Symmetry Reduction for a System of Nonlinear Evolution Equations

Transcription

1 Nonlinear Mahemaical Physics 1996, V.3, N 3 4, Symmery Reducion for a Sysem of Nonlinear Evoluion Equaions Lyudmila BARANNYK Insiue of Mahemaics of he Naional Ukrainian Academy of Sciences, 3 Tereshchenkivs ka Sree, Kyiv 4, Ukraina Absrac In his paper we obain he maximal Lie symmery algebra of a sysem of PDEs. We reduce his sysem o a sysem of ODEs, using some rank hree subalgebras of he finie-dimensional par of he symmery algebra. The corresponding invarian soluions of he PDEs are obained. 1 Inroducion Sysems of evoluion equaions describe various processes in physics, chemisry and biology [1, 2]. In he papers [3, 4] classes of sysems of evoluion equaions are seleced which are invarian wih respec o some generalizaions of he classical Galilei algebra. We consider he sysem proposed by W. Fushchych u + m 1 u v + 2m 1 u v = 0, v + 2m 1 v2 2m h2 u 1 u = 0, in four-dimensional ime-space R1, 3, where h is he Planck s consan, m R, m 0. Sysem 1 is obained from he Schrödinger equaion iψ = h 2m ψ by he subsiuion ψ = ue iv/ h. From Theorem 1 i follows ha he maximal symmery algebra of sysem 1 conains he algebra L = AG 3 3 < Z >, where AG 3 3 is he special Galilei algebra. Using he classificaion of subalgebras of he algebra AG 3 3, carried ou in [5], we obain all up o Ad L-conjugacy I-maximal rank hree subalgebras of he algebra L. For symmery reducion of sysem 1 we use only subalgebras, whose projecions ono ASL2, R belong o < T >. In some cases considered he reduced sysem can be inegraed, so ha all corresponding soluions of sysem 1 can be consruced. Copyrigh c 1996 by Mahemaical Ukraina Publisher. All righs of reproducion in any form reserved.

2 448 L. BARANNYK 2 Maximal symmery algebra of sysem 1 We use he following noaions δ = δ δ, δ a = δ δx a, δ u = δ δu, δ v = δ δv a = 1, 2, 3. Theorem 1 The maximal Lie symmery algebra of sysem 1 is generaed by he vecor fields S = 2 + x a a 3 2 u u + m 2 x 2 v, D = 2 + x a a 3 2 u u, T =, P a = a, G a = a + mx a v, J ab = x a b x b a, M = m v, Z = u u, a < b; a, b = 1, 3 and he infinie-dimensional algebra X = ρ cos θ v h δ u + h u sin θ v h δ v, 3 where pair of funcions u = ρ, x, ρ, x 0 and v = θ, x is an arbirary soluions of sysem 1 summaion over repeaed indices is assumed wih a=1,2,3. By direc calculaions i is easy o verify ha he operaors M, P a, G a, J ab a, b = 1, 2, 3, D, S, T generae he special Galilei algebra AG 3 3 of he hree-dimensional space. The operaor Z commues wih each elemen of he algebra AG 3 3. Le U =< M, P 1, P 2, P 3, G 1, G 2, G 3 >. Then AG 3 3 = U +AO3 ASL2, R. The algebra AG 3 3 conains he Galilei algebras AG j 3 j = 0, 1, 2, where AG 0 3 = U +AO3, AG 1 3 = U +AO3 < T >, AG 2 3 = U +AO3 < D, T >. The symmeries can be used o build ansazes which hen reduce he equaions of 1 o parial differenial equaions wih fewer independen variables or even o ordinary differenial equaions. These ansazes and reducions are based on subalgebra analysis of a finie-dimensional par of he symmery algebra. 3 Classificaion of I-maximal subalgebras The concep of I-maximal subalgebra was inroduced in he paper [6]. Theorem 2 The I-maximal rank hree subalgebras of he algebra L = AG 3 3 < Z >, which have zero inersecion wih < M, Z >, are up o he Ad L-conjugacy: i Subalgebras of he algebra AG 0 3 < Z >: F 0 =< P 1, P 2, P 3 > +AO3; 2

3 SYMMETRY REDUCTION FOR A SYSTEM 449 F 1 =< G 1 + αz, P 2, P 3, J 23 >, where α = 0, 1; F 2 =< G 1 + P 1 + αz, G 2 + βz, P 3 + γz > α 0, β 0; F 3 =< G 1 + αz, P 2 + Z, P 3 >; α 0 F 4 =< P 1 + Z, P 2, P 3, J 23 >. ii Subalgebras of he algebra AG 1 3 < Z > wih a nonzero projecion ono < T >: F 5 =< P 2, P 3, T + α m M + βz, J 23 >, where α = ±m or α = 0 and β = 0, ±1; F 6 =< P 2, P 3, T + G 1 + αz, J 23 >; F 7 =< P 3 + αz, J 12 + β m M + γz, T + δ Z + λz >; m F 8 = AO3 < T + α M + βz >; m F 9 =< T + G 1 + αz, P 2 + βz, P 3 > β > 0; F 10 =< T + α m M + βz, P 2 + Z, P 3 >. iii Subalgebras of he algebra AG 2 3 < Z > wih a nonzero projecion ono < D >: F 11 =< G 1, P 2, D + αm + βz >; F 12 =< P 3, D + αm + βz, T >; F 13 =< P 2, P 3, J 23, D + αm + βz >; F 14 =< J 12 + αm + βz, D + γm + δz, T >; F 15 =< P 3, J 12 + αd + βm + γz, T > α > 0; F 16 =< P 3, J 12 + αm + βz, D + γm + δz > α 0; F 17 = AO3 < D + αm + βz >. iv Subalgebras of he algebra AG 3 3 < Z >, whose projecions ono ASL2, R coincide wih < S + T >: F 18 = AO3 < S + T + αm + βz >; F 19 =< S + T + 2J 12 + αm + βz, G 1 + P 2 + 2P 3, G 2 P 1 2G 3 >.

4 450 L. BARANNYK 4 Reducion of sysem 1 by subalgebras of he algebra AG 1 3 < Z >. Exac soluions For each of he subalgebras F j j = 1, 10 we give he corresponding ansaz and he reduced sysem. In some cases we also poin ou soluions of sysem 1, which are invarian under F j. αx F 1 : u = exp ϕω, v = mx2 1 + ψω, ω =, 2 2ω ϕ + ϕ = 0, ψ h2 α 2 2mω 2 = 0. The corresponding invarian soluion of sysem 1 is of he form u = C 1 exp αx1, v = mx2 1 h2 α 2 2 2m + C 2. α 4.2. F 2 : u = exp 1 x 1 + β x 2 γx 3 ϕω, v = m 2 1 x2 1 + m 2 x2 2 + ψω, ω =, 2ωω 1 ϕ + 2ω 1ϕ = 0, ψ 2m h2 α 2 ω β2 ω 2 + γ2 = 0. In his case we obain he following invarian soluion of sysem 1: u = C exp αx1 1 + βx 2 γx 3, v = mx mx m h2 α2 1 β2 + γ2 + C 2. αx F 3 : u = exp x 2 ϕω, v = mx2 1 + ψω, ω =, 2 2ω ϕ + ϕ = 0, ψ h2 α 2 2m ω = 0. The corresponding invarian soluion is of he form: u = C 1 αx1 exp x 2, v = mx2 1 + h2 2 α 2 + C m

5 SYMMETRY REDUCTION FOR A SYSTEM F 4 : u = exp x 1 ϕω, v = ψω, ω =, ϕ = 0, ψ h2 2m = 0. The corresponding invarian soluion is u = C 1 exp x 1, v = h2 2m + C F 5 : u = expβϕω, v = α + ψω, ω = x 1. 2βmϕ + 2 ϕ ψ + ϕ ψ = 0, 2αmϕ + ϕ ψ 2 h 2 ϕ = F 6 : u = expαϕω, v = mx 1 m ψω, ω = 2 2x 1. αmϕ + 4 ϕ ψ + 2ϕ ψ = 0, m 2 ωϕ + 4ϕ ψ 2 4 h 2 ϕ = F 7 : u = exp λ αx 3 γ arcan x 1 x 2 ϕω, v = β arcan x 1 x 2 + δ + ψω, ω = x x 2 2. λm + βγϕ + 4ω ϕ ψ + 2ϕ ψ + 2ωϕ ψ = 0, β 2 h 2 α 2 ω + γ 2 + 2δmωϕ 4 h 2 ω ϕ 4 h 2 ω 2 ϕ + 4ω 2 ϕ ψ 2 = F 8 : u = expβϕω, v = α + ψω, ω = x x x 2 3. mβϕ + 4ω ϕ ψ + 3ϕ ψ + 2ωϕ ψ = 0, αmϕ + 2ωϕ ψ 2 h 2 3 ϕ + 2ω ϕ = 0.

6 452 L. BARANNYK 4.9. F 9 : u = expα βx 2 ϕω, v = m x ψω, ω = 2 2x 1. 3 αmϕ + 4 ϕ ψ + 2ϕ ψ = 0, m 2 ω + h 2 β 2 ϕ + 4ϕ ψ 2 4 h 2 ϕ = F 10 : u = expβ x 2 ϕω, v = α + ψω, ω = x 1. 2βmϕ + 2 ϕ ψ + ϕ ψ = 0, References 2αm h 2 ϕ + ϕ ψ 2 h 2 ϕ = 0. [1] Aris R., The Theory of Reacion and Diffusion in Permeable Caalys, Oxford, 1975, 300p. [2] Wilhelmsson H., Oscillaions and relaxaion o equilibrium provided by inerconnecion of emperaure and densiy in hermonuclear plasmas, Ukrain. Phys. J., 1993, V.38, N 1, [3] Fushchych W. I., Cherniha R. M., Sysems of nonlinear evoluion equaions of he second order invarian wih respec o he Galilei algebra and is exensions, Dopovidi Akademii Nauk Ukrainy Repors of he Academy of Sciences of Ukraine, 1993, N 8, [4] Fushchych W. I., Cherniha R. M., Galilei-invarian sysems of nonlinear equaions of he Hamilon- Jacobi ype, Dopovidi Akademii Nauk Ukrainy Repors of he Academy of Sciences of Ukraine, 1994, N 3, [5] Barannyk L. F., Fushchych W. I., On coninuous subgroups of he generalized Schrödinger groups, J. Mah. Phys., 1989, V.30, N 2, [6] Barannyk A. F., Barannyk L. F., Fushchych W. I., Reducion of he mulidimensional d Alember equaion o wo-dimensional equaions, Ukrain. Mah. J., 1994, V.46, N 6,