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1.
Under investigation in this paper is a fifth-order Korteweg-de Vries (fKdV) equation, which can be used to describe many nonlinear phenomena in fluid dynamics and plasma physics. Based on the binary Bell polynomials, a lucid and systematic approach is proposed to systematically study its bilinear representation, bilinear Bäcklund transformations and Lax pairs with explicit formulas, respectively. These results can be reduced to the ones of several integrable equations such as Sawada-Kotera equation, Caudrey-Dodd-Gibbon equation, Lax equation, Kaup-Kuperschmidt equation and Ito equation, etc. Furthermore, the N-solitary wave solutions formula and quasi-periodic wave solutions are obtained by using bilinear form of the fKdV equation. Finally, the relation between the periodic wave solution and solitary wave solution is rigorously established. 相似文献
2.
In this paper, nonlocal residual symmetry of a generalized (2+1)-dimensional Korteweg–de Vries equation is derived with the aid of truncated Painlevé expansion. Three kinds of non-auto and auto Bäcklund transformations are established. The nonlocal symmetry is localized to a Lie point symmetry of a prolonged system by introducing auxiliary dependent variables. The linear superposed multiple residual symmetries are presented, which give rise to the th Bäcklund transformation. The consistent Riccati expansion method is employed to derive a Bäcklund transformation. Furthermore, the soliton solutions, fusion-type -solitary wave solutions and soliton–cnoidal wave solutions are gained through Bäcklund transformations. 相似文献
3.
In this paper, we implement the exp-function method to obtain the exact travelling wave solutions of (N + 1)-dimensional nonlinear evolution equations. Four models, the (N + 1)-dimensional generalized Boussinesq equation, (N + 1)-dimensional sine-cosine-Gordon equation, (N + 1)-double sinh-Gordon equation and (N + 1)-sinhcosinh-Gordon equation, are used as vehicles to conduct the analysis. New travelling wave solutions are derived. 相似文献
4.
Nonlinear quantum theory of interaction of charged particles and monochromatic radiation in a medium
G. K. Avetisyan A. Kh. Bagdasaryan G. F. Mkrtchyan 《Journal of Experimental and Theoretical Physics》1998,86(1):24-31
We study the quantum theory of nonlinear interaction of charged particles and a given field of plane-transverse electromagnetic
radiation in a medium. Using the exact solution of the generalized Lamé equation, we find the nonlinear solution of the Mathieu
equation to which the relativistic quantum equation of particle motion in the field of a monochromatic wave in the medium
reduces if one ignores the spin-spin interaction (the Klein-Gordon equation).We study the stability of solutions of the generalized
Lamé equation and find a class of bounded solutions corresponding to the wave function of the particle. On the basis of this
solution we establish that the particle states in a stimulated Cherenkov process form bands. Depending on the wave intensity
and polarization, such a band structure describes both bound particle-wave states (capture) and states in the continuous spectrum.
It is obvious that in a plasma there can be no such bands, since bound states of a particle with a transverse wave whose phase
velocity v
ph is higher than c are impossible in this case. The method developed in the paper can be applied to a broad class of problems reducible to the
solution of the Mathieu equation.
Zh. éksp. Teor. Fiz. 113, 43–57 (January 1998) 相似文献
5.
Yu. N. Ovchinnikov 《Journal of Experimental and Theoretical Physics》1998,87(4):807-813
An explicit expression for the excitation spectrum of the stationary solutions of a nonlinear wave equation is obtained. It
is found that all branches of many-valued solutions of a nonlinear wave equation between the (2K+1,2K+2) turning points (branch points in the complex plane of the nonlinearity parameter) are unstable. Some parts of branches
between the (2K,2K+1) turning points are also unstable. The instability of the latter is related to the possibility that pairs of complex conjugate
eigenvalues cross the real axis in the κ plane.
Zh. éksp. Teor. Fiz. 114, 1487–1499 (October 1998)
Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor. 相似文献
6.
7.
《Journal of Nonlinear Mathematical Physics》2013,20(1):58-76
Abstract We construct non-localized, real global solutions of the Kadomtsev-Petviashvili-I equation which vanish for x → ?∞ and study their large time asymptotic behavior. We prove that such solutions eject (for t → ∞) a train of curved asymptotic solitons which move behind the basic wave packet. 相似文献
8.
Cheng-Shi Liu 《Foundations of Physics》2011,41(5):793-804
To find exact traveling wave solutions to nonlinear evolution equations, we propose a method combining symmetry properties
with trial polynomial solution to nonlinear ordinary differential equations. By the method, we obtain some exact traveling
wave solutions to the Burgers-KdV equations and a kind of reaction-diffusion equations with high order nonlinear terms. As
a result, we prove that the Burgers-KdV equation does not have the real solution in the form a
0+a
1tan ξ+a
2tan 2
ξ, which indicates that some types of the solutions to the Burgers-KdV equation are very limited, that is, there exists no
new solution to the Burgers-KdV equation if the degree of the corresponding polynomial increases. For the second equation,
we obtain some new solutions. In particular, some interesting structures in those solutions maybe imply some physical meanings.
Finally, we discuss some classifications of the reaction-diffusion equations which can be solved by trial equation method. 相似文献
9.
Wang Rui-min Ge Jian-ya Dai Chao-Qing Zhang Jie-Fang 《International Journal of Theoretical Physics》2007,46(1):102-115
Utilizing the extended projective Ricatti equation expansion method, abundant variable separation solutions of the (2+1)-dimensional
dispersive long wave systems are obtained. From the special variable separation solution (38) and by selecting appropriate
functions, new types of interaction between the multi-valued and the single-valued solitons, such as semi-foldon and dromion,
semi-foldon and peakon, semi-foldon and compacton are found. Meanwhile, we conclude that the solution v is essentially equivalent to the ’universal” formula (1).
PACS numbers 05.45.Yv, 02.30.Jr, 03.65.Ge 相似文献
10.
《Journal of Nonlinear Mathematical Physics》2013,20(1-2):49-61
Abstract We study integrability of a system of nonlinear partial differential equations consisting of the nonlinear d’Alembert equation □u = F (u) and nonlinear eikonal equation u xµ u x µ = G(u) in the complex Minkowski space R(1, 3). A method suggested makes it possible to establish necessary and sufficient compatibility conditions and construct a general solution of the d’Alembert-eikonal system for all cases when it is compatible. The results obtained can be applied, in particular, to construct principally new (non-Lie, non-similarity) solutions of the non-linear d’Alembert, Dirac, and Yang-Mills equations. Solutions found in this way are shown to correspond to conditional symmetry of the equations enumerated above. Using the said approach, we study in detail conditional symmetry of the nonlinear wave equation □w = F 0(w) in the four-dimensional Minkowski space. A number of new (non-Lie) reductions of the above equation are obtained giving rise to its new exact solutions which contain arbitrary functions. 相似文献
11.
《Waves in Random and Complex Media》2013,23(1):56-76
Abstract In this paper, we introduce and study rigorously a Hamiltonian structure and soliton solutions for a weakly dissipative and weakly nonlinear medium that governs two Korteweg–de vries (KdV) wave modes. The bounded solution and traveling wave solution such as cnoidal wave and solitary wave are obtained. Subsequently, the equation is numerically solved by Fourier spectral method for its two-soliton solution. These solutions may be useful to explain the nonlinear dynamics of waves for an equation supporting multi-mode weakly dispersive and nonlinear wave medium. In addition, we give an explicit explanation of the mathematics behind the soliton phenomenon for a better understanding of the equation. 相似文献
12.
《Waves in Random and Complex Media》2013,23(2):378-385
This paper obtains the soliton solutions of the Gilson–Pickering equation. The G′/G method will be used to carry out the solutions of this equation and then the solitary wave ansatz method will be used to obtain a 1-soliton solution of this equation. Finally, the invariance and multiplier approach will be applied to recover a few of the conserved quantities of this equation. 相似文献
13.
We apply the (G’/G)-expansion method to solve two systems of nonlinear differential equations and construct traveling wave solutions expressed in terms of hyperbolic functions, trigonometric functions, and rational functions with arbitrary parameters. We highlight the power of the (G’/G)-expansion method in providing generalized solitary wave solutions of different physical structures. It is shown that the (G’/G)-expansion method is very effective and provides a powerful mathematical tool to solve nonlinear differential equation systems in mathematical physics. 相似文献
14.
This paper studies the Zakharov-Kuznetsov equation in (1+3) dimensions with an arbitrary power law nonlinearity. The method
of Lie symmetry analysis is used to carry out the integration of the Zakharov-Kuznetsov equation. The solutions obtained are
cnoidal waves, periodic solutions, singular periodic solutions, and solitary wave solutions. Subsequently, the extended tanh-function
method and the G′/G method are used to integrate the Zakharov-Kuznetsov equation. Finally, the nontopological soliton solution is obtained by
the aid of ansatz method. There are numerical simulations throughout the paper to support the analytical development. 相似文献
15.
16.
S. M. Rayhanul Islam Kamruzzaman Khan K. M. Abdul Al Woadud 《Waves in Random and Complex Media》2018,28(2):300-309
The enhanced (G′/G)-expansion method presents wide applicability to handling nonlinear wave equations. In this article, we find the new exact traveling wave solutions of the Benney–Luke equation by using the enhanced (G′/G)-expansion method. This method is a useful, reliable, and concise method to easily solve the nonlinear evaluation equations (NLEEs). The traveling wave solutions have expressed in term of the hyperbolic and trigonometric functions. We also have plotted the 2D and 3D graphics of some analytical solutions obtained in this paper. 相似文献
17.
为了构造非线性发展方程的无穷序列复合型类孤子新解, 进一步研究了G'(ξ)/G(ξ) 展开法. 首先, 给出一种函数变换, 把常系数二阶齐次线性常微分方程的求解问题转化为一元二次方程和Riccati方程的求解问题. 然后, 利用Riccati方程解的非线性叠加公式, 获得了常系数二阶齐次线性常微分方程的无穷序列复合型新解. 在此基础上, 借助符号计算系统Mathematica, 构造了改进的(2+1)维色散水波系统和(2+1)维色散长波方程的无穷序列复合型类孤子新精确解.
关键词:
G'(ξ)/G(ξ)展开法')" href="#">G'(ξ)/G(ξ)展开法
非线性叠加公式
非线性发展方程
复合型类孤子新解 相似文献
18.
《Waves in Random and Complex Media》2013,23(4):444-457
In this paper, a (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko (gBK) equation is investigated, which can be used to describe the interaction of a Riemann wave propagating along y-axis and a long wave propagating along x-axis. The complete integrability of the gBK equation is systematically presented. By employing Bell’s polynomials, a lucid and systematic approach is proposed to systematically study its bilinear formalism, bilinear Bäcklund transformations, Lax pairs, respectively. Furthermore, based on multidimensional Riemann theta functions, the periodic wave solutions and soliton solutions of the gBK equation are derived. Finally, an asymptotic relation between the periodic wave solutions and soliton solutions are strictly established under a certain limit condition. 相似文献
19.
We study the existence of travelling breathers in Klein-Gordon chains, which consist of one-dimensional networks of nonlinear oscillators in an anharmonic on-site potential, linearly coupled to their nearest neighbors. Travelling breathers are spatially localized solutions which appear time periodic in a referential in translation at constant velocity. Approximate solutions of this type have been constructed in the form of modulated plane waves, whose envelopes satisfy the nonlinear Schrödinger equation (M. Remoissenet, Phys. Rev. B 33, n.4, 2386 (1986), J. Giannoulis and A. Mielke, Nonlinearity 17, p. 551–565 (2004)). In the case of travelling waves (where the phase velocity of the plane wave equals the group velocity of the wave packet), the existence of nearby exact solutions has been proved by Iooss and Kirchgässner, who have obtained exact solitary wave solutions superposed on an exponentially small oscillatory tail (G. Iooss, K. Kirchgässner, Commun. Math. Phys. 211, 439–464 (2000)). However, a rigorous existence result has been lacking in the more general case when phase and group velocities are different. This situation is examined in the present paper, in a case when the breather period and the inverse of its velocity are commensurate. We show that the center manifold reduction method introduced by Iooss and Kirchgässner is still applicable when the problem is formulated in an appropriate way. This allows us to reduce the problem locally to a finite dimensional reversible system of ordinary differential equations, whose principal part admits homoclinic solutions to quasi-periodic orbits under general conditions on the potential. For an even potential, using the additional symmetry of the system, we obtain homoclinic orbits to small periodic ones for the full reduced system. For the oscillator chain, these orbits correspond to exact small amplitude travelling breather solutions superposed on an exponentially small oscillatory tail. Their principal part (excluding the tail) coincides at leading order with the nonlinear Schrödinger approximation. 相似文献
20.
《Waves in Random and Complex Media》2013,23(4):644-655
Mathematical modeling of many autonomous physical systems leads to nonlinear evolution equations because most physical systems are inherently nonlinear in nature. The investigation of traveling wave solutions of nonlinear evolution equations plays a significant role in the study of nonlinear physical phenomena. In this article, the enhanced (G′/G)-expansion method has been applied for finding the exact traveling wave solutions of longitudinal wave motion equation in a nonlinear magneto-electro-elastic circular rod. Each of the obtained solutions contains an explicit function of the variables in the considered equations. It has been shown that the applied method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering fields. 相似文献