Wave Propagation In Nonlinear Media Research Articles

Solitary wave solutions and investigation the effects of different wave velocities of the nonlinear modified Zakharov–Kuznetsov model for the wave propagation in nonlinear media

The modified Zakharov-Kuznetsov (mZK) model convey a significant role to analyze the inner mechanism of physical compound phenomenon in the field of two-dimensional discrete electrical lattice, the electrical waves in cold plasmas, plasma physics, nonlinear optics, wave behaviors of deep oceans, etc. In this study, numerous new, advanced, and more general exact solitary travelling wave solutions are explored to the early stated nonlinear model by the aid of newly developed generalized exponential rational function (GERF) method through the traveling wave transformation. The established solutions are in terms of trigonometric functions, rational functions, hyperbolic functions, exponential rational functions, and complex-soliton solutions. The results reveal that the free parameters significantly influence the existence of traveling waves, as well as nature, shape, and stability. The established soliton solutions demonstrate that the method is effective, compatible, scientifically efficient and easily applicable to identify a variety of wave structures of nonlinear evolution models. The mathematical simulations of the results were conducted by sketching 3D and 2D structures for different values of the associated parameters by the aid of the Wolfram Mathematica program.

On soliton solutions of the modified equal width equation

PurposeThe soliton solutions are obtained by using extended rational sin/cos and sinh-cosh method. The methods are powerful and have ease of use. Applying wave transformation to the nonlinear partial differential equations (NLPDEs) and the considered equation turns into a nonlinear differential equation (NODE). According to the methods, the solution sets of the NODE are supposed to the form of the rational terms as sinh/cosh and sin/cos and the trial solutions are substituted into the NODE. Collecting the same power of the trigonometric functions, a set of algebraic equations is derived.Design/methodology/approachThe main purpose of this paper is to obtain soliton solutions of the modified equal width (MEW) equation. MEW is a form of regularized-long-wave (RLW) equation that represents one-dimensional wave propagation in nonlinear media with dispersion processes. This is also used to simulate the undular bore in a long shallow water canal.FindingsThus, the solution of the main PDE is reduced to the solution of a set of algebraic equations. In this paper, the kink, singular and singular periodic solitons have been successfully obtained.Originality/valueIllustrative plots of the solutions have been presented for physical interpretation of the obtained solutions. The methods are powerful and might be used to solve a broad class of differential equations in real-life problems.

Exploring the dynamics of soliton waves: A comparative analysis of analytical and numerical methods for the modified Equal-Width equation

The modified Equal-Width (MEW) equation is a fundamental model in the realm of partial differential equations that is used to simulate one-dimensional wave propagation in nonlinear media, incorporating dispersion processes. In this study, we have employed two contemporary analytical and numerical methods to obtain exact traveling wave solutions for this model. The derived solutions have been evaluated using a numerical technique, demonstrating excellent accuracy. Various graphical representations, including a contour plot, a two-dimensional graph, and a three-dimensional graph, have been utilized to illustrate and analyze the results. Our study offers significant contributions to the field by providing new insights into the dynamics of solitary wave solutions of the MEW model. Furthermore, our method proves to be a valuable and effective mathematical tool that can be utilized to solve various nonlinear wave problems, making it highly applicable in a range of physical and mathematical studies. By showcasing the properties of the investigated model and the primary advantages of the employed techniques, our study highlights the potential for further developments and advancements in this field.

Computational and numerical simulations of the wave propagation in nonlinear media with dispersion processes

In partial differential equations, the generalized modified equal-width (GMEW) equation is commonly used to model one-dimensional wave propagation in nonlinear media with dispersion processes. In this article, we use two modern, accurate analytical and numerical techniques to find the exact traveling wave solutions for the model we are looking at. The results are new, and at present, they can be used in many different areas of research, such as engineering and physics. The proposed numerical method is helpful because it gives an estimate on the accuracy of the solutions. Distinct graphs, such as a contour plot, a two-dimensional graph, and a three-dimensional graph, were used to show the analytical and numerical results. Using symbolic computation, we demonstrate that our approach is a powerful mathematical tool that can be applied to a wide range of nonlinear wave problems.

Correction to: 2-D P-SV and SH spectral element modelling of seismic wave propagation in non-linear media with pore-pressure effects

Correction to: 2-D P-SV and SH spectral element modelling of seismic wave propagation in non-linear media with pore-pressure effects

Wave propagation behavior in nonlinear media and resonant nonlinear interactions

The nonlinear Landau–Ginsberg–Higgs model, which depicts wave propagation in nonlinear media with a scattering system, classifies wave velocities, and effectively generates real events, is examined in this article. We use the recently enhanced couple of rising procedures to extract the important, applicable, and further general solitary wave solutions to the formerly stated nonlinear wave model via the complex traveling wave transformation. The solitons are constructed in terms of hyperbolic, exponential, rational, and trigonometric functions as well as their integration. The physical implications of the extracted wave solutions for the specific values of the resultant parameters are illustrated graphically, and the internal structure of the connected physical phenomena is analyzed using the Wolfram Mathematica program. This study shows that the method utilized is effective and may be used to find appropriate closed-form solitary solitons to a variety of nonlinear evolution equations (NLEEs).

Generalized variational principle and periodic wave solution to the modified equal width-Burgers equation in nonlinear dispersion media

The main purpose of this paper is to investigate the modified equal width-Burgers equation that is used wildly to describe long wave propagation in nonlinear media with dispersion and dissipation. With the help of the Semi inverse method, we successfully establish its generalized variational principle for the first time, which can reveal the energy conservation law of the whole solution domain. Then a novel and interesting method called He's frequency formulation, which is derived from the ancient Chinese algorithm (ACG)-Ying-Bu-Zu-Shu, is employed to construct its periodic solution. Finally, two examples are presented to illustrate the solutions in the form of 3-D and 2-D plots. The results in this work are expected to give certain guiding significance for the study of variational theory and periodic wave theory of physical equations.

Wave propagation for the nonlinear modified Kortewege–de Vries Zakharov–Kuznetsov and extended Zakharov–Kuznetsov dynamical equations arising in nonlinear wave media

In this study, we construct the new solitary wave solutions for two well known nonlinear wave equations whose describe the wave propagation in nonlinear media. We implemented a new technique which is extension of modified rational expansion method on nonlinear three-dim modified Kortewege–de Vries Zakharov–Kuznetsov and extended Zakharov–Kuznetsov equations. The new results are elliptic, hyperbolic, trigonometric, rational type which are representing to solitary waves, periodic waves, traveling waves, kink-antikink solitons, dark-bright solitons. The physical behavior of some calculated results represented graphical by using the software Mathematica. The determined results are new, more general and have numerous applications in the area of nonlinear sciences including nonlinear plasma, fiber optics, mechanics, quantum physics, laser physics, fluid physics, engineering and other types of physical sciences. The demonstrated results show that new technique is efficient, fruitful, powerful and reliable to study analytically different types of higher order complex NLPDEs arises in various types of applied sciences.

Stable wave solutions to the Landau-Ginzburg-Higgs equation and the modified equal width wave equation using the IBSEF method

The Landau-Ginzburg-Higgs equation and the modified equal width wave equation (MEWE) underscore to describe superconductivity and unidirectional wave propagation in nonlinear media with dispersion systems. In the present study, the improved Bernoulli sub-equation function method (IBSEFM) has been introduced to accomplish applicable soliton solutions to the above stated wave equations. We ascertain adequate soliton solutions, videlicet the combination of hyperbolic function, exponential function etc. and speculate the physical significance of the obtained solutions for the definite values of the included parameters through depicting graphs and interpreted the physical phenomena. It is shown that the IBSEFM is powerful, suitable, direct and provide general wave solutions to NLEEs in mathematical physics.

Exact solutions of equal-width equation and its conservation laws

Abstract In this work we investigate the equal-width equation, which is used for simulation of (1-D) wave propagation in non-linear medium with dispersion process. Firstly, Lie symmetries are determined and then used to establish an optimal system of one-dimensional subalgebras. Thereafter with its aid we perform symmetry reductions and compute new invariant solutions, which are snoidal and cnoidal waves. Additionally, the conservation laws for the aforementioned equation are established by invoking multiplier method and Noether’s theorem.

Arising wave propagation in nonlinear media for the (2+1)-dimensional Heisenberg ferromagnetic spin chain dynamical model

In this study, we study the (2+1)-dimensional nonlinear Schrödinger kind equation which illustrates the non-linear spin dynamics of (2+1)-dimensional Heisenberg ferromagnetic spin chains (HFSCs) through bilinear and anisotropic interactions in the semi-classical limit. This model also illustrates the ferromagnetic materials of magnetic ordering. Two analytical schemes are employed to study the equation namely, improved auxiliary equation and generalized Riccati mapping methods. We construct dark and bright solitons, combined bright–dark solitons, periodic waves, solitary waves and elliptic solutions to this equation. We give graphical presentations of the 2D and 3D of some achieved solutions. The obtained results of this investigation might be useful to explain the physical structure of this model. The achieved results of HFSC show the effectiveness and reliability of the proposed techniques.

2-D P-SV and SH spectral element modelling of seismic wave propagation in non-linear media with pore-pressure effects

SUMMARYIt has long been recognized that the effects of superficial geological layers, or site effects, can play a major role on the seismic ground motion at the free surface. In this study, we compute wave propagation in a 2-D asymmetrical basin considering both soil non-linearity and pore-pressure effects. Equations of elastodynamics of wave propagation are solved using the spectral element method (SEM). The geometry of the basin gives rise to basin-edge generated waves, that are different for in-plane (P-SV) and out-of-plane (SH) wave propagation and resulting in different non-linear response. Moreover, the excess-pore pressure development in superficial liquefiable layers (effective stress analysis) brings larger deformation and loss of strength than the analysis without pore-pressure effects (total stress analysis). The coupling of vertically propagating waves and the waves specifically generated in 1-D model leads to waves whose amplitude and duration are higher than the 1-D case. This multidimensional effect increases material non-linearity. Such complex wavefield provokes larger deformation and higher pore-pressure rise that cannot be predicted by 1-D modelling. Therefore, our paper suggests the use of multidimensional modelling while studying seismic wave propagation in both linear and non-linear complex media.

Exact solutions and conservation laws for the modified equal width-Burgers equation

AbstractIn this paper we study the modified equal width-Burgers equation, which describes long wave propagation in nonlinear media with dispersion and dissipation. Using the Lie symmetry method in conjunction with the (G'/G)− expansion method we construct its travelling wave solutions. Also, we determine the conservation laws by invoking the new conservation theorem due to Ibragimov. As a result we obtain energy and linear momentum conservation laws.

Modified Auxiliary Equation Method versus Three Nonlinear Fractional Biological Models in Present Explicit Wave Solutions

In this article, we present a modified auxiliary equation method. We harness this modification in three fundamental models in the biological branch of science. These models are the biological population model, equal width model and modified equal width equation. The three models represent the population density occurring as a result of population supply, a lengthy wave propagating in the positive x-direction, and the simulation of one-dimensional wave propagation in nonlinear media with dispersion processes, respectively. We discuss these models in nonlinear fractional partial differential equation formulas. We used the conformable derivative properties to convert them into nonlinear ordinary differential equations with integer order. After adapting, we applied our new modification to these models to obtain solitary solutions of them. We obtained many novel solutions of these models, which serve to understand more about their properties. All obtained solutions were verified by putting them back into the original equations via computer software such as Maple, Mathematica, and Matlab.

Phononic wave hard limiter

In this paper nonlinear Schrödinger equation (NSE) is proposed to evaluate phononic wave propagation in nonlinear media regarding mass density as the fundamental parameter of nonlinearity. This equation is used to analyze the phononic wave limiting in one dimensional periodic structure. To approach this goal, considering the analogy, the mass density in elastic wave (EW) is related to the refractive index in electromagnetic (EM) wave and thus the nonlinear terms of the refractive index in EM correspond to nonlinear coefficient of the mass density in EW. Consequently, elastic wave coupled-mode (CM) equations are derived for backward and forward waves. Eventually, the CM equations are numerically solved to demonstrate the limiting process for phononic wave.

On optimal system, exact solutions and conservation laws of the modified equal-width equation

Abstract In this paper we study the modified equal-width equation, which is used in handling simulation of a single dimensional wave propagation in nonlinear media with dispersion processes. Lie point symmetries of this equation are computed and used to construct an optimal system of one-dimensional subalgebras. Thereafter using an optimal system of one-dimensional subalgebras, symmetry reductions and new group-invariant solutions are presented. The solutions obtained are cnoidal and snoidal waves. Furthermore, conservation laws for the modified equal-width equation are derived by employing two different methods, the multiplier method and Noether approach.

A high-order discontinuous Galerkin method for 1D wave propagation in a nonlinear heterogeneous medium

We propose a nodal high-order discontinuous Galerkin method for 1D wave propagation in nonlinear media. We solve the elastodynamic equations written in the velocity-strain formulation and apply an upwind flux adapted to heterogeneous media with nonlinear constitutive behavior coupling stress and strain. Accuracy, convergence and stability of the method are studied through several numerical applications. Hysteresis loops distinguishing loading and unloading-reloading paths are also taken into account. We investigate several effects of nonlinearity in wave propagation, such as the generation of high frequencies and the frequency shift of resonant peaks to lower frequencies. Finally, we compare the results for both nonlinear models, with and without hysteresis, and highlight the effects of the former on the stabilization of the numerical scheme.

Exact traveling wave solutions to the (2+1)-dimensional Biswas–Milovic equations

In this paper, we investigate two (2+1)-dimensional Biswas–Milovic equations, which are actually a focusing nonlinear Schrödinger equation and plays a vital role in the research of the solitary wave propagation in nonlinear media. Using the method of simplest equation, we derive some exact traveling wave solutions to these two Biswas–Milovic equations. Moreover, the results in this paper can help us understand the change of the solitary waves in the condensate from a small perturbation via modulation instabilities.

LATIN: A new view and an extension to wave propagation in nonlinear media

SummaryThe LATIN (acronym of LArge Time INcrement) method was originally devised as a non‐incremental procedure for the solution of quasi‐static problems in continuum mechanics with material nonlinearity. In contrast to standard incremental methods like Newton and modified Newton, LATIN is an iterative procedure applied to the entire loading path. In each LATIN iteration, two problems are solved: a local problem, which is nonlinear but algebraic and miniature, and a global problem, which involves the entire loading process but is linear. The convergence of these iterations, which has been shown to occur for a large class of nonlinear problems, provides an approximate solution to the original problem. In this paper, the LATIN method is presented from a different viewpoint, taking advantage of the causality principle. In this new view, LATIN is an incremental method, and the LATIN iterations are performed within each load step, similarly to the way that Newton iterations are performed. The advantages of the new approach are discussed. In addition, LATIN is extended for the solution of time‐dependent wave problems. As a relatively simple model for illustrating the new formulation, lateral wave propagation in a flat membrane made of a nonlinear material is considered. Numerical examples demonstrate the performance of the scheme, in conjunction with finite element discretization in space and the Newmark trapezoidal algorithm in time. Copyright © 2017 John Wiley & Sons, Ltd.

Solutions of the cylindrical nonlinear Maxwell equations

Cylindrical nonlinear optics is a burgeoning research area which describes cylindrical electromagnetic wave propagation in nonlinear media. Finding new exact solutions for different types of nonlinearity and inhomogeneity to describe cylindrical electromagnetic wave propagation is of great interest and meaningful for theory and application. This paper gives exact solutions for the cylindrical nonlinear Maxwell equations and presents an interesting connection between the exact solutions for different cylindrical nonlinear Maxwell equations. We also provide some examples and discussion to show the application of the results we obtained. Our results provide the basis for solving complex systems of nonlinearity and inhomogeneity with simple systems.