equation that governs the evolution of DIASWs with the effects of linear resonance, in particular, linear Landau damping. The properties of the phase velocity and the Landau damping rate of DIASWs are studied with the effects of positive and negative-ion nonthermality parameters ( α p , α n ), the ratios of the positive to negative ion masses (m) and temperatures (T) as well as the dust to negative-ion number density ratio (δ). It is found that, depend on the degree of positive to negative ion mass ratios (m), both of damped and growing oscillations occur in a collisionless dusty pair-ion plasma due to the resonance phenomena between the wave and the nonthermal particles of the system. The properties of the decay rates of the amplitude of the KdV soliton with a small effect of Landau damping are also studied with the above system of parameters. The results may be useful for understanding the localization of solitary pulses and associated resonance damping and/or growing oscillations of the wave in laboratory and space plasmas, in which the number density of free electrons is much smaller than that of ions and highly charged heavy, micron seized dust grains.

Numerical Simulation for Solitary Waves of the Generalized Zakharov Equation Based on the Lattice Boltzmann Method

The generalized Zakharov equation is a widely used and crucial model in plasma physics, which helps to understand wave particle interactions and nonlinear wave propagation in plasma. The solitary wave solution of this equation provides insights into phenomena such as electron and ion acoustic waves, as well as magnetic field disturbances in plasma. The numerical simulation of solitary wave solutions to the generalized Zakharov equation is an interesting problem worth studying. This is crucial for plasma-based technology, as well as for understanding nonlinear wave propagation in plasma physics and other fields. In this study, a numerical investigation of the generalized Zakharov equation using the lattice Boltzmann method has been conducted. The lattice Boltzmann method is a new modeling and simulating method at the mesoscale. A lattice Boltzmann model was constructed by performing Taylor expansion, Chapman–Enskog expansion, and time multiscale expansion on the lattice Boltzmann equation. By defining the moments of the equilibrium distribution function appropriately, the macroscopic equation can be restored. Furthermore, the numerical experiments for the equation are carried out with the parameter lattice size m=100, time step Δt=0.001, and space step size Δx=0.4. The solitary wave solution of the equation is numerically simulated. Numerical results under different parameter values are compared, and the convergence and effectiveness of the model are numerically verified. It is obtained that the model is convergent in time and space, and the convergence orders are all 2.24881. The effectiveness of our model was also verified by comparing the numerical results of different numerical methods. The lattice Boltzmann method demonstrates advantages in both accuracy and CPU time. The results indicate that the lattice Boltzmann method is a good tool for computing the generalized Zakharov equation.

Supercontinuum generation in bulk solid-state material with bursts of femtosecond laser pulses

We report on experimental and numerical investigation of burst-mode supercontinuum generation in sapphire crystal. The experiments were performed using bursts consisting of two 190 fs, 1030 nm pulses with intra-burst repetition rates of 62.5 MHz and 2.5 GHz from an amplified 1 MHz Yb:KGW laser and revealed higher filamentation and supercontinuum generation threshold for the second pulse in the burst, which increases with the increase of intra-burst repetition rate. The experimental results were quantitatively reproduced numerically, using a developed model, which accounted for altered material response due to residual excitations remaining after propagation of the first pulse. The simulation results unveiled that residual free electron plasma and self-trapped excitons contribute to elevated densities of free electron plasma generated by the second pulse in the burst and so stronger plasma defocusing, significantly affecting its nonlinear propagation dynamics. The presented results identify the fundamental and practical issues for supercontinuum generation in solid-state materials using femtosecond pulse bursts with very high intra-burst repetition rates, which may also apply to the case of single pulses at very high repetition rate, where residual material excitations become relevant and should be accounted for.

Improvement of contrast ratio between cavitation bubbles and tissue by frequency filtering in triplet pulse ultrasound imaging

Bubble-enhanced high-intensity focused ultrasound treatment requires selective imaging of cavitation bubbles to ensure their localization to the target tissue. Previous studies have proposed the ultrasound imaging of cavitation bubbles with a filtering method, which makes bubble extraction by a triplet pulse sequence more selective. The envelope component as well as harmonic components are generated during the nonlinear propagation of an ultrasonic imaging pulse, but the triplet pulse sequence can significantly reduce only the harmonic components, and its selectivity is degraded by the residual envelope component. In this study, the effects of nonlinear propagation at various intensities of imaging pulse and the frequency filtering to obtain better selectivity are investigated for the triplet pulse sequence with a filtering method. The result shows that the imaging pulse intensity is an important parameter that determines the strength of the effect of nonlinear propagation and that the optimal frequency filtering changes according to it.

Computer simulation of a nonlinear wave model of transport phenomena in media with non-local effects

The main contribution of the presented article is that sufficient assumptions have been established for the modified Witham equation describing nonlinear wave propagation for transport phenomena in physicochemical systems. It is shown that when deriving Witham-type equations, the presence of spatial nonlocality of the medium can play a fundamental role. The article also discusses the issues of physical interpretation of the model under study. The principal presence of non-local effects in the studied systems can be justified and obtained as a result of the presence of domains with a complexly organized spatial structure, as well as the presence of heat and mass sources. Computer modeling was carried out and the results were obtained using the Java programming language.

Tracing the dynamics of superconducting order via transient terahertz third-harmonic generation.

Ultrafast optical control of quantum systems is an emerging field of physics. In particular, the possibility of light-driven superconductivity has attracted much of attention. To identify nonequilibrium superconductivity, it is necessary to measure fingerprints of superconductivity on ultrafast timescales. Recently, nonlinear THz third-harmonic generation (THG) was shown to directly probe the collective degrees of freedoms of the superconducting condensate, including the Higgs mode. Here, we extend this idea to light-driven nonequilibrium states in superconducting La2-xSrxCuO4, establishing an optical pump-THz-THG drive protocol to access the transient superconducting order-parameter quench and recovering on few-picosecond timescales. We show in particular the ability of two-dimensional TH spectroscopy to disentangle the effects of optically excited quasiparticles from the pure order-parameter dynamics, which are unavoidably mixed in the pump-driven linear THz response. Benchmarking the gap dynamics to existing experiments shows the ability of driven THG spectroscopy to overcome these limitations in ordinary pump-probe protocols.

Nonlinear generation of vector beams by using compact nonlinear fork grating

Nonlinear generation of vector beams by using compact nonlinear fork grating

Nonlinear asymmetric imaging with AlGaAs metasurface.

Nowadays, dielectric metasurfaces are a promising platform in many different research fields such as sensing, lasing, all-optical modulation and nonlinear optics. Among all the different kinds of such thin structures, asymmetric geometries are recently attracting increasing interest. In particular, nonlinear light-matter interaction in metasurfaces constitutes a valid approach for achieving miniaturized control over light. Here, we demonstrate nonlinear asymmetric generation of light in a dielectric metasurface via second harmonic generation. By inverting the illumination direction of the pump, the nonlinear emitted power is modulated by more than one order of magnitude. Moreover, we demonstrate how a properly designed metasurface can generate two completely different images at the second harmonic when the direction of illumination is reversed. Our results may pave the way to important opportunities for the realization of compact nanophotonic devices for imaging applications by densely integrating numerous nonlinear resonators.

Enhanced Gaussian-mixture-model-based nonlinear probabilistic uncertainty propagation using Gaussian splitting approach

Enhanced Gaussian-mixture-model-based nonlinear probabilistic uncertainty propagation using Gaussian splitting approach