Wave Solutions In Terms Research Articles

A novel (G′/G)-expansion method for solving the (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsov equation in mathematical physics

A novel G′/G-expansion method is presented in this article to construct more general type and new travelling wave solutions of non-linear evolution equations. The method is illustrated in its application via the 3 + 1-dimensional modified KdV-Zakharov-Kuznetsov equation which is used to model different non-linear phenomena. Through the assistance of Maple, we obtain several new and more general travelling wave solutions in terms of the hyperbolic functions, the trigonometric functions, the rational functions and their combination with parameters. The solutions obtained in this article can successfully recover previously known solutions by setting particular values of the parameters that have been found by other sophisticated methods.

Solitary wave solutions of the fourth order Boussinesq equation through the exp(-Ф(η))-expansion method.

The exp(–Ф(η))-expansion method is an ascending method for obtaining exact and solitary wave solutions for nonlinear evolution equations. In this article, we implement the exp(–Ф(η))-expansion method to build solitary wave solutions to the fourth order Boussinesq equation. The procedure is simple, direct and useful with the help of computer algebra. By using this method, we obtain solitary wave solutions in terms of the hyperbolic functions, the trigonometric functions and elementary functions. The results show that the exp(–Ф(η))-expansion method is straightforward and effective mathematical tool for the treatment of nonlinear evolution equations in mathematical physics and engineering.Mathematics subject classifications35C07; 35C08; 35P99

Exact traveling wave solutions of modified KdV–Zakharov–Kuznetsov equation and viscous Burgers equation

Mathematical modeling of many physical systems leads to nonlinear evolution equations because most physical systems are inherently nonlinear in nature. The investigation of traveling wave solutions of nonlinear partial differential equations (NPDEs) plays a significant role in the study of nonlinear physical phenomena. In this article, we construct the traveling wave solutions of modified KDV-ZK equation and viscous Burgers equation by using an enhanced (G '/G) -expansion method. A number of traveling wave solutions in terms of unknown parameters are obtained. Derived traveling wave solutions exhibit solitary waves when special values are given to its unknown parameters.Mathematics subject classification35C07; 35C08; 35P99

Effective dissipation and nonlocality induced by nonparaxiality

We investigate beam diffraction and spatial modulation instability of coherent light beams propagating in the non-paraxial regime in a nonlinear Kerr medium. We study the instability of plane wave solutions in terms of the degree of non-paraxiality, beyond the regime of validity of the nonlinear Schroedinger equation. We also numerically analyze the way non-paraxial terms breaks the integrability and affect the periodical evolution of higher order soliton solutions. We discuss non-locality and effective dissipation introduced by the non-linear coupling with evanescent waves.

The new approach of the generalized (G′/G)-expansion method for nonlinear evolution equations

In this article, we apply the new generalized (G′/G)-expansion method to solve two nonlinear partial differential equations and construct traveling wave solutions in terms of hyperbolic functions, trigonometric functions, and rational functions with arbitrary parameters. We underline the power of the new generalized (G′/G)-expansion method in providing generalized solitary wave solutions of different physical structures. It is shown that the new generalized (G′/G)-expansion method is very effective and provides a powerful mathematical tool to solve nonlinear partial differential equation systems in mathematical physics and engineering.

Exact traveling wave solutions to the Klein–Gordon equation using the novel (G′/G)-expansion method

The novel (G′/G)-expansion method is one of the powerful methods that appeared in recent times for establishing exact traveling wave solutions of nonlinear partial differential equations. Exact traveling wave solutions in terms of hyperbolic, trigonometric and rational functions to the cubic nonlinear Klein–Gordon equation via this method are obtained in this article. The efficiency of this method for finding exact solutions and traveling wave solutions has been demonstrated. It is shown that the novel (G′/G)-expansion method is a simple and valuable mathematical tool for solving nonlinear evolution equations (NLEEs) in applied mathematics, mathematical physics and engineering.

Effects of ion mobility and positron fraction on solitary waves in weak relativistic electron-positron-ion plasma

The effects of ion mobility and positron fraction on the solitary waves of the laser field envelope and the potential of the electrostatic field in weak relativistic electron-positron-ion plasma are investigated. The parameter region for the existence of solitary waves is obtained analytically, and a reasonable choice of parameters is clarified. Both cases of mobile and immobile ions are considered. It is found that the amplitudes of solitary waves in the former case are larger compared to the latter case. For small plasma density, the localized solitary wave solutions in terms of the approximate perturbation analytical method are very consistent with those by exact numerical calculations. However, as the plasma density increases the analytical method loses its validity more and more. The influence of the positron fraction on the amplitudes of solitary waves shows a monotonous increasing relation. The implications of our results to particle acceleration are also discussed briefly.

Localized nonlinear waves in combined time-dependent magnetic–optical potentials with spatiotemporally modulated nonlinearities

Abstract We construct explicit novel solutions of the nonlinear Schrodinger equation with spatiotemporal modulation of the nonlinearities and potentials. By using a modified similarity transformation we explore some localized nonlinearities and combined time-dependent magnetic–optical potentials in the form of linear-lattice ones and harmonic-lattice ones. Several families of exact localized nonlinear wave solutions in terms of Mathieu and elliptic functions corresponding to these potentials are then studied, such as snakelike solitons and breathing solitons. The stability of the obtained localized nonlinear wave solutions is investigated numerically such that some stable solutions are found.

Multi-symplectic method for the generalized (2+1)-dimensional KdV-mKdV equation

In the present paper, a general solution involving three arbitrary functions for the generalized (2+1)-dimensional KdV-mKdV equation, which is derived from the generalized (1+1)-dimensional KdV-mKdV equation, is first introduced by means of the Wiess, Tabor, Carnevale (WTC) truncation method. And then multisymplectic formulations with several conservation laws taken into account are presented for the generalized (2+1)-dimensional KdV-mKdV equation based on the multisymplectic theory of Bridges. Subsequently, in order to simulate the periodic wave solutions in terms of rational functions of the Jacobi elliptic functions derived from the general solution, a semi-implicit multi-symplectic scheme is constructed that is equivalent to the Preissmann scheme. From the results of the numerical experiments, we can conclude that the multi-symplectic schemes can accurately simulate the periodic wave solutions of the generalized (2+1)-dimensional KdV-mKdV equation while preserve approximately the conservation laws.

Quasi-periodic wave solutions for the (2+1) -dimensional generalized Calogero–Bogoyavlenskii–Schiff (CBS) equation

The Hirota bilinear method is implemented to directly construct quasi-periodic wave solutions in terms of theta functions for a Hirota bilinear equation. The result is applied to the (2+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff (CBS) equation, thereby yielding their quasi-periodic wave solutions under the Backlund transformation u = η y + 3 δ [ l n f ( x , y , t ) ] x .

Exact solutions and Painlevé analysis of a new (2+1)-dimensional generalized KdV equation

The new (2+1)-dimensional generalized KdV equation which exists the bilinear form is mainly discussed. We prove that the equation does not admit the Painleve property even by taking the arbitrary constant a=0. However, this result is different from Radha and Lakshmanan’s work. In addition, based on Hirota bilinear method, periodic wave solutions in terms of Riemann theta function and rational solutions are derived, respectively. The asymptotic properties of the periodic wave solutions are analyzed in detail.

Theta-function Solutions to the (2+l)-Dimensional Breaking Soliton Equation

A new periodic wave solution in terms of theta functions is presented for a kind of elliptic equation. Based on the results, with the help of Mathematica and the improved generalized F-expansion method, some periodic wave solutions in terms of theta functions are obtained for the (2+1)-dimensional breaking soliton equation. In addition, x-direction periodic wave solutions are derived, their properties and profiles are displayed in 3D figures. To our knowledge, these solutions are reported for the first time.

Extended Jacobi elliptic function expansion solutions of variant Boussinesq equations

With the aid of symbolic computation Maple, an extended Jacobi elliptic function expansion method is presented and successfully applied to variant Boussinesq equations. As a result, abundant periodic wave solutions in terms of the Jacobi elliptic functions are obtained. When the modulus m → 1 or m → 0, exact solitary wave solutions and trigonometric function solutions are also derived. The properties of four new solutions are graphically studied.

New approach for tanh and extended-tanh methods with applications on Hirota-Satsuma equations

In this work, we establish new exact solutions for the Hirota-Satsuma equations. New approach for the tanh is used and extended tanh methods to construct traveling wave solutions in terms of a hyperbolic tangent functions. New families of solitary wave solutions and periodic solutions are also obtained for Hirota-Satsuma equations. Our approach is reduce the size of the computational adopted in other techniques without any conditions to apply on any system of partial differential equations.

New approach for tanh and extended-tanh methods with applications on Hirota-Satsuma equations

In this work, we establish new exact solutions for the Hirota-Satsuma equations. New approach for the tanh is used and extended tanh methods to construct traveling wave solutions in terms of a hyperbolic tangent functions. New families of solitary wave solutions and periodic solutions are also obtained for Hirota-Satsuma equations. Our approach is reduce the size of the computational adopted in other techniques without any conditions to apply on any system of partial differential equations.

Exact travelling wave solutions for the Zakharov–Kuznetsov equation

The travelling wave solution for the Zakharov–Kuznetsov equation is studied by means of the extended mapping method. A diversity of travelling periodic wave solutions in terms of the Jacobi elliptic functions are obtained. Limit cases are studied, and solitary wave solutions are got.

A Study for Obtaining New and More General Solutionsof Special-Type Nonlinear Equation

The generalized algebraic method with symbolic computationis extended to some special-type nonlinear equations for constructinga series of new and more general travelling wave solutions interms of special functions. Such equations cannot be directlydealt with by the method and require some kinds of pre-processingtechniques. It is shown that soliton solutions and triangularperiodic solutions can be established as the limits of the Jacobidoubly periodic wave solutions.

On travelling wave solutions of the Burgers–Korteweg–de Vries equation**The paper was presented at the Symposium of Applied Mathematics, Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing, China, 15 May 2006, and at the 28th Texas Partial Differential Equations Conference, University of Texas–Pan American, Edinburg, TX 78541, 20 March 2005.

In this paper, we survey some recent advances in the study of travelling wave solutions to the Burgers–Korteweg–de Vries equation. Some comments are given on the existing results. A class of travelling solitary wave solutions in terms of elliptic functions with arbitrary velocity is obtained by means of the first-integral method as well as the method of compatible vector fields.

Doubly periodic wave and folded solitary wave solutions for (2+1)-dimensional higher-order Broer–Kaup equation

A general solution including three arbitrary functions is obtained for the (2 + 1)-dimensional high-order Broer–Kaup equation by means of WTC truncation method. From the general solution, doubly periodic wave solutions in terms of the Jacobian elliptic functions with different modulus and folded solitary wave solutions determined by appropriate multiple valued functions are obtained. Some interesting novel features and interaction properties of these exact solutions and coherent localized structures are revealed.

Two Classes of New Exact Solutions to (2+1)-DimensionalBreaking Soliton Equation**The project supported by the PostdoctoralScience Foundation of China

The singular manifold method is used to obtain twogeneral solutions to a (2+1)-dimensional breaking solitonequation, each of which contains two arbitrary functions. Then thenew periodic wave solutions in terms of the Jacobi ellipticfunctions are generated from the general solutions. The long wavelimit yields the new types of dromion and solitary structures.