Kinds Of Nonlinear Evolution Equations Research Articles

Analysis of cubic-quartic-nonlinear Schrödinger’s equation with cubic-quintic-septic-nonic form of self-phase modulation through different techniques

This work presents novel perception into the optical solitons of the cubic-quartic nonlinear Schrödinger’s equation (CQ-NLSE) by introducing the cubic-quintic-septic-nonic (CQSN) form of self-phase modulation (SPM). The proposed model has a significant role in physics and engineering and it is considered the equation of pulses dissemination in optical communication. To extract the optical solitons such as the bright, kink, periodic and singular solitons, the Kudryashov’s method and extended hyperbolic function method (EHFM) are applied for the first time. The results show that these methods are suitable, productive and competent to find the solutions for such kind of nonlinear evolution equations (NLEEs) in optical fibers and nonlinear sciences. Moreover, for physical illustration of obtained solutions, some 3d, contour and 2d graphs are plotted which shed light on the differences between the solutions obtained through Kudryashov’s method and extended hyperbolic function method (EHFM). Furthermore, comparison section is also given to prove the novelty of the work over the reported results.

Exact solutions to the null-geodesics in Ellis–Bronnikov wormhole spacetime via (G′/G)-expansion method

In this paper, the [Formula: see text]-expansion method is used to obtain two certain classes of one-parameter exact solutions to the null-geodesics in Ellis–Bronnikov wormhole spacetime. This method has been developed as an effective technique to construct exact analytical solutions for some kind of nonlinear evolution equations and nonlinear partial derivative equations (PDE). At the first stage of this method, a nonlinear PDE is transformed into nonlinear ordinary derivative equation (ODE) of a polynomial form. Therefore, if we initially have a nonlinear ODE of a polynomial form, say, the geodesic equation, then sometimes its solutions can be obtained following to the procedure of [Formula: see text]-expansion method. Here, this method made it possible to obtain some classes of exact analytical solutions of null-geodesic equations in the metric of Ellis–Bronnikov wormholes.

Hausdorff fractal system arising in the field of plasma physics via variational technology

Solitons and fractals are the two most significant features of all areas of science and engineering, like optics, condensed matter, plasma physics, and fluid dynamics. This article outlines a time-space fractal Hausdorff coupled non-linear Drinfeld–Sokolov model in the field of plasma physics. The diversity of new solitary wave solutions for the suggested model is achieved utilizing a variational approach. Constraint conditions by variance principle occur within the requirements of the soliton solutions. The 3D, 2D, and contour graphs for the received solutions are displayed within the set of acceptable parameter values. The solution mechanism shows that the approach to fractal problems is clear and straightforward. The variational approach is a modern strategy for fractal structures that is useful in current fields of study in mathematics and physics for other kinds of nonlinear evolution equations.

Various optical solutions to the (1+1)-Telegraph equation with space-time conformable derivatives

This paper presents a new sub-equation method based on an auxiliary equation which is implemented via the well-known generalized Kudryashov method, to construct new traveling waves to the Telegraph equation with time and space conformable derivatives. To illustrate its effectiveness, it was tested for seeking traveling wave solutions to the (1+1)-Telegraph equation with space-time conformable derivatives. With the help of Maple Software we derive some new solitary waves solutions. It can be concluded that the proposed method is an accurate tool for solving several kind of nonlinear evolution equations.

Construction of traveling and solitary wave solutions for wave propagation in nonlinear low-pass electrical transmission lines

In this study, our aim to constructed the traveling and solitary wave solutions for nonlinear evolution equation describe the wave propagation in nonlinear low-pass electrical transmission lines by implemented the modification of mathematical method. We obtained the new and more general solutions in rational, trigonometric, hyperbolic type which represent to kink and anti-kink wave solitons, bright-dark solitons and traveling waves. The physical interpretation of some results demonstrated by graphically with symbolic computation. We are hopefully determined results have numerous applications in optical fiber, geophysics, fluid dynamics, laser optics, engineering, and many other various kinds of applied sciences. The complete investigation prove that proposed technique is more reliable, efficient, straightforward, and powerful to investigate various kinds of nonlinear evolution equations involves in geophysics, fluid dynamics, nonlinear plasma, chemistry, biology, and field of engineering.

The generalized Kudryashov method for new closed form traveling wave solutions to some NLEEs

In this work, we construct closed form traveling wave solutions to some nonlinear evolution equations (NLEEs) associated with mathematical physics. This work implements the well-established generalized Kudryashov method (gKM) to compute new closed form traveling wave solutions to the Burgers-Huxley equation, the mKdV equation and the first extended fifth order nonlinear equation. Furthermore, in this investigation, we discuss the achieved results in details and portrayed some 2D and 3D figures with the aid of symbolic computation package like Mathematica. The worked-out results ascertained that the suggested generalized form of the Kudryashov method is a simple, efficient and reliable technique to deal with other kinds of NLEEs.

New optical solitons of Tzitzeíca type evolution equations using extended trial approach

The properties of Tzitzeica equations in nonlinear optics have been the discussion of many recent studies. In this article, a new and productive implementation of trial equation method is chosen to handle this class of nonlinear evolution equations. As a result, the new and more exact periodic, singular, hyperbolic and rational solutions of Tzitzeica, Dodd–Bullough–Mikhailov and Tzitzeica–Dodd–Bullough equations are formally obtained. The extended trial equation method along with the symbolic computation gives a versatile technique to deal such kinds of nonlinear evolution equations in nonlinear optics.

Exact solutions of the generalized Sinh–Gordon equation

In this paper, we successfully derive a new exact traveling wave solutions of the generalized Sinh–Gordon equation by new application of the homogeneous balance method. This method could be used in further works to establish more entirely new solutions for other kinds of nonlinear evolution equations arising in physics.

Exact Solutions to the (2+1)-Dimensional Boussinesq Equation via exp(?(?))-Expansion Method

The exp(?(?))-expansion method is applied to find exact traveling wave solutions to the (2+1)-dimensional Boussinesq equation which is an important equation in mathematical physics. The traveling wave solutions are expressed in terms of the exponential functions, the hyperbolic functions, the trigonometric functions and the rational functions. The procedure is simple, direct and constructive without the help of a computer algebra system. The applied method will be used in further works to establish more new solutions for other kinds of nonlinear evolution equations arising in mathematical physics and engineering.

Exact traveling soliton solutions for the scalar Qiao equation

We investigate exact traveling wave solutions of the scalar Qiao equation proposed by Li and Qiao (2010 J. Math. Phys. 51 042703) using the generalized auxiliary equation method. This equation is known to have bi-Hamiltonian structure and Lax pair, which imply integrability of the equation for a fixed value of k. Symmetries of the scalar Qiao equation and its solutions are also considered. The obtained solutions include kink and antikink solitons, bright and dark solitons, singular solutions and exponential solutions. This method could be used in further works to establish more entirely new solutions for other kinds of nonlinear evolution equations arising in physics. This work could be also relevant for numerical studies of the scalar Qiao equation.

All traveling wave exact solutions of three kinds of nonlinear evolution equations

In this article, we employ the complex method to obtain all meromorphic exact solutions of complex Klein–Gordon (KG) equation, modified Korteweg‐de Vries (mKdV) equation, and the generalized Boussinesq (gB) equation at first, then find all exact solutions of the Equations KG, mKdV, and gB. The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all rational and simply periodic solutions are solitary wave solutions, the complex method is simpler than other methods, and there exist some rational solutions w2r,2(z) and simply periodic solutions w1s,2(z),w2s,1(z) in these equations such that they are not only new but also not degenerated successively by the elliptic function solutions. We have also given some computer simulations to illustrate our main results. Copyright © 2014 John Wiley & Sons, Ltd.

All traveling wave exact solutions of two kinds of nonlinear evolution equations

In this article, we employ the complex method to obtain all meromorphic solutions of complex Korteweg–de Vries (KdV) equation and the modified Benjamin–Bona–Mahony (mBBM) equation at first, then find out all traveling wave exact solutions of the Eqs. (KdV) and (mBBM). The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all rational and simply periodic solutions of the Eqs. (KdV) and (mBBM) are solitary wave solutions, the complex method is simpler than other methods, and there exist some rational solutions w2r,2(z) and simply periodic solutions w2s,1(z) which are not only new but also not degenerated successively by the elliptic function solutions. We give some computer simulations to illustrate our main results.

Exact traveling wave solutions to the nonlinear Schrödinger equation

In this paper, through auxiliary equation method we investigate exact traveling wave solutions of the nonlinear Schrödinger equation. As results, we obtain miscellaneous traveling wave solutions including, bright and dark solitons, periodic solutions, singular solutions and exponential solutions. This method could be used in further works to establish more entirely new solutions for other kinds of nonlinear evolution equations arising in physics.

New Solutions of Elastic Waves in an Elastic Rod under Finite Deformation

The nonlinear wave equation of an elastic rod under finite deformation is solved by the extended mapping method. Abundant new exact traveling wave solutions for this equation are obtained, which contain trigonometric function solutions, solitary wave solutions, Jacobian elliptic function solutions, and Weierstrass elliptic function solutions. The method can be used in further works to establish more entirely new solutions for other kinds of nonlinear evolution equations arising in physics.

Blowing up Solution of Initial-Boundary Value Problem for a Kind of Nonlinear Evolution Equations

This dissertation is to discuss the initial-boundary value problem under the third nonlinear boundary condition for a kindof nonlinear evolution equations. To apply the maximum value theory and convex functional method£it is proved thatthe blowing up of solution in the definite time under some assumed conditions. The conclusion popularizes the results ofreferences [Zhongwei Zha(2003)-Zhongwei Zha(2010)].

Explicit traveling wave solutions of five kinds of nonlinear evolution equations

First of all, by using Bernoulli equations, we develop some technical lemmas. Then, we establish the explicit traveling wave solutions of five kinds of nonlinear evolution equations: nonlinear convection diffusion equations (including Burgers equations), nonlinear dispersive wave equations (including Korteweg–de Vries equations), nonlinear dissipative dispersive wave equations (including Ginzburg–Landau equation, Korteweg–de Vries–Burgers equation and Benjamin–Bona–Mahony–Burgers equation), nonlinear hyperbolic equations (including Sine–Gordon equation) and nonlinear reaction diffusion equations (including Belousov–Zhabotinskii system of reaction diffusion equations).

The presentation of explicit analytical solutions of a class of nonlinear evolution equations

In this paper, we introduce a function set Ω m . There is a conjecture that an arbitrary explicit travelling-wave analytical solution of a real constant coefficient nonlinear evolution equation is necessarily a linear (or nonlinear) combination of the product of some elements in Ω m . A widespread applicable approach for solving a class of nonlinear evolution equations is established. The new analytical solutions to two kinds of nonlinear evolution equations are described with the aid of the guess.

Three kinds of nonlinear dispersive waves in elastic rods with finite deformation

On the basis of classical linear theory on longitudinal, torsional and flexural waves in thin elastic rods, and taking finite deformation and dispersive effects into consideration, three kinds of nonlinear evolution equations are derived. Qualitative analysis of three kinds of nonlinear equations are presented. It is shown that these equations have homoclinic or heteroclinic orbits on the phase plane, corresponding to solitary wave or shock wave solutions, respectively. Based on the principle of homogeneous balance, these equations are solved with the Jacobi elliptic function expansion method. Results show that existence of solitary wave solution and shock wave solution is possible under certain conditions. These conclusions are consistent with qualitative analysis.

Some new exact Jacobian elliptic function solutions of three kinds of nonlinear evolution equations

By using the modified mapping method, we find some new exact solutions of the nonlinear Klein-Gordon equation, mKdV equation and the generalized Boussinesq equation. The solutions obtained in this paper include Jacobian elliptic function solutions, combined Jacobian elliptic function solutions, soliton-like solutions and triangular function solutions.

Application of the extended homotopy perturbation method to a kind of nonlinear evolution equations

Based on the homotopy perturbation method, we propose a new extended homotopy perturbation method. We apply the method to solve a kind of nonlinear evolution equations with the help of the symbolic computation system Maple. The success of the paper lies on the fact that we do not conform to the traditional homotopy mode but construct a new homotopy of the given equations. The results reveal that we can obtain the approximation solution of a kind of nonlinear evolution equations.