Search

A new method is developed for finding exact solitary wave solutions of a generalized Korteweg–de Vries equation with p$p$‐power nonlinearity coupled to a linear Schrödinger equation arising in many different physical applications. This method yields 22 solution families, with p=1,2,3,4$p=1,2,3,4$. No solutions for p>1$p>1$ were known previously in the literature. For p=1$p=1$, four of the solution families contain bright/dark Davydov solitons of the 1st and 2nd kind, obtained in recent literature by basic ansatz applied to the ordinary differential equation (ODE) system for traveling waves. All of the new solution families have interesting features, including bright/dark peaks with (up to) p$p$ symmetric pairs of side peaks in the amplitude and a kink profile for the nonlinear part in the phase. The present method is fully systematic and involves several novel steps that reduce the traveling wave ODE system to a single nonlinear base ODE for which all polynomial solutions are found by symbolic computation. It is applicable more generally to other coupled nonlinear dispersive wave equations  as well as to nonlinear ODE systems of generalized Hénon–Heiles form. [ABSTRACT FROM AUTHOR]

The Hirota equation, an advanced variant of the nonlinear Schrödinger equation with cubic nonlinearity, incorporates time-delay adjustments and higher-order dispersion terms, offering an enhanced approximation for wave propagation in optical fibers and oceanic systems. By utilizing the traveling wave transformation generated from Lie point symmetry analysis with the combination of generalized exponential differential rational function and modified Bernoulli sub-ODE techniques, several traveling wave solutions, such as periodic, singular-periodic, and kink solitons, emerge. To examine the solutions visually, parametric values are adjusted to create 3D, contour, and 2D illustrations. Additionally, the dynamic properties of the model are explored through bifurcation analysis. The exact results demonstrate that both techniques are practical and robust. [ABSTRACT FROM AUTHOR]

The fourth-order nonlinear Boussinesq water wave equation, which describes the propagation of long waves in the intertidal zone, is investigated in this study. The exact wave patterns of the equation were computed using the t a n h method. As stability decreased, soliton wave structures were derived using similarity transformations. Numerical simulations supported these findings. The t a n h method introduced a Galilean modification, leading to the discovery of several new exact solutions. Subsequently, the fourth-order nonlinear Boussinesq wave equation was transformed into a planar dynamical system using the travelling wave transformation. The quasi-periodic, cyclical, and nonlinear behaviors of the analyzed equation were particularly examined. Numerical simulations revealed that varying the physical parameters impacts the system's nonlinear behavior. Graphs represent all possible examples of phase portraits in terms of these parameters. Furthermore, the study was proven to be highly beneficial for addressing issues such as shock waves and highly active travelling wave processes. Sensitivity analysis theory and the Lyapunov exponent were employed, offering a wide variety of linear periodic and first-frequency periodic characteristics. Sensitivity analysis and multistability analysis of the Boussinesq water wave equation were thoroughly investigated. [ABSTRACT FROM AUTHOR]

ABSTRACT This study focuses on the manipulation of soliton parameters within optical waveguides characterized by a dual‐power law refractive index, drawing similarities with published papers. The precise management of solitons holds significant importance in the field of fiber‐optical communication. While previous research has primarily concentrated on modifying soliton attributes such as amplitude and width, comparatively less attention has been dedicated to controlling the wave speed of solitons. In order to bridge this gap, the researchers adopted the employment of the nonlinear Schrödinger equation with time‐dependent dispersion and two power‐law nonlinearities, mirroring similar approaches found in published literature. By employing this methodology, the researchers successfully achieved active control over the wave speed of solitons in non‐uniform optical waveguides. To effectively handle the inherent complexities of the nonlinear system, two well‐established techniques, namely, the generalized Kudryashov method and the simple equation method, were utilized to derive solutions. These techniques have been previously employed and demonstrated in published papers, further enhancing the validity and reliability of the research findings. We discuss the traveling wave solutions by finding the fixed points of the dynamical planar system. [ABSTRACT FROM AUTHOR]

The perturbed nonlinear Schrödinger equation plays a crucial role in various scientific and technological fields. This equation, an extension of the classical nonlinear Schrödinger equation, incorporates perturbative effects that are essential for modeling real-world phenomena more accurately. In this paper, we investigate the traveling wave solutions of the perturbed nonlinear Schrödinger equation using the bifurcation theory of dynamical systems. Graphical presentations of the phase portrait are provided, revealing the traveling wave solutions under various conditions. By employing the auxiliary equation method, we derive a variety of solutions including periodic, dark, singular and bright optical solitons. To provide comprehensive and clearer depiction of the model’s behavior 2D, contour and 3D graphical representations are offered. We also highlight specific constraint conditions that ensure the presence of these obtained solutions. This study expands the scope of known exact solutions and their stability qualities which is offering an extensive analytical technique which enhances previous research. The novelty of our research lies in its examination of bifurcation analysis and the auxiliary equation method within the context of a perturbed nonlinear Schrödinger wave equation for the first time. By integrating these two perspectives, this paper contributes to establishing the complex dynamics and stability characteristics of soliton solutions under perturbations. [ABSTRACT FROM AUTHOR]

At an altitude of 1000 - 2000 km in the Venusian upper ionospheric region, the plasma is composed of H + ions, O + ions, and solar wind (SW) protons with electrons distributed according to the kappa distribution. This research delves into the exploration of nonlinear solitary and periodic waves, including the behavior of two-soliton. The Reductive Perturbation Technique (RPT) is used to derive the Korteweg-de Vries (KdV) equation for studying these nonlinear waves. Subsequently, a travelling wave transformation is employed to establish a planar dynamical system for the obtained KdV equation. Phase portrait is drawn and the associated nonlinear waves are examined adjusting the values of the parameters temperature ratio, unperturbed number density ratios, spectral index and travelling wave speed. Two-soliton solution of the KdV equation is investigated using the Hirota's direct approach. The influence of parameters like the spectral index, unperturbed number density ratios, and temperature ratio on the two-soliton is demonstrated, including the change in the phase shift. The findings from this research provide valuable insights into the dynamics of two-soliton solutions and nonlinear waves in the upper ionosphere of Venus. [ABSTRACT FROM AUTHOR]

This study examines the extended version of the Zakharov system characterizing the dispersive and ion acoustic wave propagation in plasma. The genuine, non-dispersive field depicts a shift in plasma ion density from its equilibrium state, whereas the complex, dispersive field depicts the fluctuating envelope of a highly oscillatory field of electricity. The main focus of the analysis is on employing the expanded Fan sub-equation approach to achieve some novel travelling wave structures including the explicit, periodic, linked wave, and other new exact solutions are developed for different values of this parameter. Three dimensional graphs are utilised to examine the properties of the obtained solutions. Furthermore, ideas from planar dynamical theory are applied in this work to analyse the intricate behaviour of the analysed model. Sensitivity analysis, multistability, quasi-periodic and chaotic patterns, Poincaré map, and the Lyapunov characteristic exponent are used to analyse the dynamical features. [ABSTRACT FROM AUTHOR]

In this paper, we apply the powerful sine-Gordon expansion method (SGEM), along with a computational program, to construct some new traveling wave soliton solutions for two models, including the higher-order nonlinear Boussinesq dynamical wave equation, which is a well-known nonlinear evolution model in mathematical physics, and the (1+1)-dimensional framework of the Van der Waals gas system. This study presents some new complex traveling wave solutions, as well as logarithmic and complex function properties. The 3D and 2D graphical representations of all obtained solutions, unveiling new properties of the considered model are simulated. Additionally, several simulations, including contour surfaces of the results, are performed, and we discuss their physical implications. A comprehensive conclusion is provided at the end of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we are going to investigate the (3+1)-dimensional nonlinear modified Korteweg–de Vries-Zakharov–Kuznetsov (mKdV–ZK) equation, which governs the behavior of weakly nonlinear ion acoustic waves in magnetized electron–positron plasma. By taking advantage of the newly proposed modified Sardar sub-equation method, we derive a comprehensive set of exact soliton solutions to the mKdV–ZK equation. Additionally, we provide graphical representations of the solutions, including 2D, 3D, and contour plots, to visualize the characteristics and features of these nonlinear wave structures. These solutions encompass a diverse range of wave patterns, including traveling waves, bright solitons, periodic waves, dark–bright solitons, lump-type solitons, and multi-soliton solutions. The obtained solutions provide valuable insights into the nonlinear behaviors and dynamics exhibited by the mKdV–ZK equation. The success of the new modified Sardar sub-equation method in obtaining a diverse range of solutions for the (3+1)-dimensional mKdV–ZK equation highlights its potential for applications in the analysis of various nonlinear systems in plasma physics and beyond. Also, the study reviewed the superiority of the modified method compared to the Sardar sub-equation method. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of a class of reaction-advection-diffusion equations of the form u t − Δ u + β (y) u x = f (t , u) in cylinders, where f is a tristable or multistable nonlinearity satisfying f (⋅ , 0) = f (⋅ , 1) = 0. We establish the alternative of wave solutions, that is, either there is a unique, asymptotically stable periodic traveling wave connecting 0 to 1 directly, or there is a unique propagating terrace connecting 0 to 1. Under the assumption that all wave speeds in propagating terrace are not equal to each other, we further obtain that the propagating terrace is asymptotically stable if it exists. Moreover, the sufficient and necessary conditions for the existence of periodic traveling waves connecting 0 to 1 are given respectively. Our arguments are based on the parabolic maximum principle, sliding method and the super- and sub-solution method. [ABSTRACT FROM AUTHOR]

This paper proposes an impedance-based approach for locating short-circuit faults in active distribution networks (DNs). This topic is a crucial task for operators, especially in grids with inverter-based distributed generators (IBDGs). Various methods have been proposed in this research area, including traveling waves, impedance-based methods, and artificial intelligence (AI) techniques. Among them, the impedance-based scheme offers a straightforward and efficient feature suitable for integration with AI-based techniques. This paper introduces an enhanced fault localization method based on impedance estimation, consisting of two main components: (i) fault distance determination and (ii) faulty section identification. This method accounts for the modeling of inverter-based resources under both symmetrical and asymmetrical faults, incorporating the impact and behavior of such sources. Unlike conventional impedance-based methods, our approach does not require network information such as structure, lines, load data, or voltage and current measurements along the feeder at multiple points. It can serve as a feature in AIbased techniques, significantly enhancing accuracy and reducing the complexity of such techniques. To validate the efficacy of the proposed approach, we conducted a series of time-domain case studies and provided mathematical proofs. The results demonstrate the effectiveness of our scheme in accurately locating faults with varying resistances at different positions in the presence of IBDGs. [ABSTRACT FROM AUTHOR]

This article delves into the dynamic constructions of distinctive traveling wave solutions for wave circulation in shallow water mechanics, specifically addressing the time-fractional couple Drinfel'd–Sokolov–Wilson (DSW) equation. Introducing the previously untapped e x p (- ϕ (ξ)) -expansion method, we successfully generate a diverse set of analytic solutions expressed in hyperbolic, trigonometric, and rational functions, each with permitted parameters. Visualization through three-dimensional (3D) as well two-dimensional (2D) plots, including contour plots, reveals inherent wave phenomena in the DSW equation. These newly obtained wave solutions serve as a catalyst for refining theories in applied science, offering a means to validate mathematical simulations for the proliferation of waves in shallow water as well as other nonlinear scenarios. Obtained wave solutions demonstrate the bright soliton, periodic, multiple soliton, and kink soliton shape. The simplicity and efficacy of the implemented methods are demonstrated, providing a valuable tool for approximating the considered equation. All figures are devoted to demonstrate the complete wave futures of the attained solutions to the studied equation with the collaboration of specific selections of the chosen parameters. Moreover, it may have summarized that the attained wave solutions and their physical phenomena might be useful to comprehend the various kind of wave propagation in mathematical physics and engineering. [ABSTRACT FROM AUTHOR]

The physics of shear waves traveling through matter carries fundamental insights into its structure, for instance, quantifying stiffness for disease characterization. However, the origin of shear wave attenuation in tissue is currently not properly understood. Attenuation is caused by two phenomena: absorption due to energy dissipation and scattering on structures such as vessels fundamentally tied to the material's microstructure. Here, we present a scattering theory in conjunction with magnetic resonance imaging, which enables the unraveling of a material's innate constitutive and scattering characteristics. By overcoming a three-order-of-magnitude scale difference between wavelength and average intervessel distance, we provide noninvasively a macroscopic measure of vascular architecture. The validity of the theory is demonstrated through simulations, phantoms, in vivo mice, and human experiments and compared against histology as gold standard. Our approach expands the field of imaging by using the dispersion properties of shear waves as macroscopic observable proxies for deciphering the underlying ultrastructures. [ABSTRACT FROM AUTHOR]

The Manakov model, a framework of coupled non-linear Schrödinger equations, reflects an essential integrable model in non-linear physics, particularly used to analyze soliton behavior and wave interactions in multiple physical realms such as optical theory and plasma physics. Its distinct integrability features make it a treasure trove for investigating complex interactions between wave propagation and non-linear systems. With an emphasis on the stochastic Manakov model, this research article explores the field of stochastic partial differential equations (SPDE). This writing aims to lower the cost and time frame for experiments by using particular integration approaches to generate a wide variety of optical solitons to satisfy the said model. We complete our thrust using two innovative approaches: the Unified Ricatti Equation Algorithm which depends on the Riccati equation and its special solutions, and the New Extended Auxiliary Equation Method which builds several, more general traveling wave solutions to nonlinear partial differential equations using a more straightforward and cohesive approach. The combination of these two not only improves our capability to handle complicated SPDEs but also provides fresh perspectives on how soliton structures and stochastic effects connect within the governing model. By employing these techniques, we obtain various solitary and traveling wave solutions, including dark, singular, and bright solitons. To ensure the validity of the results, the density, contour, 3D, and 2D profiles of various attained solutions are numerically generated by selecting appropriate values for several proposed framework parameters. The presence of multiplicative white noise does not propagate to other components; it just affects the phase portion of the solitons. Research conducted on the mentioned framework involving multiplicative white noise encounters significant applications in optical communication systems. By comprehending soliton behavior and effects, we can optimize the transmission of information in optical fibers, strengthen mode-locked lasers, and advance laser-based material handling. This exclusive blend of challenging model and methodologies led to the discovery of numerous unique soliton solutions and their corresponding behaviors. [ABSTRACT FROM AUTHOR]

The (2 + 1)-dimensional Heisenberg ferromagnetic spin chain equation which is used to characterize the nonlinear behavior of magnets and describe how ferromagnetic materials are ordered is the main subject of this investigation. The system's dynamics are governed by the Schrödinger equation, which explains the variations in its quantum state over time. We implement two modern and efficient techniques, the improved auxiliary equation scheme, and the improved tanh method to find the exact solutions of HSFC model and develop travelling wave solutions in terms of rational, trigonometric, and hyperbolic functions with arbitrary parameters. Numerical findings are shown in 3-D plots, 2-D curves, and contours with carefully selected parametric values used to illustrate the dynamic wave shape using MATLAB software. The acquired results provide important insights into the physical implications of nonlinear dynamics in magnetic systems and significantly contribute to the understanding of electromagnetic wave models. The article finishes with a thorough appraisal of the physical interpretations generated from the obtained results. [ABSTRACT FROM AUTHOR]

The generalized nonlinear Schrödinger equation with full dispersion (FDNLS) is considered in the semiclassical regime. The Whitham modulation equations are obtained for the FDNLS equation with general linear dispersion and a generalized, local nonlinearity. Assuming the existence of a four-parameter family of two-phase solutions, a multiple-scales approach yields a system of four independent, first-order, quasi-linear conservation laws of hydrodynamic type that correspond to the slow evolution of the two wavenumbers, mass, and momentum of modulated periodic traveling waves. The modulation equations are further analyzed in the dispersionless and weakly nonlinear regimes. The ill-posedness of the dispersionless equations corresponds to the classical criterion for modulational instability (MI). For modulations of linear waves, ill-posedness coincides with the generalized MI criterion, recently identified by Amiranashvili and Tobisch [New J. Phys., 21 (2019), 033029]. A new instability index is identified by the transition from real to complex characteristics for the weakly nonlinear modulation equations. This instability is associated with long wavelength modulations of nonlinear two-phase wavetrains and can exist even when the corresponding one-phase wavetrain is stable according to the generalized MI criterion. Another interpretation is that while infinitesimal perturbations of a periodic wave may not grow, small but finite amplitude perturbations may grow, hence this index identifies a nonlinear instability mechanism for one-phase waves. Classifications of instability indices for multiple FDNLS equations with higher-order dispersion, including applications to finite-depth water waves and the discrete NLS equation, are presented and compared with direct numerical simulations. [ABSTRACT FROM AUTHOR]

The Serre–Green–Naghdi equations of water wave theory have been widely employed to study undular bores. In this study, we introduce a modified Serre–Green–Naghdi system incorporating the effect of an artificial term that results in dispersive and dissipative dynamics. We show that the modified system effectively approximates the classical Serre–Green–Naghdi equations over sufficiently extended time intervals and admits dispersive–diffusive shock waves as traveling wave solutions. The traveling waves converge to the entropic shock wave solution of the shallow water equations when the dispersion and diffusion approach zero in a moderate dispersion regime. These findings contribute to an understanding of the formation of dispersive shock waves in the classical Serre–Green–Naghdi equations and the effects of diffusion in the generation and propagation of undular bores. [ABSTRACT FROM AUTHOR]

In this paper, exact periodic wave solutions for the perturbed Boussinesq equation with power law nonlinearity are obtained for different nonlinear strengths n. When n = 1 , the periodic traveling wave solutions can be found by the definition of the Jacobian elliptic function. When n ≥ 1 , we construct a transformation to solve for the power law nonlinearity, and the periodic traveling wave solutions can be obtained by applying the extended trial equation method. In addition, we consider the limiting case where the periodicity of the periodic traveling wave solutions vanishes, and we obtain the soliton solution for n = 1 . Numerical simulations show the periodicity of the solution for the perturbed Boussinesq equation. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation is one of the most important physical model in optical fiber theory for comprehension of the fluctuations of optical bullet development. In this study, the exact bullet solutions for the (3+1)-dimensional Schrödinger equation which demonstrate the bullet behaviours in optical fibers can be accumulated through the Sardar sub-equation method and the unified method. The applied strategies may retrieve several kinds of optical bullet solutions within one frameworks as well as is quite simple and reliable. Mathematica are utilised for describing the dynamics of different wave structures as 3D, 2D, and contour visualisations for a given set of parameters. As a result, we are able to develop a variety of travelling wave structures namely the periodic, singular and V shaped soliton wave solutions. The stability analysis for the derived results is analysed efficiently while the modulation instability for the governing model has also been studied to demonstrate the reliability of the research. The approaches implemented here works perfectly and can be extended to deal with many advanced models in contemporary areas of science and engineering. The solutions attain by using these techniques are robust, unique and straight forward and has applications in different fields of physics, engineering and mathematical science. Specially physical applications of these obtain results are in the transmission of data in optical fibers. We also add the graphics for the better understanding of the attain solutions behaviour. [ABSTRACT FROM AUTHOR]

We study an imbibition problem for porous media. When a wetted layer is above a dry medium, gravity leads to the propagation of the water downwards into the medium. In experiments, the occurrence of fingers was observed, a phenomenon that can be described with models that include hysteresis. In the present paper we describe a single finger in a moving frame and set up a free boundary problem to describe the shape and the motion of one finger that propagates with a constant speed. We show the existence of solutions to the travelling wave problem and investigate the system numerically. [ABSTRACT FROM AUTHOR]

The second partial differential equation from the Kaup–Newell hierarchy is considered. This equation can be employed to model pulse propagation in optical fiber, wave propagation in plasma, or high waves in the deep ocean. The integrability of the explored equation in traveling wave variables is investigated using the Painlevé test. Periodic and solitary wave solutions of the studied equation are presented. The investigated equation belongs to the class of generalized nonlinear Schrödinger equations and may be used for the description of optical solitons in a nonlinear medium. [ABSTRACT FROM AUTHOR]

In this work, we use two newly powerful and efficient techniques, the unified method and the unified Riccati equation expansion (UREE) method, to derive solitary wave solutions for the (1 + 1)-dimensional longitudinal wave equation (LWE), presenting a systematic approach to reveal the underlying dynamics. By incorporating mathematical analysis and numerical simulations, we investigate the behavior and properties of these solitary wave solutions in various fields of science. The derived solutions are entirely new, and these findings show the essential complexities of the longitudinal wave equation. These obtained solutions shed light on the fundamental mechanisms governing solitary waves in the longitudinal wave equation, contributing to a deeper understanding of nonlinear wave phenomena. We present effective visualizations of the dynamical wave structures using various graphs, such as 3D, 2D, and contour plots in the obtained solutions to improve our comprehensive understanding. Consequently, numerous types of wave profiles, including periodic wave, interaction between periodic wave and kink wave, W-shape wave, kink wave, interaction between breather wave and kink wave, traveling wave, mixed periodic, singular soliton, and some new types of soliton profiles were found. Consequently, this work presents a comprehensive study of solitary wave solutions using the unified method and UREE method for the (1 + 1)-dimensional LWE equation. The outcomes of the current study manifest that the considered methods are significant and systematic in solving nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

In this work, we take into account the (2+1)-Davey Stewartson equation (DSE) and the (2+1)-complex coupled Maccari system (CCMS) and their analytical solutions. Besides, we tackle the role of the problem parameters on the soliton behavior produced by the presented DSE. Exact traveling wave solutions are highly useful in numerical and analytical theories for such equations. While numerical methods are widely used, improving analytical approaches for obtaining analytical solutions is necessary for a deeper understanding of dynamics. This study marks a significant milestone by implementing the efficient analytical approach, enhanced modified extended tanh expansion method, for the first time to the (2+1)-DSE and (2+1)-CCMS equations, thereby making a notable contribution to the existing literature. We have shown that features of the soliton solutions can represent the spread of propagation on the wavefronts and show a reasonable dependency on parameter values. Some of the solutions discovered in three- and two-dimensional arrangements can also be described in graphic representations of their behavior. With the help of the graphical depictions, bright, singular, and periodic singular soliton characters for the (2 + 1) -DSE and singular, dark, bright, and periodic singular soliton characters for the (2 + 1) -CCMS are acquired. The results show that the utilized analytical technique is easily applicable, efficient, reliable, robust, and categorical when it comes to finding analytical solutions for different nonlinear models. Moreover, the problem parameters and the coupling coefficients have significant influences on the behavior of the solitons of the DSE, and this examination is studied for the first time in this article. [ABSTRACT FROM AUTHOR]

The Gilson–Pickering equation, which describes wave propagation in plasma physics and the structure of crystal lattice theory that is most frequently used. The discussed model is converted into a bilinear form utilizing the Hirota bilinear technique. Sets of case study are kink wave solutions; breather solutions; collision between soliton and periodic waves; soliton and periodic waves. The exp (- θ (α)) -function approach is employed to discover travelling wave solutions including five classes of solutions. In addition, it has been confirmed that the established findings are stable, and it has been helpful to validate the computations. The effect of the free variables on the behavior of obtained solutions to some plotted graphs for the exact cases is also explored depending upon the nature of nonlinearities. The exact solutions are utilized to demonstrate the physical natures of 3D, density, and 2D graphs. [ABSTRACT FROM AUTHOR]

With the developing technology, magneto-optical and ferromagnetic materials are gaining importance and are used especially in magneto-optics, ferromagnetism, fluid mechanics, etc. These processes are modeled via Kadomtsev–Petviashvili-type models. In this work, a generalized (3+1)-dimensional variable-coefficient modified Kadomtsev–Petviashvili (vcmKP) system and special cases are considered that simulates electromagnetic, water, and powder-acoustic/ion-acoustic/dust-ion-acoustic waves. As to the novelty of this paper, the travelling wave, soliton solutions of the considered systems are hold by using Bernoulli method which is the well-known ansatz-based method and the analytical method. As far as we know, the obtained solutions are seen for the first time in this study and are important for the development of the use of magneto-optical and ferromagnetic materials in industry and applied sciences, fiber optic communication fields. [ABSTRACT FROM AUTHOR]

In this research, the solitary wave solutions, the periodic type, and single soliton solutions are attained. The coupled fractional Lakshmanan–Porsezian–Daniel (LPD) equation is depicted the wave pulses' physical properties in birefringent optical fibres containing two vector solitons. Here, the Paul–Painlevé operator is employed to investigate kink soliton solutions. Also, the new modified exponential Jacobi technique is used to find periodic wave and soliton, bright-dark soliton. By utilizing symbolic computation and the applied methods, the mentioned system is successfully investigated. The coupled fractional LPD model is exhibited the travelling waves, as shown by the research in the current paper. Through three-dimensional graph, contour graph, density graph, complex-plot, and two-dimensional design using Maple, the physical features of single soliton and periodic wave solutions are explained all right. The findings the investigated model's broad variety of explicit solutions are demonstrated. As a result, the exact solitary wave solutions to the studied issues, including solitary, single soliton, and periodic wave solution are found. It is shown that the approach is practical and flexible in mathematical physics. All outcomes in this work are necessary to understand the physical meaning and behavior of the explored results and shed light on the significance of the investigation of several nonlinear wave phenomena in sciences and engineering. [ABSTRACT FROM AUTHOR]

In this work, the Extended Direct Algebraic Method (EDAM) is utilized to analyze and solve the fractional (2+1)-dimensional Conformable Broer-Kaup-Kupershmit System (CBKKS) and investigate different types of traveling wave solutions and study the soliton like-solutions. Using the suggested method, the fractional nonlinear partial differential equation (FNPDE) is primarily reduced to an integer-order nonlinear ordinary differential equation (NODE) under the traveling wave transformation, yielding an algebraic system of nonlinear equations. The ensuing algebraic systems are then solved to construct some families of soliton-like solutions and many other physical solutions. Some derived solutions are numerically analyzed using suitable values for the related parameters. The discovered soliton solutions grasp vital importance in fluid mechanics as they offer significant insight into the nonlinear behavior of the targeted model, opening the way for a deeper comprehension of complex physical phenomena and offering valuable applications in the associated areas. [ABSTRACT FROM AUTHOR]

This paper reviews the latest findings on instability and subcritical transition to turbulence in wall-bounded flows (i.e., pipe Poiseuille flow, plane channel flow, and plane Couette flow). The main focus was on the early stage of transitional flow and the appearance of coherent structures. The scaling of threshold disturbance amplitude for the onset of natural transition was discussed. Generally, the scaling proved to be in the form of Ac = O(Reγ ) for Newtonian fluids where Re is the Reynolds number, γ ≤ -1, and Ac is the critical perturbation amplitude. It was noted that exploration of perturbations like vortices, streaks, and traveling waves together with their amplitudes could clarify the instability and transition process. Hence, this paper focused on physical behavior and realizations of the transitional flow. Finally, a summary of consequential implications and some open issues for future works were presented and discussed. [ABSTRACT FROM AUTHOR]

Input/output (I/O) functions of distortion-product otoacoustic emissions (DPOAEs) may contain sudden amplitude minima (notches) although they are measured in animals with a negligible reflection source. We measured DPOAEs in humans and analyzed the data by time-frequency filtering to decompose the nonlinear-distortion and coherent-reflection components of DPOAE. The presented I/O functions of the nonlinear-distortion component contain notches. We suggest that because these notches are present only in the nonlinear-distortion component, they result from destructive interference between distortion-product wavelets coming from the primary generation region. Simulations conducted with a nonlinear cochlear model showed qualitative similarities with the presented experimental results. [ABSTRACT FROM AUTHOR]

The peaked cochlear tonotopic response does not show the typical phenomenology of a resonant system. Simulations of a 2 D viscous model show that the position of the peak is determined by the competition between a sharp pressure boost due to the increase in the real part of the wavenumber as the forward wave enters the short-wave region, and a sudden increase in the viscous losses, partly counteracted by the input power provided by the outer hair cells. This viewpoint also explains the peculiar experimental behavior of the cochlear admittance (broadly tuned and almost level-independent) in the peak region. [ABSTRACT FROM AUTHOR]

This paper mainly concentrates on obtaining solutions and other exact traveling wave solutions using the generalized G-expansion method. Some new exact solutions of the coupled nonlinear Schrödinger system using the mentioned method are extracted. This method is based on the general properties of the nonlinear model of expansion method with the support of the complete discrimination system for polynomial method and computer algebraic system (AS) such as Maple or Mathematica. The nonparaxial solitons with the propagation of ultra-broad nonparaxial pulses in a birefringent optical waveguide is studied. To attain this, an illustrative case of the coupled nonlinear Helmholtz (CNLH) system is given to illustrate the possibility and unwavering quality of the strategy utilized in this research. These solutions can be significant in the use of understanding the behavior of wave guides when studying Kerr medium, optical computing and optical beams in Kerr like nonlinear media. Physical meanings of solutions are simulated by various Figures in 2D and 3D along with density graphs. The constraint conditions of the existence of solutions are also reported in detail. Finally, the modulation instability analysis of the CNLH equation is presented in detail. [ABSTRACT FROM AUTHOR]

Optical soliton is a physical phenomenon in which the waveforms and energy of optical fibers remain unchanged during propagation, which has important application value in information transmission. In this paper, the coupled nonlinear Schrödinger equations describe the propagation of optical solitons with different frequencies in sense of local fractional derivative is analyzed. The exact traveling wave solutions of the non-differentiable type defined on the Cantor sets are obtained. The characteristics of the particular solutions of fixed fractal dimension are discussed. It is proved that the local fractional coupled nonlinear Schrödinger equations can describe the interaction of fractal waves in optical fiber transmission. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to study the optical soliton solutions and single traveling wave solutions of the generalized stochastic Schrödinger–Hirota equation in magneto-optic waveguides. With the help of the complete discriminant system technique and symbolic computation, a range of new single traveling wave solutions and optcial solitons are derived, which include Jacobian elliptic function solutions, dark solitons, trigonometric function solutions, singular solitons, rational function solutions, hyperbolic function solutions, periodic wave solutions and solitary wave solutions. Lastly, in order to understand mechanisms of complex physical phenomena and dynamical processes for the generalized stochastic Schrödinger–Hirota equation in magneto-optic waveguides, two-dimensional and three-dimensional diagrams are also drawn. [ABSTRACT FROM AUTHOR]

This paper investigates (2+1)-dimensional complex modified Korteweg–de Vries (cmKdV) system equations using the improved Sadar subequation method. It uncovers analytical solutions, including dark solitons, bright solitons, and periodic waves. The dynamic behavior of these solutions is illustrated through 2D and 3D plots by adjusting parameters. The results highlight the effectiveness and simplicity of the proposed methods, providing a versatile approach to obtaining various traveling wave solutions. These findings are expected to advance the shallow water theory of ideal fluids. [ABSTRACT FROM AUTHOR]

The purpose of the presented work is to obtain the exact solutions of the Paraxial wave equation using two different methods, the modified auxiliary equation method and G ′ G , 1 G -expansion method. Results obtained by these methods are unique. The paraxial wave equation is a partial differential equation that describes the propagation of optical waves in the paraxial approximation. This equation is derived from Maxwell's equations, which describe the behavior of electromagnetic waves. The paraxial wave equation is a fundamental tool in optics for analyzing the propagation of optical waves in the paraxial approximation. The application of integrating techniques allows researchers and engineers to obtain solutions and predict the behavior of optical systems in a wide range of applications. As a result of modified auxiliary equation method and G ′ G , 1 G -expansion method, new soliton solutions of the given model are extracted. These solutions include hyperbolic solutions and periodic solutions. The necessary constraint conditions for the existence of solutions are also provided. Moreover, for appropriate parametric values, acquired findings are displayed via contour, 2D and 3D graphics that show the physical relevance and dynamical behaviors of the proposed equation. It is noticed that proposed methods are well organized and efficient in obtaining accurate solutions for many nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

It is necessary to utilize certain stochastic methods while finding the soliton solutions since several physical systems are by their very nature stochastic. By adding randomness into the modeling process, researchers gain deeper insights into the impact of uncertainties on soliton evolution, stability, and interaction. In the realm of dynamics, deterministic models often encounter limitations, prompting the incorporation of stochastic techniques to provide a more comprehensive framework. Our attention was directed towards the short-wave intermediate dispersive variable (SIdV) equation with the Wiener process. By integrating advanced methodologies such as the modified Kudrayshov technique (KT), the generalized KT, and the sine-cosine method, we delved into the exploration of diverse solitary wave solutions. Through those sophisticated techniques, a spectrum of the traveling wave solutions was unveiled, encompassing both the bounded and singular manifestations. This approach not only expanded our understanding of wave dynamics but also shed light on the intricate interplay between deterministic and stochastic processes in physical systems. Solitons maintained stable periodicity but became vulnerable to increased noise, disrupting predictability. Dark solitons obtained in the results showed sensitivity to noise, amplifying variations in behavior. Furthermore, the localized wave patterns displayed sharp peaks and periodicity, with noise introducing heightened fluctuations, emphasizing stochastic influence on wave solutions. [ABSTRACT FROM AUTHOR]

The (2 + 1) -dimensional coupled nonlinear Schrödinger equation has versatile applications for modeling nonlinear waves in different areas such as optics, atmospheric science, fluid dynamics, and plasma physics. This study focuses on the propagation of optical solitons and their interaction within various mediums such as multi-mode fiber and fiber arrays. The Sardar-subequation method is applied to achieve the bright, dark, combined bright–dark, periodic, combined dark-singular and singular optical solitons solutions. The 3D, contour and 2D profiles for some of the assimilation solutions are also plotted to show the physical behavior of the solutions. The obtained results demonstrate the effectiveness of the adopted methodology in obtaining traveling wave solutions and have many applications in applied mathematics and engineering. The novelty of our work lies in the successful application of Sardar-subequation method in obtaining diverse soliton solutions and exploring their physical behavior. This study goes beyond previous efforts in the literature by presenting a comprehensive analysis of soliton dynamics. This ability to describe and predict the behavior of optical solitons in diverse mediums opens up new opportunities for investigating the existing systems. [ABSTRACT FROM AUTHOR]

This article mainly explores the Biswas–Milovic equation containing Kudryashov's refractive index law and nonlinear perturbation terms. We take the complete discrimination system for polynomial method and the trial equation method to implement qualitative and quantitative studies for the equation. We verify that the Hamiltonian of the dynamic system is conserved and apply the complete discrimination system of polynomial method to give the associated global phase diagrams, which are qualitatively analyzed to illustrate the presence of periodic solutions and solitons. In addition, we gain a classification of all single traveling wave solutions, provide optical wave patterns with specific parameters and validate the above findings. [ABSTRACT FROM AUTHOR]

In this paper, we considered a delayed predator-prey model with stage structure and Beddington-DeAngelis type functional response. First, we analyzed the stability of the constant equilibrium points of the model by the linear stability method. Furthermore, we considered the existence of traveling wave solutions connecting the zero equilibrium point and the unique positive equilibrium point. Second, we transformed the existence of traveling wave solutions into the existence of fixed points of an operator by constructing suitable upper and lower solutions, and combined with the Schauder fixed point theorem, we gave the existence of fixed points and obtained the existence of traveling wave solutions of the model. [ABSTRACT FROM AUTHOR]

This paper investigates different kinds of exact solutions of variable-coefficients Chiral Schrödinger equation (VCCNLSE). In this equation, the fractional quantum Hall effect edge states are described. By the unified method, we successfully get the soliton, solitary and elliptic wave solutions. Through the (G ′ / G) -expansion and the exponential expansion methods, we acquire different traveling wave solutions. Using the extended F -expansion method, we attain the Jacobi elliptic function solutions. So as to get more physical information about the exact solutions of VCCNLSE, 3D, 2D , density and contour maps are described. [ABSTRACT FROM AUTHOR]

In this study, Γ-shaped rectangular resonators, which are created in a metal insulator metal (MIM) structure, have been studied analytically and numerically. The metal is silver and rectangular resonators are filled by Si and the peak of transparency profile is tuned in the communication band with proper geometrical parameters. By employing 2-D finite difference time domain method (FDTD), simulations show a plasmonic induced transparency (PIT) window in the transmission diagram. The PIT is created by instructive and destructive interference between bright and dark resonator modes. The presented structure demonstrates tunable transparency window by variation of symmetry parameter, s. The maximum group velocity is obtained 0.1 ps in communication band corresponded to 30 μm traveling of light in the vacuum which is comparable to previous articles. The proposed structure may have potential applications in optical memory and delay blocks in designing integrated optical circuits. [ABSTRACT FROM AUTHOR]

In this study, we explore a captivating (3+1)-dimensional negative-order Korteweg–de Vries Calogero–Bogoyavlenskii–Schiff equation, which combines elements of the Korteweg–de Vries equation and the Calogero–Bogoyavlenskii–Schiff equation. Our research investigates how this model characterises long-wave interactions and its relevance in mathematics, physics, and engineering. We employ unified and singular manifold methods to obtain precise travelling wave solutions expressed in various functional forms. By using Maple and Mathematica software to extract valid solutions, including kink-like soliton, singular periodic wave solution, anti-kink solutions, and singular solitons. These methodologies have shown impressive efficiency in solving complex nonlinear equations, offering precise solutions, and streamlining mathematical processes through transformations. This leads to quicker and more accurate outcomes in diverse scientific and engineering applications. Our findings underscore the model's superiority over existing methods and its importance in comprehending applied mathematical processes, as demonstrated through 3-D and 2-D graphical representations. [ABSTRACT FROM AUTHOR]

It is well known that the Korteweg-de Vries (KdV) equation and its generalizations serve as modulation equations for traveling wave solutions to generic Fermi-Pasta-Ulam-Tsingou (FPUT) lattices. Explicit approximation estimates and other such results have been proved in this case. However, situations in which the defocusing modified KdV (mKdV) equation is expected to be the modulation equation have been much less studied. As seen in numerical experiments, the kink solution of the mKdV seems essential in understanding the $\beta$-FPUT recurrence. In this paper, we derive explicit approximation results for solutions of the FPUT using the mKdV as a modulation equation. In contrast to previous work, our estimates allow for solutions to be non-localized as to allow approximate kink solutions. These results allow us to conclude meta-stability results of kink-like solutions of the FPUT. [ABSTRACT FROM AUTHOR]

Gravity currents are of high interest both for their relevance in natural scenarios and because varying horizontal buoyancy presents a canonical problem in fluid mechanics [Huppert, "Gravity currents: A personal perspective," J. Fluid Mech. 554, 299–322 (2006)]. In this paper, attention is directed to gravity currents with a full-depth lock release propagating into a linearly stratified ambient fluid. For the case of an unstratified ambient, similarity solutions are known to capture the evolving height profile of the gravity current. We will compare this solution class with numerical data from high fidelity simulations. The presence of ambient stratification (quantified by the stratification intensity, S) introduces internal gravity waves that interact with the propagating current head, which will inhibit Kelvin–Helmholtz billows, decelerate current propagation, and smooth the shape of the current head. We perform direct numerical simulations of planar two- and three-dimensional gravity currents released into stratified ambient fluid of varying S and analyze the gravity current kinematics. Our analysis complements existing findings from performed laboratory and numerical experiments [Dai et al., "Gravity currents propagating at the base of a linearly stratified ambient," Phys. Fluids 33, 066601 (2021)] that show a stratified ambient modifies the current front velocity. Previous literature employed has inconsistent Reynolds numbers and boundary conditions, complicating interpretations. In the present numerical campaign, a closer analysis clarifies influence of the top boundary condition choice on formation and structure of the internal gravity waves. While acknowledging there is no available choice for a high-accuracy simplified numerical representation of a free-surface, a family of profiles for internal wave formation emerges varying with buoyancy Reynolds number and top boundary condition selection. The subsequent results appraise similarity solutions for the distribution of the heavy fluid in the gravity current. Our results show that for unstratified and low stratification ambient fluid, height profiles permit a similarity solution but higher values of S are less amenable; these profiles suggest a continuing time dependency on the traveling internal wave. [ABSTRACT FROM AUTHOR]

The study of water waves is significant for researchers working in many branches of science. The behaviour of waves can be studied by observation or experimental means, but theoretically, mathematical modeling provides solutions to many problems in physics and engineering. Progress in this field is inevitable, with those who work in mathematics, physics, and engineering putting forth interdisciplinary studies. Jacobi elliptic functions are valuable mathematical tools that can be applied to various aspects of mathematics, physics, and ocean engineering. In this study, traveling wave solutions of the general Drinfel'd-Sokolov-Wilson (DSW) system, introduced as a model of water waves, were obtained by using Jacobi elliptic functions and the wave dynamics were examined. The extended Jacobi elliptic function expansion method is an effective method for generating periodic solutions. It has been observed that the periodic solutions obtained by using Jacobi elliptic function expansions containing different Jacobi elliptic functions may be different and some new periodic solutions can be obtained. 3D simulations were made using MapleTM to see the behaviour of the solutions obtained for different appropriate values of the parameters. 2D simulations are presented for easy observation of wave motion. In addition, we transformed the one of the exact solutions found by the extended Jacobi elliptic function expansion method into the new solution under the symmetry transformation. [ABSTRACT FROM AUTHOR]

The present study focuses on the acquisition of traveling wave solutions associated with the conformable time-fractional Zoomeron equation through the utilization of the modified exponential function method (MEFM). The solution functions derived from mathematical computations encompass hyperbolic, trigonometric, and rational functions. Various graphical representations, such as 2D, 3D, contour graphs, and density graphs, are utilized to visually depict the distinct features of the solution functions derived from the determination of suitable parameters. [ABSTRACT FROM AUTHOR]

The modified unstable nonlinear Schrödinger equation governs specific instabilities observed in modulated wave-trains and also describes the time evolution of disturbances in marginally stable or unstable media. In this article, we explore the modified unstable dynamical model analytically using three symbolic computational techniques: the positive quadratic function approach, the three-wave approach, and the double exponential approach. As a result of our analysis, we have derived novel exact solutions, including rational solitons, multi-wave solutions, and other types of wave solutions for this dynamical model. These solutions are obtained through symbolic computation and the ansatz function method, incorporating both traveling wave and logarithmic transformations. The derived wave solutions hold practical significance, contributing significantly to understanding the physical phenomena of this complex model. We also computed conservative quantities such as power, momentum, and energy associated with solitons. To evaluate the stability of this modified unstable equation, we conducted a comprehensive modulational instability analysis, confirming the stability and exactness of all soliton solutions. By selecting appropriate parameter values, we generated 3D visual representations in various forms, including breather-type waves, lump waves, multi-peak solitons, etc. Furthermore, our observations revealed intriguing phenomena arising from the interactions among these multi-waves, with applications spanning a wide range of scientific and engineering disciplines. [ABSTRACT FROM AUTHOR]

The (2+1)-dimensional extended Calogero–Bogoyavlenskii–Schiff equation appears in the mathematical description of physical phenomena in plasma physics, fluid dynamics and nonlinear optics. In this article, extended trial and modified auxiliary equation methods are utilized to observe the dynamical structures exhibiting the solitary wave behavior of the considered model. The traveling wave hypothesis is employed to extract explicit closed-form solution expressions. The presented methods show reliability and robust computational capabilities to investigate solitary waves. In order to investigate the physical behavior of these solutions, 3D, 2D and density plots are drawn for different values of parameters. The graphical observations depict kink soliton, dark–bright singular soliton and periodic wave solutions. The comparison of the outcomes of the proposed results with those obtained using prior techniques is made to confirm the usefulness of the proposed techniques. The presented study will be helpful to explain the wave propagation in many problems of plasma physics and fluid dynamics. Moreover, the reported solutions may help to understand optical wave transmission and aid in the development of optical devices. The results given in this study will contribute in the understanding of the behavior of waves in the higher-dimensional governing models. [ABSTRACT FROM AUTHOR]

In this paper, we acquire novel traveling wave solutions of the generalized seventh-order Korteweg-de Vries equation and the seventh-order Kawahara equation as a special case with physical interest. Primarily, we use the advanced exp(φ,ξ)-expansion method to find new exact solutions of the first equation, by considering two auxiliary equations. Then, we attain some exact solutions of the seventh-order Kawahara equation by using this method with another auxiliary equation, and also using the modified (G'/G)-expansion method, where G satisfies a second-order linear ordinary differential equation. Additionally, utilizing the recent scientific instruments, the 2D, 3D, and contour plots are displayed. The solutions obtained in this paper include bright solitons, dark solitary wave solutions, and multiple dark solitary wave solutions. It is shown that these two methods provide an effective mathematical tool for solving nonlinear evolution equations arising in mathematical physics and engineering. [ABSTRACT FROM AUTHOR]

Soft actuators, designed for fragile item conveyance and navigation in complex environments, have garnered recent attention. This study proposes a cost-effective soft actuator, created by weaving tubes into twill patterns, capable of transportation and movement. The actuator achieves this by inducing traveling waves on its upper and lower surfaces through sequential pressurization of tubes. Notably, its fabrication does not require specialized molds, contributing to cost efficiency. The single actuator generates traveling waves with two degrees of freedom. Conventional silicone tube-based actuators demonstrate slow transport speeds (3.5 mm/s). To address this, this study replaced silicone tubes with pneumatic artificial muscles, enhancing overall body deformation and actuator speed. Experiments involving both extensional and contractional artificial muscles demonstrated that soft actuators with extensional artificial muscles significantly improved transportation and movement speed to 8.0 mm/s. [ABSTRACT FROM AUTHOR]