共查询到20条相似文献,搜索用时 31 毫秒
1.
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. 相似文献
2.
Wenjun Yuan Fanning Meng Jianming Lin Yonghong Wu 《Mathematical Methods in the Applied Sciences》2016,39(8):2083-2092
Dedicated to Professor Yuzan He on the Occasion of his 80th Birthday In 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. 相似文献
3.
M. Molati M.P. Ramollo 《Communications in Nonlinear Science & Numerical Simulation》2012,17(4):1542-1548
We perform symmetry classification of a variable-coefficient combined KdV-mKdV equation. That is, the equation combining the Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations, or also known as the Gardner equation. The direct method of group classification is utilized to specify the forms of these time-dependent coefficients. 相似文献
4.
This paper presents a mapping approach for the construction of exact solutions to the combined KdV and mKdV equation. There exist two types of soliton solutions which will reduce back to those of the KdV and mKdV equations in some appropriate limits. Four types of the general cnoidal wave solutions are also obtained. 相似文献
5.
《Applied Mathematics Letters》2007,20(10):1039-1045
A new isospectral problem is designed and the multi-component second mKdV equation is worked out from it. It follows that two distinct types of integrable couplings of the multi-component second mKdV equation are obtained by constructing two types of new loop algebras. As its reduction, two distinct types of integrable couplings of the multi-component KdV equation, the multi-component mKdV equation and the multi-component KdV–mKdV equation are presented. 相似文献
6.
We prove the sharp global well-posedness result for the initial value problem (IVP) associated to the system of the modified
Korteweg-de Vries (mKdV) equation. For the single mKdV equation such result has been obtained by using Mirura’s Transform
that takes the KdV equation to the mKdV equation [8]. We do not know the existence of Miura’s Transform that takes a KdV system
to the system we are considering. To overcome this difficulty we developed a new proof of the sharp global well-posedness
result for the single mKdV equation without using Miura’s Transform. We could successfully apply this technique in the case
of the mKdV system to obtain the desired result. 相似文献
7.
广义组合KdV-mKdV方程的显式精确解 总被引:1,自引:0,他引:1
Abstract. With the aid of Mathematica and Wu-elimination method,via using a new generalizedansatz and well-known Riccati equation,thirty-two families of explicit and exact solutions forthe generalized combined KdV and mKdV equation are obtained,which contain new solitarywave solutions and periodic wave solutions. This approach can also be applied to other nonlinearevolution equations. 相似文献
8.
Nikolay A. Kudryashov Dariya V. Safonova 《Mathematical Methods in the Applied Sciences》2019,42(13):4627-4636
The method for constructing first integrals and general solutions of nonlinear ordinary differential equations is presented. The method is based on index accounting of the Fuchs indices, which appeared during the Painlevé test of a nonlinear differential equation. The Fuchs indices indicate us the leading members of the first integrals for the origin differential equation. Taking into account the values of the Fuchs indices, we can construct the auxiliary equation, which allows to look for the first integrals of nonlinear differential equations. The method is used to obtain the first integrals and general solutions of the KdV‐Burgers and the mKdV‐Burgers equations with a source. The nonautonomous first integrals in the polynomials form are found. The general solutions of these nonlinear differential equations under at some additional conditions on the parameters of differential equations are also obtained. Illustrations of some solutions of the KdV‐Burgers and the mKdV‐Burgers are given. 相似文献
9.
Theoretical and Mathematical Physics - We use the method of Lie symmetry analysis to investigate the properties of a (2+1)-dimensional KdV–mKdV equation. Using the Ibragimov method, which... 相似文献
10.
We establish the splitting of homoclinic orbits for a near-integrable lattice modified KdV (mKdV) equation with periodic boundary conditions. We use the Bäcklund transformation to construct homoclinic orbits of the lattice mKdV equation. We build the Melnikov function with the gradient of the invariant defined through the discrete Floquet discriminant evaluated at critical points. The criteria for the persistence of homoclinic solutions of the perturbed lattice mKdV equation are established. 相似文献
11.
It is well known that each solution of the modified Korteveg–de Vries (mKdV) equation gives rise, via the Miura transformation, to a solution of the Korteveg–de Vries (KdV) equation. In this work, we show that a similar Miura-type transformation exists also for the “good” Boussinesq equation. This transformation maps solutions of a second-order equation to solutions of the fourth-order Boussinesq equation. Just like in the case of mKdV and KdV, the correspondence exists also at the level of the underlying Riemann–Hilbert problems and this is in fact how we construct the new transformation. 相似文献
12.
《Communications in Nonlinear Science & Numerical Simulation》2012,17(6):2319-2332
A new generalized AKNS hierarchy is presented starting from a 4 × 4 matrix spectral problem with four potentials. Its generalized bi-Hamiltonian structure is also investigated by using the trace identity. Moreover, the special coupled nonlinear equation, the coupled KdV equation, the KdV equation, the coupled mKdV equation and the mKdV equation are produced from the generalized AKNS hierarchy. Most importantly, a Darboux transformation for the generalized AKNS hierarchy is established with the aid of the gauge transformation between the corresponding 4 × 4 matrix spectral problem, by which multiple soliton solutions of the generalized AKNS hierarchy are obtained. As a reduction, a Darboux transformation of the mKdV equation and its new analytical positon, negaton and complexiton solutions are given. 相似文献
13.
Cesar A. Gómez Sierra 《Journal of Computational and Applied Mathematics》2011,235(17):5330-5332
In this work, the improved tanh-coth method is used to obtain wave solutions to a Korteweg-de Vries (KdV) equation with higher-order nonlinearity, from which the standard KdV and the modified Korteweg-de Vries (mKdV) equations with variable coefficients can be derived as particular cases. However, the model studied here include other important equations with applications in several fields of physical and nonlinear sciences. Periodic and soliton solutions are formally derived. 相似文献
14.
15.
Jennifer Gorsky 《Journal of Mathematical Analysis and Applications》2005,303(2):522-529
In both the periodic and non-periodic case we construct non-analytic complex-valued solutions for the generalized KdV equation with appropriate analytic initial data. Moreover, for the KdV and mKdV we construct real-valued non-analytic solutions. 相似文献
16.
The equation hierarchy presented in this paper contains the KdV equation and the mKdV equation. By use of the concept of characteristic number, an undetermined-constant method is proposed by us, for which the polynomial Hamiltonian functions are constructed. By employing the method, the Hamiltonian structure of the equation hierarchy is established. The approach presented in the paper shares extensive applications. In addition, four explicit expressions of the travelling wave solutions to the above equation hierarchy are obtained. One of them is regular, the other three are singular. 相似文献
17.
本文从文[2]的基本方程出发,采用约化摄动方法和PLK方法,讨论了三阶非线性和色散效应相平衡的修正的KdV(mKdV)孤立波迎撞问题.这些波在流体密度比等于流体深度比平方的两流体系统界面上传播.我们求得了二阶摄动解,发现在不考虑非均匀相移的情况下,碰撞后孤立波保持原有的形状,这与Fornberg和whitham[6]的追撞数值分析结果一致,但当考虑波的非均匀相移后,碰撞后波形将变化. 相似文献
18.
Fu-mei Yin Peng Chen Guang-sheng Wang Deng-yuan Chen 《Chaos, solitons, and fractals》2009,39(2):926-935
The generalized double Wronskian solutions of the third-order isospectral AKNS equation are obtained. Thus we found rational solutions, Matveev solutions, complexitons and interaction solutions. Moreover, rational solutions of the mKdV equation and KdV equation in double Wronskian form are constructed by reduction. 相似文献
19.
In this paper, a new method to solve space–time‐dependent non‐linear equations is proposed. After considering the variable coefficient of a non‐linear equation as a new dependent variable, some special types of space–time‐dependent equations can be solved from corresponding space–time‐independent equations by using the general classical Lie approach. The rich soliton solutions of space–time‐dependent KdV equation and mKdV equation are given with the help of the approach. Copyright © 2004 John Wiley & Sons, Ltd. 相似文献
20.
In this work we devise an algebraic method to uniformly construct solitary wave solutions and doubly periodic wave solutions of physical interest for the Kersten–Krasil’shchik coupled KdV–mKdV system. This system as the classical part of one of superextension of the KdV equation was proposed very recently. The complete integrability, singular analysis and Lax pairs for this system have been found, but its exact solution are still unknown. 相似文献