NONLINEAR DYNAMICAL SYSTEMS IN VARIOUS SPACE-TIME DIMENSIONS

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1 NONLINEAR DYNAMICAL SYSTEMS IN VARIOUS SPACE-TIME DIMENSIONS R. CIMPOIASU, V. CIMPOIASU, R. CONSTANTINESCU Universiy of Craiova, 3 A.I. Cuza, Craiova, Romania Received January, 009 The paper invesigaes he connecions beween he symmeries of he same dynamical sysem defined in spaces wih various number of degrees of freedom. More precisely, we will consider a sysem defined in a n-dimensional space, we will apply he similariy reducion procedure, and we will compare is soluions wih he ones of he same sysem projeced in a n dimensional space. Despie he fac ha boh cases lead o equaions depending on n independen variables, here are imporan differences in he behavior of he sysem. Some similariies can be also noiced. As a concree eample for susaining hese conclusions, he Burgers-Koreweg de Vries equaion in dimensions will be considered. Key words: symmeries of PDEs sysems, similariy reducion, Burgers-KdV model. PACS: a,.30.Na. INTRODUCTION Many naural phenomena are described by a sysem of nonlinear parial differenial equaions (PDEs) which is ofen difficul o be analyically solved, as here is no eising general heory for compleely solving nonlinear PDEs. The modern approach for finding special soluions of nonlinear PDEs was pioneered by Lie []. Over he years, Lie s mehod has been proven o be a powerful ool for sudying a remarkable number of PDEs arising in mahemaical physics [], [3], [4]. Known oday as symmery analysis, his heory links differenial geomery o PDEs heory [5], symbolic compuaion [6], numerical analysis [7], and recenly, o image processing [8]. The classical Lie mehod, he nonclassical mehod (due o Bluman and Cole [9]) and he direc mehod (due o Clarkson and Kruskal [0]) are some of he basic mehods for finding special classes of soluions for nonlinear PDEs. The classical Lie mehod gives he one-parameer groups of ransformaions (called classical symmeries or Lie poin symmeries) which map he whole se of analyical Paper presened in he Naional Conference of Physics, Sepember 0 3, 008, Buchares, Măgurele, Romania. Rom. Journ. Phys., Vol. 55, Nos., P. 5 35, Buchares, 00

2 6 R. Cimpoiasu, V. Cimpoiasu, R. Consaninescu soluions o iself. The nonclassical mehod (also denominaed by some auhors as he mehod of condiional symmeries), allows o derive an one-parameer groups of ransformaions ha leave invarian a subse of he se of all analyical soluions only. Any classical symmery is a nonclassical one, bu no reciprocally. Arrigo, Broadbridge and Hill sudied he connecion beween he direc mehod and he nonclassical mehod []. If a PDEs sysem represens he Euler-Lagrange equaions associaed wih a variaional problem, hen he classical symmeries leads o variaional symmeries [5]. The variaional symmeries allows, by Noeher s heorem, o deermine conservaion laws: for each one-parameer variaional symmery here is an associaed conservaion law. In a igh connecion wih inegrabiliy is he problem of isolaing he consans of moion for a given physical sysem. Two invesigaion ways are possible for ha: (i) direcly, by imposing a specific form of he invarian (linear, quadraic, ec, in velociies) he poenials which accep such ype of invarian could be deermined; (ii) indirecly, saring from a concree poenial he invarian quaniies can be found. For eample, in [] wo dynamical sysems wih polynomial poenials, he Yang-Mills and he Hénon-Heiles sysems, were analyzed on he direc line. The indirec mehod for consrucing he invarians consis in deermining hem from he symmeries found for he analyzed sysem. In [3],[4] we invesigaed he Lie symmeries and associaed invarians for he D Ricci flow model and he D nonlinear hea equaion, respecively. This paper will focus on he relaion beween he symmeries of a non-linear dynamical sysem, is invarians and he similariy reducion mehod. I is organized as follows: in secion, a monographic one, we presen how he invarians of an arbirary nonlinear dynamical sysem associaed o he general symmery operaor, could be deermined using he characerisic and he similariy reducion mehods; he secion 3 conains he auhors resuls concerning he Lie symmery analysis for he D and D Burgers-KdV models. Afer deducing he symmeries for he wo cases, he similariy reducion mehod is applied o he D Burgers-KdV equaion in order o compare he resuling D equaion wih he one direcly given in he firs subsecion of his hird secion. The paper will end wih some concluding remarks.. DETERMINATION OF THE INVARIANTS BY THE CHARACTERISTIC METHOD Le us consider a dynamical sysem described by an n-h order parial differenial equaion: ( n) ( u ), [ ] ()

3 3 Nonlinear dynamical sysems 7 where { i, i=, p} represen he independen variables, while u { u α, α =, q} he dependen ones. The noaion u (n) designaes he se of variables which includes u and he parial derivaives of u up o n-h order. The symmery group associaed o he equaion () consiss in one-parameer group of ransformaions which leave he se of all is analyic soluions invarian. From he general symmery heory [5], he infiniesimal symmery operaor has he form: p q i U = ξ (, u) + φ (, u i α ) α i= α = u () and is n-h eension is given by: q ( n) J n U = U + φα (, u ) (3) α α = J uj where m α α u uj =. (4) j j jm In epression (3), he second summaion refers o all he muli-indices J = (j, J j m ), wih jm p, m n. The coefficien funcions φα are given by he following formula: p p J i ( n) i α i α φα (, u ) = DJ φα ξ ui + ξ uj, i, α =, q (5) i= i= in which α α u ui =, i=, p (6) i α m+ α α uj u uji, = = (7) i i j j jm m d DJ = Dj D... j D j = (8) m j j jm d d d The Lie invariance condiion is: ( n ) [ ] 0 (9) U = The characerisic equaions associaed o he general symmery generaor () have he form: p q d d d du du du = =... = =... = (0) p ξ ξ ξ φ φ φq By inegraing he characerisic sysem of ordinary differenial equaions (0) he invarians H r, r =, ( p q ) + of he analyzed sysem could be found.

4 8 R. Cimpoiasu, V. Cimpoiasu, R. Consaninescu 4 They are idenified hrough he consans of inegraion. Following his way, we can use H r as a se of new variables, he similariy variables, in erm of which he original sysem of PDEs wih p independen variables could be reduced o a sysem of PDEs wih p independen variables. 3. THE BURGERS-KdV MODEL 3.. LIE SYMMETRIES AND INVARIANTS FOR THE D MODEL The evoluion equaion of he model has he form: u + uu + au 3 bu () I is imporan o remark ha he equaion () arises from many differen physical cones as a nonlinear model equaion incorporaing he effecs of dispersion, dissipaion and nonlineariy. In [5], he equaion () is derived as he governing equaion for waves propagaing in a liquid-filled elasic ube in which he weak effecs of dispersion, dissipaion and nonlineariy are presen. I was used as a nonlinear model in he flow of liquids conaining gas bubbles and urbulence [6], [7], using a seady sae version of () o describe a weak shock profile in plasmas. I is imporan o remark ha, by using compuaional mehod, he D equaion () has a soluion of he form: 3 3b + 50a c 6 b b u(, ) = + anh c c 5ab 5 a + 0a () 3 b b anh c c 5 a + 0a wih a, b, c, c arbirary consans. This soluion will be compared in he ne secion o he soluion of he D equaion generaed by similariy reducion applied o he D model. The general epression of he classical Lie operaor which leaves () invarian is: U( u,, ) = ϕ( u,, ) + ξ( u,, ) + φ( u,, ) (3) u Wihou loss he generaliy we could impose ϕ = cons. =. Because () is a parial differenial equaion of hird order, he invariance condiion for he D Burgers-KdV model is given by he relaion: ( 3 ) [ ] U u + uu + au bu = (4) 3 0 where U (3) is he eension of hird order of he Lie operaor (3). I has he general epression:

5 5 Nonlinear dynamical sysems 9 ( 3) U = U + φ + φ + φ + φ + φ + u u u u u φ +... φ u3 u3 The invariance condiion (4) is equivalen wih he equaion: φu φ u φ aφ bφ (5) (6) 3 The funcions φ, φ, φ, φ will be deermined using he general formulas from [5]: φ = Dφ ( Dξ) u, φ = Dφ ( Dξ) u [ ] ( ), [ ] ( 3) φ = D φ u ξu + u + ξu φ = D φ u ξu + u + ξu where D is he oal derivaive operaor. By eending he relaions (7), subsiuing hem ino he condiion (6) and hen cancelling he coefficien funcions of he various monomials in parial derivaives of u, we would find a sysem wih 0 parial differenial equaions. I could be reduced o he following parial differenial sysem: ξ ξu φu (8) φu φ ξ φ + uφ + aφ bφ 3 We solved his sysem hrough Maple and obained he following soluions: ξ = c + c φ = c (9) where c i, i =, are arbirary consans. In conclusion, he general Lie generaor (3) has he concree epression: U = + ( c + c) + c u (0) The independen symmery operaors associaed o he soluion (9) are: U = + + u, U = + () The operaor U generaes a ime ranslaion and a galilean boos, U represens ime and space ranslaions. The characerisic equaions associaed o he symmery operaor (0) have he form: d = d = du () c+ c c (7)

6 30 R. Cimpoiasu, V. Cimpoiasu, R. Consaninescu 6 By inegraing () we obain he invarians. They have he following epressions: c H = + c (3) c H = u c or H = u + c (4) 3.. LIE SYMMETRIES AND INVARIANTS FOR THE D MODEL The D Burgers-KdV equaion is a -dimensional generalizaion of equaion () and has he following form [8]: u + u + u u u + u + u = (5) α 3 β 4 γ y 0 Similarly wih he D case, we could apply numerical mehods in order o obain soluions for he equaion (5). Such a soluion will have he epression: 4 3 3α + 50αβ c γβ c3 u(,, y) = 5α β 6 α α anh c c3y c4 5 β β (6) 3 α α anh c c3y c4 5 β β where α, βγ,, c, c 3, c 4 are arbirary consans. Passing now o he problem of symmeries of (5), he corresponding Lie symmery operaor has he general form: U(, y,, u) = ϕ(, y,, u) + ξ(, y,, u) + (7) + η( yu,,, ) + φ( yu,,, ) y u The governing equaion (5) would be invarian under he acion of he symmery generaor (7) iff he following condiion should saisfied: ( 4 ) ( α β γ y ) U u + u + u u u + u + u = (8) where U (4) represens he fourh eension of he Lie generaor (7). The above invarian condiion (8) has he equivalen form: φ + φ + φ + φ + γφ αφ + βφ = (9) y 3 4 u u u 0

7 7 Nonlinear dynamical sysems 3 y 3 4 where he funcions φ, φ, φ, φ, φ, φ are given by he epressions: φ = Dφ ( Dξ) u ( Dη) uy, φ = D φ u ξu ηu y + + u + ξu + ηu ( ) ( ) y φ = D φ u ξu ηu y+ u ( ) + ξu3 + ηu( ), y y φ = Dy φ u ξu ηuy + u( y) + ξu( y) + ηu 3y 3 φ = D 3 φ u ξu ηu y+ u ( 3 ) + ξu4 + ηu( 3 ), y 4 φ = D 4 φ u ξu ηu y + u ( 4 ) + ξu5 + ηu( 4 ). y Subsiuing he eended relaions (30) ino he condiion (9) and hen equaing wih zero he coefficien funcions of various monomials in derivaives of u, he following parial differenial sysem is obained: ξu = ξ = ξy, ηu = η = ηy (3) φ = φu, φ ξ, φ y The previous PDEs sysem has he soluion: ξ = F( ) d+ c, η = F( ), φ = F( ) (3) where F (), F () are arbirary funcions. Thereby, he D Burgers-KdV model has a classical (Lie) symmery generaor of he form: U = + ( F() d+ c) + F() y + F() u (33) Inegraing he characerisic equaions: d = ( d = () ) dy = du (34) F F() F() d+ c he following invarians resul: F( ) F( ) H = G() d+ c, H = y, H3 = y u. (35) G + c F () () (30) 3.3. THE SIMILARITY REDUCTION FOR D MODEL By inegraing he characerisic sysem (34) we obain he following similariy variables: τ = G( ) d+ c, τ = F( ) d y, (36) H = G u, G = F d () () ()

8 3 R. Cimpoiasu, V. Cimpoiasu, R. Consaninescu 8 wih heir forms depending on he original variables (, y,, u). Under he above similariy ransformaions, he original equaion (5) in 3 independen variables (, y, ) becomes reduced o a PDE in independen variables (τ, τ ) which has he form: () ch + F H + H + HH H H H = (37) τ ττ τ τ α 3τ β 4τ γ τ 0 Ne we will consider he paricular case: F () = cons =. (38) The evoluion equaion (37) has he equivalen epression: ( chτ + H ) ( ) τ + HH τ α H τ βh 3τ = G τ, τ (39) τ τ where we inroduce inser he noaion G( τ, τ) = γh τ dτ. From (39) we should obain he following reduced equaion: Hτ + HH τ α H τ β H 3τ (40) iff he following condiion would be saisfied: ch τ = γ H τ (4) The condiion (4) imposes for he reduced equaion (40) he following class of soluions: γ c H( τ, τ) = f τ τ g τ τ + + (4) c γ wih f and g arbirary consans of heir argumens. Through compuaional approach, he reduced equaion (40) offers, in erms of he similariy variables τ, τ he soluion: 3 3α + 50β c3 6 α α H( τ, τ) = + anh c τ c3τ 5αβ 5 β + 0β + (43) 3 α α + anh c τ c3 τ 5 β + 0β wih α, β, c, c 3 arbirary consans. I is imporan o remark he cases of compaibiliy beween (4) and (43). Two such cases can be considered: (i) If c, c 3 =, α = 5β, γ = c/4, hen he soluion (43) is of he form (4) wih g. (ii) If c, c 3 =, α = 5β, γ = c/4, hen he soluion (43) is of he form (4) wih f. These are he cases ha should be aken ino accoun in he forhcoming consideraions, when we will come back o he D Burgers-KdV equaion and o he original variables. By doing ha, from he relaions (3), (36) and (4) we

9 9 Nonlinear dynamical sysems 33 could wrie he soluion of he D model (5) in erms of symmery coefficiens ξ, η and he independen variables,, y. This soluion has he epression: u(,, y) = ξ 4γ f d d y ξ + η (44) g ξd ηd+ y Le us consider he case (i) and le us poin ou some paricular forms of he general soluion (44). (a) le us firs consider he choice: ξ = c0 + c= cons, η =, φ F( ), F( ) = (45) For his simples choice for F and F, he soluion of he D Burgers-KdV model is: c0 + c u(,, y) = c0 + 3β + 6β anh + y (46) c 0+ c 3β anh + y which is similar wih he soluion (6) obained by a direc compuaion applied o he equaion (5). (b) A slighly complicaed choice for he symmery coefficiens could be: ξ = m + c, η =, φ = m = cons F( ) = m, F( ) = (47) In his case (44) akes he form: m c u(,, y) = m+ 3β + 6β anh + + y 4 (48) m c 3β anh + + y 4 I is already differen from he direcly compued soluion (6) (c) Le us consider he case (i) and he paricular epressions: n n n ξ = e + c, η =, φ = e, n = cons F () = e, n F () = We discover a new soluion of (5), namely: n n c u(,, y) = e 3β + 6β anh e y n + + n n c 3β anh e + y + n (49) (50)

10 34 R. Cimpoiasu, V. Cimpoiasu, R. Consaninescu 0 (d) In he general case, when he symmery coefficiens ξ and η have he form (3), he soluion (44) of he D Burgers-KdV model akes he form: u,, y = F d+ 3β + ( ) ( ) ( () ) c 6β anh F d d + + y (5) c 3β anh ( F () d) d + + y A similar analysis could be made for he case (ii). We do no follow his compuaion, bu we will remark ha (44) obained before afer he similariy reducion represens a larger class of soluions for he D Burgers-KdV equaion han (6), which was obained by direc compuaion. 4. CONCLUDING REMARKS The main objecive of his paper was o presen how he similariy mehod could be applied in order o reduce he number of independen variables of a general nonlinear dynamical sysem. We were also ineresed in poining ou he connecion beween he original sysem, defined in erms of n independen variables, and he reduced sysem of n variables which are in fac he invarians given by he research of symmeries for he iniial sysem. As a general conclusion, we remark ha, despie he fac ha boh he sysem iself and he same sysem projeced in a (n )-dimensional space by applying he similariy reducion, lead o equaions depending on n independen variables, here are imporan differences in he behavior of he sysem. Anoher imporan conclusion is ha a bilaeral approach could be considered: (i) saring from he n-dimensional sysem one can generae he associaed (n )-dimensional sysem, sysem which is someimes very general and does no appear by a simple cancelling of one variable; (ii) he sar from he (n )-dimensional sysem gives (even in paricular cases) deails abou he possible invarians of he n-dimensional sysem, by using already known invarians for he aached (n )-dimensional case. To susain hese conclusions, we effecively compared wo D Burgers-KdV models: he one given by he D equaion () and he one generaed by applying he similariy reducion o he D equaion (5). A firs conclusion is ha he D Burgers-KdV model sudied by using he similariy reducion approach is described by he equaion (40), which is of prey similar form wih he equaion () governing he D case. Consequenly, looking o he form of he invarians (3) and (4) of (), we are epecing o obain for (40) invarians wih he form: ( ) τ τ H= g H or H = H c (5)

11 Nonlinear dynamical sysems 35 wih () τ ( ) τ = f + c, = y, H = f u (53) On he oher hand, we saw ha he soluion of he equaion (5) could ake, wih suiable choices, he form of he soluion () of (). Bu he class of numerical soluions for he reduced equaion (40) is significanly larger han wha he same numerical procedure spoed for he D case. Moreover, coming back from he reduced variables o he original ones, we were able o noe a larger class of soluions (see (5)) for he D equaion (5) as he direc soluion (6). I is a resul proving he imporance of he similariy reducion procedure in he sudy of he dynamical sysems described by parial derivaive equaions. Acknowledgemens. Auhors acknowledge for financial suppor o he Romanian Minisry of Educaion, Research and Innovaion, represened by he CNCSIS council, ino framework Ideas 008, gran code ID 48. REFERENCES. S. Lie, Gesammele Abhandlungen, Band 4, B.G. Teubner, Leipzig, Germany, 99, W.F. Ames, Nonlinear Parial Differenial Equaions in Engineering, Academic Press, New York, vol. I, 965, vol. II, G.W. Bluman and S. Kumei, Symmeries and Differenial Equaions, Appl. Mah. Sci., 8, Springer-Verlag, New York, P.E. Hydon, Symmery Mehods for Differenial Equaions, Cambridge Tes in Applied Mahemaics, Cambridge Universiy Press, P.J. Olver, Applicaions of Lie Groups o Differenial Equaions, Grad. Tes in Mah., 07, Springer-Verlag, New York, V.A. Grundland and G. Tafel, J. Mah. Phys., 36, 46 (995). 7. C.J. Budd and M.D. Piggo, Geomeric inegraion and is applicaions, in Handbook of Numerical Analysis, XI, Norh Holland, Amserdam, 003, F. Cao, Geomeric Curve Evoluion and Image Processing, Lecure Noes in Mahemaics, Springer-Verlag, G.W. Bluman and J.D. Cole, J. Mah. Mech., 8(969), P.A. Clarkson and M. Kruskal, J. Mah.Phys., 30(989), D.J. Arrigo, P. Broadbridge and J.M. Hill, J. Mah. Phys., 34, 0(993), R. Cimpoiasu, R. Consaninescu, V.M. Cimpoiasu, Rom.J. Phys., vol. 50, nr. 3 4, (005), R. Cimpoiasu, R. Consaninescu, J. Nonlin. Mah. Phys., vol. 3, no., (006), R. Cimpoiasu., R. Consaninescu, Nonlinear Analysis Series A: Theory, Mehods & Applicaions, vol. 68 (008), R.S. Jhonson, J. Fluid Mech., 4 (970), G. Gao, Sci. Sinica Ser. A 8 (985), S.D. Liu, S.K. Liu, Sci. Sinica Ser. A 35 (99), Z. Heng, X. Wang, Phys. Leers A 308 (003), 73.

12 36 R. Cimpoiasu, V. Cimpoiasu, R. Consaninescu