Zhiber Shabat Equation Research Articles

Propagation of novel traveling wave envelopes of Zhiber–Shabat equation by using Lie analysis

In this paper, we aim to find novel forms of wave structures by employing some innovative ideas. Various solitary wave solutions of the Zhiber–Shabat equation have been extracted using the Lie symmetry analysis and the extended direct algebraic method. In the mathematical community, the considered model has several applications, notably in integral quantum field theory, fluid dynamics, and kink dynamics. First of all, the Lie symmetry has been used to determine the corresponding similarity reductions through similarity variables and wave transformation with the help of optimal systems. Afterward, the method described has been used to create new complex, hyperbolic, rational, and trigonometric forms of solutions to the problem. Depending on the strength of the propagating pulse, these solutions reflect dark, bright, kink-type, and periodic solitary wave envelopes. Further, two-dimensional (2D), three-dimensional (3D), as well as contour 2D graphics of the results have been analyzed by giving some specific values to parameters. At last, sensitivity analysis of the evolution equation has been observed.

Construction of Different Types Analytic Solutions for the Zhiber-Shabat Equation

In this paper, a new solution process of ( 1 / G ′ ) -expansion and ( G ′ / G , 1 / G ) -expansion methods has been proposed for the analytic solution of the Zhiber-Shabat (Z-S) equation. Rather than the classical ( G ′ / G , 1 / G ) -expansion method, a solution function in different formats has been produced with the help of the proposed process. New complex rational, hyperbolic, rational and trigonometric types solutions of the Z-S equation have been constructed. By giving arbitrary values to the constants in the obtained solutions, it can help to add physical meaning to the traveling wave solutions, whereas traveling wave has an important place in applied sciences and illuminates many physical phenomena. 3D, 2D and contour graphs are displayed to show the stationary wave or the state of the wave at any moment with the values given to these constants. Conditions that guarantee the existence of traveling wave solutions are given. Comparison of ( G ′ / G , 1 / G ) -expansion method and ( 1 / G ′ ) -expansion method, which are important instruments in the analytical solution, has been made. In addition, the advantages and disadvantages of these two methods have been discussed. These methods are reliable and efficient methods to obtain analytic solutions of nonlinear evolution equations (NLEEs).

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.

An application of the new function method to the Zhiber–Shabat equation

This paper applies a new approach including the trial equation based on the exponential function in order to find new traveling wave solutions to Zhiber-Shabat equation. By the using of this method, we obtain a new elliptic integral function solution. Also, this solution can be converted into Jacobi elliptic functions solution by a simple transformation.

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.

Traveling wave solutions along microtubules and in the Zhiber–Shabat equation

The aim of this paper is to apply the exponential rational function method for solving nonlinear equations arising in various physical models such as nonlinear electrical transmission lines, optical fibers, DNA, to mention a few. New exact solutions are obtained and can help to well understand the process of those systems. The method used is very effective and simple and can be applied to other types of nonlinear equations.

Solutions of Zhiber-Shabat and Related Equations Using a Modified tanh-coth Function Method

The modified tanh-coth function method is used to obtain new exact travelling wave solutions for Zhiber-Shabat equation and the related equations: Liouville equation, sinh-Gordon equation, Dodd-Bullough-Mikhailov equation, and Tzitzeica-Dodd-Bullough equation. Exact travelling wave solutions for each equation are derived and expressed in terms of hyperbolic functions, trigonometric functions and rational functions. The modified tanh-coth function method is easy to execute, brief, efficient, and can be used to solve many other nonlinear evolution equations.

An analysis of the Zhiber-Shabat equation including Lie point symmetries and conservation laws

We address the integrability aspects of the Zhiber-Shabat equation that serves as a generalised version of the $$\Phi -4$$ equation which is related to relativistic quantum mechanics. Three methods of integration are applied to this nonlinear evolution equation, namely the travelling-wave approach, Lie group method and the invariance-multiplier approach.

Algebro-Geometric Solution to the Bullough-Dodd-Zhiber-Shabat Equation

Based on solutions of the stationary zero-curvature equation associated with a 3×3 matrix spectral problem, we introduce a trigonal curve related to the Bullough–Dodd–Zhiber–Shabat equation and define Baker–Akhiezer functions and appropriate meromorphic function on the trigonal curve. Resorting to the algebro-geometric characters of the trigonal curve and properties of the three kinds of Abel differentials, we arrive at the explicit Riemann theta function representations of the Baker–Akhiezer functions and the meromorphic function. The algebro-geometric solution for the Bullough–Dodd–Zhiber–Shabat equation is obtained with the help of the asymptotic properties of the meromorphic function.

Some New Exact Traveling Wave Solutions for the Zhiber-Shabat Equation

In this paper, we implemented a very important method to solve nonlinear partial differential equations known as the )) ( exp( ξ φ - -expansion method. Recently, there are several methods being constructed for finding analytical solutions of nonlinear pa rtial differential equations. However, the )) ( exp( ξ φ - -expansion method is more effective and useful for solving the nonlinear evolution equations. With the help of this method, w e are investigated the exact traveling wave solutions of the Zhiber-Shabat equation. The ob taining exact solutions of this equation are describe many physical phenomena in mathematical physics such as solid state physics, plasma physics, nonlinear optics, chemical ki netics and quantum field theory. Further, three-dimensional plots of the solutions such as solitons, cus pon, periodic, singular kink and bell type are also given to visualize the dynamics of the equation.

G'/G-Expansion method for related equations to the Zhiber–Shabat equation

In this work G'/G-expansion method has been employed for related equations to ZhiberShabat equation: Liouville equation, sinh-Gordon equation, DoddBulloughMikhailov (DBM) equation. It is shown that G'/G- expansion method provides a very effective and a powerful mathematical tool, for solving these equations.

Classifying Traveling Wave Solutions to the Zhiber-Shabat Equation

By the complete discrimination system for polynomials, we classify exact traveling wave solutions to the Zhiber-Shabat equation, and compute some new traveling wave solutions.

Bell-polynomial approach and soliton solutions for the Zhiber–Shabat equation and (2+1)-dimensional Gardner equation with symbolic computation

Under investigation in this paper are the Zhiber–Shabat and (2+1)-dimensional Gardner equations in quantum fields, fluids and plasmas. Via the Hirota method and symbolic computation, the Bell-polynomial approach is performed to directly bilinearize those equations. For the Zhiber–Shabat equation, based on the bilinear form with an auxiliary variable, the bell-shaped soliton, upside-down bell-shaped soliton and breather-like solutions are obtained. Figures are plotted to illustrate the elastic interactions between two upside-down bell-shaped solitons and the interaction between the breather-like. As to the (2+1)-dimensional Gardner equation, bilinear form, Backlund transformation, one- and two-shock wave solutions are derived. Amplitude-compression and amplification interactions are investigated analytically and graphically.

A improved F‐expansion method and its application to the Zhiber–Shabat equation

Based on the F‐expansion method and Exp‐function method, an improved F‐expansion method is introduced. As illustrative examples, the exact solutions expressed by exponential function, hyperbolic functions, logarithmic function, and other type of functions for the Zhiber–Shabat equation are derived. Some previous results are extended. Copyright © 2012 John Wiley & Sons, Ltd.

Application of the ( [formula omitted])-expansion method for the Zhiber–Shabat equation and other related equations

In this paper, the exact traveling wave solutions of the Zhiber–Shabat equation and the related equations: Liouville equation, Dodd–Bullough–Mikhailov (DBM) equation, sinh–Gordon equation, and Tzitzeica–Dodd–Bullough (TDB) equation are studied by using the ( G ′ G )-expansion method. Solitons and periodic solutions for these equations are formally derived.

A CLASS OF SPECIAL EXACT SOLUTIONS OF SOME HIGH DIMENSIONAL NON-LINEAR WAVE EQUATIONS

In this paper, the separation transformation approach is extended to some high dimensional non-linear wave equations, such as the (N+1)-dimensional Zhiber–Shabat equation, the generalized (N+1)-dimensional complex non-linear Klein–Gordon equation and the generalized (N+1)-dimensional Toda lattice equation. As a result, a class of special exact solutions of these equations are obtained. The solutions obtained contain one or two arbitrary functions which may lead to abundant structures of the high dimensional non-linear wave equations.

New exact bounded travelling wave solutions for the Zhiber–Shabat equation

In this paper, the Zhiber–Shabat equation is studied by using the bifurcation theory and the method of phase portraits analysis. Many smooth solitary wave solutions, compacton solutions and smooth periodic wave solutions are obtained. Some previous results are extended.

Qualitative behavior and exact travelling wave solutions of the Zhiber–Shabat equation

In this paper, the qualitative behavior and exact travelling wave solutions of the Zhiber–Shabat equation are studied by using qualitative theory of polynomial differential system. The phase portraits of system are given under different parametric conditions. Some exact travelling wave solutions of the Zhiber–Shabat equation are obtained. The results presented in this paper improve the previous results.

Variational Principles for Some Nonlinear Wave Equations

Using the semi-inverse method proposed by Ji-Huan He, variational principles are established for some nonlinear wave equations arising in physics, including the Pochhammer-Chree equation, Zakharov-Kuznetsov equation, Korteweg-de Vries equation, Zhiber-Shabat equation, Kawahara equation, and Boussinesq equation.

The tanh method for travelling wave solutions to the Zhiber–Shabat equation and other related equations

The tanh method and the extended tanh method are used for handling the Zhiber–Shabat equation and the related equations: Liouville equation, sinh-Gordon equation, Dodd–Bullough–Mikhailov (DBM) equation, and Tzitzeica–Dodd–Bullough equation. Travelling wave solutions of different physical structures are formally derived for each equation.