Combined KdV mKdV Equation Research Articles

Exact solutions for the KdV–mKdV equation with time-dependent coefficients using the modified functional variable method

In this article, the functional variable method (fvm for short) is introduced to establish new exact travelling solutions of the combined KdV–mKdV equation. The technique of the homogeneous balance method is used in second stage to handle the appropriated solutions. We show that, the method is straightforward and concise for several kinds of nonlinear problems. Many new exact travelling wave solutions are successfully obtained.

Exact structures for the KdV–mKdV equation with variable coefficients via the functional variable method

In this study we introduce the functional variable method (fvm for short) for obtaining new exact travelling solutions of the combined KdV–mKdV equation. The technique of the homogeneous balance method is used in second stage to handle the appropriated solutions. We show that, the method is straightforward and concise for several kind of nonlinear problems. Many new exact traveling wave solutions are successfully obtained.

New solutions for conformable fractional Boussinesq and combined KdV-mKdV equations using Jacobi elliptic function expansion method

In this paper, the Jacobi elliptic function expansion method is proposed for the first time to construct the exact solutions of the time conformable fractional two-dimensional Boussinesq equation and the combined KdV-mKdV equation. New exact solutions are found. This method is based on Jacobi elliptic functions. The results obtained confirm that the proposed method is an efficient technique for analytic treatment of a wide variety of nonlinear conformable time-fractional partial differential equations.

New interaction solutions to the combined KdV–mKdV equation from CTE method

The consistent tanh expansion (CTE) method is developed for the combined KdV–mKdV equation. The combined KdV–mKdV equation is proved to be CTE solvable. New exact interaction solutions such as soliton–cnoidal wave solutions, soliton–periodic wave solutions for the combined KdV–mKdV equation are given out analytically and graphically.

All meromorphic solutions of an ordinary differential equation and its applications

Dedicated to Professor Yuzan He on the Occasion of his 80th BirthdayIn this paper, we employ the complex method to obtain all meromorphic solutions of an auxiliary ordinary differential equation at first and then find out all meromorphic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations. Our result shows that all rational and simply periodic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations are solitary wave solutions, the method is more simple than other methods, and there exist some rational solutions wr,2(z) and simply periodic solutions ws,2(z) that are not only new but also not degenerated successively by the elliptic function solutions. Copyright © 2016 John Wiley & Sons, Ltd.

Analytical Treatment of the Evolutionary (1 + 1)-Dimensional Combined KdV-mKdV Equation via the Novel (G'/G)-Expansion Method

The novel (G'/G)-expansion method is a powerful and simple technique for finding exact traveling wave solutions to nonlinear evolution equations (NLEEs). In this article, we study explicit exact traveling wave solutions for the (1 + 1)-dimensional combined KdV-mKdV equation by using the novel (G'/G)-expansion method. Consequently, various traveling wave solutions patterns including solitary wave solutions, periodic solutions, and kinks are detected and exhibited.

Analytic Solutions of the Space-Time Fractional Combined KdV-mKdV Equation

The fractional mapping method is proposed to solve fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional combined KdV-mKdV equation. Many types of exact analytical solutions are obtained. The solutions include generalized trigonometric and hyperbolic functions solutions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time.

All exact traveling wave solutions of the combined KdV-mKdV equation

Abstract In this article, we employ the complex method to obtain all meromorphic solutions of complex combined Korteweg-de Vries-modified Korteweg-de Vries equation (KdV-mKdV equation) at first, then we find all exact traveling wave solutions of the combined KdV-mKdV equation. The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all rational and simply periodic exact traveling wave solutions of the combined KdV-mKdV equation are solitary wave solutions, the complex method is simpler than other methods, and there exist some rational solutions w r , 2 ( z ) and simply periodic solutions w s , 2 ( z ) such that they are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role in finding exact solutions in mathematical physics. We also give some computer simulations to illustrate our main results. MSC: Primary 30D35; secondary 34A05. PACS Codes: 02.30.Jr; 02.70.Wz; 02.30.-f.

Enhanced group classification of Gardner equations with time-dependent coefficients

We classify the Lie symmetries of variable coefficient Gardner equations (called also the combined KdV–mKdV equations). In contrast to the particular results presented in Molati and Ramollo (2012) we perform the exhaustive group classification. It is shown that the complete results can be achieved using either the gauging of arbitrary elements of the class by the equivalence transformations or the method of mapping between classes. As by-product of the second approach the complete group classification of a class of variable coefficient mKdV equations with forcing term is derived. Advantages of the use of the generalized extended equivalence group in comparison with the usual one are also discussed.

Some new exact traveling wave solutions to the simplified MCH equation and the (1 + 1)-dimensional combined KdV–mKdV equations

The generalized (G′/G)-expansion method is thriving in finding exact traveling wave solutions of nonlinear evolution equations (NLEEs) in mathematical physics. In this paper, we bring to bear the generalized (G′/G)-expansion method to look for the exact solutions via the simplified MCH equation and the (1+1)-dimensional combined KdV–mKdV equations involving parameters. When the parameters take special values, solitary wave solutions are originated from the traveling wave solutions. It is established that the generalized (G′/G)-expansion method offers a further influential mathematical tool for constructing the exact solutions of NLEEs in mathematical physics.

Symmetry reduction, exact group-invariant solutions and conservation laws of the Benjamin–Bona–Mahoney equation

We show that the Benjamin–Bona–Mahoney (BBM) equation with power law nonlinearity can be transformed by a point transformation to the combined KdV–mKdV equation, that is also known as the Gardner equation. We then study the combined KdV–mKdV equation from the Lie group-theoretic point of view. The Lie point symmetry generators of the combined KdV–mKdV equation are derived. We obtain symmetry reduction and a number of exact group-invariant solutions for the underlying equation using the Lie point symmetries of the equation. The conserved densities are also calculated for the BBM equation with dual nonlinearity by using the multiplier approach. Finally, the conserved quantities are computed using the one-soliton solution.

Analytic treatment of nonlinear evolution equations using first integral method

In this paper, we show the applicability of the first integral method to combined KdV–mKdV equation, Pochhammer–Chree equation and coupled nonlinear evolution equations. The power of this manageable method is confirmed by applying it for three selected nonlinear evolution equations. This approach can also be applied to other nonlinear differential equations.

A LATTICE BGK MODEL FOR SIMULATING SOLITARY WAVES OF THE COMBINED KdV–MKDV EQUATION

A lattice BGK model for simulating solitary waves of the combined KdV–MKDV equation, ut+αuux-βu2ux+δuxxx= 0, is established. The tunable parameters in Chapman–Enskog expansion of the local equilibrium distribution function are determined by the coefficient of the combined KdV–MKDV equation. Simulating results fit close in with the theoretical results.

The single solitary wave with double kinks of the combined KdV equation and its stability

Based on the ideas of the hyperbola function expansion method, we obtained some analytical solutions of the combined KdV-mKdV (cKdV) equations by introducing new expansion functions.One of the single soliton solutions has the kink-antikink structure, and this solution reduces to the kink-like solution and the bell-like solution under different limitations. Theoretical analysis shows that the cKdV equation has both propagated-type and non-propagated-type solitary wave solutions. We also investigated the stability of the single solitary wave solution with double kinks numerically. The results indicate that the solution may be stable or unstable, depending on different sets of parameters.

A generalized [formula omitted]-expansion method and its applications to nonlinear evolution equations

In this paper, a generalized ( G ′ / G ) -expansion method and its algorithm are proposed by studying Wang’s ( G ′ / G ) -expansion method and constructing a first order nonlinear ordinary differential equation with a third-degree nonlinear term. Being concise and straightforward, the method is applied to combined KdV–MKdV equation and coupled Jaulent–Miodek equation. As a result, some new exact travelling wave solutions are obtained which include solitary wave solutions, triangular periodic wave solutions, exponential solutions, complex travelling solutions and rational solutions. This method can also be applied to other nonlinear evolution equations in mathematical physics.

A note on the Exp-function method and its application to nonlinear equations

An improved Exp-function method is described and used for finding a unified solution of a nonlinear wave equation. The combined KdV–MKdV equation is selected to illustrate the effectiveness and simplicity of the method. A generalized solitary solution with free constants is obtained. This method can be applied also to many other equations.

Bifurcations of travelling wave solutions for the combined [formula omitted]–[formula omitted] equation

By using the theory of bifurcations of dynamical systems to the combined kdv–mkdv equation, the existence of solitary wave, kink and anti-kink wave solutions and uncountably infinite many smooth periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.

Generalized Solitary Solution and Periodic Solution of the combined KdV-mKdV Equation with Variable Coefficients using the Exp-function Method

Article Generalized Solitary Solution and Periodic Solution of the combined KdV-mKdV Equation with Variable Coefficients using the Exp-function Method was published on June 1, 2009 in the journal International Journal of Nonlinear Sciences and Numerical Simulation (volume 10, issue 6).

On traveling wave solutions to combined KdV–mKdV equation and modified Burgers–KdV equation

In this work, we established exact solutions for some nonlinear evolution equations. The extended tanh method was used to construct solitary and soliton solutions of nonlinear evolution equations. The extended tanh method presents a wider applicability for handling nonlinear wave equations.

Variational iteration method for solving non-linear partial differential equations

In this paper, we shall use the variational iteration method to solve some problems of non-linear partial differential equations (PDEs) such as the combined KdV–MKdV equation and Camassa–Holm equation. The variational iteration method is superior than the other non-linear methods, such as the perturbation methods where this method does not depend on small parameters, such that it can fined wide application in non-linear problems without linearization or small perturbation. In this method, the problems are initially approximated with possible unknowns, then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory.