Exact Traveling Wave Solutions Research Articles

Real-World Applications of Analytic Travelling Wave Solutions of a (3+1)-Dimensional Hirota–Satsuma–Ito-Like System Via Polynomial Complete Discriminant System and Elementary Integral Technique

Models reveal the dynamic character of wave motion, delineated in shallow waters alongside fluid dynamics; an example is the Hirota–Satsuma–Ito model equation. Therefore, this paper showcases the detailed analytical investigations of a (3+1)-dimensional Hirota–Satsuma–Ito-like system. In order to reduce the equation to a nonlinear ordinary differential system of equations, a traveling plane wave transformation is engaged. Thereafter, the direct integration technique is adopted to solve the model, thus culminating in obtaining Jacobi elliptic integral function solutions. Moreover, to attain more various solitonic solutions of diverse structures, a standard approach called the polynomial complete discriminant system and elementary integral technique is engaged. This provides exact traveling wave solutions of diverse known functions in the form of periodic, trigonometric, dark, mixed bright, and topological kink, as well as singular soliton solutions. These are found to appear in the form of Jacobi elliptic, trigonometric, as well as hyperbolic functions. Furthermore, some of these solutions are further examined by investigating their wave nature via numerical simulations.

Construction of the Closed Form Wave Solutions for TFSMCH and (1 1) Dimensional TFDMBBM Equations via the EMSE Technique+

The purpose of this study is to investigate a series of novel exact closed form traveling wave solutions for the TFSMCH equation and (1 + 1) dimensional TFDMBBM equation using the EMSE technique. The considered FONLEEs are used to delineate the characteristic of diffusion in the creation of shapes in liquid beads arising in plasma physics and fluid flow and to estimate the external long waves in nonlinear dispersive media. These equations are also used to characterize various types of waves, such as hydromagnetic waves, acoustic waves, and acoustic gravity waves. Here, we utilize the Caputo-type fractional order derivative to fractionalize the considered FONLEEs. Some trigonometric and hyperbolic trigonometric functions have been used to represent the obtained closed form traveling wave solutions. Furthermore, here, we reveal that the EMSE technique is a suitable, significant, and dominant mathematical tool for finding the exact traveling wave solutions for various FONLEEs in mathematical physics. We draw some 3D, 2D, and contour plots to describe the various wave behaviors and analyze the physical consequence of the attained solutions. Finally, we make a numerical comparison of our obtained solutions and other analogous solutions obtained using various techniques.

Dynamics and soliton solutions of the perturbed Schrödinger-Hirota equation with cubic-quintic-septic nonlinearity in dispersive media

<p>This study systematically investigates the dynamics of the perturbed Schrödinger-Hirota equation with cubic-quintic-septic nonlinearity under spatiotemporal dispersion, providing insights into soliton propagation in dispersive media. We begin by examining the system's phase portrait and chaotic behavior, followed by the derivation of exact traveling wave solutions, including optical solitons and periodic solutions, using an enhanced algebraic method. The findings are vividly illustrated through three-dimensional and two-dimensional graphical simulations, which analyze the impact of key parameters on the solutions. This study not only presents a variety of optical soliton solutions, but also clarifies the underlying dynamics, offering theoretical guidance for fiber optic communication systems and holding significant applied value for achieving more efficient and reliable optical communications.</p>

A modified Jacobi elliptic functions method for optical soliton solutions of a conformable nonlinear Schrödinger equation

In this paper, we study precise and exact traveling wave solutions of the conformable differential nonlinear Schrödinger equation. Then, we transform the given equation into an integer order differential equation by utilizing the wave transformation and the characteristics of the conformable derivative. To extract optical soliton solutions, we divide the wave profile into amplitude and phase components. Further, we introduce a new extension of a modified Jacobi elliptic functions method to the conformable differential nonlinear Schrödinger equation with group velocity dispersion and coefficients of second-order spatiotemporal dispersion.

Novel insights into high-order dispersion and soliton dynamics in optical fibers via the perturbed Schrödinger–Hirota equation

This study offers a comprehensive analysis of the Perturbed Schrödinger -Hirota Equation (PSHE), crucial for understanding soliton dynamics in modern optical communication systems. We extended the traditional Nonlinear Schrödinger Equation (NLSE) to include higher-order nonlinearities and spatiotemporal dispersion, capturing the complexities of light pulse propagation. Employing the modified auxiliary equation method and Adomian Decomposition Method (ADM), we derived a spectrum of exact traveling wave solutions, encompassing exponential, rational, trigonometric, and hyperbolic functions. These solutions provide insights into soliton behaviors across diverse parameters, essential for optimizing fiber optic systems. The precision of our analytical solutions was validated through numerical solutions, and we explored modulation instability, revealing conditions for soliton formation and evolution. The findings have significant implications for the design and optimization of next-generation optical communication technologies.

Exact solutions and bifurcations of the new (3+1)-dimensional generalized Kadomstev-Petviashvili equation

Abstract The exact solutions of the new generalized (3+1)-dimensional Kadomtsev-Petviashvili (ngKP) equation by using the modified extended tanh-function method, the improved ( G ′ / G ) method and the generalized Riccati equation mapping method are considered. Different types of exact traveling wave solutions are obtained, enriching the exact solutions of this equation. In addition, the dynamical behaviors of the different types of solutions obtained by the equation are graphically presented using 3D, 2D and density plots. Then, the phase portraits of the equations are obtained through the bifurcation theory to verify the validity of the previous three methods. Finally, the comparison between our results and the known results is given, it can be concluded that we have provided a large number of new travelling wave solutions to the equation.

The Exact Traveling Wave Solutions of a KPP Equation

We obtain the exact analytical traveling wave solutions of the Kolmogorov–Petrovskii–Piskunov equation, with the reaction term belonging to the class of functions, which includes that of the (generalized) Fisher equation, for the particular values of the wave’s speed. Additionally we obtain the exact analytical traveling wave solutions of the generalized Burgers–Huxley equation.

Multi-stability and dynamical analysis of Zhanbota-IIA equation with couple of integrating architectures

Abstract This research delves into exploring the Zhanbota-IIA equation, which has applications in medical imaging and diagnostics, power systems, and fuzzy logic processing. This study employs two analytical techniques, namely the variational method that emerges from the variational principle and the improved Cham method, to find various types of exact traveling wave solutions. These algorithms yield periodic, dark, singular, and bright soliton solutions as well as combinations of these forms Appropriate parametric values are chosen to make 2D and 3D plots to illustrate how the retrieved solutions behave. Chaotic phenomena are analyzed using different initial conditions and constant parameters through multi-stability analysis. The extracted results illustrate the efficiency of the outlined techniques in resolving exact solutions in nonlinear areas.

Bifurcations and Exact Traveling Wave Solutions for the Generalized Alexeyev’s $$A\pm $$ Equation

Bifurcations and Exact Traveling Wave Solutions for the Generalized Alexeyev’s $$A\pm $$ Equation

Bifurcations and Exact Traveling Wave Solutions for the Model of Slightly Dispersive Quasi-Incompressible Hyperelastic Materials

Bifurcations and Exact Traveling Wave Solutions for the Model of Slightly Dispersive Quasi-Incompressible Hyperelastic Materials

Dynamics of optical solitons and sensitivity analysis in fiber optics

The nonlinear Schrödinger equation (NLSE) and its various forms have significant applications in the field of soliton theory. The Fokas-Lenells (FL) equation stands as a cornerstone in deepening our understanding of nonlinear wave dynamics within optical systems, particularly concerning the behavior of ultrashort pulses across different media. Its significance lies in providing a comprehensive framework to study and analyze complex phenomena, ultimately contributing to advancements in optical technology and applications. The FL equation is an integrable extension of the NLSE that provides a description of the nonlinear propagation of pulses in optical fiber. This paper seeks to discover optical soliton solutions for the FL equation by employing a modified sub-equation method. Additionally, the sensitivity analysis is described by using the various initial conditions. The main novelty of this paper lies in conducting a sensitivity analysis of the FL equation by examining the effects of various initial conditions, providing deeper insights into how these conditions influence the behavior of soliton solutions. For the physical behavior of the models, some solutions are graphically shown in 2D, 3D, and contour graphs by assigning specific values to the parameters under the provided situation at each solution. As a result, we discovered several new families of exact traveling wave solutions, such as bright solitons, dark solitons, and combined bright and dark solitons. This research opens numerous avenues for further exploration in the field of nonlinear wave dynamics and optical soliton theory. The discovery of exact soliton solutions for the FL equation through a modified sub-equation method paves the way for deeper investigations for newcomer researchers. The results of this study will contribute further to the field of mathematical physics, particularly in enhancing the understanding of nonlinear wave propagation and soliton theory in optical and other physical systems.

The Improved tan (\(\frac{\Phi(\xi)}{2}\)) -Expansion Method for The Non-dissipative Double Dispersive Equation in Microstructured Solids

In microstructured solids, the non-dissipative double dispersive equation is a fourth-order non-linear partial differential equation that arises in the study of non-dissipative strain wave propagation. Seeking the exact solution of a nonlinear partial differential equations with a physical background is helpful to understand the motion law of matter and to explain the corresponding physical phenomena scientifically. This equation has been studied widely, and there have been a lot of research methods to find exact solutions to this equation. In this manuscript, we try to study the non-dissipative double dispersive equation by the improved tan (\(\frac{\Phi(\xi)}{2}\)) -expansion method. As far as our known, no one has used this method to study the non-dissipative double dispersive equation, partly because of the complicated calculation. We obtain a lot of exact traveling wave solutions, including hyperbolic function solutions, trigonometric function solutions, exponential function solutions, and rational function solutions. Compared with the other methods, some solutions obtained in this manuscript are consistent with existing solutions, which shows the effectiveness of the improved tan (\(\frac{\Phi(\xi)}{2}\)) -expansion method. Through this method, Our manuscript obtains more general and new exact solutions. These solutions may play an important role in engineering and physics. What's more, we plot the 3D graphs of some of the solutions obtained in this manuscript, which helps us understand the physical phenomenon of the non-dissipative double dispersive equation. In the future, we will continue to explore its physical significance from the new analytical solution we have obtained.

The Construction of Analytical Exact Soliton Waves of Kuralay Equation

Abstract The primary objective of this work is to examine the Kuralay equation, which is a complex integrable coupled system, in order to investigate the integrable motion of induced curves. The soliton solutions derived from the Kuralay equation are thought to be the supremacy study of numerous significant phenomena and extensive applications across a wide range of domains, including optical fibres, nonlinear optics and ferromagnetic materials. The inverse scattering transform is unable to resolve the Cauchy problem for this equation, so the analytical method is used to produce exact travelling wave solutions. The modified auxiliary equation and Sardar sub-equation approaches are used to find solitary wave solutions. As a result, singular, mixed singular, periodic, mixed trigonometric, complex combo, trigonometric, mixed hyperbolic, plane and combined bright–dark soliton solution can be obtained. The derived solutions are graphically displayed in 2-D and 3-D glances to demonstrate how the fitting values of the system parameters can be used to predict the behavioural responses to pulse propagation. This study also provides a rich platform for further investigation.

Boundary excitation of nonlinearly modulated bending strain waves in a metamaterial

A non-linear modulation of bending waves in a metamaterial mass-in-mass lattice model is studied. Various kinds of non-linear wave modulation are obtained using a harmonic boundary excitation. It is shown, that these waves can be described by an asymptotic simplification of the equations of motion resulting in a non-linear modulation equation for the displacements. Exact periodic traveling wave solutions to the equation demonstrate a dependence on the sign of the equation coefficients or the elastic parameters of the original model. Dispersion relation for the carrier part of the solution is obtained as a function of wave number versus frequency, it is found how its solutions relate to the acoustic and optic branches and the band gap. This form of the dispersion relation is further used for a description of numerical results on the wave modulation by a harmonic boundary excitation.

New Exact Solutions for Generalized of Combined with Negative Calogero-Bogoyavlenskii Schiff and Generalized Yu–Toda–Sassa–Fukuyama Equations

In this paper, we present generalized model of combined Calogero-Bogoyavlenskii Schiff and negative-order Calogero-Bogoyavlenskii Schiff G(CBS-nCBS) equation and generalized Yu–Toda–Sassa–Fukuyama g(YTSF) equation. We apply the extended hyperbolic function method, to solve generalized models. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions, trigonometric functions, rational functions solutions of these equations from the method with the aid of the computer program Maple.

The analysis of traveling wave solutions and dynamical behavior for the stochastic coupled Maccari's system via Brownian motion

The stochastic coupled Maccari's system (MS) is a kind of important nonlinear partial differential equations to describe fluid flow, plasma physics, nonlinear optics and so on. In this article, the dynamical behavior and some new exact traveling wave solutions of the system are investigated. By means of complex traveling wave transformation, the system is transformed into a nonlinear ordinary differential equation. The dynamical behavior of the system as well as its perturbation case are illustrated by bifurcation theory. And then, some new stochastic traveling wave solutions of the system are extracted based on the theory of polynomial complete discrimination system. To show the effect of stochastic factor on the solutions, their structures under different Brownian motion amplitudes are compared by several sets of graphs. The results obtained in this paper have supplemented the study of the system, and the technique used to exploit the traveling wave solutions are effective.

Exploring New Traveling Wave Solutions for the Spatiotemporal Evolution of a Special Reaction–Diffusion Equation by Extended Riccati Equation Method

In this work, we aim to explore new exact traveling wave solutions for the reaction–diffusion equation, which describes complex nonlinear phenomena such as cell growth and chemical reactions in nature. Obtaining exact solutions to this equation is crucial for understanding aspects such as reaction activity and the diffusion coefficient. We solve the reaction–diffusion equation by using the Riccati equation as an auxiliary equation. By controlling the parameters in the Riccati equation, we obtained a large number of traveling wave solutions, many of which were not formerly recorded in other documents. Numerical simulations demonstrate the evolution of various traveling waves of the reaction–diffusion equation in time and space. These rich exact solutions and wave phenomena help to expand our knowledge of this equation.

Diverse variety of exact solutions for some nonlinear models via the [formula omitted]-expansion method

In this article, we explore several significant nonlinear physical models, including the Benjamin–Bona–Mahony–Peregrine–Burgers (BBMPB) equation, the Burgers–Korteweg–De Vries (BK) equation, the one-dimensional Oskolkov (OSK) equation, the Klein–Gordon (KG) equation with quadratic non-linearity, and the improved Boussinesq (IB) equation. Utilizing the (G′G)-expansion method ansatz, we derive new exact traveling wave solutions for these models. These solutions, expressed in the forms of rational, hyperbolic, and trigonometric functions, present a novel contribution distinct from existing literature. The physical dynamics of these solutions are elucidated through Mathematica simulations.

Exploring New Traveling Wave Solutions to the Nonlinear Integro-Partial Differential Equations with Stability and Modulation Instability in Industrial Engineering

In this research article, we demonstrate the generalized expansion method to investigate nonlinear integro-partial differential equations via an efficient mathematical method for generating abundant exact solutions for two types of applicable nonlinear models. Moreover, stability analysis and modulation instability are also studied for two types of nonlinear models in this present investigation. These analyses have several applications including analyzing control systems, engineering, biomedical engineering, neural networks, optical fiber communications, signal processing, nonlinear imaging techniques, oceanography, and astrophysical phenomena. To study nonlinear PDEs analytically, exact traveling wave solutions are in high demand. In this paper, the (1 + 1)-dimensional integro-differential Ito equation (IDIE), relevant in various branches of physics, statistical mechanics, condensed matter physics, quantum field theory, the dynamics of complex systems, etc., and also the (2 + 1)-dimensional integro-differential Sawda–Kotera equation (IDSKE), providing insights into the several physical fields, especially quantum gravity field theory, conformal field theory, neural networks, signal processing, control systems, etc., are investigated to obtain a variety of wave solutions in modern physics by using the mentioned method. Since abundant exact wave solutions give us vast information about the physical phenomena of the mentioned models, our analysis aims to determine various types of traveling wave solutions via a different integrable ordinary differential equation. Furthermore, the characteristics of the obtained new exact solutions have been illustrated by some figures. The method used here is candid, convenient, proficient, and overwhelming compared to other existing computational techniques in solving other current world physical problems. This article provides an exemplary practice of finding new types of analytical equations.

New solutions of the general elliptic equation and its applications to the new (3 + 1)-dimensional integrable Kadomtsev-Petviashvili equation

In this article, we introduce many new Jacobi elliptic function solutions to the general elliptic equation. Consequently, the Jacobi elliptic function expansion method is improved to accommodate the general elliptic equation and its new solutions for constructing exact traveling wave solutions of nonlinear partial differential equations (NLPDEs). Moreover, the improved method is used to obtain new explicit solutions for the (3+1)-dimensional integrable Kadomtsev-Petviashvili (KP) equation. This method can be applied to many other NLPDEs as well for obtaining new exact solutions.