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1.
Given a class F of differential equations, the symmetry classification problem is to determine for each member f ∈ F the structure
of its Lie symmetry group Gf or, equivalently, of its Lie symmetry algebra. The components of the symmetry vector fields of the Lie algebra are solutions
of an associated overdetermined "defining system" of differential equations. The usual computer classification method which
applies a sequence of total derivative operators and eliminations to this associated system often fails on problems of interest
due to the excessive size of expressions generated in intermediate computations. We provide an alternative classification
method which exploits the knowledge of an equivalence group G preserving the class. A noncommutative differential elimination
procedure due to Lemaire, Reid, and Zhang, where each step of the procedure is invariant under G, can be applied and an existence
and uniqueness theorem for the output used to classify the structure of symmetry groups for each f ∈ F. The method is applied
to a class of nonlinear diffusion convection equations vx = u, vt = B(u) ux - K(u) which is invariant under a large but easily determined equivalence group G. In this example the complexity of the
calculations is much reduced by the use of G-invariant differential operators. 相似文献
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(3+1)维带有源项的反应扩散方程的不变集和精确解 总被引:2,自引:0,他引:2
讨论了(3+1)维带有源项的反应扩散方程ut=A1(u)uxx+A2(u)uyy+A3(u)uzz+B1(u)ux^2;+B2(u)uy^2+B3(u)uz^2+Q(u).通过构建函数不变集的思想方法.得到了上述方程的几个新精确解.该方法也可以用来解N+1维反应扩散方程. 相似文献
4.
S. V. Khabirov 《Siberian Mathematical Journal》2002,43(5):942-954
We consider three-dimensional subalgebras admitted by the equations of gas dynamics having time as an invariant and containing no rotation operator. For such subalgebras we seek for irregular partially invariant solutions of rank 2 and defect 1. The representation for solutions has the form which generalizes motion of a gas with a linear velocity field. We show that partially invariant solutions exist for each subalgebra. We describe the set of these solutions. We find solutions with the indicated representation that are not partially invariant. The solutions reducible to invariant solutions are generalized to new submodels. 相似文献
5.
Jan Seidler 《Czechoslovak Mathematical Journal》1997,47(2):277-316
The ergodic behaviour of homogeneous strong Feller irreducible Markov processes in Banach spaces is studied; in particular, existence and uniqueness of finite and -finite invariant measures are considered. The results obtained are applied to solutions of stochastic parabolic equations. 相似文献
6.
Wen-Xiu Ma 《中国科学 数学(英文版)》2012,55(9):1769-1778
The invariant subspace method is refined to present more unity and more diversity of exact solutions to evolution equations.The key idea is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit.A two-component nonlinear system of dissipative equations is analyzed to shed light on the resulting theory,and two concrete examples are given to find invariant subspaces associated with 2nd-order and 3rd-order linear ordinary differential equations and their corresponding exact solutions with generalized separated variables. 相似文献
7.
Larry Bates 《Proceedings of the American Mathematical Society》2007,135(10):3039-3040
An explicit example is given of a smooth function invariant under a linear group action that is not a smooth function of the invariant polynomials.
8.
利用经典李群方法得到了Landau-Lifshitz方程不变群的无穷小生成元,验证其对换位运算构成一个七维的李代数,得到了对应的群不变解,建立了Landau-Lifshit,z新解和旧解之间的关系.同时利用对称和共轭方程组求得了Landau-Lifshitz方程的守恒律. 相似文献
9.
We consider Lipschitz smoothness of an arbitray invariant potential U on the unit ball B in
. We establish some Lipschitz estimates for both U and its gradient vector field U with respect to the Bergman metric. These estimates are taken with respect an invariant distance on B and shown to hold outside on open sets with arbitrarily small Hausdorff conttent. We also prove that for an M-subharmonic function u which satisfies Littelwood's integrability condition, there are such open sets , such that u is Lipschitz smooth on B\. 相似文献
10.
研究(2+1)维拟线性扩散方程的精确解问题.运用推广的不变集方法,给出(2+1)维拟线性扩散方程的一些特殊解.此方法是(1+1)维拟线性扩散方程的推广. 相似文献
11.
On the complete group classification of the one‐dimensional nonlinear Klein–Gordon equation with a delay 下载免费PDF全文
This research gives a complete Lie group classification of the one‐dimensional nonlinear delay Klein–Gordon equation. First, the determining equations are derived and their complete solutions are found. Then the complete group classification and representations of all invariant solutions are obtained. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
12.
We have obtained in this paper the existence of weak solutions to the Cauchy problem for a special system of quasillnear equations with physical interest of the form {\frac{∂}{∂t}(u + qz) + \frac{∂}{∂x}f(u) = 0 \frac{∂z}{∂t} + kφ(u)z = 0 for the assumed smooth function φ(u) by employing the viscosity method and the theory of compensated compactness. 相似文献
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本文研究带非奇扰动项的(2,p)-Laplace方程{u=0,-△u-△pu=a(x)|u|q-2u+f(x,u)x∈ЭΩ,x∈Ω,其中ΩСRN是有界光滑区域,1
相似文献
14.
In this paper, one-dimensional optimal system of group invariant solutions of (2 + 1)-dimensional Klein–Gordon system is constructed. Then the classification of group invariant solutions is given out and the corresponding two-dimensional symmetry reduced equations are obtained. At last some symmetry transformations are gained in detail. Especially, we obtain the most general solution from a given solution by use of six variable one-parameter subgroups transformations. 相似文献
15.
利用不变子空间方法研究了(3+1)维短波方程的不变子空间和精确解.在(2+1)维短波方程增加一维的情形下,构造了更加广泛的精确解,同时也得到了超曲面的爆破解.主要结果不仅推广了不变子空间理论在高维非线性偏微分方程中的应用,而且对研究高维方程的动力系统有重要意义. 相似文献
16.
设0∈Ω∈RN,(N≥2)为有界光滑区域,利用山路定理,考虑如下一类含Hardy位势的拟线性椭圆型方程非平凡解的存在性:-△u-u△(|u|N,(N≥2)为有界光滑区域,利用山路定理,考虑如下一类含Hardy位势的拟线性椭圆型方程非平凡解的存在性:-△u-u△(|u|2)=μu/|x|2)=μu/|x|2+λg(x,u),x∈Ω,其中μ>0,λ>0为常数,g(x,u)为Caratheodory函数. 相似文献
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We integrate the equations of gas dynamics in finite form for the solutions in which the thermodynamic parameters depend only on one spatial variable. The corresponding motion of gas represents the nonlinear superposition of the one-dimensional gas motion corresponding to the invariant system and the two-dimensional motion determined by noninvariant functions. These motions are called 2.5-dimensional. We reduce the invariant system to a first-order implicit ordinary differential equation. We study various solutions of the latter. We construct some continuous and discontinuous solutions to the equations of gas dynamics and give their physical interpretation. 相似文献
19.
P.Y. Picard 《Journal of Mathematical Analysis and Applications》2008,337(1):360-385
We use the symmetry reduction method based on Lie group theory to obtain some exact solutions, the so-called invariant solutions, of the ideal magnetohydrodynamic equations in (3+1) dimensions. In particular, these equations are invariant under a Galilean-similitude Lie algebra for which the classification by conjugacy classes of r-dimensional subalgebras (1?r?4) was already known. We restrict our study to the three-dimensional Galilean-similitude subalgebras that give us systems composed of ordinary differential equations. Here, some examples of these solutions are presented with a brief physical interpretation. 相似文献
20.
Y. Charles Li 《Journal of Mathematical Analysis and Applications》2006,315(2):642-655
Zero dispersion and viscosity limits of invariant manifolds for focusing nonlinear Schrödinger equations (NLS) are studied. We start with spatially uniform and temporally periodic solutions (the so-called Stokes waves). We find that the spectra of the linear NLS at the Stokes waves often have surprising limits as dispersion or viscosity tends to zero. When dispersion (or viscosity) is set to zero, the size of invariant manifolds and/or Fenichel fibers approaches zero as viscosity (or dispersion) tends to zero. When dispersion (or viscosity) is nonzero, the size of invariant manifolds and/or Fenichel fibers approaches a nonzero limit as viscosity (or dispersion) tends to zero. When dispersion is nonzero, the center-stable manifold, as a function of viscosity, is not smooth at zero viscosity. A subset of the center-stable manifold is smooth at zero viscosity. The unstable Fenichel fiber is smooth at zero viscosity. When viscosity is nonzero, the stable Fenichel fiber is smooth at zero dispersion. 相似文献