Group classification via mapping between classes: an example of semilinear reaction-diffusion equations with exponential nonlinearity

Group classificati o n via mapping b et w een classes: an example o f semilinear reaction– diffusion equations with exp onen tial no nlinearity Olena V ane eva Institute o f Mathemat ics of NAS of Ukr aine, 3 T er eshchenkivska Str., K yiv-4, 01601 Ukr aine vane eva@ima th.kiev.ua The group classificatio n of a cla ss o f se milinea r r eaction–diffusion equa tions with exp onen- tial nonlinea rity is car ried out using the technique of mapping b etw een classes, which was recently prop osed in [O .O. V a neev a, R.O. Popovych a nd C. Sopho c le o us, A cta Appl. Math. , doi:10.10 07/s1 0440-008-9280-9, arXiv:070 8.3457 ]. 1 In tro duction There exist relativ ely few equ ations d escribing natural phenomena among a great num b er of partial d ifferential equations (PDEs). This b egs the qu estion w hat mathematical prop erties differ equations d escribing physic al pro cesses from other p ossible ones? It app ears that large ma jorit y of equations of mathematical physics has non trivial symmetry prop erties (see a n um b er of examples e.g. in [1]). I t means that manifolds of their solutions are in v arian t with resp ect to m ulti-parameter group s of contin uous transf ormations (Lie group s of transformations) with a n umb er of parameters. Therefore, the pr esence of nont rivial symmetry pr op erties is one of such distinctiv e features (and v ery imp ortan t one)! In some cases the requ iremen t o f in v aria nce of equations under a group enables us to select these equations from a wide s et of other admissible ones. F or example, there is the only one system of Poincar ´ e-in v aria nt partial differen tial equations of first order for t w o real ve ctors E ( x 0 , x ) and B ( x 0 , x ), and this is the system of Ma xwell equations [1]. The prob lem arises to single out equati ons having h igh symmetry prop erties from a giv en class of PDEs. A solution of so-calle d gr oup cla ssific ation pr oblem giv es an exhaustiv e solution of this p r oblem. There exist t wo main ap p roac hes of solving group classification p roblems. The firs t one is more algebraic and b ased on su bgroup analysis of the equiv alence group of a class of different ial equations under consid er ation (see [2, 3, 4, 5] for details). The second app roac h inv olv es the inv estig ation of compatibilit y and the d irect inte gration of determining equations implied b y the infi nitesimal in v ariance criterion [6]. Unfortunately it is efficien t only for classes of a simple stru cture, e.g., which hav e a few arbitrary elemen ts of one or t w o same arguments. A n um b er of results on group classification problems in vest igated within the framew ork of this appr oac h are collected in [7] and other b o oks on the su b ject. T o solv e more group classification p r oblems differen t to ols h a v e b een recen tly d evelo p ed. One of them is to carry out grou p classification using appropriate mapp ing of a giv en c lass to a on e h a ving a simp ler stru cture. See the theoretical bac kground of this approac h and th e first example of its implement ation in [8]. In this p ap er we p erform the group classification of the class of semilinear reaction–diffusion equations with exp onen tial nonlinearit y f ( x ) u t = ( g ( x ) u x ) x + h ( x ) e mu (1) in the framework of this appr oac h. Here f = f ( x ), g = g ( x ) and h = h ( x ) are arbitrary smo oth functions of the v ariable x , f g h 6 = 0, m is an arbitrary constan t. The linear case is excluded from consid eration as well- inv estig ated (i.e., we assume m 6 = 0). 1 2 Equiv alence transformations and mapping of class (1) to a simpler one It is essent ial for group classification problems to derive the transformations which preserv e differen tial structure of a class under consideration and transform only arbitrary elemen ts. Such transformations are called e quivalenc e ones and form a group [6]. There exist sev eral kind s of equiv alence groups. Th e simplest one is giv en b y usu al equiv ale nce groups wh ic h consist of th e nondegenerate p oint transform ations of ind ep endent and d ep endent v aria bles as w ell as transformations of arbitrary element s of a class. Here transformations of indep end en t and d ep endent v ariables do not dep end on arb itrary elemen ts. If su c h dep endence arises then the corresp onding equ iv alence group is called gene r alize d . If new arb itrary element s are expressed via old ones in some nonp oint, p ossibly nonlo cal, wa y (e.g. n ew arbitrary elemen ts are determined via in tegrals of old ones) then the equiv alence transformations are called extende d ones. The first examples of a generali zed equiv al ence group and of an extended equiv alence group are presente d in [9] an d [10], resp ectiv ely . See a num b er of examples of different equiv alence groups and their role in solving complicated group classification problems, e.g., in [8, 11, 12]. Theorem 1. The gener alize d extende d e quivalenc e gr o up ˆ G ∼ of class (1 ) c o nsists of the tr ans- formations ˜ t = δ 1 t + δ 2 , ˜ x = ϕ ( x ) , ˜ u = δ 3 u + ψ ( x ) , ˜ f = δ 0 δ 1 ϕ x f , ˜ g = δ 0 ϕ x g , ˜ h = δ 0 δ 3 ϕ x e − mψ ( x ) δ 3 h, ˜ m = m δ 3 , wher e ϕ ( x ) is an arbitr ary smo oth function, and ψ ( x ) = δ 4 R dx g ( x ) + δ 5 . H er e δ j , j = 0 , 1 , . . . , 5 , ar e arbitr ary c onsta nts, δ 0 δ 1 δ 3 6 = 0 . The ab o v e transformations with δ 4 = 0 form the u sual equ iv alence group of class (1). The pr esence of the arb itrary fun ction ϕ ( x ) in the equiv alence transformations from ˆ G ∼ allo ws us to s im p lify the group classification problem of class (1) via reducing the num b er of arbitrary elemen ts and making its more con ve nient for mappin g to another class. Th us, the transformation from th e equiv alence group ˆ G ∼ ˜ t = sign( f ( x ) g ( x )) t, ˜ x = Z     f ( x ) g ( x )     1 2 dx, ˜ u = m u, (2) connects (1) with the class ˜ f ( ˜ x ) ˜ u ˜ t = ( ˜ f ( ˜ x ) ˜ u ˜ x ) ˜ x + ˜ h ( ˜ x ) e ˜ u , with the new arbitrary elemen ts ˜ f ( ˜ x ) = ˜ g ( ˜ x ) = sign( g ( x )) | f ( x ) g ( x ) | 1 2 , ˜ h ( ˜ x ) = m    g ( x ) f ( x )    1 2 h ( x ), ˜ m = 1. Without loss of generalit y , we can restrict ourselv es to the stu dy of the class f ( x ) u t = ( f ( x ) u x ) x + h ( x ) e u , (3) since all results on symmetries and exact solutions f or this class can b e extended to class (1) with tr an s formation (2). It is easy to deduce the generalized extend ed equiv alence group for class (3) from Th eorem 1 b y setting ˜ f = ˜ g , f = g and ˜ m = m = 1. Th e results are summarized in the follo wing theorem. Theorem 2. The ge ner alize d extende d e quivalenc e gr oup ˆ G ∼ 1 of c lass (3) is forme d by the tr ansformations ˜ t = δ 2 1 t + δ 2 , ˜ x = δ 1 x + δ 3 , ˜ u = u + ψ ( x ) , ˜ f = δ 0 δ 2 1 f , ˜ h = δ 0 e − ψ ( x ) h, wher e ψ ( x ) = δ 4 R dx f ( x ) + δ 5 ; δ j , j = 0 , 1 , . . . , 5 , ar e arbitr a ry c o nstants, δ 0 δ 1 6 = 0 . 2 The n ext step is to c hange the dep end en t v ariable in class (3): v ( t, x ) = u ( t, x ) + ω ( x ) , where ω ( x ) = ln | f ( x ) − 1 h ( x ) | . (4) As a result, we obtain the class v t = v xx + F ( x ) v x + εe v + H ( x ) , (5) where ε = sign( f ( x ) h ( x )) and the new arbitrary elemen ts F and H are expressed via the formulas F = f x f − 1 , H = − ω xx − ω x F . (6) All results on Lie s ymmetries and exact solutions of class (5) can b e extend ed to class (3) by the in v ersion of transformation (4). See the theoretical bac kground in [8 ]. 3 Lie symmetries In the previous section the group classification problem of class (1) has b een reduced to the similar but simpler p roblem for class (5). In this section we inv estig ate Lie sy m metry prop erties of class (5). Then th e obtained results are used to derive the group classification of class (3) that is equiv alen t to class (1) with resp ect to transform ation (2) fr om ˆ G ∼ . The group classification problem for class (5) is solv ed in the framew ork of the classical approac h [6]. All necessary ob jects (the equiv ale nce group, the k ernel and all inequ iv a lent extensions of maximal Lie in v ariance algebras) are found. The u sual equ iv alence group G ∼ of class (5) is formed by the transformations ˜ t = δ 2 1 t + δ 2 , ˜ x = δ 1 x + δ 3 , ˜ v = v − ln δ 2 1 , ˜ F = δ − 1 1 F , ˜ H = δ − 2 1 H , where δ j , j = 1 , 2 , 3 , are arbitrary constan ts, δ 1 6 = 0. The generalized extended equiv alence grou p of class (5) degenerates to the usual one. The kernel of the maximal Lie inv ariance algebras of equations from class (5) coincides with the one-dimensional algebra h ∂ t i . It means that any equation from class (5) is inv arian t with resp ect to translations by t . All p ossible G ∼ -inequiv ale nt c ases of extension of the maximal Lie in v ariance algebras in class (5 ) are exhausted by ones pr esen ted in T able 1. T able 1. The group classification of class (5 ) N F ( x ) H ( x ) Basis of A max 1 αx − 1 + µx β x − 2 + 2 µ ∂ t , e − 2 µt ( ∂ t − µx∂ x + 2 µ∂ v ) 2 αx − 1 β x − 2 ∂ t , 2 t∂ t + x∂ x − 2 ∂ v 3 µx γ ∂ t , e − µt ∂ x 4 λ γ ∂ t , ∂ x 5 µx 2 µ ∂ t , e − µt ∂ x , e − 2 µt ( ∂ t − µx∂ x + 2 µ∂ v ) 6 λ 0 ∂ t , ∂ x , 2 t∂ t + ( x − λt ) ∂ x − 2 ∂ v Here λ ∈ { 0 , 1 } mo d G ∼ , µ = ± 1 mo d G ∼ ; α, β , γ are arbitrary constan ts, α 2 + β 2 6 = 0. In case 3 γ 6 = 2 µ , in case 4 γ 6 = 0. 3 No w we are able to deriv e the group classification of class (3) using the resu lts of T able 1. T o find the cases of extension of the maxi mal Lie inv ariance algebras in class (3 ) we should , at first, to solv e ODEs (6) for eac h pair of fu nctions F and H from T able 1. I n suc h a wa y w e will obtain the fun ctions f and ω . Then all corresp onding h can b e easily foun d from the f orm ula h ( x ) = δ f ( x ) e ω ( x ) , δ = ± 1 . In T able 2 w e list the general solutions of (6 ) wh ic h are connected with six p airs of fun ctions F and H present ed by cases 1–6 of T able 1. T able 2. The general solutions of equations (6 ) N f ( x ) ω ( x ) 1 c 0 x α e µ 2 x 2 R  c 1 − R ( β x − 2 + 2 µ ) x α e µ 2 x 2 dx  x − α e − µ 2 x 2 dx + c 2 2 | α 6 =1 c 0 x α β 1 − α ln x + c 1 x 1 − α + c 2 2 | α =1 c 0 x − β 2 ln 2 x + c 1 ln x + c 2 3 c 0 e µ 2 x 2 R  c 1 − γ R e µ 2 x 2 dx  e − µ 2 x 2 dx + c 2 4 | λ =1 c 0 e x − γ x + c 1 e − x + c 2 4 | λ =0 c 0 − γ 2 x 2 + c 1 x + c 2 5 c 0 e µ 2 x 2 R  c 1 − 2 µ R e µ 2 x 2 dx  e − µ 2 x 2 dx + c 2 6 | λ =1 c 0 e x c 1 e − x + c 2 6 | λ =0 c 0 c 1 x + c 2 Note that R e µ 2 x 2 dx = √ π √ − 2 µ Erf  1 2 √ − 2 µx  , where E r f ( z ) is the error function. c i , i = 0 , 1 , 2 , are arbitrary constan ts, c 0 6 = 0. T r ansformation (4) is not a bijectio n since the preimage set of e ac h equation from class (5) is a tw o-paramet ric family of equations from class (3). Every suc h family consists of equations whic h are equiv alen t with resp ect to the group ˆ G ∼ 1 from Th eorem 2 (see the pr o of in [8]). A classification list for class (3) can b e obtained from a classificatio n list for class (5) by means of taking a single p reimage for eac h elemen t of the latter list w ith resp ect to the mapping realized b y transformation (4). It means that w e sh ould c hoose partial solutions of equations (6) from the general ones p resen ted in T able 2 in ord er to obtain th e group cl assification of class (3) up to ˆ G ∼ 1 -equiv a lence. Example 1. The equation v t = v xx + v x + e v + γ from cla ss (5) is the image of the family of equations from class (3) e x u t = ( e x u x ) x + e − γ x + c 1 e − x + c 2 e u (7) with r esp ect to th e transformation v = u − (1 + γ ) x + c 1 e − x + c 2 . The s implest represent ativ e of this family is the equation e ˜ x ˜ u ˜ t = ( e ˜ x ˜ u ˜ x ) ˜ x + e − γ ˜ x e ˜ u . (8) Theorem 2 implies that equations (7) and (8) are equiv alen t with resp ect to the transformation ˜ t = t , ˜ x = x , ˜ u = u + c 1 e − x + c 2 from ˆ G ∼ 1 . Hence, kno wing th e maximal Lie in v ariance algebra or exact solutions of (8), one ca n deriv e the basis el ements of the m aximal Lie in v ariance algebra and exact solutions of equation (7) that h as more complicated co efficien ts. 4 Therefore, to complete the group classification of class (3 ) with resp ect to its equiv alence group ˆ G ∼ 1 , w e should set, e.g., c 1 = c 2 = 0 , c 0 = 1 in th e fun ctions f and h and construct the basis o p erators of the maximal Lie in v ariance algebras for equations from (3) w ith s u c h f an d h using the formula X = τ ∂ t + ξ ∂ x + ( η − ξ ω x ) ∂ u . Here τ , ξ and η are co efficien ts of ∂ t , ∂ x and ∂ v in infinitesimal generators from T able 1. ω x = dω dx , where the corresp onding v alues of ω connected with f and h via (4) are listed in T able 2 . The obtained results are collected in T able 3. The fir s t num b er of eac h case indicates the asso ciated case of T able 1. T able 3. The group classification of class (3 ) N f ( x ) h ( x ) Basis of A max 1 x α e µ 2 x 2 δ x α e µ 2 x 2 + ω 1 ∂ t , e − 2 µt  ∂ t − µx∂ x + µ  2 + xω 1 x  ∂ u 2 . 1 x α δ x α + β 1 − α ∂ t , 2 t∂ t + x∂ x −  2 + β 1 − α  ∂ u 2 . 2 x δ x 1 − β 2 ln x ∂ t , 2 t∂ t + x∂ x − (2 − β ln x ) ∂ u 3 e µ 2 x 2 δ e µ 2 x 2 + ω 3 ∂ t , e − µt ∂ x − e − µt ω 3 x ∂ u 4 . 1 e x δ e ρx ∂ t , ∂ x + (1 − ρ ) ∂ u 4 . 2 1 δ e − γ 2 x 2 ∂ t , ∂ x + γ x∂ u 5 e µ 2 x 2 δ e µ 2 x 2 + ω 5 ∂ t , e − µt ∂ x − e − µt ω 5 x ∂ u , e − 2 µt  ∂ t − µx∂ x + µ  2 + xω 5 x  ∂ u 6 . 1 e x δ e x ∂ t , ∂ x , 2 t∂ t + ( x − t ) ∂ x − 2 ∂ u 6 . 2 1 δ ∂ t , ∂ x , 2 t∂ t + x∂ x − 2 ∂ u Here δ = ± 1, µ = ± 1 mo d ˆ G ∼ 1 ; α, β , γ , ρ are arbitrary constan ts, ρ 6 = 1, α 2 + β 2 6 = 0. ω 1 = − R x − α e − µ 2 x 2 R ( β x − 2 + 2 µ ) x α e µ 2 x 2 dx dx , ω 3 = − γ R e − µ 2 x 2 R e µ 2 x 2 dx dx , ω 5 = ω 3 | γ =2 µ , ω i x = dω i dx , i=1,3,5. In case 2.1 α 6 = 1. In case 3 γ 6 = 2 µ . In case 4.2 γ 6 = 0. The ke rn el of the maximal Lie in v ariance algebras of equations from class (3) coincides with the one-dimensional algebra h ∂ t i . 4 Construction of exact solutions via reduction metho d In this s ection w e present an example of finding exact sol utions of equ ations from class (3) via reduction m etho d. This tec hnique is we ll known and quite algorithmic (see, e.g., [6, 13]). As sho wn in the pr evious section, equation (8) with γ 6 = − 1 (Case 4.1 of T able 3 with ρ = − γ and δ = 1) admits the tw o-dimensional (comm utativ e) Lie inv ariance algebra g generated by the op erators X 1 = ∂ ˜ t , X 2 = ∂ ˜ x + (1 + γ ) ∂ ˜ u . A complete list of inequiv alen t non-zero su balgebras of g is exhaus ted by th e algebras h X 1 i , h X 2 i and h X 1 , X 2 i . 5 Lie reduction of equation (8) to an algebraic equation can b e made with the t wo -dimensional subalgebra h X 1 , X 2 i whic h coincides with the whole alg ebra g . Th e asso ciated ansatz and the reduced algebraic equ ation h a v e the form h X 1 , X 2 i : ˜ u = (1 + γ ) ˜ x + C , (1 + γ ) + e C = 0. The real solution of the reduced equation exists only for γ < − 1. Substituting th e solution C = ln | 1 + γ | of the r educed algebraic equation into the ansatz, we constru ct the exact solution ˜ u = (1 + γ ) ˜ x + ln | 1 + γ | (9) of equation (8) for γ < − 1. The an s atzes and reduced equations corresp onding to the one-dimen s ional su balgebras fr om the optimal sys tem are the follo wing: h X 1 i : ˜ u = z ( y ) , y = ˜ x ; z y y + z y + e − (1+ γ ) y e z = 0; h X 2 i : ˜ u = (1 + γ ) ˜ x + z ( y ) , y = ˜ t ; z y = (1 + γ ) + e z . The solution of the latter reduced equation is z = ln     ± (1 + γ ) e − ( y + c )(1+ γ ) ∓ 1     , where c is an arb itrary constan t. T h en ˜ u = (1 + γ ) ˜ x + ln     ± (1 + γ ) e − ( ˜ t + c )(1+ γ ) ∓ 1     (10) is the corresp onding solution of equation (8). Applying the equ iv alence transformation adduced in Examp le 1 to (9) and (10) exact solutions of equation (7) with complicated co efficients can b e easily constructed. 5 Conclusion The complete solution of the group classification problem f or class (1) b ecame p ossible only due to using of the metho d based on simulta neous application of a m apping b et wee n classes and equiv al ence transformations. T his metho d can b e applied for solving of similar problems for other classes of d ifferential equations and extended, e.g., to inv estiga tions of reduction op erators (nonclassical symmetries), conserv ation laws and p oten tial symmetries. The u sage of transfor- mations from the generalized extended equiv alence group allo ws us to present the final result in concise form. Ac kno wledgmen ts The author thanks the Organizing Committee of the 5th Mathematical P h ysics Meeting and esp ecially Prof. Branko Drago vic h for h ospitalit y and giving an opp ortunit y to giv e a talk. Her participation in the conference was partially supp orted b y CEI and ICTP . The author is also grateful to Prof. Roman Pop o vyc h for useful discussions. References [1] W.I. F ushc hich and A.G. Nikitin, Symmetries of Equations of Quantum M e chanics , Allerton Press I nc., New Y ork, 1994. [2] R. Zhdanov and V. Lahn o, Group classification of heat conductivity equations with a nonlinear source, J. Phys. A: Math. Gen. 32 (1999) 7405–74 18. [3] P . Basarab-Horwath, V. Lahn o and R. Zhdanov, The structure of Lie algebras and the classificatio n problem for partial differential equations, A cta A ppl . Math. 69 (2001) 43–94. 6 [4] V. Lahno, R. Zhdan ov and O. Magda, Group classification and exact solutions of nonlinear wa v e equations, A cta Appl. Math. 91 (2006) 253–313. [5] R. Zhdanov and V. Lahno, Group classificati on of the general second-order evolution equation: semi-simple inv ariance groups, J. Phys. A: M ath. The or. 40 ( 2007) 5083–5103. [6] L.V. Ovsianniko v, Gr oup analysis of differ ential e quations , Academic Press, New Y ork, 1982. [7] N.H. Ibragimo v (Editor), Lie gr oup analysis of differ ent ial e quations — symmetries, exact solutions and c onser vation laws , V ol. 1,2, CRC Press, Bo ca Raton, FL, 1994. [8] O.O. V aneev a, R .O. P op ovyc h and C. Sopho cleous, Enhanced Group A n alysis and Exact Solutions of V ari- able Co efficient S emilinear Diffusion Equations with a Po wer Source, A cta Appl. Math. , doi:10. 1007/s1044 0- 008-9280-9, 46 p., [9] S.V. Meleshko , Group classification of equations of tw o-dimensional gas motions, Prikl. Mat. Mekh. 58 (1994) 56–62 (in Ru ssian); translation in J. Appl. Math. Me ch. 58 (1994) 629–635. [10] N.M. Iv ano v a, R.O. Popovyc h and C. Sopho cleous, Conserv ation la ws of v ariable coefficient diffusion– conv ection equations, Pr o c. of T enth International Confer enc e in Mo dern Gr oup Analysis (L arnac a, Cyprus, 2004) (2005) 107–113; arXiv:math-p h/0505015. [11] O.O. V aneev a, A.G. Johnpillai, R.O. Popovych and C. Sopho cleous, Enhanced group analysis and conserv a- tion la ws of v ariable coefficient reaction–diffusion equations with p ow er nonlinearities, J. Math. Anal. Appl . 330 (2007) 1363–1386; arXiv :math-ph/0605081. [12] N.M. I v ano v a, R.O. Popovyc h, C. Sopho cleous, Group analysis of v ariable co efficient d iffusion–con vection equations. I . En h anced group classification, 24 p., arXiv:0710.27 31 . [13] P . Olver, Applic ations of Lie gr oups to di ffer ent ial e quations , Springer-V erlag, New Y ork, 1986. 7