Explicit Traveling Wave Solutions Research Articles

Modified (G′/G)-expansion method for finding traveling wave solutions of the coupled Benjamin–Bona–Mahony-KdV equation

Using the Lie symmetry approach, the author has examined traveling wave solutions of coupled Benjamin–Bona–Mahony-KdV equation. The coupled Benjamin–Bona–Mahony-KdV equation is reduced to nonlinear ordinary differential equations for all optimal subalgebras by using Lie classical symmetries and various solutions are obtained by the modified (G′/G)-expansion method. Further, with the aid of solutions of the nonlinear ordinary differential equations, more explicit traveling wave solutions of the coupled Benjamin–Bona–Mahony-KdV equation are found out. The traveling wave solutions are expressed by rational function.

New analytic solutions of the space-time fractional Broer–Kaup and approximate long water wave equations

In the present paper, the exp(−ϕ(ξ)) expansion method is applied to the fractional Broer–Kaup and approximate long water wave equations. The explicit approximate traveling wave solutions are obtained by using this method. Here, fractional derivatives are defined in the conformable sense. The obtained traveling wave solutions are expressed by the hyperbolic, trigonometric, exponential and rational functions. Simulations of the obtained solutions are given at the end of the paper.

Exact and explicit traveling wave solutions to two nonlinear evolution equations which describe incompressible viscoelastic Kelvin-Voigt fluid

Two nonlinear evolution equations, namely the Kadomtsev-Petviashvili (KP) equation which describes the dynamics of soliton and nonlinear wave in the field of fluid dynamics, plasma physics and the Oskolkov equation which describes the dynamics of an incompressible visco-elastic Kelvin-Voigt fluid are investigated. We deliberate exact traveling wave solutions, specially kink wave, cusp wave, periodic breather waves and periodic wave solutions of the models applying the modified simple equation method. The solutions can be expressed explicitly. The dynamics of obtained wave solutions are analyzed and illustrated in figures by selecting appropriate parameters. The modified simple equation method is reliable treatment for searching essential nonlinear waves that enrich variety of dynamic models arises in engineering fields.

The solitons and periodic travelling wave solutions for Dodd–Bullough–Mikhailov and Tzitzeica–Dodd–Bullough equations in quantum field theory

In this paper, new types of Jacobi elliptic function solutions of Dodd–Bullough–Mikhailov(DBM) and Tzitzeica–Dodd–Bullough(TDB) equations have been obtained using a new extended auxiliary equation method. A new family of explicit travelling wave solutions is derived. The solitary wave solutions and periodic solutions for these equations are formally derived from the Jacobi elliptic function solutions. The proposed method has been efficiently applied to solve the DBM and TDB equations.

Fractional space–time nonlinear reaction–convection–diffusion

We comment the fractional space–time reaction–convection–diffusion defined by Grunwald–Letnikov, Riesz and Riesz–Feller fractional derivatives. We find the explicit travelling wave solution, using the similarity reduction. We plot its graphics to illustrate the anomalous behavior and investigate its properties associated with memory and distance effects.

Analytical study of exact solutions of the nonlinear Korteweg–de Vries equation with space–time fractional derivatives

This paper gives an analytical study of dynamic behavior of the exact solutions of nonlinear Korteweg–de Vries equation with space–time local fractional derivatives. By using the improved [Formula: see text]-expansion method, the explicit traveling wave solutions including periodic solutions, dark soliton solutions, soliton solutions and soliton-like solutions, are obtained for the first time. They can better help us further understand the physical phenomena and provide a strong basis. Meanwhile, some solutions are presented through 3D-graphs.

满足变系数方程的G展开法及RLW-Burgers方程的新精确解

本文借鉴(G'/G)展开法的基本思路，构造了一类新的G展开法，并令其中的函数G满足一类变系数Bernoulli方程。用此法对RLW-Burgers方程进行了求解，得到了该方程的多个新的显式行波解。事实证明，这类满足变系数方程的G展开法对于求解非线性偏微分方程的精确解是有效可行的。 Based on the basic idea of the (G'/G) expansion method, we construct a kind of New G method, and make the function G satisfy a class of variable coefficient Bernoulli equation. The RLW-Burgers equation is solved by this method, and several new explicit traveling wave solutions of the equation are obtained. It has been proved that this kind of satisfying variable coefficient equation G expansion method for solving nonlinear partial differential equations solutions is feasible and effective.

New Generalized (G′/G)-expansion Method to the Zhiber-Shabat Equation and Liouville Equations

This paper presents the implementation of the new extended -expansion method to the Zhiber-Shabat equation and the Liouville equation. Using this method, a variety of explicit travelling wave solutions has been generated. Results also show that the obtained solutions are exclusively new and, the implemented scheme has significant improvement against other methods. Finally, the work presents here the profound implications for future studies of nonlinear evolution equations in visualizing the dynamics of the equations and may help to describe physical phenomena in mathematical physics.

Various Exact Soliton Solutions and Bifurcations of a Generalized Dullin–Gottwald–Holm Equation with a Power Law Nonlinearity

In this paper, we study a model of generalized Dullin–Gottwald–Holm equation, depending on the power law nonlinearity, that derives a series of planar dynamical systems. The study of the traveling wave solutions for this model derives a planar Hamiltonian system. By investigating the dynamical behavior and bifurcation of solutions of the traveling wave system, we derive all possible exact explicit traveling wave solutions, under different parametric conditions. These results completely improve the study of traveling wave solutions to the mentioned model stated in [Biswas & Kara, 2010].

Bell and kink type soliton solutions in birefringent nano-fibers

In this paper we have obtained the explicit Jacobian elliptic periodic traveling wave solutions in birefringent fibers with spatio-temporal dispersion with the help of F-expansion method. Two nonlinear media are used that are Kerr and parabolic law. There are constraint conditions that evolve with the solution structure.

Travelling Waves in Space-Fractional Nonlinear Diffusion with Linear Convection

In this paper we investigate anomalous diffusion coupled with linear convection, using fractional calculus to describe the anomalous associated memory effects in diffusive term. We get an explicit travelling wave solution, wavefront, with finite propagation. We comment the properties of the solution, including the stationary case.

RLW-Burgers方程的新显式行波解

通过运用一类含参数的G 展开法对RLW-Burgers方程进行了研究，求得了该方程的多种函数形式的新显式行波解。事实证明，此类含参数的G 展开法不仅可以得到非线性偏微分方程的精确解，而且由于所含参数的任意性，可以得到非线性偏微分方程更多类型的显式行波解。 We studied the RLW-Burgers equation by using a class of parametric G expansion method, and obtained many new explicit traveling wave solutions for the various functional forms of the equa-tion. In fact, the parameters of the G expansion method can not only obtain the exact solutions of nonlinear partial differential equations, but also because of the arbitrariness of parameters, we can obtain more explicit traveling wave solutions for nonlinear partial differential equations.

Jacobian elliptic periodic traveling wave solutions for Biswas–Milovic equation

The Biswas–Milovic equation can also be treated as the generalized version of usual nonlinear Schrodinger's equation. With the help of F-expansion method, the explicit Jacobian elliptic periodic traveling wave solutions for Biswas–Milovic equations are constructed. Here we will discuss the two nonlinear media, those are Kerr law and the power law.

Jacobian elliptic periodic traveling wave solutions in the negative-index materials

The aim of this work is to present an analytical study on optical solitons in nonlinear negative-index materials. Three types of nonlinearities that are Kerr law, power law and parabolic law are taken into account. With the help of F-expansion method, the explicit Jacobian elliptic periodic traveling wave solutions are constructed.

Symmetry Reductions and Exact Solutions to the Kudryashov–Sinelshchikov Equation

Abstract In this article, based on the compatibility method, some nonclassical symmetries of Kudryashov–Sinelshchikov equation are derived. By solving the corresponding characteristic equations associated with symmetry equations, some new exact explicit solutions of Kudryashov–Sinelshchikov equation are obtained. For the exact explicit traveling wave solutions, the exact parametric representations are investigated by the integral bifurcation method.

Exact Solutions for a Local Fractional DDE Associated with a Nonlinear Transmission Line

Of recent increasing interest in the area of fractional calculus and nonlinear dynamics are fractional differential-difference equations. This study is devoted to a local fractional differential-difference equation which is related to a nonlinear electrical transmission line. Explicit traveling wave solutions (kink/antikink solitons, singular, periodic, rational) are obtained via the discrete tanh method coupled with the fractional complex transform.

Three kinds of periodic wave solutions and their limit forms for a modified KdV-type equation

A modified KdV-type equation is studied by using the bifurcation theory of dynamical system. By investigating the dynamical behavior with phase space analysis, all possible explicit exact traveling wave solutions including peakon solutions, kink and anti-kink wave solutions, blow-up wave solutions, smooth periodic wave solutions, periodic cusp wave solutions, and periodic blow-up wave solutions are obtained. When the first integral varies, we also show the convergence of the periodic wave solutions, such as the smooth periodic wave solutions converge to the kink and anti-kink wave solutions, the periodic cusp wave solutions converge to the peakon solution, the periodic blow-up wave solutions converge to the blow-up wave solution, the blow-up wave solutions converge to the blow-up wave solution, and the periodic blow-up wave solutions converge to the periodic blow-up wave solution.

Traveling wave solutions of new coupled Konno-Oono equation

In this paper, the recently developed tanh-function method and extended tanh-function method are applied to explore the traveling wave solutions of new coupled Konno-Oono equation. We have successfully derived some explicit traveling wave solutions of new coupled Konno-Oono equation.

Exact traveling wave solutions for system of nonlinear evolution equations.

In this work, recently deduced generalized Kudryashov method is applied to the variant Boussinesq equations, and the (2 + 1)-dimensional breaking soliton equations. As a result a range of qualitative explicit exact traveling wave solutions are deduced for these equations, which motivates us to develop, in the near future, a new approach to obtain unsteady solutions of autonomous nonlinear evolution equations those arise in mathematical physics and engineering fields. It is uncomplicated to extend this method to higher-order nonlinear evolution equations in mathematical physics. And it should be possible to apply the same method to nonlinear evolution equations having more general forms of nonlinearities by utilizing the traveling wave hypothesis.

Exact traveling wave solutions and bifurcations of the generalized derivative nonlinear Schrödinger equation

In this paper, we consider a model of generalized derivative of nonlinear Schrodinger equation. By investigating the dynamical behavior and bifurcation of solutions of the traveling wave system, we derive all possible exact explicit traveling wave solutions under different parametric conditions. These results completely improve the study of traveling wave solutions for the mentioned model stated in Wang et al. (Commun Theor Phys 50:39–42, 2008).