Quantum Theory and Symmetries IV ed. V.K. Dobrev, Heron Press, Sofia, 2006
Reduction Operators of Nonlinear Filtration Equation Olena Vaneeva1 1
Institute of Mathematics of National Academy of Sciences of Ukraine, 3, Tereshchenkivska Str., Kyiv-4, 01601, Ukraine
[email protected]
Abstract The nonclassical symmetries of the nonlinear filtration equation having the special value 1/vx of the filtration coefficient are completely classified with respect to the Lie symmetry group of this equation. As a result, new wide classes of nonclassical symmetries and non-Lie exact solutions of this equation are constructed.
1
Introduction
Investigation of nonlinear filtration equations by means of symmetry methods was started in 1987 with work [1], where authors carried out the group classification of the class of equations of the form vt = h(vx )vxx . This class of equations are known to describe the motion of a non-Newtonian, weakly compressible fluid in a porous medium with a nonlinear filtration law h(vx ), h 6= const (see [2] and references therein). In the present paper an equation from the above class vt =
vxx vx
(1)
is studied from the symmetry point of view. The transformation u = vx connects equations (1) with the so-called fast diffusion equation ut = (u−1 ux )x (2) which has remarkable mathematical properties and arises in different physical applications [10]. Review of results on symmetries, exact solutions and conservation laws of the class of nonlinear diffusion equations is given e.g. in [6]. Thus, results obtained in the paper can be used also for finding exact solutions of equation (2). 1
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Reduction Operators of Nonlinear Filtration Equation Lie symmetries and exact solutions
The Lie invariance algebra A2 = h∂t , ∂x , ∂v , t∂t + v∂v , x∂x − v∂v i of equation (1) and the corresponding connected Lie symmetry group are quite ordinary for nonlinear filtration equations. However, equation (1) is distinguished for its discrete symmetries since it possesses, besides two usual involutions of alternating sign in the sets {t, v} and {x, v}, the hodograph transformation t˜ = t, x ˜ = v, v˜ = x. These three involutive transformations with continuous oneparameter transformation groups having infinitesimal operators from A2 generate the complete Lie invariance group G2 of (1). All G2 -inequivalent invariant solutions constructed previously in closed form by means of classical Lie method are exhausted by the following ones [9]: x − t −x t x ; 1) v = − ln |e + εe |; 2) v = e + t; 3) v = ln x + t Z dϑ x 2t 4) v = ln |t| + ; 5) v = 2 arctan ; 6) v = − ; −ϑ ϑ − 1 + µe ϑ=x/t t x 7) v = 2t tan x;
8) v = −2t coth x;
9) v = −2t tanh x.
Here ε and µ are arbitrary constants, ε ∈ {−1, 0, 1} mod G2 . 3
Nonclassical symmetries
The notion of a nonclassical symmetry was introduced in [3] in 1969. A precise and rigorous definition was suggested later (see e.g. [5, 11]). Consider a r-th order differential equation L of the form L(x, v(r) ) = 0 for the unknown functions v of n independent variables x = (x1 , . . . , xn ). Here v(r) denotes the set of all the derivatives of the functions v with respect to x of order no greater than r, including v as the derivatives of the zero order. Below the index µ runs from 1 to n, and the summation convention for repeated indices is used. Definition 1 The differential equation L is called conditionally invariant with respect to the operator Q = ξ µ (x, v)∂xµ + η(x, v)∂v (∃ µ : ξ µ 6= 0) if the relation Q(r) L(x, v(r) ) L, Q(r) = 0 holds, which is called as the conditional invariance criterion. In definition 1 the symbol Q(r) stands for the standard r-th prolongation of the operator Q and Q(r) denotes the set of differential consequences of the equation Q[v] = 0 up to the (r − 1)-th order i.e. of such consequences which have, as differential equations, orders no greater than r, where Q[v] := η − ξ µ vµ is the characteristic of the operator Q. The operator Q is called a conditional symmetry operator (or a Qconditional symmetry operator, a nonclassical symmetry operator etc) of the equation L. Conditional invariance of the equation L with respect to the operator Q implies that the ansatz constructed with this operator reduces L to a differential
Olena Vaneeva
3
equation with n − 1 independent variables [11]. So, we will shortly call conditional symmetry operators as reduction operators of L. Further we will consider inequivalent reduction operators of equation (1). The definitions of the different kinds of equivalence of conditional symmetry operators were investigated in [7]. Definition 2 The reduction operators Q1 and Q2 are equivalent if there exists such function λ = λ(x, u) 6= 0 that Q1 = λQ2 . Definition 3 The reduction operators Q1 and Q2 are equivalent with respect to the invariance group G of L if after action of a transformation g from G the transformed operator g∗ Q1 is equivalent to Q2 (in the sense of definition 2). For equation (1) reduction operators have the general form Q = τ ∂t +ξ∂x +θ∂v , where τ , ξ and θ are functions of t, x and v. In [8] it was shown that for every evolution equation the problem of finding the conditional symmetry operators with the vanishing coefficient of ∂t is reduced to solving the initial equation. Since (1) is an evolution equation, there are two principal different cases of finding Q: τ 6= 0 and τ = 0. In the case τ 6= 0 we can assume τ = 1 up to usual equivalence of reduction operators [11]. Then the determining equations for coefficients ξ and θ have the form θxx = θθx ,
θt = 2θxv − ξxx − ξx θ + ξθx − θθv ,
ξvv = ξξv ,
ξt = 2ξxv − θvv − θv ξ + θξv − ξξx .
Theorem 1 Up to equivalence with respect to the complete invariance group G2 , there exist the following classes of non-Lie reduction operators of equation (1), namely, 1.
∂t + ε∂x + f (ω)∂v
(ω = x + εt);
2.
∂t + f (ω)(∂x + ∂v )
(ω = x + v);
3.
∂t + ξ∂x + (φt + φx ξ)∂v : −2 , φ ∈ {t + ex , tf (x)}; 3.1. ξ = v+φ 3.2. ξ = −2 cot(v + φ), where φ ∈ {− arctan(tan 2t tanh x), − arctan(coth 2t cot x)}; ξ = −2 coth(v + φ) or ξ = −2 tanh(v + φ), nx o 1 where φ ∈ +t+ ln(cosh(x−2t)), − arctanh χ , 2 2 n e2x tanh 2t + 1 o χ ∈ tanh 2t tanh x, tanh 2t coth x, coth 2t coth x, 2x . e − tanh 2t Here ε ∈ {0, 1}, f ∈ {−2/ω, −2 cot ω, −2 tanh ω, −2 coth ω} mod G2 . 3.3.
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Reduction Operators of Nonlinear Filtration Equation
The list of G2 -inequivalent non-Lie solutions of equation (1) which can be constructed by the reduction with the above nonclassical symmetry operators are exhausted by the following ones: v = − arctan(tan 2t tanh x); v = − arctan(coth 2t cot x); v = − arctanh χ, where χ is the same as in Theorem 1. 4
Conclusions
In this paper classification of reduction operators of nonlinear filtration equation (1) and the list of constructed exact solutions are presented. Since the equation (1) is a potential form for equation (2), all the nonclassical symmetry operators from the Theorem 1 are potential nonclassical symmetries of the fast diffusion equation (2) [4]. In the future we plan to investigate the problem of complete classification of nonclassical symmetry operators for the whole class of nonlinear filtration equation vt = h(vx )vxx . Acknowledgments I would like to thank the Organizing Committee of the IV International Symposium “Quantum Theory and Symmetries” and especially Prof. Vladimir Dobrev for hospitality and giving an opportunity to give a talk. The participation in the Symposium was partially supported by a QTS-4 UNESCO-ROSTE short-term Fellowship. I am also grateful to Roman Popovych and Nataliya Ivanova for useful discussions. References [1] I. Sh. Akhatov, R. K. Gazizov and N. H. Ibragimov, Dokl. Akad. Nauk. S.S.S.R., 293, 1033 (1987). (English translation in Soviet Math. Dokl., 35, 384 (1987)) [2] I. Sh. Akhatov, R. K. Gazizov and N. H. Ibragimov, in Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, 34, VINITI, Moscow, 3 (1989). (English translation in J. Soviet Math. 55, 1401 (1991)) [3] G. W. Bluman, J. D. Cole, J. Math. Mech., 18, 1025 (1969). [4] G. W. Bluman, S. Kumei and G. J. Reid, J. Math. Phys., 29, 806 (1988). [5] W. I. Fushchych, I. M. Tsyfra, J. Phys. A: Math. Gen., 20, L45 (1987). [6] N. H. Ibragimov (Editor), Lie group analysis of differential equations: V.1. Symmetries, exact solutions and conservation laws, Boca Raton, FL, CRC Press (1994). [7] R. O. Popovych, in Proc. of the Third International Conference “Symmetry in Nonlinear Mathematical Physics”, Proc. of Institute of Mathematics of Acad. of Sci. Ukraine, 30, Part 1, Kyiv, 184 (2000). [8] R. O. Popovych, in Symmetry and Analytic Methods in Mathematical Physics, Proc. of Institute of Mathematicsof Acad. of Sci. Ukraine, 19, Kyiv, 194 (1998) (in Ukrainian). [9] R. O. Popovych, N. M. Ivanova , J. Phys.A., 38, 3145 (2005). [10] P. Rosenau, Phys. Rev. Lett., 74, 1056 (1995). [11] R. Z. Zhdanov, I. M. Tsyfra and R. O. Popovych, J. Math. Anal. Appl., 238, 101 (1999).
