With the aid of Mathematica and Wu elimination method,via using a new generalized ansatz and well known Riccati equation,thirty two families of explicit and exact solutions for the generalized combined KdV and mKdV...With the aid of Mathematica and Wu elimination method,via using a new generalized ansatz and well known Riccati equation,thirty two families of explicit and exact solutions for the generalized combined KdV and mKdV equation are obtained,which contain new solitary wave solutions and periodic wave solutions.This approach can also be applied to other nonlinear evolution equations.展开更多
Bifurcation, bistability and solitary waves of the combined KdV and mKdV equation are investigatedsystematically. At first, bifurcation and bistability are analyzed by selecting an integral constant as the bifurcation...Bifurcation, bistability and solitary waves of the combined KdV and mKdV equation are investigatedsystematically. At first, bifurcation and bistability are analyzed by selecting an integral constant as the bifurcationparameter. Then, different conditions expressed in terms of the bifurcation parameter are obtained for the existence ofbreather-like, algebraic, pulse-like solitary waves, and shock waves. All types of the solitary wave and shock wave solutionsare given by direct integration. Finally, an approximate analytic method by employing the interpolation polynomials iscomplete and the theoretical methods are the simplest hitherto.展开更多
Some new exact solutions of an auxiliary ordinary differential equation are obtained, which were neglected by Sirendaoreji et al in their auxiliary equation method. By using this method and these new solutions the com...Some new exact solutions of an auxiliary ordinary differential equation are obtained, which were neglected by Sirendaoreji et al in their auxiliary equation method. By using this method and these new solutions the combined Korteweg-de Vries （KdV） and modified KdV （mKdV） equation and （2＋1）-dimensional Broer-Kaup-Kupershmidt system are investigated and abundant exact travelling wave solutions are obtained that include new solitary wave solutions and triangular periodic wave solutions.展开更多
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 sol...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.展开更多
In this paper the ( G'/G )-expansion method is used to find exact travelling wave solutions for a combined KdV and Schwarzian KdV equation. As a result, multiple travelling wave solutions with arbitrary parameters...In this paper the ( G'/G )-expansion method is used to find exact travelling wave solutions for a combined KdV and Schwarzian KdV equation. As a result, multiple travelling wave solutions with arbitrary parameters are obtained, which are expressed by hyperbolic functions, trigonometric functions and rational functions. When the parameters are taken as special values, the solitary waves are derived from the travelling waves. The (G'/G)-expansion method presents a wider applicability for handling nonlinear wave equations.展开更多
This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries （KdV） equation and a coupled modified Korteweg-de Vries （mKdV）...This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries （KdV） equation and a coupled modified Korteweg-de Vries （mKdV） equation. This method provides a sequence Of functions which converges to the exact solution of the problem and is based on the use of Lagrange multiplier for identification of optimal values of parameters in a functional. Some examples are given to demonstrate the reliability and convenience of the method and comparisons are made with the exact solutions.展开更多
The authors develop a direct approach to the soliton perturbation based on the separation of variables. With the aid of approach, the first-order effects of perturbation on a KdV-MKdV soliton are derived, both the slo...The authors develop a direct approach to the soliton perturbation based on the separation of variables. With the aid of approach, the first-order effects of perturbation on a KdV-MKdV soliton are derived, both the slow time-dependence of the soliton parameters and the first-order correction are obtained.展开更多
The exact solution for the combined KS and KdV equation is obtained via introducing a simple and effective nonlinear transformations.This method is very concise and primary and can be applied to other unintegrable non...The exact solution for the combined KS and KdV equation is obtained via introducing a simple and effective nonlinear transformations.This method is very concise and primary and can be applied to other unintegrable nonlinear evolution equations.It is common knowledge that the Korteweg de Vries(KdV) equation [1] (1)has been proposed as model equation for the weakly nonlinear long waves which occur in many different physical systems; the Kuramoto-Sivashinsky (KS) equationis one of the simplest nonliaear partial differential equations that exhibit Chaotic behavior frequently encounted in the study of continous media [2-4] . Many interesting mathematical and physical properties of eqs. (1) and (2) have been studied widely. But, in several problems where a lonq wavelength oscilatory instability is found, the noulineai evolution of the perturbations near rriticality is governed by the dispersion modified Kuramoto-Sivashi nsky equation(3)ft is clear that this equation is a combination of the KdV and展开更多
In this paper, we consider small perturbations of the KdV-mKdV equation u_t =-u_(xxx) + 6 uu_x + 6 u^2 u_x on the real line with periodic boundary conditions. It is shown that the above equation admits a Cantor family...In this paper, we consider small perturbations of the KdV-mKdV equation u_t =-u_(xxx) + 6 uu_x + 6 u^2 u_x on the real line with periodic boundary conditions. It is shown that the above equation admits a Cantor family of small amplitude quasi-periodic solutions under such perturbations. The proof is based on an abstract infinite dimensional KAM theorem.展开更多
基金Supported by the National Key Basic Research Project Foundation of China(G1 9980 30 60 0 ) and theHigher Education Doctoral Fo
文摘With the aid of Mathematica and Wu elimination method,via using a new generalized ansatz and well known Riccati equation,thirty two families of explicit and exact solutions for the generalized combined KdV and mKdV equation are obtained,which contain new solitary wave solutions and periodic wave solutions.This approach can also be applied to other nonlinear evolution equations.
文摘Bifurcation, bistability and solitary waves of the combined KdV and mKdV equation are investigatedsystematically. At first, bifurcation and bistability are analyzed by selecting an integral constant as the bifurcationparameter. Then, different conditions expressed in terms of the bifurcation parameter are obtained for the existence ofbreather-like, algebraic, pulse-like solitary waves, and shock waves. All types of the solitary wave and shock wave solutionsare given by direct integration. Finally, an approximate analytic method by employing the interpolation polynomials iscomplete and the theoretical methods are the simplest hitherto.
基金Project supported by the National Natural Science Foundation of China （Grant No 10472029）.
文摘Some new exact solutions of an auxiliary ordinary differential equation are obtained, which were neglected by Sirendaoreji et al in their auxiliary equation method. By using this method and these new solutions the combined Korteweg-de Vries （KdV） and modified KdV （mKdV） equation and （2＋1）-dimensional Broer-Kaup-Kupershmidt system are investigated and abundant exact travelling wave solutions are obtained that include new solitary wave solutions and triangular periodic wave solutions.
文摘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.
基金Supported by the Natural Science Foundation of Education Department of Henan Province（2011Bl10013） Supported by the Youth Science Foundation of Henan University of Science and Tech- nology（2008QN026）
文摘In this paper the ( G'/G )-expansion method is used to find exact travelling wave solutions for a combined KdV and Schwarzian KdV equation. As a result, multiple travelling wave solutions with arbitrary parameters are obtained, which are expressed by hyperbolic functions, trigonometric functions and rational functions. When the parameters are taken as special values, the solitary waves are derived from the travelling waves. The (G'/G)-expansion method presents a wider applicability for handling nonlinear wave equations.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.10771019 and 10826107)
文摘This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries （KdV） equation and a coupled modified Korteweg-de Vries （mKdV） equation. This method provides a sequence Of functions which converges to the exact solution of the problem and is based on the use of Lagrange multiplier for identification of optimal values of parameters in a functional. Some examples are given to demonstrate the reliability and convenience of the method and comparisons are made with the exact solutions.
基金the National Science Foundation of China(19775013)
文摘The authors develop a direct approach to the soliton perturbation based on the separation of variables. With the aid of approach, the first-order effects of perturbation on a KdV-MKdV soliton are derived, both the slow time-dependence of the soliton parameters and the first-order correction are obtained.
文摘The exact solution for the combined KS and KdV equation is obtained via introducing a simple and effective nonlinear transformations.This method is very concise and primary and can be applied to other unintegrable nonlinear evolution equations.It is common knowledge that the Korteweg de Vries(KdV) equation [1] (1)has been proposed as model equation for the weakly nonlinear long waves which occur in many different physical systems; the Kuramoto-Sivashinsky (KS) equationis one of the simplest nonliaear partial differential equations that exhibit Chaotic behavior frequently encounted in the study of continous media [2-4] . Many interesting mathematical and physical properties of eqs. (1) and (2) have been studied widely. But, in several problems where a lonq wavelength oscilatory instability is found, the noulineai evolution of the perturbations near rriticality is governed by the dispersion modified Kuramoto-Sivashi nsky equation(3)ft is clear that this equation is a combination of the KdV and
基金Supported by NSFC(11601036,11401041)Science and Technology Foundation of Shandong Province(J16LI52)
文摘In this paper, we consider small perturbations of the KdV-mKdV equation u_t =-u_(xxx) + 6 uu_x + 6 u^2 u_x on the real line with periodic boundary conditions. It is shown that the above equation admits a Cantor family of small amplitude quasi-periodic solutions under such perturbations. The proof is based on an abstract infinite dimensional KAM theorem.
