Explicit Traveling Wave Solutions Research Articles

Explicit and exact travelling wave solutions for Hirota equation and computerized mechanization.

By using the power-exponential function method and the extended hyperbolic auxiliary equation method, we present the explicit solutions of the subsidiary elliptic-like equation. With the aid of the subsidiary elliptic-like equation and a simple transformation, we obtain the exact solutions of Hirota equation which include bright solitary wave, dark solitary wave, bell profile solitary wave solutions and Jacobian elliptic function solutions. Some of these solutions are found for the first time, which may be useful for depicting nonlinear physical phenomena. This approach can also be applied to solve the other nonlinear partial differential equations.

Dynamical analysis and explicit traveling wave solutions to the higher-dimensional generalized nonlinear wave system

For dealing with exact solutions and properties of PDEs, the higher-dimensional equations are far more complicated than the lower dimensional ones. In the current paper, by using the dynamical system method, we study the (n+1)-dimensional generalized nonlinear coupled KdV (c-KdV) system, which includes a lot of important c-KdV types of systems as its special case. The bifurcations and phase portraits of the nonlinear PDE system are investigated, then we give the explicit parametric representations of all possible solitary wave solutions and periodic wave solutions under different parametric conditions.

Study of explicit travelling wave solutions of nonlinear (2 + 1)-dimensional Zoomeron model in mathematical physics

Abstract This article studeis the nonlinear (2 + 1)-dimensional Zoomeron equation by utilizing the various prominent analytical approaches namely the unified method and the extended hyperbolic function approach. The analysis in the current paper demonstrates the presence of travelling wave solutions. The applied methods are utilized as powerful tools to investigate and solve the model. The results obtained through these analytical methods reveal insightful patterns in the behavior of the Zoomeron equation. The significance of our work lies in the uniqueness of the methods employed. The two methods are applied to systematically analyze the equation, revealing hidden patterns and structures within its solution space. This leads to the discovery of a collection of solitary wave solutions such as kink waves, singular kink waves, periodic waves and dark soliton using contour plots, 3D and 2D graphics. In this article, we definitely prove that as the free parameters change, the wave amplitude changes as well. It is shown that the applied strategies are more effective and may be implemented to a variety of contemporary nonlinear evolution models emerging in mathematical physics.

Wave solutions of the time-space fractional complex Ginzburg-Landau equation with Kerr law nonlinearity

In this paper, the bifurcation theory of dynamical system is applied to investigate the time-space fractional complex Ginzburg-Landau equation with Kerr law nonlinearity. We mainly consider the case of $\alpha\neq 2\beta$ which is not discussed in previous work. By overcoming some difficulties aroused by the singular traveling wave system, such as bifurcation analysis of nonanalytic vector field, tracking orbits near the full degenerate equilibrium and calculation of complicated elliptic integrals, we give a total of 20 explicit exact traveling wave solutions of the time-space fractional complex Ginzburg-Landau equation and classify them into 11 categories. Some new traveling wave solutions of this equation are obtained including the compactons and the bounded solutions corresponding to some bounded manifolds.

On the dynamics of a dual space–time fractional nonlinear Schrödinger model in optical fibers

The nonlinear Schrödinger equation (NLSE) is a significant nonlinear complex model known as the most important struture to represent light pulse propagation in optical fibers, which plays a key role in demonstrating the proliferation specifically short pulse in optical fibers and have large appliances in telecommunication system and ultra fast signal routing in nonlinear optics. In this study, we look into explicit solutions of a coupled fractional NLSE for a dual space–time, which reflect numerous explicit travelling wave solutions, as well as their physical interpretation. We use the fractional complex transform to obtain multiple exact answers using the unified technique and the extended simple equation method. These exact answers include trigonometric function solutions as well as dark, bright, singular, periodic, and optical soliton solutions. For a set of appropriate parameters, Mathematica is used to represent the dynamics of various wave structures as 3D, 2D, and contour visualizations. These findings show that the proposed methodologies are simple, prolific, and capable of studying the nature of numerous physical models of complex phenomena in modern science.

Traveling edge states in massive Dirac equations along slowly varying edges

Abstract Topologically protected wave motion has attracted considerable research interest due to its chirality and potential applications in many applied fields. We construct quasi-traveling wave solutions to the 2D Dirac equation with a domain wall mass in this work. It is known that the system admits exact and explicit traveling wave solutions, which are termed edge states if the interface is a straight line. By modifying such explicit solutions, we construct quasi-traveling-wave solutions if the interface is nearly straight. The approximate solutions in two scenarios are given. One is the circular edge with a large radius, and the second is a straight line edge with the slowly varying along the perpendicular direction. We show the quasi-traveling wave solutions are valid in a long lifespan by energy estimates. Numerical simulations are provided to support our analysis both qualitatively and quantitatively.

Study of explicit travelling wave solutions of nonlinear evolution equations

The Bernoulli Sub-ODE approach is used in this study to look for comprehensive travelling wave solutions to the nonlinear evolution equations (NLEEs). The analysis in the present paper shows the existence of travelling waves for the time-regularized long-wave (TRLW) equation, the modified Korteweg–de Vries –Zakharov–Kuznetsov (mKdV–ZK) equation, and the (2+1)-dimensional Zoomeron equation. The outcomes demonstrate the richness of explicit solutions of the studied models. As a result, precise solitary wave solutions to the studied problems, such as kink waves, singular kink waves, dark soliton, and periodic waves are found. The phase plane is briefly examined after the determination of the Hamiltonian function. Using Maple 13, we validated the accuracy of the obtained solutions by reintroducing them into the original equation. We will demonstrate how the amplitudes and wave profiles are impacted by free parameters. In this article, we firmly establish that the wave amplitude varies as the free parameters change. It is demonstrated that the technique is efficient and applicable to several different NLEEs in mathematical physics.

Exact and explicit traveling wave solutions to the generalized Gardner and BBMB equations with dual high-order nonlinear terms

In this paper, the first integral method is used to solve the generalized Gardner and Benjamin–Bona–Mahoney–Burgers (BBMB) equations with dual high-order nonlinear terms. Consequently, the exact and explicit traveling wave solutions are obtained. For some specific choice of parameters, obtained solutions are reduced to the solutions of classical Gardner and BBM equations. Comparisons between our results and the well-known results are given. In addition, 3-D and 2-D graphical representations of the exact solutions are depicted. • The first integral method is utilized to find the exact traveling wave solutions for generalized Gardner and BBMB equations. • For some specific choice of parameters, the exact solutions are reduced to the solutions of classical Gardner and BBM equations. • The obtained exact solutions are compared with the other existing results.

New explicit exact traveling wave solutions of Camassa–Holm equation

In this paper, we construct two families of new explicit exact traveling wave solutions of the Camassa–Holm equation which solve an open problem in the previous paper [Zhang G, Qiao ZJ, Liu F. Cuspons and smooth solitons of the Camassa-Holm equation under an inhomogeneous boundary condition. Pac J Appl Math. 2007;1(1):113–130]. With the help of the numerical graphs and a profound mathematical analysis, we discovered the analytic formulas for those two explicit exact solutions.

Coupled Fractional Traveling Wave Solutions of the Extended Boussinesq–Whitham–Broer–Kaup-Type Equations with Variable Coefficients and Fractional Order

In this paper, we propose the extended Boussinesq–Whitham–Broer–Kaup (BWBK)-type equations with variable coefficients and fractional order. We consider the fractional BWBK equations, the fractional Whitham–Broer–Kaup (WBK) equations and the fractional Boussinesq equations with variable coefficients by setting proper smooth functions that are derived from the proposed equation. We obtain uniformly coupled fractional traveling wave solutions of the considered equations by employing the improved system method, and subsequently their asymmetric behaviors are visualized graphically. The result shows that the improved system method is effective and powerful to find explicit traveling wave solutions of the fractional nonlinear evolution equations.

Explicit Travelling Wave Solutions to Nonlinear Partial Differential Equations Arise in Mathematical Physics and Engineering

To describe the interior phenomena of the mysterious problems around the real world, non-linear partial differential equations (NLPDEs) plays a substantial role, for which construction of analytic solutions of those is most important. This paper stands for a goal to find fresh and wide-ranging solutions to some familiar NLPDEs namely the non-linear cubic Klein-Gordon (cKG) equation and the non-linear Benjamin-Ono (BO) equation. A wave variable transformation is made use to convert the mentioned equations into ordinary differential equations. To acquire the desired precise exact travelling wave solutions to the above-stated equations, the rational -expansion method is employed. Consequently, three types of equipped solutions are successfully come out in the forms of hyperbolic, trigonometric and rational functions in a compatible way. To analyse the physical problems arisen relating to nonlinear complex dynamical systems, our obtained solutions might be most helpful. So far we know, these achieved solutions are different than those in the literature. The applied method is efficient and reliable which might further be used to find different and novel solutions to many other NLPDEs successfully in research field.

New explicit and exact traveling wave solutions of (3+1)-dimensional KP equation

<p style='text-indent:20px;'>In this paper, we investigate explicit exact traveling wave solutions of the generalized (3+1)-dimensional KP equation <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \ (u_{t}+\alpha uu_{x}+\beta u_{xxx})_{x}+\gamma u_{yy}+\delta u_{zz} = 0, \ \ \ \ \beta&gt;0 \;\;\;\;\;\;(1) \ \end{equation}$ \end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>describing the dynamics of solitons and nonlinear waves in the field of plasma physics and fluid dynamics, where <inline-formula><tex-math id="M1">\begin{document}$ \alpha, \beta, \gamma, \delta $\end{document}</tex-math></inline-formula> are nonzero constants. By using the simplified homogeneous balance method, we get one single soliton solution and one double soliton solution of (1). Moreover, we use the extended tanh method with a Riccati equation and the simplest equation method with Bernoulli equation to obtain seven sets of explicit exact traveling wave solutions. When <inline-formula><tex-math id="M2">\begin{document}$ \delta = 0 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M3">\begin{document}$ \gamma = 0 $\end{document}</tex-math></inline-formula>, (1) reduces to (2+1)-dimensional KP equation. Therefore, we can get some exact traveling wave solutions of (2+1)-dimensional KP equation.

BIFURCATIONS AND EXACT TRAVELLING WAVE SOLUTIONS FOR A NEW INTEGRABLE NONLOCAL EQUATION

<abstract> By using the method of dynamical systems, we consider the dynamical behavior of travelling wave solutions for a new integrable nonlocal equation. All possible exact explicit travelling wave solutions under different parameter conditions are given, including solitary wave solutions, periodic wave solutions and pseudo-peakon wave solutions. </abstract>

Kinky periodic pulse and interaction of bell wave with kink pulse wave propagation in nerve fibers and wall motion in liquid crystals

In this article, we apply a direct influential approach namely enhanced modified simple equation (EMSE) method to integrate the Burgers–Huxley (BH) and FitzHugh–Nagumo (FHN) equations which explain nerve pulse propagation in nerve fibers, circuit theory as well as wall motion in liquid crystals. The EMSE technique usually used to obtain pulse wave with time variety constants. In fact, we present a set of explicit traveling wave solutions for BH model and three sets of such solutions for FHN model using the technique. Choosing various particular values of the free parameters, we determine kinky periodic pulse, single solitonic pulse, bell type pulse in the kink waves. In particular, we obtain both bright and dark type kink and bell pulse waves from the solutions and interactions between time varying periodic waves with the solitons. We incorporate some 2D and 3D plots to illustrate the wave characteristics (pulses, directions and amplitudes) of the obtained solutions. The solutions of BH and FHN equations can explain nerve pulse propagation in nerve fibers in the population genetics, circuit theory as well as wall motion in liquid crystals.

Exact optical solutions for the regularized long-wave Kadomtsev-Petviashvili equation

Exploring new wave soliton solutions to nonlinear partial differential equations has always been one of the most challenging topics in different disciplines of physics, applied mathematics, and engineering. In this paper, we examine the regularized long-wave Kadomtsev-Petviashvili (KP) equation using the generalized exponential rational function method (GERFM). Through the application of this method, fourteen explicit traveling wave solutions are formally generated. Furthermore, some graphical representations of the acquired solutions are also included to indicate that all parameters can drastically influence their nature, profile, and structures. The method employed in this paper is very simple and straightforward to use and at the same time with much lower computational costs compared to other known methods in the field. So, it can be considered direct and powerful mathematical tools to derive exact soliton wave solutions of other nonlinear models. All symbolic manipulations are done in Mathematica software.

BIFURCATIONS AND EXACT TRAVELLING WAVE SOLUTIONS OF M-N-WANG EQUATION

By using the method of dynamical systems to Mikhailov-Novikov-Wang Equation, through qualitative analysis, we obtain bifurcations of phase portraits of the traveling system of the derivative $\phi(\xi)$ of the wave function $\psi(\xi)$. Under different parameter conditions, for $\phi(\xi)$, exact explicit solitary wave solutions, periodic peakon and anti-peakon solutions are obtained. By integrating known $\phi(\xi)$, nine exact explicit traveling wave solutions of $\psi(\xi)$ are given.

Minimal travelling wave speed and explicit solutions in monostable reaction-diffusion equations

We investigate the connection between the existence of an explicit travelling wave solution and the travelling wave with minimal speed in a scalar monostable reaction-diffusion equation.

Detection of new multi-wave solutions in an unbounded domain

We deduce new explicit traveling wave solutions for Zoomeron evolution equation and (3[Formula: see text]+[Formula: see text]1)-dimensional shallow water wave equation. The reduction process using Lie vectors leads in some cases to ordinary differential equations (ODEs) that having no quadrature. The integrating factor property has been used to derive several new solutions for these nonsolvable ODEs. These solutions have been illustrated with three dimensions plots. Comparison with other works are presented.

Comment on “the solitons and periodic travelling wave solutions for Dodd-Bullough-Mikhailov and Tzitzeica-Dodd-Bullough equations in quantum field theory, Optik 168 (2018) 807-816”

In the recently published article, S.Saha Ray investigated Dodd-Bullough-Mikhailov (DBM) and Tzitzeica-Dodd-Bullough (TDB) equations via his new extended auxiliary equation method. And he claimed that new types of Jacobi elliptic function solutions had been obtained and meanwhile a new family of explicit travelling wave solutions were derived. In this comment, we show that all of exact traveling solutions presented by S.Saha Ray do not satisfy the new extended auxiliary equation method and therefore are incorrect. In addition, general traveling wave solutions of the two differential equation are presented by making full use of certain results published previously.

Explicit exact traveling wave solutions and bifurcations of the generalized combined double sinh–cosh-Gordon equation

This paper concerns the qualitative behavior and exact traveling wave solutions of the generalized double combined sinh–cosh-Gordon equation. This equation generalizes sinh-Gordon equation, Liouville equation, Dodd–Bullough–Mikhailov equation, Tzitzeica–Dodd-Bullough equation and Zhiber-Shabat equation as special cases. Thus, this paper presents a unified analysis to find the exact solutions of these known equations. Our results generalize many previous known results (Chen et al., 2009; Fan et al., 2011; Geng et al., 2007; He and Meng, 2017; He et al. 2014; Li and Li 2005; Seadawy et al. 2017; Tang and Huang 2007). We find different kinds of exact solutions such as bright soliton, dark soliton, kink wave, anti-kink wave solutions and periodic wave solutions. Moreover, the explicit expressions of the bounded exact traveling wave solutions are given. Finally, a conclusion ends this paper.