非线性扩散方程的广义条件对称
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摘要
非线性现象广泛地呈现在物理、化学、生命、社会、经济等领域。随着科学的发展,对非线性系统的研究日趋深入。而描述非线性系统的非线性方程的求解研究成为研究者的重要热点课题之一。众所周知与群论相关的对称约化方法,例如李点对称方法、广义对称方法、条件对称方法、广义条件对称方法等,是研究非线性偏微分方程精确解的有效工具。
最先由Zhdanov和Fokas以及Liu提出的广义条件对称方法是对称群方法中最行之有效的方法之一。该方法己成功地应用于寻求某些非线性偏微分方程的精确解和对称约化的研究中,这些解一般不能由古典对称方法或条件对称方法求得。
在本文中,被看作是条件对称方法推广的广义条件对称方法被用于讨论以幂指数规律扩散的非线性扩散方程u_t=[u~m(u_x)~n]_x+P(u)u_x+Q(u),该方程有很多物理应用背景。我们分别得出了该方程允许二阶广义条件对称η=u_(xx)+H(u)u_x~2+G(u)u_x+F(u)和η=u_(xx)+H(u)u_x~2+G(u)(u_x)~(2-n)+F(u)(u_x)~(1-n)的条件。又对允许广义条件约化的方程的反应系数和热源项的函数形式作了分类。并且依据广义条件对称和所考虑方程的相容性求得了相应方程的群不变解。这些解(包括熄灭解、爆破解和周期解等)为对方程进行更深刻的讨论提供了很多重要信息。
广义条件对称方法必然也能用于讨论高阶的扩散型方程,例如四阶的薄膜方程或者分数阶的扩散方程以及这些方程的高维形式。
The nonlinear phenomena widely appear in almost all the scientific fields such as physics, chemistry, biology, society, economy and so on. With the development of science, researches on the nonlinear systems are making more and more progress. As a result, to deal with the nonlinear equations that describe the nonlinear systems becomes one of the main and hot topics for the researchers. It's well known that the symmetry reduction method related to group theory, such as Lie point symmetry method, generalized symmetry, conditional symmetry, generalized conditional symmetry and etc, is an effective tool to the study of the exact solutions for the nonlinear Partial Differential Equations.It is Zhdanov, Fokas and Liu who first presented the generalized conditional symmetry method, which is one of the most effective symmetry method. The method has been used to study the symmetry reduction and exact solutions of some nonlinear differential equations, the solutions here generally can't be obtained via the classical symmetry method or the conditional symmetry method.In this paper, the generalized conditional symmetry method, which can be considered as a generalization of the conditional symmetry method, is used to study the nonlinear diffusion equations u_t = [u~m(u_x)~n]_x+P(u)u_x+Q(u) with power law diffusivity, which has quite a large number of physical applications. We obtain conditions under which the equations admit the second-order generalized conditional symmetries η = u_(xx) + H(u)u)x~2 + G(u)u_x + F(u) and η = u_(xx) + H(u)u)x~2 + G(u)(u_x)~(2-n) + F(u)(u_x)~(1-n). A complete classification of the functional forms of the reaction coefficients and source terms is presented when the equation admits the generalized conditional symmetry reduction. Exact solutions to the resulting equations are constructed via the compatibility of the generalized conditional symmetry and the considered equation. These group invariant solutions, including extinction solution, blow-up solution and periodic solution, provide some important information to further study the equations.It's certain that the generalized conditional method can be also used to study the high-order diffusion-like equations, such as the fourth-order thin film equation or the fractional diffusion equation and their high-dimensional forms.
最先由Zhdanov和Fokas以及Liu提出的广义条件对称方法是对称群方法中最行之有效的方法之一。该方法己成功地应用于寻求某些非线性偏微分方程的精确解和对称约化的研究中,这些解一般不能由古典对称方法或条件对称方法求得。
在本文中,被看作是条件对称方法推广的广义条件对称方法被用于讨论以幂指数规律扩散的非线性扩散方程u_t=[u~m(u_x)~n]_x+P(u)u_x+Q(u),该方程有很多物理应用背景。我们分别得出了该方程允许二阶广义条件对称η=u_(xx)+H(u)u_x~2+G(u)u_x+F(u)和η=u_(xx)+H(u)u_x~2+G(u)(u_x)~(2-n)+F(u)(u_x)~(1-n)的条件。又对允许广义条件约化的方程的反应系数和热源项的函数形式作了分类。并且依据广义条件对称和所考虑方程的相容性求得了相应方程的群不变解。这些解(包括熄灭解、爆破解和周期解等)为对方程进行更深刻的讨论提供了很多重要信息。
广义条件对称方法必然也能用于讨论高阶的扩散型方程,例如四阶的薄膜方程或者分数阶的扩散方程以及这些方程的高维形式。
The nonlinear phenomena widely appear in almost all the scientific fields such as physics, chemistry, biology, society, economy and so on. With the development of science, researches on the nonlinear systems are making more and more progress. As a result, to deal with the nonlinear equations that describe the nonlinear systems becomes one of the main and hot topics for the researchers. It's well known that the symmetry reduction method related to group theory, such as Lie point symmetry method, generalized symmetry, conditional symmetry, generalized conditional symmetry and etc, is an effective tool to the study of the exact solutions for the nonlinear Partial Differential Equations.It is Zhdanov, Fokas and Liu who first presented the generalized conditional symmetry method, which is one of the most effective symmetry method. The method has been used to study the symmetry reduction and exact solutions of some nonlinear differential equations, the solutions here generally can't be obtained via the classical symmetry method or the conditional symmetry method.In this paper, the generalized conditional symmetry method, which can be considered as a generalization of the conditional symmetry method, is used to study the nonlinear diffusion equations u_t = [u~m(u_x)~n]_x+P(u)u_x+Q(u) with power law diffusivity, which has quite a large number of physical applications. We obtain conditions under which the equations admit the second-order generalized conditional symmetries η = u_(xx) + H(u)u)x~2 + G(u)u_x + F(u) and η = u_(xx) + H(u)u)x~2 + G(u)(u_x)~(2-n) + F(u)(u_x)~(1-n). A complete classification of the functional forms of the reaction coefficients and source terms is presented when the equation admits the generalized conditional symmetry reduction. Exact solutions to the resulting equations are constructed via the compatibility of the generalized conditional symmetry and the considered equation. These group invariant solutions, including extinction solution, blow-up solution and periodic solution, provide some important information to further study the equations.It's certain that the generalized conditional method can be also used to study the high-order diffusion-like equations, such as the fourth-order thin film equation or the fractional diffusion equation and their high-dimensional forms.
引文
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[2] G.W.Bluman and S.Kumei, Symmetries and Differential Equations, Springer, New York, 1989.
[3] P.Olver, Applications of Lie Groups to Differential Equations, Second edition, Springer, New York, 1993.
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[7] L.V.Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982.
[8] V.A.Galaktionov, V.A.Dorodnitsyn, G.G.Elenin, S.P.Kurdyumovand and A.A .Samarskii,A quasilnear equation of heat conduction with a source,J.Sov.Math. 41, 1222-1292, 1988.
[9] P.G.Leach, First integrals for the modified Emden equation q + a(t)q + qn = 0, J.Math.Phys.26, 2510-2514, 1985.
[10] C.Z.Qu, Allowed transformations and symmetry classes of variable coefficient Burgers equations, IMA J.Appl.Math.54, 203-225, 1995.
[11] R.M.Cherniha, M.Serov, Symmetries, ansatze and exact solutions of nonlinear second- order evolution equations with convection terms, Eur. J. Appl. Math.9, 527-542, 1998.
[12] E.Noether, Invariante variations probleme, Nachr. Konig.Gesell.Wissen.Gottingen, Math.Phys.l, 235-257,1918. (see Transport Theory and Stat.Phya.l, 186-207,1917 for an English translation)
[13] G.W.Bluman and J.D.Cole, The general similarity solution of the heat equation, J.Math.Mech.18, 1025-1042,1969.
[14] W.I.Fushchych and I.M.Tsyfra, On a reduction and solutions of nonlinear wave equations with broken symmetry, J.Phys.A.20, 45-48,1987.
[15] P.A.Clarkson and M.Kruskal, New similarity reductions of the Boussineq equation, J.Math.Phys.30, 2201-2213,1989.
[16] A.S.Fokas and Q.M.Liu, Nonlinear interaction of travelling waves of nonintegrable equations, Phys.Rev.Lett.72, 3293-3296, 1994.
[17] A.S.Fokas and Q.M.Liu, Generalized conditional symmetries and exact solutions of nonintegrable equations, Theo.Math.Phys.99, 263-277, 1994.
[18] R.Z.Zhdanov, Conditional Lie-Backlund symmetry and reductions of evolution equations, J. Phys A.128, 3841-3850, 1995.
[19] M.Leo, R.A.Leo, Soliani, L.Solombrino and G.Mancarella, Lie-Backlund symmetries for the Harry Dym equation, Phys.Rev.D.27, 1983.
[20] N.H.Ibragimov and R.L.Anderson, Lie-Backlund tangent transformations, J.Math.Anal.Appl.59, 145-162, 1997.
[21] J.D.Cole, On a quasilinear parabolic equation used in cerodynamics, Quart.Appl.Math.9, 225-236, 1951.
[22] L.V.Ovsiannikov, Group Properties of Differential Equations, Siberian Branch of the USSR Academy of Sciences, Novosibirsk, 1962(Russian).
[23] N.H.Ibragimov and A.B.Shabat, Koreweg-de Vries equation from the guoup- theoretical point of view, Dokl.Akad.Nauk.USSR.244.1, 57-61,1979. English transl., Soviet Phys.Dokl.24.1, 15-17, 1979.
[24] R.L.Anderson and N.H.Ibragimov, Lie-Backlund transformations in applications, SIAM.l, Philadelphia, 1979.
[25] A.S.Fokas, Invariants Lie-Backlund operators and Backlund transformations, Ph.D.Thesis, California Institute of Technology, 1979.
[26] T.Sen and M.Tabor, Lie symmetries of the Lorenz model, Phys.D.44, 313-339, 1990.
[27] C.Z.Qu and P.C.Niu, Generalized Lie symmetries and the integrability of generalized Holmes-Rand nonlinear oscilillator, Int.J.Nonl.Mech.32.6, 1187-1192, 1997.
[28] A.V.Mikhailov, A.B.Shabat and V.V.Sokolov, The Symmetry Approach to Classidi-cation of Integrable Equations in What is Integrability, edited by V.E.Zakharov, Springer Series on Nonlinear Dynamics(Springer, Berlin, 1991).
[29] G.W.Bluman, S.Kumei and G.J.Reid, New classes of symmetries for partial differential equations, J.Math.Phys.29, 806-811, 1988.
[30] W.I.Fushchych, W.M.Shtelen and N.I.Serov, Symmetry Analysis and Exact solutions of Nonlinear Equations of Mathematical Physics, Naukova Dumka,Kiev. 1989(translated into English by Kluwer Academic Publishers, Dordrecht, 1993).
[31] D.J.Arrigo, P.Broadbridge and J.M.Hill, Nonclassical symmetry reduction of the linear diffusion equation with a nonlinear source, IMA J.Appl.Math.52, 1-24, 1994.
[32] P.A.Clarkson and E.L.Mansfield, Symmetry reductions and exact solutions of a class of nonlinear heat equations, Phys.D.70, 250-288, 1993.
[33] M.C.Nucci and P.A.Clarkson, The nonclassical method is more general than the di- rect method for symmetry reductions, An example of the Hugh-Nagumo equation, Phys.Lett.A.164, 49-56, 1992.
[34] E.Pucci, Similarity reductions of differential equations, J.Phys.A.25, 2631-2640, 1992.
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