Nonlinear Physics Research Articles

Compliant Mechanism Synthesis Using Nonlinear Elastic Topology Optimization With Variable Boundary Conditions

ABSTRACTIn topology optimization of compliant mechanisms, the specific placement of boundary conditions strongly affects the resulting material distribution and performance of the design. At the same time, the most effective locations of the loads and supports are often difficult to find manually. This substantially limits topology optimization's effectiveness for many mechanism design problems. We remove this limitation by developing a method which automatically determines optimal positioning of a prescribed input displacement and a set of supports simultaneously with an optimal material layout. Using nonlinear elastic physics, we synthesize a variety of compliant mechanisms with large output displacements, snap‐through responses, and prescribed output paths, producing designs with significantly improved performance in every case tested. Compared to optimal designs generated using manually designed boundary conditions used in previous studies, the mechanisms presented in this paper see performance increases ranging from 47% to 380%. The results show that nonlinear mechanism responses may be particularly sensitive to boundary condition locations and that effective placements can be difficult to find without an automated method.

The extended Adomian decomposition method and its application to the rotating shallow water system for the numerical pulsrodon solutions

The rotating shallow water system is an important physical model, which has been widely used in many scientific areas, such as fluids, hydrodynamics, geophysics, oceanic and atmospheric dynamics. In this paper, we extend the application of the Adomian decomposition method from the single equation to the coupled system to investigate the numerical solutions of the rotating shallow water system with an underlying circular paraboloidal basin. By introducing some special initial values, we obtain interesting approximate pulsrodon solutions corresponding to pulsating elliptic warm-core rings, which take the form of realistic series solutions. Numerical results reveal that the numerical pulsrodon solutions can quickly converge to the exact solutions derived by Rogers and An, which fully shows the efficiency and accuracy of the proposed method. Note that the method proposed can be effectively used to construct numerical solutions of many nonlinear mathematical physics equations. The results obtained provide some potential theoretical guidance for experts to study the related phenomena in geography, oceanic and atmospheric science.

The exact analytical solutions of the (2+1)-dimensional extended Korteweg–de Vries equation using bilinear neural network method and bilinear residual network method

The [Formula: see text]-dimensional extended Korteweg–de Vries (KdV) equation is predominantly utilized to elucidate the propagation of waves that exhibit both dispersive and nonlinear characteristics within the domain of nonlinear physics. This paper employs the bilinear neural network method (BNNM) to derive the exact analytical solutions of the equation. By constructing various bilinear neural network models, we obtain lump solution, breather solution and periodic interaction solution of the equation. The bilinear residual network method (BRNM) is an extension of BNNM. We apply BRNM under specific constraints to obtain breather solution of the equation, thereby offering a broader conceptual framework. Subsequently, various 3D plots, contour plots, density plots and x-curves are used to illustrate the physical properties and dynamic behaviors of these waves.

Quantifying the role of electron plateau distribution on the chorus gap formation using one-dimension PIC simulations

Although ubiquitous in the Earth’s inner magnetosphere, how chorus waves with a gap in intensity near half the electron cyclotron frequency, 0.5fce, is formed remains to be understood. One hypothesis invokes the often-observed feature in electron phase space density where two anisotropic populations of warm and hot temperatures, respectively, are separated by a relatively isotropic beam-like component, called the parallel plateau distribution. According to linear theory, its mean velocity is such that the plateau population can lead to a sufficient cyclotron damping in a narrow frequency range near 0.5fce. To test this hypothesis more quantitatively, we carry out a series of one-dimensional particle-in-cell simulations in a parabolic magnetic field with observationally constrained plateau density and mean velocity. We find that even though the chorus generation involves nonlinear physics, the gap formation is largely determined by the linear properties of plateau electrons. In addition, given a sufficiently large plateau density with a right combination of warm and/or hot source populations, our simulations were able to generate lower-band-only chorus, upper-band-only chorus, banded chorus with partial damping at the gap, and banded chorus with a strong power gap. When it comes to comparing with observations, however, we find that (1) the minimum plateau density required to generate banded chorus is somewhat larger than that observed; (2) even with the largest plateau density used, the gap formation in the simulations is not consistent; and (3) most importantly, the dependence of the gap frequency on the plateau velocity in our parametric analysis differs quite substantially from that of the recent statistical results. Provided that the simplifying assumptions in our simulations remain valid, these discrepancies would indicate that the electron plateau distribution is not the major contributor to the chorus gap formation.

An optimization-based equilibrium measure describes fixed points of non-equilibrium dynamics: application to edge of chaos

Abstract Understanding neural dynamics is a central topic in machine learning, non-linear physics and neuroscience. However, the dynamics is non-linear, stochastic and particularly non-gradient, i.e., the driving force can not be written as gradient of a potential. These features make analytic studies very challenging. The common tool is the path integral approach or dynamical mean-field theory, but the drawback is that one has to solve the integro-differential or dynamical mean-field equations, which is computationally expensive and has no closed form solutions in general. From the aspect of associated Fokker-Planck equation, the steady state solution is generally unknown. Here, we treat searching for the steady states as an optimization problem, and construct an approximate potential related to the speed of the dynamics, and find that searching for the ground state of this potential is equivalent to running an approximate stochastic gradient dynamics or Langevin dynamics. Only in the zero temperature limit, the distribution of the original steady states can be achieved. The resultant stationary state of the dynamics follows exactly the canonical Boltzmann measure. Within this framework, the quenched disorder intrinsic in the neural networks can be averaged out by applying the replica method, which leads naturally to order parameters for the non-equilibrium steady states. Our theory reproduces the well-known result of edge-of-chaos, and further the order parameters characterizing the continuous transition are derived, and the order parameters are explained as fluctuations and responses of the steady states. Our method thus opens the door to analytically study the steady state landscape of the deterministic or stochastic high dimensional dynamics.

A multi-step calibration strategy for reliable parameter determination of salt rock mechanics constitutive models

The storage of renewable hydrogen in salt caverns requires fast injection and production rates to cope with the imbalance between energy production and consumption. This raises concerns about the mechanical stability of salt caverns under such operational conditions. The use of appropriate constitutive models for salt mechanics is an important step in investigating this issue, therefore many constitutive models with several parameters have been presented in the literature. However, a robust calibration strategy to reliably determine which model and parameter set represents the given rock, based on stress–strain data sets, remains an unsolved challenge. In this paper, for the first time in the community, we present a multi-step strategy to determine a single parameter set based on many deformation data sets for salt rocks. Towards this end, we first develop a comprehensive constitutive model able to capture all relevant nonlinear deformation physics of transient, reverse, and steady-state creep. The determination of the single set of representative material parameters is then achieved by framing the calibration process as an optimization problem, for which the global Particle Swarm Optimization algorithm is employed. To allow for dynamic data integration, a multi-step calibration strategy is developed for a situation where experiments are included one at a time, as they become available. Additionally, due to the existing mild heterogeneity in the experimental rock samples, our optimization strategy is made flexible to allow for this slight heterogeneity. The devised optimization strategy, based on the multi-physics comprehensive constitutive modeling framework, results in a single set of representative material properties of all the deformation data sets. As a rigorous mathematical analysis for the presented method and the lack of relevant experimental data sets, we consider a wide range of synthetic experimental data sets, inspired by the existing sparse relevant data in the literature. The results of our performance analyses show that the proposed calibration strategy is robust. Moreover, the results become increasingly more accurate as more data sets become available.

Soliton solutions of nonlinear coupled Davey–Stewartson Fokas system using modified auxiliary equation method and extended (G′/G2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(G'/G^{2})$$\\end{document}-expansion method

The captivating realm of the nonlinear coupled Davey–Stewartson Fokas system is explored in this research paper. As a powerful tool, the proposed system is utilized for the realistic representation of various non-linear dynamical mechanisms in different fields of sciences and engineering including non-linear optical fibers, plasma physics and water waves theory. Two distinct exact methods, namely the modified auxiliary equation method and the extended (G′/G2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(G'/G^{2})$$\\end{document}-expansion method, are utilized to acquire the exact soliton solutions of the non-linear coupled Davey–Stewartson Fokas system. A plethora of novel soliton solutions containing anti-kink, kink, bright, dark, dark-bright, bright-dark and some other singular soliton solutions, have been obtained using the employed exact methods. The significance of proposed manuscript lies in the novelty of obtained solutions. Kink, dark and bright solitons have wide applications in optical fiber communications, plasma physics and water waves dynamics. The acquired nontrivial exact solutions contain exponential, trigonometric, rational and hyperbolic functions. Some obtained solutions are visually represented through graphical simulations of 3D, 2D-contour and 2D-line plots, providing a comprehensive visualization of the soliton dynamics.The modulation instability of the proposed nonlinear system has been investigated, which ensures the stability of the system.

Nonlinear physics of moiré superlattices.

Nonlinear physics is one of the most important research fields in modern physics and materials science. It offers an unprecedented paradigm for exploring many fascinating physical phenomena and realizing diverse cutting-edge applications inconceivable in the framework of linear processes. Here we review the recent theoretical and experimental progress concerning the nonlinear physics of synthetic quantum moiré superlattices. We focus on the emerging nonlinear electronic, optical and optoelectronic properties of moiré superlattices, including but not limited to the nonlinear anomalous Hall effect, dynamically twistable harmonic generation, nonlinear optical chirality, ultralow-power-threshold optical solitons and spontaneous photogalvanic effect. We also present our perspectives on the future opportunities and challenges in this rapidly progressing field, and highlight the implications for advances in both fundamental physics and technological innovations.

Inverse Scattering Integrability and Fractional Soliton Solutions of a Variable-Coefficient Fractional-Order KdV-Type Equation

In the field of nonlinear mathematical physics, Ablowitz et al.’s algorithm has recently made significant progress in the inverse scattering transform (IST) of fractional-order nonlinear evolution equations (fNLEEs). However, the solved fNLEEs are all constant-coefficient models. In this study, we establish a fractional-order KdV (fKdV)-type equation with variable coefficients and show that the IST is capable of solving the variable-coefficient fKdV (vcfKdV)-type equation. Firstly, according to Ablowitz et al.’s fractional-order algorithm and the anomalous dispersion relation, we derive the vcfKdV-type equation contained in a new class of integrable fNLEEs, which can be used to describe the dispersion transport in fractal media. Secondly, we reconstruct the potential function based on the time-dependent scattering data, and rewrite the explicit form of the vcfKdV-type equation using the completeness of eigenfunctions. Thirdly, under the assumption of reflectionless potential, we obtain an explicit expression for the fractional n-soliton solution of the vcfKdV-type equation. Finally, as specific examples, we study the spatial structures of the obtained fractional one- and two-soliton solutions. We find that the fractional soliton solutions and their linear, X-shaped, parabolic, sine/cosine, and semi-sine/semi-cosine trajectories formed on the coordinate plane have power–law dependence on discrete spectral parameters and are also affected by variable coefficients, which may have research value for the related hyperdispersion transport in fractional-order nonlinear media.

The impact of standard Wiener process on the optical solutions of the stochastic nonlinear Kodama equation using two different methods

The stochastic nonlinear Kodama (SNLK) equation forced in the Stratonovich sense by multiplicative noise is considered here. New elliptic, hyperbolic, trigonometric, and rational stochastic solutions are acquired using ( G′/ G)-expansion method and mapping method. Because the SNLK equation is extensively used in is extensively used in nonlinear optics, fluid dynamics, and plasma physics, the obtained solutions may be used to study a broad range of relevant physical phenomena. In order to interpret the effects of multiplicative noise, the dynamic performances of the various obtained solutions are displayed utilizing 3D and 2D graphs. We infer that multiplicative noise impacts and stabilizes the solutions of SNLK equation.

In plasma physics and fluid dynamics: Symbolic computation on a (2+1)-dimensional variable-coefficient Sawada-Kotera system

Investigations on the real world have been facilitating the development of nonlinear science, and today, fluid dynamics and plasma physics attract people’s attention. This Letter studies a (2+1)-dimensional variable-coefficient Sawada-Kotera system in plasma physics and fluid dynamics. We build up a family of the similarity reductions, connecting that system with a known ordinary differential equation. Our similarity reductions depend on the plasma/fluid variable coefficients in that system, under certain variable-coefficient constraints.

Analytic soliton solutions to the shallow water dispersive long gravity wave equations: the first integral approach in nonlinear physics

In this article, we investigate the (2+1)-dimensional dispersive long water wave equation and the (1+1)-dimensional Phi-four equation, which describe the behavior of long gravity waves with small amplitudes, long wave propagation in oceans and seas, coastal structures and harbor design, effects of wave motion on sediment transport, quantum field theory, phase transitions of matter, ferromagnetic systems, liquid-gas transitions, and the structure of optical solitons. We use the first integral technique and obtain new and generic solutions for the models under consideration. By setting definite values for the associated parameters, various types of richly structured solitons are generated. The solitons include kink, flat kink, bell-shaped, anti-bell-shaped, and singular kink formations. These solutions allow for a profound understanding of the behavior and properties of the phenomena, offering new insights and potential applications in the associated field. The first integral technique is simpler, directly integrates the models, and the solutions offer clear insights into the underlying phenomena without requiring intermediate steps, making it widely applicable to various other models, including nonlinear equations and those that are challenging to solve using other standard techniques.

Nonlinear lower hybrid wave equations in collisional tokamak plasmas

A new set of coupled integro-differential nonlinear lower hybrid (LH) wave equations is derived within the framework of a kinetic theory coupled to the Maxwell equations to study the parametric instabilities (PIs) produced by LH waves in collisional tokamak plasma. Previous models of nonlinear LH wave equations have been significantly improved. The wave equations derived overcome the limits and incorrectness of the standard theory of the PI in inhomogeneous plasma. They allow us to treat the full spectrum in the parallel and poloidal wavenumber of the coupled LH power wave, diffraction effects and possible cascade phenomena, which are elements of the nonlinear LH physics ignored in the standard PI theory. Numerical solutions of the new nonlinear LH wave equations are proposed. The relevant LH frequency spectra produced by PI are calculated, exhibiting characteristic features of PI observed in LH experiments. It is shown that the LH sideband amplification can be overestimated by orders of magnitude by the standard theory of PI. A benchmark of the new model is provided for spatially homogeneous plasmas. The role of the collisions for PI has been assessed. We demonstrate that previous analyses significantly overestimated their stabilization effect.

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.

Bifurcation study, phase portraits and optical solitons of dual-mode resonant nonlinear Schrodinger dynamical equation with Kerr law non-linearity

This study investigates the dynamic characteristics of the dual-mode resonant non-linear Schrodinger equation with a Bhom potential. Hydrodynamics, nonlinear optical fibre communication, elastic media, and plasma physics are just a few of the mathematical physics and engineering applications for this model. The study aims to achieve two main objectives: first, to discuss bifurcation analysis, and second, to extract optical soliton solutions using the extended hyperbolic function method. The study successfully derives various wave solutions, including bright, singular, periodic singular and dark solitons, based on the governing model. The findings conferred in this article show a crucial advancement in understanding the propagation of waves in non-linear media. Additionally, bifurcation of phase portraits of ordinary differential equation consistent with the partial differential equation under consideration is conducted. We also highlight specific constraint conditions that ensure the presence of these obtained solutions. The existing literature shows that these methods are first time applied on this model.

Nonlinear optical physics at terahertz frequency

Abstract Terahertz (THz) waves have exhibited promising prospects in 6G/7G communications, sensing, nondestructive detection, material modulation, and biomedical applications. With the development of high-power THz sources, more and more nonlinear optical effects at THz frequency and THz-induced nonlinear optical phenomena are investigated. These studies not only show a clear physics picture of electrons, ions, and molecules but also provide many novel applications in sensing, imaging, communications, and aerospace. Here, we review recent developments in THz nonlinear physics and THz-induced nonlinear optical phenomena. This review provides an overview and illustrates examples of how to achieve strong THz nonlinear phenomena and how to use THz waves to achieve nonlinear material modulation.

Nonlinear lattices from the physics of ecosystems: The Lefever–Lejeune nonlinear lattice in ℤ2

We argue that the spatial discretization of the strongly nonlinear Lefever–Lejeune partial differential equation defines a nonlinear lattice that is physically relevant in the context of the nonlinear physics of ecosystems, modelling the dynamics of vegetation densities in dry lands. We study the system in the lattice , which is especially relevant because of its natural dimension for the emergence of pattern formation. Theoretical results identify parametric regimes for the system that distinguish between extinction and potential convergence to nontrivial states. Importantly, we analytically identify conditions for Turing instability, detecting thresholds on the discretization parameter for the manifestation of this mechanism. Numerical simulations reveal the sharpness of the analytical conditions for instability and illustrate the rich potential for pattern formation even in the strongly discrete regime, emphasizing the importance of the interplay between higher dimensionality and discreteness.

Dynamics of transformed nonlinear waves for the (2+1)-dimensional pKP-BKP equation: interactions and molecular waves

In this paper, the dynamical behaviors of transformed nonlinear waves for the (2+1)-dimensional combined potential Kadomtsev-Petviashvili and B-type Kadomtsev-Petviashvili (pKP-BKP) equation are investigated, which can be used to reveal the nonlinear wave phenomena in nonlinear optics, plasma physics and hydrodynamics. The breath-wave and the lump solutions are constructed by means of the soliton solutions. The conversion mechanism for the breath-wave is systematically analyzed, which leads to several new kink-shaped nonlinear waves. The gradient relationships of these transformed waves are revealed by a Riemannian circle. Through the analysis of the nonlinear superposition between the periodic wave component and the kink solitary wave component, the dynamical characteristics including the formation mechanism, oscillation and locality for the nonlinear waves are investigated. The time-varying properties of transformed waves are shown by the study of time variables. By virtue of the two breath-wave solutions, several interactions including elastic and inelastic collisions between two nonlinear waves are studied. In particular, some transformed molecular waves encompassing the non-, semi- and full-transition modes are presented with the aid of velocity resonance. The results can help us further understand the complex nonlinear waves existing in the integrable systems.

Novel dynamics of the Fokas-Lenells model in Birefringent fibers applying different integration algorithms

Abstract The Fokas-Lenells model has the broad applications in nonlinear physics to study various soliton phenomena. Employing the direct algebraic scheme, the modified rational sine-cosine technique, and the (1/G′) expansion scheme, the analytical solutions to this model are derived. Double periodic waves, bright soliton, dark soliton, single and multiple breather waves, and periodic breather waves are extracted from this model using symbolic computation. The dynamic behaviors of the acquired outcomes are vividly illustrated through density, two-dimensional (2D), and three-dimensional (3D) graphical representations. These discoveries are strategically positioned to significantly contribute to the advancement of exploring nonlinear models, standing as a fundamental pillar for forthcoming research endeavors.

Inverse design of coherent supercontinuum generation using free-form nanophotonic waveguides

Many key functionalities of optical frequency combs, such as self-referencing and broad spectral access, rely on coherent supercontinuum generation (SCG). While nanophotonic waveguides have emerged as a compact and power-efficient platform for SCG, their geometric degrees of freedom have not been fully utilized due to the underlying complex nonlinear and noise-dependent stochastic physics. Here, we introduce inverse design to unlock free-form waveguides for coherent SCG. The efficacy of our design is numerically and experimentally demonstrated on Si3N4 waveguides, producing flat and coherent spectra from visible to mid-infrared wavelengths. Our work has direct applications in developing chip-based broadband light sources for spectroscopy, metrology, and sensing across multiple spectral regimes.