SYMMETRY ANALYSIS AND LINEARIZATION OF THE (2+1) DIMENSIONAL BURGERS EQUATION

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1 SYMMETRY ANALYSIS AND LINEARIZATION OF THE 2+ DIMENSIONAL BURGERS EQUATION M. SENTHILVELAN Cenre for Nonlinear Dynamics, School of Physics, Bharahidasan Universiy, Tiruchirapalli , India M. TORRISI Diparimeno di Maemaica e Informaica, Universià di Caania, 9525 Caania, Ialy orrisi@dmi.unic.i We invesigae group invariance properies of a 2 + dimensional Burgers equaion. We show ha i is one of he higher dimensional nonlinear parial differenial equaions which does no admi Kac-Moody-Virasoro ype sub-algebras. Through Lie symmery analysis we deduce a wide class of ineresing soluions. Furher, we deduce a ransformaion which ransforms he 2+ dimensional Burgers equaion o a + dimensional linear equaion.. Inroducion During he pas hree decades considerable progress has been made in undersanding he mahemaical properies of higher dimensional nonlinear parial differenial equaions PDEs []. In paricular several 2 + dimensional nonlinear evoluion equaions have been shown o be inegrable and possess a rich variey of soluions. To explore hese soluions several approaches have been adoped. The mos widely used echniques are Inverse Scaering Transform, Hiroa mehod, Painlevé analysis, separaion of variables, Lie group analysis and direc mehod [ 7]. Among hese he Lie group analysis is playing a vial role. The mehod provides symmeries, Lie algebras, similariy reduced PDEs and heir underlying geomery [8 0]. As far as 2 + dimensional nonlinear PDEs are concerned he mehod has been used no only o explore he soluions bu also o poin ou he exisence of infinie dimensional Lie algebras in paricular Kac-Moody-Virasoro ype algebras. In his direcion i has shown

2 2 ha many inegrable 2 + -dimensional nonlinear PDEs admi infiniedimensional Lie algebras, ofen of he Kac-Moody-Virasoro ype. Typical examples include Kadomsev-Peviashvili equaion, Davey-Sewarson equaion, Nizhnik-Novikov-Veselov equaion, nonlinear Schrödinger equaion inroduced by Fokas and he 2+ dimensional sine-gordon equaion. However, here are cerain 2+ dimensional inegrable PDEs hough admi infinie- dimensional Lie algebras, do no admi Virasoro-ype algebras. Examples of his caegory are breaking solion equaion and he nonlinear Schrödinger equaion inroduced by Srachn [3, 4]. In his paper we consider he following 2+ dimensional generalized Burgers equaion [] u + u xy + uu y + u x x u y dx = 0. Eq. reduces o he Burgers equaion when y = x. I describes wave propagaion in 2 + dimensional space. Eq. passes Painlevé es and admis several solion ype soluions [, 2]. In his paper we perform Lie symmery analysis o and poin ou ha i does no admi Virasoro ype sub-algebras evenhough i is inegrable. Furher, we show ha i is ransformable o one dimensional linear equaion. The plan of he paper is as follows. In he following secion we sudy Lie symmeries and Lie algebra of Eq.. In Sec. 3, we consruc general similariy reduced + dimensional PDEs and some special soluions. We invesigae he group invariance properies of he similariy reduced PDEs and deduce general similariy soluions in Sec. 4. We perform he linearizaion in Sec. 5. Finally, we presen our conclusions in Sec Lie algebra To sudy classical Lie symmeries of Eq. we inroduce he ransformaion u y = v x so ha can be rewrien as u + u xy + uv x + vu x = 0, u y = v x. 2 The invariance of Eq. 2 under he one parameer Lie group of infiniesimal poin ransformaions leads o he following expressions for he infiniesi-

3 3 mals, ha is, ξ = c x c 2 x + α, ξ 2 = c y c 3 y + c 4 + c 5, ξ 3 = c 2 + c 3 c 2 + c 6, φ = c + c 2 u c y + c 4, φ 2 = c + c 3 v c x + α, 3 where ξ i s and φ js, i =, 2, 3 and j =, 2, are he infiniesimal componens associaed wih he variables x, y, and u, v respecively. In he above, c is, i =,..., 6, are arbirary consans, α is an arbirary funcion of and do denoes differeniaion wih respec o. The presence of an arbirary funcion α in he infiniesimal symmeries leads o an infinie dimensional Lie algebra. Thus he sysem admis a infinie dimensional Lie vecor fields of he form where V = V + V 2 + V 3 + V 4 + V 5 + V 6 + V 7 α, 4 V = y, V 2 =, V 3 = y + u, V 4 = x x + u u, V 5 = y x + v v, V 7α = α x + α v, V 6 = x x y y 2 + u y u + v x v. 5 by The non-zero commuaion relaions beween he vecor fields are given [V, V 5 ] = V, [V, V 6 ] = V 3, [V 2, V 3 ] = V, [V 2, V 4 ] = V 2, [V 2, V 5 ] = V 5, [V 2, V 6 ] = V 4 + V 5, [V 2, V 7 ] = V 7 α, [V 3, V 4 ] = V 3, [V 3, V 5 ] = V 3, [V 4, V 6 ] = V 6, [V 4, V 7 ] = V 7 α + α, [V 5, V 6 ] = V 6, [V 5, V 7 ] = V 7 α, [V 6, V 7 ] = V 7 α 2 α. 6 The commuaion relaion beween V 7 α and V 7 α 2 urns ou o be [V 7 α, V 7 α 2 ] = 0 7 which is no of Virasoro ype.

4 4 3. General similariy reducions While solving he characerisic equaions associaed wih he infiniesimal symmeries we observed ha hey can be inegraed under he following hree differen parameric resricions only, namely, i c 2 + c 3 2 4c c 6 = 0, ii c 2 + c 3 2 4c c 6 > 0, iii c 2 + c 3 2 4c c 6 < 0. We consider each case separaely and derive he similariy variables. As he calculaions are sraighforward in he following we omi some deails. Case c 2 + c 3 2 = 4c c 6 The resricion c 2 +c 3 2 = 4c c 6 gives he following similariy variables α g η = xg f 2 + where g = exp and f 3 2 c 2 c 3 c 2 + c 3 + 2c ũη, ζ = c ζ f 2 u g d, ζ = ṽη, ζ = c η + vgf 2 + αg f 2 y gf 2 c 4 c 5 g f 3 2 d,, f = c 6 + c 2 + c 3 + c 2, d c 4 g f 3 2 c c 4 c 5 g f 3 2 d d, + c α g d d f 3 2 c 2 + c α g + d. 8 f 3 2 Case 2 c 2 + c 3 2 4c c 6 > 0 In his case we ge η = ˆF α x + ˆF c 2 + k + c 6 d, where ζ = y ˆF c 2 + k + c 6 + c 4 + c 5 ˆF /2 c 2 + k + c 6 2 d, 2c + k + ˆF = k c 2 c k 2 c 2 + k + c 6 2 2c + k k, 2 k = c 2 + c 3, k2 = c 2 + c 3 2 4c c 6 2,

5 5 and ũη, ζ = c ζ + u ˆF + c c 4 + c 5 d d ˆF c 2 + k + c 6 2 +c 4 ˆF c 2 + k + c 6 d, ṽη, ζ = c 2 + k + c 6 ˆF α v + c ˆF c 2 + k + c 6 d d α d c η c 2 + k + c 6 ˆF. 9 Case 3 c 2 + c 3 2 4c c 6 < 0 The parameric resricion c 2 + c 3 2 < 4c c 6 provides us η = F α x + F c 2 + k + c 6 d, ζ = y F c 2 + k + c 6 + c 4 + c 5 F c 2 + k + c 6 2 d, where exp c3 c 2 k 2 an 2c +k k 2 F =, k c 2 + k + c 6 = c 2 + c 3, k 2 = 4c c 6 k 2 2, 2 and ũη, ζ = u F + c c ζ + c 4 c 4 + c 5 F c 2 + k + c 6 2 d d F c 2 + k + c 6 d, [ ṽη, ζ = c 2 + k + c 6 F α v + c F ] c 2 + k + c 6 d d α c η + F c 2 + k + c 6 d. 0 Subsiuing he similariy ransformaions 8 ino 2 we obain he following + dimensional PDE ũ ηζ + ũ c 3 ζũ ζ + ṽ c 2 ηũ η c 2 ũ + c c 6 ζ = 0, ũ ζ ṽ η = 0.

6 6 One may noe ha he similariy variables 9 and 0 also provide exacly he same equaion. The resuls of he above derivaions are quie general. However, for cerain special values of he parameers he similariy variables could loss heir presen forms. These special cases are o be reaed separaely. We consider some of hese cases in Secs. 4 and Special soluions In his subsecion we show ha one can generae some nonrivial soluions for he sysem 2 from a rivial soluion of hrough he similariy ransformaions 8-0. For example, le us consider c 6 = 0 in. In his case he laer possesses he rivial soluion ũ = ṽ = 0. One may noe ha he condiion c 6 = 0 imposes a furher resricion c 2 = c 3 in he Case. Subsiuing ũ = ṽ = 0 and c 2 = c 3 in 8 we obain he following soluion for he PDE, namely, 2 u = c y f + c g c 4 c 5 f 2 g f 3 2 d c 4g f 2 c g c 4 c 5 f 2 g f 3 2 d d, d g f 3 2 v = c x f + c α g d 2 gf 2 f 3 2 c α g d d gf 2 f 3 2 c c 2 + c α g d, gf 2 f 3 2 c2 c g = e, f = c 2. In he Case 2 we observed ha here is no resricion on he parameers c 2 and c 3 when c 6 = 0. As a resul we obain anoher class of soluion of he form u = c F c 2 + k + c 6 y + c 4 + c 5 /2 F c 2 + k d + c 6 /2 c 4 F d F c 2 + k + c 6,

7 7 v = c F α F x + F F c F α F c 2 + k + c 6 d d + c α F d. 3 F Since he parameers c 2 and c 3 are real i is no possible o exrac any soluion for he Case Group invarian soluions The invariance of under he one parameer Lie group of infiniesimal ransformaions leads o he following symmeries ξ = η + c b, ξ 2 = ζ, φ = ũ, φ 2 = c 2 c b + ṽ, 4 where and c b are arbirary consans. In he above, ξ i, φ i, i =, 2, are he infiniesimals associaed wih he independen and dependen variables respecively. The corresponding Lie vecor fields are V a = η η + ζ ζ + ũ ũ + ṽ ṽ, V b = η. 5 Solving he characerisic equaions by assuming 0 associaed wih he symmeries 4 we obain he following similariy ransformaion z = ζ η + c b, ũ = ζw z, ṽ = w 2z ζ which ransforms Eq. o he following sysem of ODEs c 2c b. 6 zw + zw w + 2 w + w 2 + w 2 w αw αzw + α2 4 = 0, w 2 zw w = 0, α = c 2 + c 3, 7 where prime denoes differeniaion wih respec o he new independen variable z. Inegraing he second equaion in 7 we ge w 2 = zw + I, 8 where I is an inegraion consan. Subsiuing 8 ino he firs equaion and inegraing he resulan equaion we obain zw + zw 2 c2 + c I c 2 + c 3 z w + z I 2 = 0, 9 4

8 8 where I 2 is he second inegraion consan and we have aken c = for convenience. Now, inroducing he ransformaion w = p z pz, eq. 9 can be brough o a linear second order ODE of he form p z c 2 + c 3 + I p z+ I 2 z s + c 2 + c 3 2 pz = The general soluion of 20 can be wrien in erms of confluen hypergeomeric funcion. However, cerain ineresing special soluions can be exraced by appropriaely choosing he arbirary consans. For example, le us choose, I = 0, I 2 = and c 2 + c 3 = 2. Then he resulan ODE can be shown o admi he following general soluion pz = I 3 + I 4 log ze z, 2 where I 3 and I 4 are inegraion consans, which in urn provides an explici soluion for w and w 2 of he form, w = + I 4 zi 3 + I 4 log z, w I 4 2 = z +, 22 zi 3 + I 4 log z respecively. Subsiuing 22 ino 6 we ge a soluion for he similariy reduced PDE of he form [ ũ = ζ + ṽ = η + c b I 4 ζη + c b I 3 + I 4 logζc 8 η + c 9 [ + ], I 4 ζη + c b I 3 + I 4 logζc 8 η + c 9 ] c 2 c b. 23 Going back o he original variables we arrive a he following soluion for he 2 + dimensional Burgers equaion u = c y f + c g c 4 c 5 f 2 g f 3 2 d c g c 4 c 5 f 2 g f 3 2 d d c 4g d f 2 g f 3 2 g I 4 ζ +, f 2 ζη + c b I 3 + I 4 logζη + c b

9 9 v = c x f + c gf 2 α g d c 2 + c α g d f 3 2 gf 2 f 3 2 I 4 + ζgf 2 η + c b I 3 + I 4 log[ζη + c b + η + c b gf 2 α f c α g d d. 24 gf 2 f 3 2 We menion here ha he soluion 24 exis in all hree cases. 5. Some special similariy reducions Resricing some of he parameers, c is in he infiniesimal symmeries one can derive a large class of paricular soluions including he ravelling wave soluion. In he following we discuss one such ineresing case. 5.. Classical self-similar soluions We noe ha his case is a sub-case of Case 2 of he general similariy reducion discussed in Sec.2. Choosing c 2 =, α = c = c 3 = c 4 = c 5 = c 6 = 0 in he similariy variables we ge where ũ ηζ + ũṽ ζ + ṽũ ζ ζũ ζ ũ = 0, ũ η = ṽ ζ. 25 η = y, ζ = x, ũ = u, ṽ = v. 26 Eq. 25 is invarian under he following infiniesimal symmeries ξ =, ξ 2 = c b, φ = 0, φ 2 = c b. 27 Solving he characerisic equaion associaed wih he infiniesimal symmeries 27 we obain he following similariy variables, ha is, z = η + c b ζ, u = w z, v = ζ + w 2 z, 28 which ransforms 25 o he following ODE w + c b w w + w 2 w + c b w = 0, w c b w 2 = Inegraing he second equaion in 29 and subsiuing i ino he firs equaion we arrive a w + 2c b w w c b zw I c b w c b w = 0, 30

10 0 which can be again ransformed o he Bessel equaion. However, he subcase I = 0 provides us an explici soluion w = w 2 = expi 2 + c b 2 zz I 3 + cb expi2 + c b 2 zzdz cb expi 2 + c b 2 zz I 3 + c b c expi2 a + c b 2 zzdz. 3 Rewriing 3 in erms of original variables we obain an exponenial soluion for he 2 + dimensional Burgers equaion., 6. Linearizaion In his secion we show ha he sysem 2 can be ransformed o a + dimensional linear equaion. To illusrae his we consider he following infiniesimal symmeries ξ = x, ξ 2 = y, ξ 3 = 0, φ = u, φ 2 = v. 32 Using 32 we can ransform Eq. 2 o he following form: where ũ ζ + ηũ ηη + ηũṽ η + ηṽũ η = 0, ũ η ṽ ηṽ η = 0, 33 η = xy, ζ =, u = Inegraing he second equaion in 33 we ge ũη, ζ, v = xṽη, ζ. 34 x ũ = ηṽ + φζ, 35 where φζ is an arbirary funcion of ζ. equaion we obain Subsiuing i ino he firs ṽ ζ + ηṽ ηη + 2ηṽṽ η + 2ṽ η + ṽ 2 + φṽ η + φ η where prime denoes differeniaion wih respec o ζ. rewrien in he form = 0, 36 Eq. 36 can be ṽ ζ + ṽ + ηṽ η + ηṽ 2 + φ logη + φṽ η = By inroducing an auxiliary variable wη, ζ eq. 37 can be spli ino w η = ṽ, w ζ = ṽ + ηṽ η + ηṽ 2 + φ logη + φṽ. 38

11 Seeking Lie symmeries o his equaion we find 38 is invarian under he following infiniesimal ransformaion ξ = 0, ξ 2 = 0, φ = ṽf η, ζ + F η η, ζe w, φ 2 = c d + F η, ζe w, 39 where F η, ζ is a soluion of he following linear PDE ηf ηη + + φf η + F ζ + φ logηf = The vecor field associaed wih he infiniesimal componens 39 is V = [ ṽf η, ζ + F η η, ζ] e w + F η, ζe w ṽ w. 4 Taking ino accoun ha 4 generaes an infinie dimensional algebras, in agreemen wih he algorihm given in [8], we consruc he following overdeermined sysem of PDEs for he unknown funcion φ, ha is, ṽφṽ + φṽ = 0, φṽ = Le us choose wo paricular soluions of eq. 42 as new independen variables z and z 2 z = η, z 2 = ζ. 43 The new dependen variables can be obained from he soluion, ψ ṽ, w and ψ 2 ṽ, w, of he following sysem of equaions Solving 44 and 45 we ge ṽψ ṽ + ψ we w =, ψ ṽ = 0, 44 ṽψ 2 ṽ + ψ 2 w = 0, e w ψ 2 ṽ =. 45 w = log ψ, ṽ = ψ 2 ψ, 46 where ψ and ψ 2 are new dependen variables. Subsiuing 46 and 43 ino 38 we can ransform he laer ino he following linearized sysem ψ z = ψ 2, ψ z2 = ψ 2 + z ψ 2z + ψ φ z 2 logz + φz 2 ψ Choosing φ = in 47, we ge ψ 2z2 + z ψ 2zz = 0.

12 2 7. Conclusions In his paper we have sudied he group invariance properies of he 2+ dimensional generalized Burgers equaion. We have poined ou ha Eq. does no admi Virasoro ype algebras evenhough i is inegrable. As we menioned in he inroducion he 2+ dimensional breaking solion equaion and nonlinear Schrödinger equaion inroduced by Srachn are he only equaions which do no admi Virasoro ype algebras in higher dimensions. Equaion is anoher example in his caegory. Furher, we have explored several ineresing soluions for his equaion which are all new o his problem. In addiion o he above, we have shown ha 2+ dimensional Burgers equaion can be wrien as a + dimensional linear equaion. Acknowledgmens The work of MS forms par of a Deparmen of Science and Technology, Governmen of India, sponsored research projec. The work of MT is suppored by P.R.A ex 60 % of Universiy of Caania, by INdAM hrough G.N.F.M. and by M.I.U.R. PRIN 2003/05 Nonlinear Mahemaical Problems of Wave Propagaion and Sabiliy in Models of Coninuous Media. References. M. J. Ablowiz and P. A. Clarkson, Solions: Nonlinear Evoluion Equaions and Inverse Scaering, Cambridge Universiy Press, Cambridge, V. B. Maveev and M. A. Salle, Darboux ransformaions and solions, Springer, New York, J. Weiss, M. Tabor and G. Carnevale, J. Mah. Phys., 24, S. Y. Lou and L. L. Chen, J. Mah. Phys., 40, S. Y. Lou, Phys. Le., 277A, M. L. Wang, Phys. Le., 99A, E. G. Fan and H. Q. Zhang, Phys. Le., 246A, G. W. Bluman and S. Kumei, Symmeries and Differenial Equaions, Springer-Verlag, New York, P. J. Olver, Applicaions of Lie Groups o Differenial Equaions, Spring, New York N. H. Ibragimov, CRC Hanbook of Lie Group analysis of Differenial Equaions, Boca, Raon, Z. Yan, Chinese J. Phys., 40, Z. Yan, J. Phys. A. Mah. Gen., 35, M. Lakshmanan and M. Senhil Velan, J. Nonlinear Mah. Phys. 3, M. Senhil Velan and M. Lakshmanan, J. Nonlinear Mah. Phys. 5,