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1.
YAO Ruo-Xia JIAO Xiao-Yu LOU Sen-Yue 《理论物理通讯》2009,51(5):785-788
Starting from Lie symmetry theory and combining with the approximate symmetry method, and using the package LieSYMGRP proposed by us, we restudy the perturbed Kuramoto-Sivashinsky (KS) equation. The approximate symmetry reduction and the infinite series symmetry reduction solutions of the perturbed KS equation are constructed. Specially, if selecting the tanh-type travelling wave solution as initial approximate, we not only obtain the general formula of the physical approximate similarity solutions, but also obtain several new explicit solutions of the given equation, which are first reported here. 相似文献
2.
Approximate direct reduction method: infinite series reductions to the perturbed mKdV equation
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The approximate direct reduction method is applied to the perturbed mKdV
equation with weak fourth order dispersion and weak dissipation. The
similarity reduction solutions of different orders conform to formal
coherence, accounting for infinite series reduction solutions to the
original equation and general formulas of similarity reduction
equations. Painlevé II type equations, hyperbolic secant and
Jacobi elliptic function solutions are obtained for zero-order
similarity reduction equations. Higher order similarity reduction
equations are linear variable coefficient ordinary differential
equations. 相似文献
3.
LIU Xi-Zhong 《理论物理通讯》2010,54(5):797-802
The perturbed Kaup-Kupershmidt equation is investigated in terms of the approximate symmetry perturbation method and the approximate direct method. The similarity reduction solutions of different orders are obtained for both methods, series reduction solutions are consequently derived. Higher order similarity reduction equations are linear variable coefficients ordinary differential equations. By comparison, it is find that the results generated from the approximate direct method are more general than the results generated from the approximate symmetry perturbation method. 相似文献
4.
以同伦近似对称法为理论依据研究了远场模型方程, 通过归纳各阶相似约化解和各阶相似约化方程的通式构造相应的同伦级数解. 各阶相似约化方程均为线性变系数常微分方程, 并且可以从零阶开始依次求解. 同伦模型中的辅助参数影响同伦级数解的收敛性.
关键词:
同伦近似对称法
远场模型方程
同伦级数解 相似文献
5.
An extension of the direct method and similarity reductions of a generalized Burgers equation with an arbitrary derivative function 总被引:2,自引:0,他引:2
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In this paper,we extend the well-known direct method proposed by Clarkson and Kruskal for finding similarity reductions of partial differential equations.It follows that some new similarity reductions of the generalized Burgers equation,such as travelling wave reduction,logarithmic reduction,power reduction,rational fractional reduction,etc,are derived,in which some of these cannot be obtained solely by using the direct method.The similarity reductions obtained are interpreted by the nonclassical symmetry Lie group. 相似文献
6.
An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the sixth-order Boussinesq equation,which arises from fluid dynamics.We summarize the general formulas for similarity reduction solutions and similarity reduction equations of different orders,educing the related homotopy series solutions.Zero-order similarity reduction equations are equivalent to the Painlevé IV type equation or Weierstrass elliptic equation.Higher order similarity solutions can be obtained by solving... 相似文献
7.
Approximate homotopy similarity reduction for the generalized Kawahara equation via Lie symmetry method and direct method
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<正>This paper studies the generalized Kawahara equation in terms of the approximate homotopy symmetry method and the approximate homotopy direct method.Using both methods it obtains the similarity reduction solutions and similarity reduction equations of different orders,showing that the approximate homotopy direct method yields more general approximate similarity reductions than the approximate homotopy symmetry method.The homotopy series solutions to the generalized Kawahara equation are consequently derived. 相似文献
8.
将行波变换替换为更一般的函数变换,推广了修正的Jacobi椭圆函数展开方法.给出了非线性 Klein-Gordon方程新的周期解.当模m→1或m→0时,这些解退化成相应的孤立波解、三 角函数解和奇异的行波解.对于某些非线性方程,在一定条件下一般变换退化为行波约化.
关键词:
Jacobi椭圆函数
非线性发展方程
精确解 相似文献
9.
Based on the general direct method developed by Lou et al. [J. Phys. A: Math. Gen. 38 (2005) L129], the symmetry group theorem is obtained,from that both the Lie point groups and the non-Lie symmetry groups of the Konopelchenko-Dubrovsky (KD) equation are obtained. From the theorem, some exact solutions of KD equation are derived from a simple travelling wave solution and a multi-soliton solution. 相似文献
10.
11.
LIU Xi-Zhong 《理论物理通讯》2010,54(1):31-34
The Kawahara equation is studied through the approximate homotopy symmetry method. Under this method we get the similarity reduction solutions of the Kawahara equation, leading to the corresponding homotopy series solutions. Furthermore, the similarity solutions of the corresponding reduced linear ordinary differential equations are also considered. 相似文献
12.
In this paper, by using the symmetry method, the relationships between new explicit solutions and old ones of the (2+1)-dimensional Kaup-Kupershmidt (KK) equation are presented. We successfully obtain more general exact travelling wave solutions for (2+1)-dimensional KK equation by the symmetry method and the (G, /G)-expansion method. Consequently, we find some new solutions of (2+1)-dimensional KK equation, including similarity solutions, solitary wave solutions, and periodic solutions. 相似文献
13.
Classification of All Single Travelling Wave Solutions to Calogero-Degasperis-Focas Equation 总被引:1,自引:0,他引:1
LIU Cheng-Shi 《理论物理通讯》2007,48(4):601-604
Under the travelling wave transformation, Calogero-Degasperis-Focas equation is reduced to an ordinary differential equation. Using a symmetry group of one parameter, this ODE is reduced to a second-order linear inhomogeneous ODE. Furthermore, we apply the change of the variable and complete discrimination system for polynomial to solve the corresponding integrals and obtained the classification of all single travelling wave solutions to Calogero- Degasperis-Focas equation. 相似文献
14.
An extended Boussinesq equation that models weakly nonlinear and
weakly dispersive waves on a uniform layer of water is studied in
this paper. The results show that the equation is not
Painlev\'e-integrable in general. Some particular exact travelling
wave solutions are obtained by using a function expansion method. An
approximate solitary wave solution with physical significance is
obtained by using a perturbation method. We find that the extended
Boussinesq equation with a depth parameter of $1/\sqrt 2$ is able to
match the Laitone's (1960) second order solitary wave solution of
the Euler equations. 相似文献
15.
We study a third-order nonlinear evolution equation, which can be transformed to the modified KdV equation, using the Lie symmetry method. The Lie point symmetries and the one-dimensional optimal system of the symmetry algebras are determined. Those symmetries are some types of nonlocal symmetries or hidden symmetries of the modified KdV equation. The group-invariant solutions, particularly the travelling wave and spiral wave solutions, are discussed in detail, and a type of spiral wave solution which is smooth in the origin is obtained. 相似文献
16.
G. Rosen 《Physics letters. A》1973,45(3):263-264
It is shown that within the manifold of exact solutions a system of reaction-diffusion equations admits only travelling waves with planar symmetry. A derivation of the generic form of approximate (asymptotic) cylindrical and spiral travelling periodic wave solutions is given. If an exact solution homogeneous in space and periodic in time is admitted by the system of reaction-diffusion equations, then travelling periodic spiral waves are admissble as approximate solutions. This is the theoretical explanation for the travelling periodic waves of chemical activity observed in recent experiments. 相似文献
17.
Making use of the direct method proposed by Lou et al. and symbolic computation, finite symmetry transformation groups for a (2+1)-dimensional cubic nonlinear Schrödinger (NLS) equation and its corresponding cylindrical NLS equations are presented. Nine related linear independent infinitesimal generators can be obtained from the finite symmetry transformation groups by restricting the arbitrary constants in infinitesimal forms. Some exact solutions are derived from a simple travelling wave solution. 相似文献
18.
In this paper, Lie symmetry is investigated for a new integrable
coupled Korteweg--de Vries (KdV) equation system. Using some symmetry
subalgebra of the equation system, we obtain five types of the
significant similarity reductions. Abundant solutions of the coupled
KdV equation system, such as the solitary wave solution, exponential
solution, rational solution and polynomial solution, etc. are
obtained from the reduced equations. Especially, one type of
group-invariant solution of reduced equations can be acquired by
means of the Painlev\'e I transcendent function. 相似文献
19.
研究了一类扰动Vakhnemko方程.给出了改进的渐近方法.首先, 对原模型系统对应的典型方程得到对应的行波解.其次, 引入一个泛函, 建立迭代关系式,将求解非线性问题转化为求解一系列的迭代序列.然后, 逐次地求出对应的解的近似式, 最后,得到了原扰动Vakhnemko模型行波解的任意次精度的近似展开式,并讨论了它的精度.
关键词:
泛函
行波解
Vakhnemko方程 相似文献
20.
The classification of travelling wave solutions and superposition of multi-solutions to Camassa-- Holm equation with dispersion 总被引:1,自引:0,他引:1
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Under the travelling wave transformation, the Camassa--Holm equation
with dispersion is reduced to an integrable ordinary differential
equation
(ODE), whose general solution
can be obtained using the trick of one-parameter group. Furthermore,
by using a complete discrimination system for polynomial, the
classification of all single travelling wave solutions to the
Camassa--Holm equation with dispersion is obtained.
In particular, an affine subspace structure in the set of the
solutions of the reduced ODE is
obtained. More generally, an implicit linear structure in the Camassa--Holm equation with
dispersion is found. According to the linear structure, we obtain
the superposition of multi-solutions to Camassa--Holm equation with
dispersion. 相似文献