Group Classification of Systems of Non-Linear Reaction-Diffusion Equations

Group Classification of Systems of Non-Linear Reaction-Diffusion Equations

The completed group classification of systems of two coupled nonlinear reaction-diffusion equation with general diffusion matrix is carried out. The simple and convenient method for deduction and solution of classifying equations is presented.

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1. Verfasser: Nikitin, A.G.
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Veröffentlicht: Інститут прикладної математики і механіки НАН України 2005
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spelling irk-123456789-1245872017-09-30T03:04:02Z Group Classification of Systems of Non-Linear Reaction-Diffusion Equations Nikitin, A.G. The completed group classification of systems of two coupled nonlinear reaction-diffusion equation with general diffusion matrix is carried out. The simple and convenient method for deduction and solution of classifying equations is presented. 2005 Article Group Classification of Systems of Non-Linear Reaction-Diffusion Equations / A.G. Nikitin // Український математичний вісник. — 2005. — Т. 2, № 2. — С. 149-200. — Бібліогр.: 29 назв. — англ. 1810-3200 2000 MSC. 35K55, 35K57, 35Q55. http://dspace.nbuv.gov.ua/handle/123456789/124587 en Український математичний вісник Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description The completed group classification of systems of two coupled nonlinear reaction-diffusion equation with general diffusion matrix is carried out. The simple and convenient method for deduction and solution of classifying equations is presented.
format Article
author Nikitin, A.G.
spellingShingle Nikitin, A.G.
Group Classification of Systems of Non-Linear Reaction-Diffusion Equations
Український математичний вісник
author_facet Nikitin, A.G.
author_sort Nikitin, A.G.
title Group Classification of Systems of Non-Linear Reaction-Diffusion Equations
title_short Group Classification of Systems of Non-Linear Reaction-Diffusion Equations
title_full Group Classification of Systems of Non-Linear Reaction-Diffusion Equations
title_fullStr Group Classification of Systems of Non-Linear Reaction-Diffusion Equations
title_full_unstemmed Group Classification of Systems of Non-Linear Reaction-Diffusion Equations
title_sort group classification of systems of non-linear reaction-diffusion equations
publisher Інститут прикладної математики і механіки НАН України
publishDate 2005
url http://dspace.nbuv.gov.ua/handle/123456789/124587
citation_txt Group Classification of Systems of Non-Linear Reaction-Diffusion Equations / A.G. Nikitin // Український математичний вісник. — 2005. — Т. 2, № 2. — С. 149-200. — Бібліогр.: 29 назв. — англ.
series Український математичний вісник
work_keys_str_mv AT nikitinag groupclassificationofsystemsofnonlinearreactiondiffusionequations
first_indexed 2025-07-09T01:40:14Z
last_indexed 2025-07-09T01:40:14Z
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fulltext Український математичний вiсник Том 2 (2005), № 2, 149 – 200 Group Classification of Systems of Non-Linear Reaction-Diffusion Equations Anatoly G. Nikitin Abstract. The completed group classification of systems of two cou- pled nonlinear reaction-diffusion equation with general diffusion matrix is carried out. The simple and convenient method for deduction and solution of classifying equations is presented. 2000 MSC. 35K55, 35K57, 35Q55. Key words and phrases. Nonlinear Schrödinger equation, Ginzburg- Landau equation, coupled systems of diffusion equations, general diffu- sion matrix. 1. Introduction Group classification of differential equations is one of corner stones of group analysis. Such classification specifies the origin of possible appli- cations of powerful group-theoretical tools such as constructing of exact solutions, group generation of solution families starting with known ones, etc. A very important result of group classification consists in a pri- ori description of mathematical models with a desired symmetry (e.g., relativistic invariance). One of the most impressive results in group classification belongs to S. Lie who had completely classified second order ordinary differential equations [17]. It was Lie also who first presented the group classification of partial differential equations (PDE), namely, he had classified linear equations including two independent variables [18]. Using the classical Lie approach whose excellent presentation was given in [25] it is not difficult to derive determining equations for possible symmetries admitted by equations of interest. Moreover, to describe Lie symmetries for a fixed (even if very complicated) equation is a purely technical problem which is easily solved using special software packages. However, the situation is changing dramatically whenever we try to search Received 11.01.2005 ISSN 1810 – 3200. c© Iнститут прикладної математики i механiки НАН України 150 Group Classification of Systems... for Lie symmetries for an equation including an arbitrary element which is not a priory specified, i.e., when we are interested in group classification of an entire class of differential equations. The main problem of group classification of a substantially extended class of partial differential equations (PDEs) consists in effective solv- ing of determining equations for coefficients of generators of symmetry group. In general the determining equations are rather complicated sys- tems whose variables are not necessarily separable. A nice result in group classification of PDEs belongs to Dorodnitsyn [26] who had classified nonlinear (but quasi linear) heat equations ut − uxx = f(u) (1.1) where f is an arbitrary function of the dependent variable u, the sub- scripts denote derivations w.r.t. the corresponding variables, i.e., ut = ∂u/∂t and uxx = ∂2u/∂x2. Moreover, in paper [26] more general equa- tions ut − (Kux)x = f(u) were classified. The related determining equa- tions appears to be easily integrable, which made it possible to specify all non-equivalent non-linearities f (which are power, logarithmic and exponential ones) which correspond to different symmetries of equation (1.1). The non-classical (conditional) symmetries of (1.1) were described by Fushchych and Serov [10] and Clarkson and Mansfield [6]. The results of group classification of equations (1.1) play an important role in constructing of their exact solutions and qualitative analysis of the nonlinear heat equation, refer, e.g. to [28]. In the present paper we perform the group classification of systems of the nonlinear reaction-diffusion equations ∂u1 ∂t − ∆(a11u1 + a12u2) = f1(u1, u2), ∂u2 ∂t − ∆(a21u1 + a22u2) = f2(u1, u2) (1.2) where u = ( u1 u2 ) are function of t, x1, x2, . . . , xm, symbols a11, a12, a21, a22 denote real constants and ∆ is the Laplace operator in Rm. We shall write (1.2) also in the matrix form: ∂u ∂t −A∆u = f (1.3) where A is a matrix whose elements are a11, . . . a22 and f = ( f1 f2 ) . Mathematical models based on equations (1.2) are widely used in mathematical physics and mathematical biology. Some of these models A. G. Nikitin 151 are discussed in [23] and in Section 12 of the present paper, the entire collection of such models is presented in [20]. Thus the symmetry analysis of equations (1.2) has a large application value and can be used, e.g., to construct exact solutions for a very extended class of physical and biological systems. The comprehensive group analysis of systems (1.2) is also a nice “internal” problem of the Lie theory which admits exact general solution for the case of arbitrary number of independent variables x1, x2, . . . , xm. Symmetries of equation (1.3) for the case of a diagonal (and invert- ible) matrix A were investigated by Yu. A. Danilov [7]. Unfortunately, the results presented in [7] and cited in the handbook [14] are neither complete nor correct. We discuss these results in detail in Section 12. Symmetry classification of equations (1.3) with a diagonal diffusion matrix was presented in paper [3], then some results missing in [3] were added in Addendum [4] and paper [5]. However, we shall demonstrate that the results given in [3]–[5] are still incomplete, and add the list of non-equivalent equations given in these papers. We notice that symmetries of equations (1.3) with a diagonal diffusion matrix was partly described in paper [15] were symmetries of more general class of diffusion equations where studied. Equations (1.3) with arbitrary invertible matrix A were investigated in paper [23], the related results were announced in [24]. Unfortunately, mainly due to typographical errors made during publishing procedure, presentation of classification results in [23] was not satisfactory 1. In the present paper we give the completed group classification of coupled reaction-diffusion equations (1.3) with an arbitrary diffusion ma- trixes A. Moreover, we present a straightforward and easily verified pro- cedure of solution of the determining equations which guarantees the completeness of the obtained results. We also indicate clearly the equiva- lence relations used in the classification procedure. In addition, we extend the results obtained in [23] to the case of non-invertible matrix A. The additional aim of this paper is to present a rather straightforward and conventional algorithm for investigation of symmetries of a class of partial differential equations which includes (1.3) as a particular case. We will show that the classical Lie approach (refer, for example, [11],[25]) when applied to systems (1.3) admits a rather simple formulation which can be used even by such investigators which are not experts in group analysis of differential equations. Furthermore the algorithm may be 1The tables presenting the results of group classification have been deformed and cut off. It is necessary to stress that it was the authors fault, one of whom signed the paper proofs without careful reading. 152 Group Classification of Systems... used to search for conditional symmetries of (1.3) [23] (for definition of conditional symmetries see [12]). There exist two non-equivalent 2× 2 matrices with zero determinant, namely, the diagonal matrix with the only non-zero element and the Jordan cell. We will consider the following generalized versions of the related equation (1.2) ∂tu1 − ∆u1 = f1(u1, u2), ∂tu2 − pµ∂µu1 = f2(u1, u2) (1.4) and ∂tu1 − pµ∂µu2 = f1(u1, u2), ∂tu2 − ∆u1 = f2(u1, u2). (1.5) Here pµ are arbitrary constants and summation is imposed over repeating µ = 1, 2, · · · ,m . Moreover, without loss of generality we set p1 = p2 = · · · = pm−1 = 0, pm = p. (1.6) In the case p ≡ 0 equations (1.4) and (1.5) are nothing but particular cases of (1.2), which include such popular models of mathematical biology as the FitzHung-Naguno [9] and Rinzel-Keller [27] ones. In addition, (1.5) can serve as a potential equation for the nonlinear D’alembert equation. The determining equations for symmetries of equations (1.2) are rath- er complicated systems of PDE including two arbitrary elements, i.e., unknown functions f1 and f2. To handle them we use the approach developed in paper [29] , whose main idea is to make a priori classification of realizations of the related Lie algebras. In fact this method has roots in works of S. Lie who used his knowledge of vector field representations of Lie algebras in space of two variables to classify second order ordinary equations [17]. In the case of partial differential equations we have no hope to classify all related realizations of vector fields . However, for some fixed classes of PDEs it appears to be possible to make this classification restricting ourselves to realizations which are compatible with equations of interest [29]. We notice that analogous technique was used earlier [13] to classify the nonlinear Schrödinger equations with cubic nonlinearity and variable coefficients. In Section 2 we present the general equivalence transformations for equations (1.3) which are valid for arbitrary nonlinearities f1 and f2, and give the list of additional equivalence transformations which are valid for some fixed nonlinearities. In Section 3 the simplified algorithm for investigation of symmetries of systems of reaction-diffusion equation is presented. A. G. Nikitin 153 In Section 4 we deduce determining equations for symmetries admit- ted by equations (1.3) and specify the general form of the related group generators. In Section 5 we present the kernel of symmetry group for equations (1.3) and give definitions of main and extended symmetries. In Sections 6–8 the results of group classification of equations (1.4) and (1.5) are presented. Equations (1.3) with invertible diffusion matrix are classified in Sections 9 and 10, the case of nilpotent diffusion matrix is studied in Section 11. In Section 12 we discuss the results of group classification and present some important model equations which appear to be particular subjects of our analysis. The Appendix includes a priori classification of realizations of low dimension Lie algebras which are used in the main text to solve the determining equations. 2. Equivalence Transformations The problem of group classification of equations (1.2)–(1.5) will be solved up to equivalence transformations. We say the equation ũt − Ã∆ũ = f̃(ũ) (2.1) be equivalent to (1.3) if there exist an invertible transformation u → ũ = G(u, t, x), t → t̃ = T (t, x, u), x → x̃ = X(t, x, u) and f → f̃ = F (u, t, x, f) which connects (1.3) with (2.1). In other words the equiva- lence transformations should keep the general form of equation (1.3) but can change concrete realizations of matrix A and non-linear terms f1 and f2. Let us note that there are six ad hoc non-equivalent classes of equa- tions (1.3) corresponding to the following forms of matrices A I. A = ( 1 0 0 a ) , I∗. A = ( 1 0 0 1 ) , II. A = ( a −1 1 a ) , III. A = ( 1 0 a 1 ) , IV. A = ( 1 0 0 0 ) , V. A = ( 0 0 1 0 ) (2.2) where a is an arbitrary parameter. Indeed any 2 × 2 matrix A can be reduced to one of the forms (2.2) using linear transformations of depen- dent variables and scaling independent variables in (1.3). For matrices I and III it is possible to restrict ourselves to the cases a 6= 0, 1 and a 6= 0 154 Group Classification of Systems... respectively, but we prefer to reserve the possibility to treat version I∗ as a particular case of versions I and III. The group of equivalence transformations for equation (1.3) can be found using the classical Lie approach and treating f1 and f2 as addi- tional dependent variables. In addition to the obvious symmetry trans- formations t→ t′ = t+ a, xµ → x′µ = Rµνxν + bµ (2.3) where a, bµ and Rµν are arbitrary parameters satisfying RµνRµλ = δµλ, this group includes the following transformations ua → Kabub + ba, fa → λ2Kabf b, t→ λ−2t, xa → λ−1xa (2.4) where Kab are elements of an invertible constant matrix K commuting with A, λ 6= 0 and ba are arbitrary constants. Let us specify the form of matrices K. By definition, K commutes with A, so for the versions I–V present in (2.2) we have I∗ : K = ( K11 K12 K21 K22 ) , K11K22 −K21K12 6= 0; (2.5) I, IV : K = ( K1 0 0 K2 ) , K1K2 6= 0; (2.6) II : K = ( K1 −K2 K2 K1 ) , K2 1 +K2 2 6= 0; (2.7) III, V : K = ( K1 0 K2 K1 ) , K1 6= 0. (2.8) In addition, for the Case I there is one more transformation (2.4) with K = ( 0 1 1 0 ) , λ2 = a. (2.9) Such transformations reduce to the change a → 1 a in the related matrix A, i.e., to scaling the parameter a. Equivalence transformations (2.4) are valid also for equations (1.4) and (1.5) . The related matrices K are given in (2.6) and (2.8). It is possible to show that there is no more extended equivalence re- lations valid for arbitrary nonlinearities f1 and f2. However, if functions f1, f2 are fixed, the class of equivalence transformations is more extended. In addition to transformations (2.4) it includes symmetry transformations A. G. Nikitin 155 which does not change the form of equation (1.3). Moreover, for some classes of functions f1, f2 equation (1.3) admits additional equivalence transformations (AET). The corresponding set of equivalence transfor- mations for equation (1.3) can be found using the classical Lie approach and treating f1 and f2 as additional dependent variables constrained by the relations specifying the dependence of f1, f2 on u1 and u2. In spite of the fact that we search for AET after description of sym- metries of equations (1.3) and specification of functions f1, f2, for conve- nience we present the list of the additional equivalence transformations in the following formulae: 1. u1 → exp(ωt)u1, u2 → exp(ρt)u2, 2. u1 → u1 + ωt+ λaxa + µx2, u2 → u2, 3. u1 → u1, u2 → u2 + ρt+ λaxa + µx2, 4. u1 → u1 + ρt, u2 → u2 exp(ρt), 5. u1 → exp(ωt)u1, u2 → u2 + ωt, 6. u1 → u1, u2 → u2 + ρtu1, 7. u1 → exp(ωt)u1, u2 → u2 + ω t2 2 , 8. u1 → exp(ωt)u1, u2 → u2 + κtu1 + ρ t2 2 , 9. u1 → u1, u2 → u2 − ρtu1 + ρλ t2 2 , 10. u1 → exp(ρt)u1, u2 → u2 − κρt, 11. u1 → exp(ρt)u1, u2 → exp(ρt) ( u2 + ερ t2 2 u1 ) , 12. u1 → u1 + ρt+ νx2, u2 → u2 − ρt− νx2, 13. u1 → u1 + ρt, u2 → e− ρ ν tu2, 14. u1 → u1 + ρt, u2 → u2 + ρtu1 + ρ t2 2 , 15. u1 → u1 cosωt− u2 sinωt, u2 → u2 cosωt+ u1 sinωt, 16. u1 → exp(ωt)u1, u2 → exp(ωt)(u2 − ωtu1) 17. Transformations (11.2) valid for equations with matrix A of type V only. (2.10) Here the Greek letters denote parameters which are either arbitrary or specified in the tables presented below. We stress once more that in contrast with (2.4), equivalence transformations (2.10) are admitted by some particular equations (1.3), which will be specified in the following. 156 Group Classification of Systems... 3. An Algorithm for Description of Symmetries for the Systems (1.3)–(1.5) Let us investigate Lie symmetries of systems (1.3)–(1.5), i.e., find all continuous groups of transformations for u, t, x which keep these equa- tions invariant. In contrast with the equivalence transformations, sym- metry transformations do not change functions f1 and f2. In as much as any term in (1.3) does not depend on t and x explic- itly, this equation with arbitrary functions f1 and f2 admits obvious symmetry w.r.t. translations of all independent variables and rotations of spatial variables present in (2.3). For equations (1.4) and (1.5) such symmetries also have the form (2.3) where the indices of Rµν runs over the values 1, 2, . . . ,m− 1. To find all Lie symmetries we require form-invariance of the systems of reaction diffusion equations with respect to the one-parameter groups of transformations: t→ t′(t, x, ε), x→ x′(t, x, ε), u→ u′(t′, x′, ε), (3.1) where ε is a group parameter. In other words, we require that u′(t′, x′, ε) satisfies the same equation as u(t, x): L′u′ = f(u′), (3.2) where L are the linear differential expressions involved into equations (1.3)–(1.5), i.e., L = ∂ ∂t −A ∑ i ∂2 i ∂x2 i −B ∂ ∂xm , L′ = ∂ ∂t′ −A ∑ i ∂2 i ∂x′i 2 −B ∂ ∂x′m . Here B is the zero matrix for equation (1.3), B = ( 0 0 p 0 ) for equations (1.4) and B = ( 0 0 0 p ) for equation (1.5). Starting with the infinitesimal transformations: t→ t′ = t+ ∆t = t+ εη, xa → x′a = xa + ∆xa = xa + εξa, ua → u′a = ua + ∆ua = ua + επa (3.3) we obtain the following representation for the operator L′: L′ = [ 1 + ε ( η ∂ ∂t + ξa ∂ ∂xa )] L [ 1 − ε ( η ∂ ∂t + ξa ∂ ∂xa )] +O(ε2). (3.4) A. G. Nikitin 157 Using the Lie algorithm one can find find the determining equations for the functions η, ξa and πa which specify the generator X of the symmetry group: X = η ∂ ∂t + ξa ∂ ∂xa − πb ∂ ∂ub (3.5) where a summation from 1 to m and from 1 to n is assumed over repeated indices a and b respectively. We will obtain these determining equations directly. First we notice that without loss of generality it is possible to restrict ourselves to such functions η, ξa, πa which satisfy the conditions ∂η ∂ua = 0, ∂ξa ∂ub = 0, ∂2πa ∂uc∂ub = 0. (3.6) This is nothing but a consequence of results of paper [2] were PDE are classified whose symmetries satisfy (3.6). These results admit a straight- forward generalization to the case of systems (2.2) with invertible matrix A. Substituting (3.3), (3.4) into (3.2), using (1.3)–(1.5) and neglecting the terms of order ε2 we find that: [Q,L]u− Lω = πf + ∂f ∂ua ( −πabub − ωa ) , (3.7) where Q = η ∂ ∂t + ξa ∂ ∂xa + π [Q,L] = QL − LQ is a commutator of operators Q and L and π is a matrix whose elements are πab, so that [23] πa = πabub + ωa, with πab and ωa being functions of independent variables t, x. Equation (3.7) is compatible with (1.3)–(1.5) and does not impose new nontrivial conditions for u if the commutator [Q,L] admits the rep- resentation: [Q,L] = ΛL+ ϕ (3.8) where Λ and ϕ are 2 × 2 matrices dependent on t, x. Substituting (3.8) into (3.7) the following classifying equations for f are obtained: ( Λkb + πkb ) f b + ϕkbub + (Lω)k = (ωa + πabub) ∂fk ∂ua . (3.9) Thus, to find all non-linearities fk generating Lie symmetries for equa- tions (1.3)–(1.5) it is necessary to solve operator equations (3.8) and find the general form of matrices Λ, π, ϕ and functions η, ξ. In the second 158 Group Classification of Systems... step we find the non-linearities fa solving the system (3.9) with its known coefficients. We stress that the described procedure of group classification of equa- tions (1.3)–(1.5) is equivalent to the standard Lie algorithm but is more straightforward. In addition, it is rather convenient, and till an appro- priate moment all equations (1.3) with non-singular matrices A can be analyzed in a parallel way. 4. Determining Equations Evaluating the commutator in (3.8) and equating the coefficients for linearly independent differential operators we obtain the determin- ing equations: ( ∂ξa ∂xb + ∂ξb ∂xa ) A = −δab(ΛA+ [A, π]), ∂2η ∂t∂xa = 0, ∂η ∂t = Λ, (4.1) ∂ξa ∂t − 2 ∂ ∂xa Aπ − ∆Aξa = 0, ϕ = ∂π ∂t − ∆Aπ (4.2) where δab is the Kronecker symbol. The general expressions for coefficient functions η, ξa and π of sym- metry X (3.5) can be obtained evaluating determining equations (4.1) and (4.2). We shall not reproduce this procedure here but present the general form of the related generator (3.5) found in [23]: X = λK + σαGα + ωαĜα + µD − 2(Cabub +Ba) ∂ ∂ua + Ψµνxµ∂ν + ν∂t + ρµ∂µ (4.3) where the Greek letters denote arbitrary constants moreover Ψµν = −Ψνµ, Ba are functions of t, x, and Cab are functions of t satisfying CabAbk −AabCbk = 0 (4.4) and K = 2t ( t ∂ ∂t + xµ ∂ ∂xµ ) − x2 2 (A−1)abub ∂ ∂ua − tm ( u1 ∂ ∂u1 + u2 ∂ ∂u2 ) , Gα = t∂α + 1 2 xα(A−1)abub ∂ ∂ua , Ĝα = eγt ( ∂α + 1 2 γxα(A−1)abub ∂ ∂ua ) , D = 2t ∂ ∂t + xµ ∂ ∂xµ . (4.5) A. G. Nikitin 159 If a = 0 then the related generator X again has the form (4.3) where however λ = σµ = ωµ = C2 = 0 and B2 is a function of t, x and u. Formula (4.3) presents a symmetry operator for equation (1.2) iff the related classifying equations (3.9) for f1 and f2 are satisfied, i.e., (λt(m+ 4) + µ)fa + (1 2 λx2 + σµxµ + γeγtωµxµ ) (A−1)abf b + Cabf b + Cabt ub +Ba t − ∆AabBb = ( Bs + Csbub + λtmus + (1 2 λx2 + σµxµ + γeγtωµxµ ) (A−1)skuk ) ∂ ∂us fa. (4.6) Thus the group classification of equations (1.3) with a non-singular matrix A reduces to solving equation (4.6) where λ, µ, σν , ων , γ are arbi- trary parameters, Ba and Cab are functions of (t, x) and t respectively. Moreover, matrix C with elements Cab should commute with A. We notice that relations (4.3)–(4.6) are valid for group classification of systems (1.3) of coupled reaction-diffusion equations including arbitrary number n of dependent variables u = (u1, u2, . . . un) provided the related n × n matrix A be invertible [23]. In this case indices a, b, s, k in (4.3)– (4.6) run over the values 1, 2 . . . n. Consider now equation (1.4) and the related symmetry operator (3.5). The determining equations for η, ξµ and πa are easily obtained using (3.8), (3.9) and have the following form ηtt = ηxµ = ∂η ∂ua = 0, ξµt = ∂ξµ ∂ua = 0, ∂2πa ∂ub∂uc = 0, ∂πaxµ ∂ub = 0, ∂π1 ∂u2 = ∂π2 ∂u1 = 0; ∂π1 ∂u1 − ∂π2 ∂u2 = 1 2 ηt, if p 6= 0; ξµxν + ξνxµ = −δµνηt, µ 6= m (4.7) where subscripts denote derivatives w.r.t. the corresponding independent variable, i.e., ηt = ∂η ∂t , ξ µ xν = ∂ξµ ∂xν , etc. Integrating system (4.7) we obtain the general form of operator X: X = ν∂t+ρν∂ν+Ψµν∂νxµ+µD−2Ba ∂ ∂ua −2Fu1 ∂ ∂u1 −2Gu2 ∂ ∂u2 ; (4.8) µ = 2(F −G) if p 6= 0 (4.9) 160 Group Classification of Systems... where Ba are functions of (t, x), F and G are functions of t and summa- tion over the indices µ, ν is assumed with µ, ν = 1, 2, · · · , n− 1. The classifying equations (3.9) reduce to the following ones (µ+ F )f1 + Ftu1 + (∂t − ∆)B1 = ( B1 ∂ ∂u1 +B2 ∂ ∂u2 + Fu1 ∂ ∂u1 +Gu2 ∂ ∂u2 ) f1, (µ+G)f2 +Gtu2 +B2 t − pB2 xm = ( B1 ∂ ∂u1 +B2 ∂ ∂u2 + Fu1 ∂ ∂u1 +Gu2 ∂ ∂u2 ) f2. (4.10) Relations (4.8)–(4.10) are valid for p 6= 0 and p = 0 as well (in the last case condition (4.9) should be omitted). Solving (4.10) we specify both the coefficients of infinitesimal operator (4.8) and the related non- linearities f1 and f2. For equations (1.5) we obtain in analogous way that generator (3.5) reduces to X = µ ( 3t∂t + 2xν∂ν − u2 ∂ ∂u2 ) − F ( u1 ∂ ∂u1 + u2 ∂ ∂u2 ) −Ba ∂ ∂ua (4.11) while the classifying equations are (3µ+ F )f1 + Ftu1 +B1 t − pB2 xm = ( B1 ∂ ∂u1 +B2 ∂ ∂u2 + Fu1 ∂ ∂u1 + (F + µ)u2 ∂ ∂u2 ) f1, (4µ+ F )f2 + Ftu2 +B2 t − ∆B1 = ( B1 ∂ ∂u1 +B2 ∂ ∂u2 + Fu1 ∂ ∂u1 + (F + µ)u2 ∂ ∂u2 ) f2 (4.12) where F and B1, B2 are unknown functions of t and t, x respectively. The determining equations for symmetries of equation (1.5) with p = 0 are qualitatively different for the cases, when the number m of spatial variables x1, x2, . . . xm ism = 1, m = 2 andm > 2. The related generator (3.5) has the form X = αD + (∫ (N −M)dt ) ∂ ∂t + 2mHa ∂ ∂xa − ( N + (m− 2) ∂Ha ∂xa ) u1 ∂ ∂u1 − ( M + (m+ 2) ∂Ha ∂xa ) u2 ∂ ∂u2 −B1 ∂ ∂u1 −B2 ∂ ∂u2 −B3u1 ∂ ∂u2 (4.13) A. G. Nikitin 161 where summation from 1 to m is imposed over repeating indices, the Greek letters denote arbitrary parameters,M,N are functions of t, B1, B2 are functions of t, x, B3 is a function of t, x, u1 and Ha = 2λbxbxa−x2λa for m > 2 . For m = 2 Ha are arbitrary functions satisfying the Caushy- Rieman conditions ∂H1 ∂x1 = ∂H2 ∂x2 , ∂H1 ∂x2 = −∂H2 ∂x1 ; for m = 1 H1 is a function of x and the sums with respect to a in (4.13) are degenerated to one terms. The corresponding classifying equations have the form (α 2 + 2N −M + (m− 2) ∂Ha ∂xa ) f1 +Ntu1 +B1 t = ( B1 ∂ ∂u1 +B2 ∂ ∂u2 +B3u1 ∂ ∂u2 + ( N + (m− 2) ∂Ha ∂xa ) u1 ∂ ∂u1 + ( M + (m+ 2) ∂Ha ∂xa ) u2 ∂ ∂u2 ) f1, (4.14) (α 2 +N + (m+ 2) ∂Ha ∂xa ) f2 +B3f1 +Mtu2 +B3 t u1 +B2 t − ∆B1 + (2 −m) ( ∆ ∂Ha ∂xa ) u1 = ( B1 ∂ ∂u1 +B2 ∂ ∂u2 +B3u1 ∂ ∂u2 + ( N + (m− 2) ∂Ha ∂xa ) u1 ∂ ∂u1 + ( M + (m+ 2) ∂Ha ∂xa ) u2 ∂ ∂u2 ) f2. We notice that in this case symmetry classification appears to be rather complicated and cumbersome. Nevertheless, the classifying equa- tions can be effectively solved using the approach outlined in the following sections. Thus the group classification of equations (1.3), (1.4) and (1.5) re- duces to searching for general solutions of equations (4.6), (4.10), (4.12) and (4.14). To solve these equation it is necessary to make an effective separation of independent variables. To do this we will use an approach which includes a priori specification and simplification of possible forms of generators X (4.3), (4.8), (4.11) and (4.13) using the condition that X belong to n-dimensional Lie algebra with n = 1, 2, . . .. This specification will be based on classification of algebras of 3×3 matrices of special form. 5. Basic, Main and Extended Symmetries Let us start with equation (1.3). The general form for the related symmetries and the classifying equation for nonlinearities f1, f2 are given by relation (4.3) and (4.6) respectively. 162 Group Classification of Systems... Equation (4.6) does not include parameters Ψµν , ν and ρν present in (4.3) thus for any f1 and f2 equation (1.3) admits symmetries generated by the following operators P0 = ∂t, Pλ = ∂λ, Jµν = xµ∂ν − xν∂µ. (5.1) For some classes of nonlinearities f1 and f2 the invariance algebra of equation (1.3) is more extended but includes (5.1) as a subalgebra. We will refer to (5.1) as to basic symmetries. Operators (5.1) generate the maximal local Lie group which is admit- ted by equations (1.3) for any functions f1 and f2. In other words the basic symmetries generate the kernel of the invariance group of equation (1.3). Let us specify main symmetries for equation (1.3), whose generator X̃ has the form (4.3) with Ψµν = ν = ρν = σν = ων = 0, i.e., X̃ = µD + Cabub ∂ ∂ua +Ba ∂ ∂ua . (5.2) The classifying equation for symmetries (5.2) can be obtained from (4.6) by setting µ = σa = ωa = 0. As a result we get (µδab + Cab)f b + Cabt ub +Ba t − ∆AabBb = (Cnbub +Bn) ∂fa ∂un . (5.3) Operator (5.2) is a particular case of (4.3). Moreover, it is easily veri- fied that operators (5.2) and (5.1) form a Lie algebra which is a subalgebra of symmetries for equation (1.3). On the other hand, if equation (1.3) ad- mits a more general symmetry (4.3) with σa 6= 0 or (and) λ 6= 0, ωµ 6= 0 then it has to admit symmetry (5.2) also. To prove this we will calculate multiple commutators of (4.3) with the basic symmetries (5.1) and use the fact that such commutators have to belong to symmetries of equation (1.3). Let equation (1.3) admits extended symmetry (4.3) with σν 6= 0, Ψµν = ρµ = ν = λ = ωk = 0, i.e., X = σαGα + µD + (Cabub +Ba) ∂ ∂ub . (5.4) Commuting Y with Pα we obtain one more symmetry Yα = −σα 2 (A−1)abub ∂ ∂ua +Ba xα ∂ ∂ua + µPxα . (5.5) The latest term belongs to the basic symmetry algebra (5.1) and so can be omitted. The remaining terms are of the type (5.2). A. G. Nikitin 163 Thus supposing the extended symmetry (5.4) is admissible we con- clude that equation (1.3) has to admit the main symmetry also. Commuting (5.5) with P0 and Pλ we come to the following symme- tries: Yµν = Ba xµxν ∂ ∂ua , Yµt = Ba xµt ∂ ∂ua . (5.6) Any symmetry (5.4)–(5.6) generates this own system (4.6) of classify- ing equations. After straightforward but rather cumbersome calculations we conclude that all these systems are compatible provided the following condition is satisfied (A−1)abf b = (A−1)nbub ∂fa ∂un . (5.7) If (5.7) is satisfied equation (1.3) admits symmetry (5.4) with µ = Cab = Ba = 0, i.e., Galilei generators Gα of (4.5). In analogous way, supposing that equation (1.3) admits extended sym- metry (4.3) with λ 6= 0 and ωa = 0 we prove that it has to admit also symmetry (5.4) with µ 6= 0 and σν 6= 0. The related functions f1 and f2 should satisfy relations (5.7) and (5.3). Moreover, analyzing possible de- pendence of Cab and Ba in the corresponding relations (4.6) on t we con- clude that they should be ether scalars or linear in t, i.e., Cab = µabt+νab. Moreover, up to equivalence transformations (2.4) we can choose Ba = 0, and reduce (5.3) to the following system: (m+ 4)fa + µabf b = (µkbub +muk) ∂fa ∂uk , νabf b + µabub = νkbub ∂fa ∂uk (5.8) where the parameters νab and µab are distinct from zero in the case of the diagonal matrix A only. Finally for general symmetry (4.3) it is not difficult to show that the condition ων 6= 0 leads to the following equation for fa (A−1)kb(f b + γub) = (A−1)abub ∂fk ∂ua . (5.9) We notice that relations (5.7) and (5.9) are particular cases of (5.3) for µ = 0, Cab = (A−1)ab and µ = 0, Cab = eγt(A−1)ab respectively. Thus if relation (5.7) is valid then, in addition to Gα (4.5) equation (1.3) admits the symmetry X = (A−1)abub ∂ ∂ua . (5.10) 164 Group Classification of Systems... Alternatively, if (5.9) is satisfied, equation (1.2) admits symmetry Ĝα (2.6) and also the following one X = eγt(A−1)abub ∂ ∂ua , γ 6= 0. (5.11) Thus it is reasonable first to classify equations (1.3) which admit main symmetries (5.2) and then specify all cases when these symmetries can be extended. The conditions when system (1.3) admits extended symmetries are given by relations (5.7)–(5.9). Concerning equations (1.4) and (1.5) we notice that in accordance with (4.8) and (4.11) they admit basic symmetries only. Now we are ready to search for solutions to classifying equations (4.10), (4.12) and (5.3). To present clearly main details of our approach we start with group classification of systems (1.4), because this problem appears to be essentially more simple than other ones considered here. 6. Symmetry Algebras of Equations (1.4) Consider equations (1.4) and suppose that parameter p is nonzero. Then scaling dependent and independent variables we can reduce its value to p = 1. To solve rather complicated classifying equations (4.10), (4.12) and (5.3) we use the main algebraic property of the related symmetries, i.e., the fact that they should form a Lie algebra. In other words, instead of going throw all non-equivalent possibilities arising via separation of variables in the classifying equations we first specify all non-equivalent realizations of the invariance algebra for our equations whose elements are defined by relations (5.2), (4.8) and (4.11) up to arbitrary constants and arbitrary functions. Then using the one-to-one correspondence between these algebras and classifying equations (4.10), (4.12), (5.3) we easily solve the group classification problems for equations (1.3)–(1.5). Let us start with classifying equations (4.10) and the related symme- tries (4.8). For any functions f1 and f2 equations (1.4) admit symmetries (5.1) where the indices µ, ν and λ run over the values 1, 2, . . .m− 1 and 1, 2, . . .m respectively. In accordance with (4.8) any symmetry generator extending algebra (5.1) has the following form X = µD − 2Ba ∂ ∂ua − 2Fu1 ∂ ∂u1 + (µ− 2F )u2 ∂ ∂u2 . (6.1) A. G. Nikitin 165 Let X1 and X2 be operators of the form (6.1) then the commutator [X1, X2] is also a symmetry whose general form is given by (6.1). Thus operators (6.1) form a Lie algebra which we denote as A. Let us specify algebras A which can appear in our classification pro- cedure. First consider one-dimensional A , i.e., suppose that equation (1.4) admits the only symmetry of the form (6.1). Then any commutator of operator (5.1) with (6.1) should be reducible to a linear combination of operators (5.1) and (6.1). This condition presents us the following possibilities only: X = X1 = µD − 2αa ∂ ∂ua − 2βu1 ∂ ∂u1 − (2β − µ)u2 ∂ ∂u2 , X = X2 = eνt ( αa ∂ ∂ua + βu1 ∂ ∂u1 + βu2 ∂ ∂u2 ) , X = X3 = eνt+ρ·xαa ∂ ∂ua (6.2) where the Greek letters again denote arbitrary parameters and ρ · x = ρµxµ. All the other choices of arbitrary functions F and Ba in (6.1) corre- spond to algebras A whose dimension is larger than one. The next step is to specify all non-equivalent sets of arbitrary con- stants in (6.2) using the equivalence transformations (2.4). If the coefficient for ua ∂ ∂ua (a is fixed) is non-zero then translating ua we reduce to zero the related coefficient αa in X1 and X2; then scaling ua we can reduce to ±1 all non-zero αa in (6.2). In addition, all op- erators (6.2) are defined up to constant multipliers. Using these simple arguments we come to the following non-equivalent versions of operators (6.2) belonging to one-dimensional algebras A: X (1) 1 = µD − u1 ∂ ∂u1 + (µ− 1)u2 ∂ ∂u2 , X (2) 1 = D + u2 ∂ ∂u2 + ν ∂ ∂u1 , X (3) 1 = D − u1 ∂ ∂u1 − ∂ ∂u2 , X (ν) 2 = eνt+ρ2·x ( u1 ∂ ∂u1 + u2 ∂ ∂u2 ) ; X (1) 3 = eσ1t+ρ1·x ( ∂ ∂u1 + ∂ ∂u2 ) , X (2) 3 = eσ2t+ρ2·x ∂ ∂u1 , X (3) 3 = eσ3t+ρ3·x ∂ ∂u2 . (6.3) 166 Group Classification of Systems... To describe two-dimensional algebras A we represent one of the re- lated basis element X in the general form (6.1) and calculate the com- mutators Y = [P0, X] − 2µP0, Z = [P0, Y ], W = [X,Y ] where P0 is operator given in (5.1), Y,Z and W are symbols denoting the terms in the r.h.s.. After simple calculations we obtain Y = Ft ( u1 ∂ ∂u1 + u2 ∂ ∂u2 ) +Ba t ∂ ∂ua , Z = Ftt ( u1 ∂ ∂u1 + u2 ∂ ∂u2 ) +Ba tt ∂ ∂ua , W = 2µtZ + µxbB a txb ∂ ∂ua . (6.4) By definition, Y , Z and W belong to A. Let Ft 6= 0 than it follows from (6.4) that µ 6= 0 : Ba tt = Ftt = Ba tb = 0, (6.5) µ = 0 : Ftt = αFt + γaBa t , Ba tt = γaFt + βabBb t , (6.6) otherwise the dimension of A is larger than 2. The Greece letters in (6.5) and (6.6) denote arbitrary parameters. Starting with (6.5) we conclude that up to translations of t the coef- ficients F and Ba have the following form F = σt or F = β; Ba = νat+ αa if µ 6= 0. If F = σt then the change ua → uae −σt − νa µ t (6.7) reduces the related operator (4.8) to the following form: X = µ ( D + u2 ∂ ∂u2 ) − 2αa ∂ ∂ua , (6.8) i.e., X coincides with X1 of (6.2) for β = 0. Moreover it is possible to show that (6.7) gives the equivalence transformation for the related equations (1.4) (i.e., for equations (1.4) which admit symmetry (6.8)). A. G. Nikitin 167 The choice F = β corresponds to the following operator (6.1) X = X4 = X1 − 2tαa ∂ ∂ua (6.9) where X1 is given in (6.2). Thus if one of basis elements of two dimension algebra A is of general form (6.1) with µ 6= 0 then it can be reduced to (6.8) or (6.9). We denote such basis element as e1. Without loss of generality the second basis element e2 of A is a linear combination of operators X(ν) 2 and X(a) 3 (6.3). Going over possible pairs (e1, e2) and requiring [e1, e2] = α1e1 + α2e2 we come to the following two dimensional algebras A1 = 〈 D + u2 ∂ ∂u2 , X (0) 2 〉 , A2 = 〈X(2) 1 , X (3) 3 〉, A3 = 〈X(3) 1 , X (3) 3 〉, A4 = 〈X(1) 1 , X (3) 3 〉, A5 = 〈X(1) 1 , X (3) 3 〉, A6 = 〈 D + 2u2 ∂ ∂u2 + u1 ∂ ∂u1 + νt ∂ ∂u2 , X (2) 3 〉 A7 = 〈 D + 2u1 ∂ ∂u1 + 3u2 ∂ ∂u2 + 3νt ∂ ∂u1 , X (1) 3 〉 . (6.10) The form of basis elements in (6.10) is defined up to transformations (6.7) (2.4). If A does not include operators (6.1) with non-trivial parameters µ then in accordance with (6.7) its elements are of the following form ea = F(a) ( u1 ∂ ∂u1 + u2 ∂ ∂u2 ) +Bb (a) ∂ ∂ub , a = 1, 2 (6.11) where F(α) and Bb (a) are solutions of (6.6). Formulae (6.10), (6.11) define all non-equivalent two-dimensional al- gebras A which have to be considered as possible symmetries of equations (1.4). We will see that asking for invariance of (1.4) w.r.t. these algebras the related arbitrary functions fa are defined up to arbitrary constants, and it is impossible to make further specification of these functions by extending algebra A. 7. Group Classification of Equations (1.4) To classify equations (1.4) which admit one- and two- dimension ex- tensions of the basis invariance algebra (5.1) it is sufficient to solve clas- sifying equations (4.10) for fa with known coefficient functions Ba and F of symmetries (6.1). These functions are easily found comparing (4.8) with (6.3), (6.10) and (6.11). 168 Group Classification of Systems... Let us present an example of such calculation which corresponds to algebra A1 whose basis elements are X1 = 2t∂t + xa∂a + u2 ∂ ∂u2 and X (0) 2 = u1 ∂ ∂u1 + u2 ∂ ∂u2 , refer to (6.10). Operator X(0) 2 generates the following form of equation (4.10): fa = ( u1 ∂ ∂u1 + u2 ∂ ∂u2 ) fa, a = 1, 2 whose general solution is f1 = u1F1 ( u2 u1 ) , f2 = u1F2 ( u2 u1 ) . (7.1) Here F1 and F2 are arbitrary functions of u2 u1 . Equations (1.4) with non-linearities (7.1) admit symmetry X (0) 2 . In order this equation be invariant w.r.t. X1 also, functions f1, f2 have to satisfy equation (4.10) with F = 0, i.e., f1 = −u1 ∂f1 ∂u1 ; f2 = −1 2 u1 ∂f2 ∂u1 . (7.2) It follows from (7.1), (7.2) that f1 = αu3 1u −2 2 , f2 = λu2 1u −1 2 . (7.3) Thus equation (1.4) admits symmetries X(2) 0 and X1 which form al- gebra A1 (6.10) provided f1 and f2 are functions given in (7.3). These symmetries are defined up to arbitrary constants α and λ. If one of them is nonzero, than it can be reduced to +1 or −1 by scaling independent variables. In analogous way we solve equations (4.10) corresponding to other symmetries indicated in (6.3) and (6.10). For one-dimension algebras (6.3) the related non-linearities f1 and f2 are defined up to arbitrary functions F1 and F2 while for two dimension algebras (6.10) functions f1 and f2 are defined up to two integration constants. We shall not reproduce the related rather routine calculations but present their results in Table 1. A. G. Nikitin 169 Table 1. Non-linearities and symmetries for equation (1.4) with p=1 No Non-linearities Arguments Symmetries of F1 F2 1. f1 = u2µ+1 1 F1, f2 = uµ+1 1 F2 u2u µ−1 1 µD − u1 ∂ ∂u1 + (µ− 1)u2 ∂ ∂u2 2. f1 = F1u −2 2 , f2 = F2u −1 2 u1 − ν lnu2 D + u2 ∂ ∂u2 + ν ∂ ∂u1 3. f1 = u1(F1 + λ lnu1), f2 = u2(F2 + λ lnu1) u2 u1 eλt ( u1 ∂ ∂u1 + u2 ∂ ∂u2 ) 4. f1 = u3 1F1, f2 = u2 1F2 u2 − lnu1 D − u1 ∂ ∂u1 − ∂ ∂u2 5. f1 = F1, f2 = F2 + νu2 u2 eνtΨ(x) ∂ ∂u2 6. f1 = αu1 + F1, f2 = λu+ F2 u1 eλt+νxmΨ̃µ(x̃) ∂ ∂u1 , µ = λ− ν2 − α 7. f1 = σu+ F1, f2 = λv + F2 u− v eλt e xm+t 2 Ψµ(x̃, xm + t) ( ∂ ∂u1 + ∂ ∂u2 ) , µ = λ− σ + 1 4 8. f1 = αu3 1u −2 2 , f2 = βu2 1u −1 2 D + u2 ∂ ∂u2 , u1 ∂ ∂u1 + u2 ∂ ∂u2 9. f1 = αe−2u1 , f2 = λe−u1 D + u2 ∂ ∂u2 + ∂ ∂u1 , Ψ(x) ∂ ∂u2 10. f1 = λe3u2 , f2 = αe2u2 D − u1 ∂ ∂u1 − ∂ ∂u2 , Φ̃0(t, x̃) ∂ ∂u1 11. f1 = αu2µ+1 1 , f2 = λuµ+1 1 µD − u1 ∂ ∂u1 + (µ− 1)u2 ∂ ∂u2 , Ψ(x) ∂ ∂u2 12. f1 = λu3µ−2 2 , f2 = αu2µ−1 2 (µ− 1)D − µu1 ∂ ∂u1 − u2 ∂ ∂u2 , Φ̃0(t, x̃) ∂ ∂u1 13. f1 = α u1 , f2 = lnu1 D + 2u2 ∂ ∂u2 + u1 ∂ ∂u1 + t ∂ ∂u2 , Ψ(x) ∂ ∂u2 14. f1 = lnu2, f 2 = αu 1 3 2 D + 2u1 ∂ ∂u1 + 3u2 ∂ ∂u2 + 3t ∂ ∂u1 , Φ̃0(t, x̃) ∂ ∂u1 170 Group Classification of Systems... HereD is the dilatation operator given in (4.5), x̃=(x1, x2, . . . , xm−1), Ψ(x) is an arbitrary function of spatial variables; Ψ̃µ(x̃),Ψµ(x̃, xm + t) and Φ̃µ(t, x̃) are solutions of the Laplace and linear heat equations ∆̃Ψ̃µ = µΨ̃µ, ∆Ψµ = µΨµ, ( ∂ ∂t − ∆̃)Φ̃0 = 0, ∆̃ = ∂2 ∂x2 1 + ∂2 ∂x2 2 + · · · + ∂2 ∂x2 m−1 , ∆ = ∆̃ + ∂2 ∂x2 m . (7.4) We notice that equations (1.4) with non-linearities 5, 6, 9-14 of Table 1 admit infinite-dimension algebras A because the related symmetries are defined up to arbitrary functions Ψ(x) or arbitrary solutions of equations (7.4). Nevertheless, the form of these non-linearities was fixed requiring invariance w.r.t. one- and two-dimension algebras enumerated in (6.3), (6.10). The second note is that equations (1.4) with non-linearities given in Item 8 of Table 1 admit additional equivalence transformations uα → eσtuα while for Items 9, 11, 13 and 10, 12, 14 we have in our disposal transformations 3 and 2 respectively from the list (2.10). 8. Group Classification of Equations (1.5) Like (1.4), equations (1.5) with arbitrary functions f1 and f2 ad- mit the basic symmetries (5.1) were µ, ν = 1, 2, . . . ,m − 1. To classify equations admitting other symmetries it is sufficient to find the general solution for equations (4.12). We will solve (4.12) using the technique applied in Sections 5 and 6. Comparing (4.11) and (6.1) we conclude that generators of extended symmetry for equations (1.4) and (1.5) are rather similar and so we can essentially exploit the algebra classification scheme used in Section 5. As a result we easily come to the following list of one-dimension algebras A (compare with (6.3)) X̃ (1) 1 = µD̃ − u1 ∂ ∂u1 − u2 ∂ ∂u2 , X̃ (2) 1 = D̃ − ν ∂ ∂u1 , X̃ (ν) 2 = eνt ( u1 ∂ ∂u1 + u2 ∂ ∂u2 ) , X̃ (3) 1 = D̃ + u1 ∂ ∂u1 + u2 ∂ ∂u2 + ν ∂ ∂u2 , X̃ (3) 3 = eσ3t+ρ3x ( ∂ ∂u1 + ∂ ∂u2 ) , X̃ (j) 3 = eσit+ρi·x ∂ ∂uj , j = 1, 2 (8.1) A. G. Nikitin 171 where D̃ = 3t∂t + 2xν∂ν − u2 ∂ ∂u2 . The two-dimension algebras are given by the following relations (compare with (6.10)): Ã1 = 〈D̃, X̃(0) 2 〉, Ã2 = 〈X̃(2) 1 , X (3) 3 〉, Ã3 = 〈X̃(3) 1 , X̃ (1) 3 〉, Ã4 = 〈X̃(1) 1 , X̃ (2) 3 〉, Ã5 = 〈X̃(1) 1 , X̃ (1) 3 〉, Ã6 = 〈 D̃ + 4 ( u2 ∂ ∂u2 + u1 ∂ ∂u1 + t ∂ ∂u2 ) , X (2) 3 〉 , Ã7 = 〈 D̃ + 3 ( u1 ∂ ∂u1 + u2 ∂ ∂u2 + t ∂ ∂u1 ) , X (1) 3 〉 . (8.2) Using (8.1), (8.2) and solving the related classifying equations (4.12) we find non-linearities f1, f2 which are given in Table 2. In six cases enumerated in the table the corresponding equations (1.5) admit infinite dimension symmetry algebras whose generators are defined up to arbi- trary functions, see Items 5–7, 9–14 here. Table 2. Non-linearities and symmetries for equation (1.5) with p = 1 No Non-linearities Arguments Symmetries of F1 F2 1. f1 = u1+3µ 1 F1, f2 = u1+4µ 1 F2 u2u −µ−1 1 µD̃ − u1 ∂ ∂u1 − u2 ∂ ∂u2 2. f1 = u3 2F1, f2 = u4 2F2 u1 − ν lnu2 D̃ − ν ∂ ∂u1 3. f1 =u1(F1+ν lnu1), f2 =u2(F2+ν lnu1) u2 u1 eνt ( u1 ∂ ∂u1 + u2 ∂ ∂u2 ) 4. f1 = u−2 1 F1, f2 = u−3 1 F2 u2 − ν lnu1 D̃ + u1 ∂ ∂u1 + u2 ∂ ∂u2 + ν ∂ ∂u2 5. f1 = λu1 + F1, f2 = −µu1 + F2 u2 eλtΨµ(x) ∂ ∂u1 6. f1 = νu2 + F1, f2 = λu2 + F2 u1 eλt−νxmΨ(x̃) ∂ ∂u2 7. f1 = αu1 + F1, f2 = σu2 + F2 u1 − u2 eλt e xm−t 2 Ψµ(x̃, xm + t) ( ∂ ∂u1 + ∂ ∂u2 ) , µ = λ− σ + 1 4 172 Group Classification of Systems... 8. f1 = αu−2 1 u3 2, f2 = νu−3 1 u4 2 D̃, u1 ∂ ∂u1 + u2 ∂ ∂u2 9. f1 = αe3u1 , f2 = νe4u1 D̃ − ∂ ∂u1 , Ψ(x̃) ∂ ∂u2 10. f1 = αe−2u2 , f2 = νe−3u2 D̃ + u1 ∂ ∂u1 + u2 ∂ ∂u2 + ∂ ∂u2 , Ψ0(x) ∂ ∂u1 11. f1 = αu3µ+1 1 , f2 = νu4µ+1 1 µD̃ − u1 ∂ ∂u1 − u2 ∂ ∂u2 , Ψ(x̃) ∂ ∂u2 12. f1 = αu2ν+1 2 , f2 = νu3ν+1 2 νD̃ − u1 ∂ ∂u1 − u2 ∂ ∂u2 , Ψ0(x) ∂ ∂u1 13. f1 = αu 1 4 1 , f2 = ν lnu1 D̃ + 3u2 ∂ ∂u2 + 4u1 ∂ ∂u1 + 4νt ∂ ∂u2 , Ψ(x̃) ∂ ∂u2 14. f1 = ν lnu2, f2 = α√ u2 D̃ + 3u1 ∂ ∂u1 + 2u2 ∂ ∂u2 + 2νt ∂ ∂u1 , Ψ0(x) ∂ ∂u1 Here Ψµ(x) and Ψµ(x̃, xm + t) are arbitrary solutions of the Laplace equation ∆Ψµ = µΨµ, µ, ν and λ are arbitrary parameters satisfying νλ 6= 0. Equations (1.5) with the non-linearities given in Item 8 of Table 2 admit additional equivalence transformation uα → eσtuα. Besides, for Items 9,11,13 and 10,12, 14 we have transformations 3 and 2 from the list (2.10) respectively. 9. Group Classification of Equations (1.3) with Invertible Diffusion Matrices In this Section we present the group classification of systems of cou- pled reaction-diffusion equations (1.3) with invertible matrix A. In ac- cordance with the plane outlined in Section 4 we first describe the main symmetries generated by operators (5.2) and then indicate extensions of these symmetries. Like in Sections 5, 7 the first step of our analysis consists in description of realizations of Lie algebras A generating basic symmetries of equation (1.3). However, the basis elements of A are now of the general form (5.2) while in Sections 5 and 7 we were restricted to the representations (6.1) and (4.11) respectively which are particular cases of (5.1). A. G. Nikitin 173 Thus the first step of our analysis is to describe non-equivalent re- alizations of finite dimension algebras A whose basis elements have the form (5.2). Let us specify all non-equivalent “tails” of operators (5.2), i.e., the terms π = Cabub ∂ ∂ua +Ba ∂ ∂ua . (9.1) These terms can either be a constituent part of a more general sym- metry (5.2) or represent a particular case of (5.2) corresponding to µ = 0. If equation (1.3) admits a one-dimensional invariance algebra A then commutators of π with the basic symmetries P0 and Pa should be equal to a linear combination of π and operators (5.1). In other words, there are three possibilities: 1. Cab = µab, Ba = µa, (9.2) 2. Cab = eλtµab, Ba = eλtµa, (9.3) 3. Cab = 0, Ba = eλt+ω·xµa (9.4) where µab, µa, λ, and ω are constants. In any case the problem of classification of one-dimension algebras A includes the subproblem of classification of non-equivalent linear com- binations (9.1) with constant coefficients µab and µa. To describe such linear combinations we will use the isomorphism of (9.1) with 3 × 3 ma- trices of the following form g =   0 0 0 B1 C11 C12 B2 C21 C12   ∼   0 0 0 µ1 µ11 µ12 µ2 µ21 µ12   . (9.5) Equations (1.3) admit equivalence transformations (2.4) which change the term π (9.1) and can be used to simplify it. The corresponding transformation for matrix (9.5) can be represented as g → g′ = UgU−1 (9.6) where U is a 3 × 3 matrix of the following special form U =   1 0 0 b1 K11 K12 b2 K21 K22   . (9.7) 174 Group Classification of Systems... We will use relations (9.2)–(9.4) and equivalence transformations (9.6) to construct basis elements of basic symmetry algebras. For different forms of matrix A specified in (2.2) the transformation matrix (9.7) needs further specification in accordance with (2.5)–(2.8). The obtained non-equivalent realizations of low dimension algebras A are present in Appendix. Starting with these realizations one easily solves the related determining equations (4.6) for non-linearities f1 and f2 and specify all cases when the main symmetries can be extended (i.e., when relations (5.7)–(5.9) are satisfied). In addition we have to control all cases when basis elements of A depend on arbitrary solutions Ψ of the linear heat equation. Such algebras (whose basis elements can be obtained from (A.1.10), (A.1.11), (A.1.15)–(A.1.18) changing g5 and g3 by Ψg5 and Ψg3) are infinite dimensional but generate the same number of determining equations as the low-dimension algebras. 10. Classification Results We will not reproduce the related exact calculations but present the results of group classification in Tables 3-9. In addition to equations with invertible diffusion matrix we present here the results of classification which are related to the diffusion matrix of type IV while the type V is will be considered separately (see (2.2) for classification of diffusion matrices). The Tables 3–9 present the classification results for different types of equations (1.3) corresponding to non-equivalent diffusion matrices enu- merated in (2.2). The type of diffusion matrix is indicated in the fourth columns of Tables 3, 4 and third columns of Tables 5 and 6. In Tables 7-9 the results of symmetry classification of special equations are presented; these equations are indicated in the table titles. In the last columns of Tables 3, 5 and 6 the additional equivalence transformations (AET) are specified, which are possible for the related class of non-linearities. Fi- nally, the symbols D, Gα, Ĝα denote generators (4.5), ψµ denotes an arbitrary solution of the linear heat equation ∂ ∂tψµ − ∆ψµ = µψµ, ψ̃ν = { ψν for Class III eνtΨ(x) for Class IV and Ψ(x), Ψν(x) have the same meaning as in Tables 1,2. The results of group classification are briefly discussed in Section 12. A. G. Nikitin 175 Table 3. Non-linearities with arbitrary functions and extendible symmetries for equations (1.3), (2.2) No Nonlinear Argu- Type Main Addi- AET terms ments of sym- tional (2.10) of F1 F2 matrix metries sym- A metries 1. f1 = uν+1 1 F1, f2 = uν+µ1 F2 u2 uµ1 I, IV, µ 6= 1; I − IV, µ = 1 ν 2 D −u1 ∂ ∂u1 −µu2 ∂ ∂u2 For I : Gα, if ν = 0, aµ = 1 1, ρ = µω if ν = 0 2. f1 = u1(F1 +ε lnu1), f2 = u2(F2 +εµ lnu1) u2 uµ1 I, IV, µ 6= 1; I − IV, µ = 1 eεt ( u1 ∂ ∂u1 +µu2 ∂ ∂u2 ) For I : Ĝα, if aµ = 1 3. f1 = u1F1 + νu2, f2 = ν u2 u1 (u1 +u2) + u1F2 +u2F1, ν 6= 0 u1e − u2 u1 I∗, III eνt ( u1 ∂ ∂u2 +u1 ∂ ∂u1 +u2 ∂ ∂u2 ) For III : Ĝα, if a = −1 4. f1 = uν+1 1 F1, f2 = uν1 (F1u2 +F2u1) u1e − u2 u1 I∗, III ν 2 D −u1 ∂ ∂u2 −u1 ∂ ∂u1 −u2 ∂ ∂u2 For III : Gα, if ν = 0, a = −1 5. f1 = e ν u2 u1 F1u1, f2 = e ν u2 u1 (F1u2 +F2) u1 I∗, III ν 2 D −u1 ∂ ∂u2 For I∗ : u2 ∂ ∂u2 , if ν = 0, F2 = 0 6 if ν = 0 For I∗ : ψ0 ∂ ∂u2 , D +2u2 ∂ ∂u2 , if F1 = 0, ν = 0 3,6 6. f1 = u1(F1 − ν), f2 = F1u2 + F2, ν 6= 0 u1 I∗, III eνtu1 ∂ ∂u2 ψµ ∂ ∂u2 , if F1 = µ 3 if F1 =0 7. f1 = u1F1 + u2F2 −νz(µu1 + u2), f2 = u2F1 − u1F2 +νz(u1 − µu2); Reµz I∗, II eνt ( µR ∂ ∂R − ∂ ∂z ) For II : Ĝα, if µ = a, ν 6= 0; 15 if R=(u2 1 + u2 2) 1 2, z = tan−1 ( u2 u1 ) Gα, if µ = a, ν = 0 µ = 0 176 Group Classification of Systems... Table 4. Non-linearities with arbitrary functions and non-extendible symmetries for equations (1.3), (2.2) No Nonlinear terms Argu- ments of Fa Type of mat- rix A Symmetries and AET (2.10) [in square brackets] 1. f1 = uν2F1, f2 = uν+1 2 F2 u2e u1 I, IV νD − 2u2 ∂ ∂u2 + 2 ∂ ∂u1 [4 if ν = 0] 2. f1 = F1 + εu1, f2 = F2u2 + εu1u2 u2e u1 I, IV eεt ( u2 ∂ ∂u2 − ∂ ∂u1 ) , 3. f1 = eνu1F1, f2 = eνu1(F2 + F1u1) 2u2 −u2 1 I∗, III νD − 2u1 ∂ ∂u2 − 2 ∂ ∂u1 4. f1 = νu1 + F1, f2 = νu2 1 + F1u1 + F2 2u2 −u2 1 I∗, III ψ̃ν ( u1 ∂ ∂u2 + ∂ ∂u1 ) 5. f1 = νu1 + F1, f2 = −µu1 + F2 u2 II, III For II : e(ν−aµ)tΨµ ∂ ∂u1 , For III : e(ν+σa)tΨσ, µ = σa 6. f1 =eνz (F1u2+F2u1), f2 =eνz (F2u2−F1u1) Re−µz I∗, II νD−2µ ( u1 ∂ ∂u1 +u2 ∂ ∂u2 ) −2 ( u1 ∂ ∂u2 − u2 ∂ ∂u1 ) 7. f1 = 0, f2 = F u2 I, IV ψ0 ∂ ∂u1 , u1 ∂ ∂u1 , [ 2; 1, ρ = 0] 8. f1 = 0, f2 = F u1 I, a 6= 1, IV D + 2u2 ∂ ∂u2 , ψ̃0 ∂ ∂u2 , [3, 6] 9. f1 = F1, f2 = F2 + νu2 u1 I, III, IV ψ̃ν ∂ ∂u2 10. f1 = F1 + (ν − µ)u1, f2 = F2 + (ν − aµ)u2 u2−u1 I, a 6= 1 IV eνtΨµ(x) ( ∂ ∂u1 + ∂ ∂u2 ) 11. f1 = αu1 + µ, f2 = νu2 + F, αµ = 0 u1 I∗, III ψ̃ν ∂ ∂u2 , e(ν−α)t (u1 − µt) ∂ ∂u2 12. f1 = u2 1, f2 = u1u2 + νu2 + F, u1 I∗, III eνtu1 ∂ ∂u2 , eνt ( ∂ ∂u2 + tu1 ∂ ∂u2 ) 13. f1 = ( u2 1 − 1 ) , f2 = (u1 + ν)u2 + F u1 I∗, III e(ν+1)t ( u1 ∂ ∂u2 + ∂ ∂u2 ) , e(ν−1)t ( u1 ∂ ∂u2 − ∂ ∂u2 ) 14. f1 = ( u2 1 + 1 ) , f2 = (u1 + ν)u2 + F u1 I∗, III eνt(cos tu1 ∂ ∂u2 −sin t ∂ ∂u2 ), eνt(sin tu1 ∂ ∂u2 +cos t ∂ ∂u2 ) 15. f1 = eνu2F1, f2 = eνu2F2 µu2 −u1 I, IV µ 6= 0; II, III µ = 0 νD − 2 ( µ ∂ ∂u1 + ∂ ∂u2 ) 16. f1 = eνu1F1, f2 = eνu1F2 u2 III νD − 2 ∂ ∂u1 A. G. Nikitin 177 Table 5. Non-linearities with arbitrary parameters and extendible symmetries for equations (1.3), (2.2) No Nonlinear terms Type of mat- rix A Main symmetries Additional symmetries AET (2.10) 1. f1 =λuν+1 1 uµ2 , f2 =σuν1u µ+1 2 I, IV µD−2u2 ∂ ∂u2 , νD−2u1 ∂ ∂u1 Gα if aν = −µ 6= 0 & K if ν = 4 m(1−a) , a 6= 1; 1, νω +µρ = 0 ψ0 ∂ ∂u1 if σ = 0, ν = −1 & Gα if µ = a 6= 0 & K if a = 1 + m 4 ; 2; 1, νω +µρ = 0 ψ0 ∂ ∂u1 , u2 ∂ ∂u1 if σ = 0, ν = −1, a = 1; 2; 1, νω +µρ = 0 Gα,K if σ = 1, λ 6= 1 µ = −ν = a = 1; 1, νω +µρ = 0 u1 ∂ ∂u2 if µ = 0, λ = σ, a = 1 & ∂ ∂u2 + tu1 ∂ ∂u2 if ν = 1 1, νω +µρ = 0 2. f1 = λuν+1 1 , f2 = σuν+µ1 , I, IV νD−2u1 ∂ ∂u1 −2µu2 ∂ ∂u2 , (u1 − λt) ∂ ∂u2 if −ν = a = 1; 3 λσ 6= 0 ψ̃0 ∂ ∂u2 Gα if ν = 0, aµ = 1; 3 e−λtu1 ∂ ∂u2 if ν = 0, a = 1, & eλt ( σu1 ∂ ∂u2 +λ ∂ ∂u1 ) if µ = 2 3; 1, ρ =µω 3. f1 = λu1, f2 = σuµ1 III ψ0 ∂ ∂u2 , e−λtu1 ∂ ∂u2 eλt ( u1 ∂ ∂u2 + λ ∂ ∂u1 ) if µ = 2 3; 6 if λ=0 4. f1 =λeνu1 , f2 =σe(ν+1)u1, I, IV νD − 2u2 ∂ ∂u2 −2 ∂ ∂u1 , u2 ∂ ∂u2 if σ = 0 & u2 ∂ ∂u1 if a = 1; 3; 1, ω=0 λ 6= 0 ψ̃0 ∂ ∂u2 (u1 − λt) ∂ ∂u2 if ν = 0, a = 1 3,4 5. f1 = λeu1 , f2 = σeu1 III D − 2 ∂ ∂u1 , ψ̃0 ∂ ∂u2 u1 ∂ ∂u2 if λ = 0 3; 6 if λ=0 6. f1 = λeu2 , f2 = σeu2 , λ 6= 0 I, IV D − 2 ∂ ∂u2 , ψ0 ∂ ∂u1 u1 ∂ ∂u1 + ∂ ∂u2 if σ = 0 & for I u2 ∂ ∂u1 if a = 1 2; 5 if σ = 0 178 Group Classification of Systems... 7. f1 = λuν+1 1 e µ u2 u1, I∗, III µD − 2u1 ∂ ∂u2 , For I∗ : Gα if ν = 0; 1, ρ = ω; 6 if µ = 0 f2 = e µ u2 u1 (λu2 +σu1)u ν 1 νD − 2u1 ∂ ∂u1 −2u2 ∂ ∂u2 For III : Gα if µ = aν & K if ν = 4 m 1, ρ = ω if ν = 0; 6 if µ = 0 8. f1 = eµzRν(λu1 −σu2), I∗, II νD − 2u1 ∂ ∂u1 −2u2 ∂ ∂u2 , For I∗ : Gα if ν = 0; 1, ρ = ω f2 = eµzRν(λu2 +σu1) µD − 2u1 ∂ ∂u2 +2u2 ∂ ∂u1 For II : Gα if µ = aν & K if ν = 4 m 1, ρ = ω if ν = 0 9. f1 = εuµ+1 1 , f2 = εuµ1 (u2 − lnu1), µ 6= 0, I∗ µD−2u1 ∂ ∂u1 −2 ∂ ∂u2 , u1 ∂ ∂u2 ∂ ∂u2 + tu1 ∂ ∂u2 if µ = 1 6 10. f1 = λ, f2 = ε lnu1 I − IV 1 2 D + u1 ∂ ∂u1 +u2 ∂ ∂u2 +εt ∂ ∂u2 , For I, a 6= 1, IV : u1 ∂ ∂u1 + εt ∂ ∂u2 if λ = 0; 3, 7, 9 (for II : 3, 7) ψ̃0 ∂ ∂u2 For I∗, III (u1 − λt) ∂ ∂u2 ; & (for I∗) u1 ∂ ∂u1 + εt ∂ ∂u2 if λ = 0 3, 9; & 6, 7 if λ = 0 (7 for I∗ only) 11. f1 = 0, f2 = εu2 + lnu1 I, IV µu1 ∂ ∂u1 −ε ∂ ∂u2 , ψ̃ε ∂ ∂u2 eεtu1 ∂ ∂u2 if a = 1 10, κ = ε 12. f1 = λu1 lnu1, f2 = νu2 + lnu1 I, IV ψ̃ν ∂ ∂u2 eνt ( u1 ∂ ∂u1 +t ∂ ∂u2 ) if ν = λ; 10, κ = ν eλt ( (λ− ν)u1 ∂ ∂u1 + ∂ ∂u2 ) if ν 6= λ 10, κ = ν 13. f1 = λuµ+1 1 , f2 = σuµ+1 1 , λσ = 0 III µD − 2u1 ∂ ∂u1 −2u2 ∂ ∂u2 , ψ̃0 ∂ ∂u2 u1 ∂ ∂u2 if λ = 0 3; 6 if λ = 0 Here and in the following ε = ±1, K is generator defined in (4.6), K = K + 2 λ−1 [ t ( λu1 ∂ ∂u1 + (2 − λ)u2 ∂ ∂u2 ) + u1 ∂ ∂u2 ] . In the following table Q = 2 ( (µ− aν)t− ν 2mx 2 ) for version II and Q = 2 ( (µ− ν)t− ν 2amx 2 ) , a 6= 0 for version III. A. G. Nikitin 179 Table 6. Non-linearities with arbitrary parameters and non-extendible symmetries for equations (1.3), (2.2) No Nonlinear terms Type of mat- rix A Symmetries AET (2.10) 1. f1 = λuν+1 2 , f2 = µuν+1 2 II, III νD − 2u1 ∂ ∂u1 −2u2 ∂ ∂u2 , Ψ0(x) ∂ ∂u1 2 2. f1 = λ (u1 + u2) ν+1 , f1 = µ (u1 + u2) ν+1 I, a 6= 1 IV νD − 2u1 ∂ ∂u1 −2u2 ∂ ∂u2 , Ψ0(x) ( ∂ ∂u1 − ∂ ∂u2 ) 12 3. f1 = λuν+1 1 , f2 = uν1 (λu2 + µuσ1 ) , ν + σ 6= 0, 1, µ 6= 0 I∗ νD − 2u1 ∂ ∂u1 −2σu2 ∂ ∂u2 , u1 ∂ ∂u2 6 4. f1 = λeu2 , f2 = σeu2 II, III D − 2 ∂ ∂u2 , Ψ0(x) ∂ ∂u1 2 5. f1 = λe(u1+u2), f2 = σe(u1+u2) I, a 6= 1, IV D − 2 ∂ ∂u2 , Ψ0(x) ( ∂ ∂u1 − ∂ ∂u2 ) 12 6. f1 = λuν2e u1 , f2 = σuν+1 2 eu1 , ν2 + (a− 1)2 6= 0 I, IV D − 2 ∂ ∂u1 , u2 ∂ ∂u2 − ν ∂ ∂u1 13 if σ = 0 7. f1 = λeu1 , f2 = σu1e u1 I∗, III D − 2 ∂ ∂u1 − 2u1 ∂ ∂u2 , ψ0 ∂ ∂u2 (& u2 ∂ ∂u2 for I∗) 3; 6 if λ = 0 8. f1 = εeu1 , ε = ±1, f2 = λu1 I, IV D + 2u2 ∂ ∂u2 − 2 ∂ ∂u1 −2λt ∂ ∂u2 , ψ̃0 ∂ ∂u2 3 9. f1 = νeλ(2u2−u2 1), f2 = (νu1 + µ) eλ(2u2−u2 1) I∗, III λD − ∂ ∂u2 , ∂ ∂u1 + u1 ∂ ∂u2 14 10. f1 = λ ln(2u2 − u2 1), f2 = σ(2u2 − u2 1) +λu1 ln(2u2 − u2 1) I∗ D + 2u1 ∂ ∂u1 + 4u2 ∂ ∂u2 +4λt ( ∂ ∂u1 +u1 ∂ ∂u2 ) , ∂ ∂u1 +u1 ∂ ∂u2 14 11. f1 = µ lnu2, f2 = ν lnu2 II, III Ψ0(x) ∂ ∂u1 , D + 2u1 ∂ ∂u1 +2u2 ∂ ∂u2 +Q ∂ ∂u1 2 12. f1 = ε ln (u1 + u2) , f2 = ν ln (u1 + u2) I, a 6= 1, IV, a = 0 Ψ0(x) ( ∂ ∂u1 − ∂ ∂u2 ) , (a− 1) ( D + 2u1 ∂ ∂u1 +2u2 ∂ ∂u2 ) + (2(aε+ ν)t + ε+ν m x2 ) ( ∂ ∂u1 − ∂ ∂u2 ) 12 180 Group Classification of Systems... 13. f1 = λuν+1 1 , f2 = lnu1, λ(ν + 1) 6= 0 I, IV ν ( D + 2u2 ∂ ∂u2 ) −2u1 ∂ ∂u1 − 2t ∂ ∂u2 , ψ̃0 ∂ ∂u2 3 14. f1 = λuν+1 1 , f2 = λuν+1 1 lnu1 I∗, III νD − 2 ( u1 ∂ ∂u1 +u2 ∂ ∂u2 + u1 ∂ ∂u2 ) , ψ0 ∂ ∂u2 3 15. f1 = λuν+1 1 , f2 = λuν1u2 + u1 lnu1, λ(ν − 1) 6= 0 I∗ νD − 2u1 ∂ ∂u1 −2tu1 ∂ ∂u2 −2(1 − ν)u2 ∂ ∂u2 , u1 ∂ ∂u2 6 16. f1 =λ ( 2u2 − u2 1 )ν+ 1 2 , f2 =λu1 ( 2u2 − u2 1 )ν+ 1 2 +µ ( 2u2 − u2 1 )ν+1 I∗ νD − u1 ∂ ∂u1 − 2u2 ∂ ∂u2 , ∂ ∂u1 + u1 ∂ ∂u2( & 2λt ( ∂ ∂u1 + u1 ∂ ∂u2 ) + ∂ ∂u2 if µ=0, ν= 1 2 ) 14; 1, ρ = 2ω if ν = 0 17. f1 = 2νu1 lnu1 + u1u2, f2 = −(ν − µ)2 lnu1 +2µu2 I, IV X = e(µ+ν)t ( u1 ∂ ∂u1 +(µ− ν) ∂ ∂u2 ) , tX + e(µ+ν)t ∂ ∂u2 10, κ = 2ν if µ+ ν = 0 18. f1 = 2νu1 lnu1 + u1u2, f2 = 2µu2 + ( 1 − (ν − µ)2 ) lnu1 I, IV X± = eλ±t ( u1 ∂ ∂u1 +(λ± − 2ν) ∂ ∂u2 ) , λ± = µ+ ν ± 1 10, κ = 2ν if µ+ ν = ±1 19. f1 = 2νu1 lnu1 + u1u2, f2 = 2µu2 − ( 1 + (ν − µ)2 ) lnu1 I, IV e(µ+ν)t [ cos tu1 ∂ ∂u1 −(sin t+ (ν −µ) cos t) ∂ ∂u2 ] , e(µ+ν)t [ sin tu1 ∂ ∂u1 +(cos t+ (µ −ν) sin t) ∂ ∂u2 ] 20. f1 = ε(2u2 − u2 1), f2 = (µ+ εu1) ( 2u2 − u2 1 ) −µ2 2 εu1, µ 6= 0 I∗, III X1 = eµt ( 2 ∂ ∂u1 +2u1 ∂ ∂u2 + εµ ∂ ∂u2 ) , tX1 + εeµt ∂ ∂u2 21. f1 = ε(2u2 − u2 1), f2 = (µ+ εu1) ( 2u2 − u2 1 ) + 1−µ2 2 εu1 I∗, III X± = eµ±1 ( 2 ∂ ∂u1 +2u1 ∂ ∂u2 +ε(µ± 1) ∂ ∂u2 ) 14 if µ2 = 1 22. f1 = ε(2u2 − u2 1), f2 = − 1+µ2 2 εu1 +(µ+ εu1) ( 2u2 − u2 1 ) I∗, III eµt ( 2ε cos t ( ∂ ∂u1 +u1 ∂ ∂u2 ) + (µ cos t − sin t) ∂ ∂u2 ) , eµt ( 2ε sin t ( ∂ ∂u1 +u1 ∂ ∂u2 ) + (µ sin t + cos t) ∂ ∂u2 ) A. G. Nikitin 181 Table 7. Symmetries of equations (1.3) with diagonal matrix A and non-linearities f1 = u1 (µ lnu1 + λ lnu2) , f 2 = u2 (ν lnu2 + σ lnu1) No Conditions Main symmetries Additional for coefficients symmetries and notations 1. λ = 0, µ = ν eµtu2 ∂ ∂u2 , eµt ( u1 ∂ ∂u1 + σtu2 ∂ ∂u2 ) Ĝα, if a 6= 0, σ = 0, µ 6= 0 2. λ = 0, µ 6= ν eνtu2 ∂ ∂u2 , Ĝα if µ 6= 0, µ− ν = aσ eµt ( (µ− ν)u1 ∂ ∂u1 +σu2 ∂ ∂u2 ) Gα if aσ = −ν, µ = 0; ψ0 ∂ ∂u2 if σ = ν = 0; ψ0 ∂ ∂u1 if σ = µ = 0; u1 ∂ ∂u2 , Ĝα if a = 1, ν = 0 µ = σ 6= 0 3. δ = 1 4 (µ− ν)2 +λσ = 0, µ+ ν = 2ω0 X2 = eω0t ( 2λu1 ∂ ∂u1 + (ν − µ)u2 ∂ ∂u2 ) , Ĝα if ν 6= −µ, 2λ = a(ν − µ) λσ 6= 0 eω0t2u2 ∂ ∂u2 + tX2 Gα if λ = aν µ = −ν 6= 0 4. λσ 6= 0, δ = 1, eω+t ( λu1 ∂ ∂u1 + (ω+ − µ)u2 ∂ ∂u2 ) , Ĝα if µν 6= λσ, λ = a(ν − µ+ aσ) ω± = ω0 ± 1 eω−t ( λu1 ∂ ∂u1 + (ω− − µ)u2 ∂ ∂u2 ) Gα if νµ = λσ, λ = −aµ 5. δ = −1 eω0t ( 2λ cos tu1 ∂ ∂u1 + ((ν − µ) cos t −2 sin t)u2 ∂ ∂u2 ) , eω0t ( 2λ sin tu1 ∂ ∂u1 + ((ν − µ) sin t +2 cos t)u2 ∂ ∂u2 ) none Equations (1.3) with the nonlinearities present in Table 7 admit equiv- alence transformation 1 from the list (2.10) provided µν = λσ. The re- lated parameters ρ and ω should satisfy µω + λρ = 0. In addition, the equations corresponding to the last version enumerated in Item 2 admit additional equivalence transformation 6 given by formula (2.10). 182 Group Classification of Systems... Table 8. Symmetries of equations (1.3) with matrix A of type I∗, II and non-linearities f1 = (µu1 − σu2) lnR+ z(λu1 − νu2), f2 = (µu2 + σu1) lnR+ z(λu2 + νu1) No Conditions Main symmetries Additional for coefficients symmetries 1. λ = 0, µ = ν eµt ∂ ∂z , For II : Ĝα if aσ = 0, µ 6= 0 eµt ( R ∂ ∂R + σt ∂ ∂z ) For II : Gα if a = ν = 0, σ 6= 0 For I∗ : Ĝα if σ = 0, µ 6= 0 2. λ = 0, µ 6= ν, eνt ∂ ∂z , eµt ( σ ∂ ∂z + (µ− ν)R ∂ ∂R ) For II : Ĝα if aσ=ν − µ, µ 6= 0 or a = 0, µ 6= 0 For II : Gα if aσ = ν, µ = 0 For I∗ : Ĝα if µ 6= 0, σ = 0 For I∗ : Gα if µ = 0, σ = 0 3. δ = 0, λ 6= 0 X3 = eω0t ( 2λR ∂ ∂R + (ν − µ) ∂ ∂z ) , For II : Ĝα if µ 6= −ν a(µ− ν) = 2λ For II : Gα if aν = −λ, ω0 = 0 For I∗ : Ĝα if µ = ν 6= 0 For I∗ : Gα if µ = ν = 0 4. λ 6= 0, δ = 1 eω+t ( λR ∂ ∂R + (ω+ − µ) ∂ ∂z ) , Ĝα if µν 6= λσ, λ = a(ν − µ+ aσ) eω−t ( λR ∂ ∂R + (ω− − µ) ∂ ∂z ) Gα if νµ = λσ, λ = −aµ For I∗ : Ĝα if σ = 0, µ 6= 0 For I∗ : Gα if σ = µ = 0 5. δ = −1 exp(ω0t) [ 2λ cos tR ∂ ∂R + ((ν − µ) cos t −2 sin t) ∂ ∂z ] , exp(ω0t) [ 2λ sin tR ∂ ∂R + ((ν − µ) sin t +2 cos t) ∂ ∂z ] none A. G. Nikitin 183 All equations enumerated in Table 8 admit additional equivalence transformations 15 from the list (2.10). Table 9. Symmetries of equations (1.3) with non-linearities f1 = λu2 + µu1 lnu1, f 2 = λ u2 2 u1 + (σu1 + µu2) lnu1 + νu2 and matrices A of type III (and I∗ if a = 0) No Conditions Main symmetries Additional for coefficients symmetries 1. λ = 0, µ 6= ν eνtu1 ∂ ∂u2 , ψν ∂ ∂u2 if µ = 0, & Gα if aν = σ 6= 0 eµt ( (µ− ν)R ∂ ∂R + σu1 ∂ ∂u2 ) Ĝa, if µ 6= 0, σ = a(ν − µ) 6= 0 2. λ = 0, µ = ν eµtu1 ∂ ∂u2 , eµt ( R ∂ ∂R + σtu1 ∂ ∂u2 ) ψ0 ∂ ∂u2 if µ = 0, σ 6= 0 & D + u2 ∂ ∂u2 if a = 0 Ĝa if σ = 0, µ 6= 0 3. σ = 0, µλ 6= 0, eνt (λR∂R + (µ− ν)u∂v) Ga if ν = 0, µ = −λ µ 6= ν, a = 1 eµtR∂R Ĝa if ν − µ = λ 4. δ = 0 µ+ ν = 2ω0, λ 6= 0 X4 = eω0t ( 2λR ∂ ∂R +(ν − µ)u1 ∂ ∂u2 ) , Ga if ω0 = 0, ν = −aλ & D + 2u1 ∂ ∂u1 if a = 0 2eω0tu1 ∂ ∂u2 + tX4 Ĝa, if ω0 6= 0, 2aλ = µ− ν 5. λ 6= 0, δ = 1, eω+t ( λR ∂ ∂R +(ω+ − µ)u1 ∂ ∂u2 ) , Ga, if µ = aλ, µν = λσ ω± = ω0 ± 1 eω−t ( λR ∂ ∂R +(ω− − µ)u1 ∂ ∂u2 ) Ĝα, if µν 6= λσ, µ− ν = λ− σ, a = 1 or σ = a = 0, µ 6= 0 6. δ = −1, eω0t[2λ cos tR ∂ ∂R +((ν−µ) cos t− 2 sin t)u1 ∂ ∂u2 ], eω0t[2λ sin tR ∂ ∂R +((ν−µ) sin t+ 2 cos t)u1 ∂ ∂u2 ] none If λ = µ = 0 or λ = ν = 0 then the related equation (1.3) admits additional equivalence transformations 16 or 6 from the list (2.10) corre- spondingly. Tables 3–9 present results of group classification of equations (1.3) 184 Group Classification of Systems... with invertible diffusion matrix A. The results present in Tables 3–7 are valid for equations with the singular matrix A of type IV also but do not exhaust all non-equivalent non-linearities for such equations. Moreover, the equations with singular diffusion matrix admit strong equivalence transformations u1 → u1, u2 → ε(u2) where ε(u2) is an arbitrary function of u2 which reduce the number of non-equivalent symmetries in Tables 3–9 for a = 0. The completed group classification of equations (1.3) with matrix A of type IV is given in paper [21] 11. Classification of Reaction-Diffusion Equations with Nilpotent Diffusion Matrix To complete the classification of systems (1.3) we need to consider the remaining class of these equations when matrix A belongs to type V , i.e., is nilpotent. The procedure of classification of such equations appears to be more complicated then in the case of invertible or diagonalizable dif- fusion matrix. The general form of symmetry admitted by this equation is given by equation (4.13) while the classifying equations take the form (4.14). A specific feature of symmetries (4.13) is that the coefficient B3 can be a function of u1. One more specific point in the classification of equations with matrix A of type V is that they admit powerful equivalence relations u1 → u1, u2 → u2 + Φ(u1) (11.1) and u1 → u1, u2 → u2 + Φ̂(u1, t, x) (11.2) which did not appear in our analysis presented in the previous sections. Transformation (11.1) (where Φ(u1) is an arbitrary function of u1) are admitted by any equation (1.3) with matrix A of type V . Transformations (11.2) are valid for the cases when f1 does not depend on u2 and at the same time f2 is linear in u2. Moreover, the related functions Φ̂(u1, t, x) should satisfy the following system of equations f2 u2 Φ̂t − Φ̂tt − f1Φ̂tu1 = 0, f2 u2 Φ̂xν − Φ̂txν − f1Φ̂u1xν = 0 (11.3) Thus the group classification of equation (1.3) with the nilpotent diffu- sion matrix is reduced to solving the classifying equations (4.14) with ap- plying the equivalence transformations discussed in Section 2 and trans- formations (11.1), (11.2) as well. To do this we again use the analysis A. G. Nikitin 185 of low dimension algebras A whose results are given in the Appendix. We will not reproduce the related routine calculations but present the classification results in Tables 8–10. In Tables 8–10 we use without explanations all the notations applied in Tables 1–9. In addition, a number of classified equations appear a specific symmetry W∂u2 where W is a function of t, x and u1 which solve the following equation: f2 u2 −Wt −Wu1f 1 = 0. Table 10. Non-linearities with arbitrary functions for equations (1.3) with nilpotent diffusion matrix No Nonlinear terms Argu- ments of Fα Symmetries 1. f1 = F1u µ−ν 1 , f2 = F2u µ 1 u ν+1 1 u2 Q1 = (µ− 1)D − νt ∂ ∂t −u1 ∂ ∂u1 − (ν + 1)u2 ∂ ∂u2 & (m− 2)x2 ∂ ∂xa − xaQ1 if ν(m− 2) = 4, µ(m− 2) = m+ 2, m 6= 2 2. f1 = F1u1u µ−1 2 , f2 = F2u µ 2 , F2 6= 0 u1 µD − t ∂ ∂t − u2 ∂ ∂u2 & eW ∂ ∂u2 if µ = 1 & Ha ∂ ∂xa −Hb xb u2 ∂ ∂u2 if m = 2 3. f1 = F1u −1 2 , f2 = F2 + νu2 u1 eνt ( ∂ ∂t + νu2 ∂ ∂u2 ) & eW ∂ ∂u2 if F1 = 0 4. f1 = F1u µ−1 2 , f2 = F2u µ 2 u2e u1 µD − t ∂ ∂t − u2 ∂ ∂u2 + ∂ ∂u1 5. f1 = F1 u2 + ν, f2 = F2 + νu2 u2e u1 eνt ( ∂ ∂t + νu2 ∂ ∂u2 − ν ∂ ∂u1 ) 6. f1 = 0, f2 = F2 u2 Ψ0(x) ∂ ∂u1 , xa ∂ ∂xa + 2u1 ∂ ∂u1 7. f1 = F1, f 2 = 0 u1 eW ∂ ∂u2 , xa ∂ ∂xa − 2u2 ∂ ∂u2 8. f1 = ν µ−1 u1 + F1u 2−µ 1 , f2 = µν µ−1 u2 + F2u1, µ 6= 1 u2u −µ 1 eνt ( (1 − µ)t ∂ ∂t − νu1 ∂ ∂u1 −νµu2 ∂ ∂u2 ) 186 Group Classification of Systems... 9. f1 = u1F1, m = 1 f2 = u2F2 + u1 u2u 3 Q2 =cos(2x) ( u1 ∂ ∂u1 − 3u2 ∂ ∂u2 ) + sin(2x)x ∂ ∂x , Q3 = (Q2)x 10. f1 = u1F1, m = 1 f2 = u2F2 − u1 u2u 3 1 Q4 =e2x ( ∂ ∂x + u1 ∂ ∂u1 − 3u2 ∂ ∂u2 ) , Q5 =e−2x ( ∂ ∂x − u1 ∂ ∂u1 + 3u2 ∂ ∂u2 ) 11. f1 = F1, f2 = u2F2, m = 2 u2e u1 Ha ∂ ∂xa −Hb xb ( u2 ∂ ∂u2 − ∂ ∂u1 ) 12. f1 = νe u2 u1 , f2 = e u2 u1 F u1 D − u1 ∂ ∂u2 13. f1 = F1, F2 = u2F2 + F3 u1 eW ∂ ∂u2 14. f1 = eνu2F1, f2 = eνu2F2 u1 νD − ∂ ∂u2 15. f1 = eνu1F1, f2 = eνu1F2 u2 νD − ∂ ∂u1 16. f1 = νu1 + F1, f2 = µu1 + F2 u2 e(ν−aµ)tΨµ(x) ∂ ∂u1 17. f1 = u1(F1 + ν lnu1), f2 = u2(F2 + ν lnu1), ν 6= 0 u1 u2 eνt(u1 ∂ ∂u1 + u2 ∂ ∂u2 ) 18. f1 = u1F1 − νu2, f2 = ν u2 u1 (u2 − u1) +u1F2 − u2F1 u1e u2 u1 eνt ( u1 ∂ ∂u2 − u1 ∂ ∂u1 − u2 ∂ ∂u2 ) & Ĝα if a = 1 19. f1 = uν+1 1 F1, f2 = uν1 (F2u1 − F1u2) u1e u2 u1 νD + u1 ∂ ∂u2 − u1 ∂ ∂u1 − u2 ∂ ∂u2 & Gα if ν = 0, a = 1 20. f1 = uµ+1 1 F1, f2 = uµ+1 1 F2 u2 u1 µD − u1 ∂ ∂u1 − u2 ∂ ∂u2 For the non-linearities enumerated in Items 2 (when µ = 1), 3 (when F1 = 0), 4 and 8 of Table 8 the related equation (1.3) admits additional equivalence transformations (11.2). In addition, transformations (2.4) and (11.1) and some equivalence transformations from the list (2.10) are admissible, namely, transformations 9 for the non-linearities given in Item 1 (when ν = −1, µ = 0) and Item 6, Item 6,Item 18, Item 19 and Item 20 transformations 1 with ρ = ω for the non-linearities from Item I (when ν = 1, µ = 0), Item 20 (when µ = 0) and Items 18, 19. Finally for f1 and f2 present in Item 7 transformation 3 of (2.10) is admissible. A. G. Nikitin 187 Table 11. Non-linearities with arbitrary parameters and extendible symmetries for equations (1.3) with nilpotent diffusion matrix No Non-linearities Main symmetries Additional symmetries AET (2.10) 1. f1 = λuν+1 1 uµ2 , f2 = σuν1u µ+1 2 (µ+ ν)t ∂ ∂t −(µ+ 1)u1 ∂ ∂u1 +(ν − 1)u2 ∂ ∂u2 xaQ6 − 2κx2 ∂ ∂xa if κ(m+ 2) = ν, κ(2 −m) = µ Q6 = 2µu1 ∂ ∂u1 +(µ+ ν)xa ∂ ∂xa −2νu2 ∂ ∂u2 eW ∂ ∂u2 if λ = 0, µ = −1 & 2x2 ∂ ∂xa −(m− 2)xaQ6 if ν = m+2 m−2 , m 6= 2 17, 3, 6 Ψ0(x) ∂ ∂u1 , if σ = 0, ν = −1, & xaQ6 + 2 m+2 x2 ∂ ∂xa if µ = m−2 m+2 , m 6= 2 17, 9 Ψ0(x) ∂ ∂u1 if λ = ν = 0 17, 9 eW ∂ ∂u2 if µ = 0 & Ha ∂ ∂xa − ∂Ha ∂xa u2 ∂ ∂u2 if m = 2 17 & 6 if λ = σ 2. f1 = λuν+1 1 u−1 2 , f2 = σuν1 + εu2, eεt ( ∂ ∂t + εu2 ∂ ∂u2 ) , Q′ 6 = Q6|µ=−1 xaQ ′ 6 − 2 m−2 x2 ∂ ∂xa if ν = m+2 m−2 , m 6= 2 λ 6= 0 Ψ0(x) ∂ ∂u1 if σ = 0, ν = −1 9 u2 ∂ ∂u2 + u1 ∂ ∂u1 if σ = 0, ν = 1 1 3. f1 = λeνu1 , f2 = σe(ν+1)u1 , (ν + 1)D − u2 ∂ ∂u2 −t ∂ ∂t − ∂ ∂u1 , eW ∂ ∂u2 ν ( u2 ∂ ∂u2 + t ∂ ∂t ) − ∂ ∂u1 if σ = 0 17, 3 λσ = 0 u2 ∂ ∂u2 + t ∂ ∂t if λ = 0 10, 3, 6 4. f1 = λe(ν+1)u2 , f2 = σeνu2 (ν − 1)D − u1 ∂ ∂u1 +t∂t − ∂ ∂u2 , Ψ0(x) ∂ ∂u1 ν ( u1 ∂ ∂u1 − t ∂ ∂t ) + ∂ ∂u2 if λ = 0 9 for any λ 5. f1 = λuµ−1 2 eu1 , f2 = σuµ2 e u 1 D − ∂ ∂u1 , t∂t+u2 ∂ ∂u2 −µ ∂ ∂u1 eW if µ = 1 & Ha ∂ ∂xa − ∂Ha ∂xa u2 ∂ ∂u2 if m = 2 17 if µ = 1 6. f1 = λ lnu2, f2 = σu µ+1 2 2 µD − µ+1 2 t ∂ ∂t − 1−µ 2 u1 ∂ ∂u1 −u2 ∂ ∂u2 − λt ∂ ∂u1 , Ψ0(x) ∂ ∂u1 xa ∂ ∂xa + 2u1 ∂ ∂u1 , u1 ∂ ∂u1 + 2u2 ∂ ∂u2 +t∂t + 2λt ∂ ∂u1 if σ = 0 9 188 Group Classification of Systems... Table 12. Non-linearities with arbitrary parameters and non extendible symmetries for equations (1.3) with a = 0 No Non-linearities Condi- tions Symmetries AET (2.10) 1. f1 = λu3µ+1 1 uµ2 , f2 = σu3µ 1 uµ+1 2 − αu1, µ 6= 0, m = 1, α = −1 Q7 = 4µt ∂ ∂t −(µ+ 1)u1 ∂ ∂u1 +(3µ− 1)u2 ∂ ∂u2 , Q2, Q3 µ 6= 0, m = 1, α = 1 Q4, Q5, Q7 2. f1 = λu−2 1 u−1 2 , m = 1, α = −1 eεt ( ∂ ∂t + εu2 ∂ ∂u2 ) , Q2, Q3 17 if f2 = σu−3 1 + εu2 − αu1 m = 1, α = 1 eεt ( ∂ ∂t + εu2 ∂ ∂u2 ) , Q4, Q5 λ=0 3. f1 = λuµ+1 2 , f2 = σuµ−ν+1 2 µ 6= −1 (µ− 2ν)D + νt ∂ ∂t −u2 ∂ ∂u2 − (ν + 1)u1 ∂ ∂u1 , Ψ0(x) ∂ ∂u1 9 4. f1 = λu2, f 2 = e−u2 λ 6= 0 2D − t∂t + u1 ∂ ∂u1 + ∂ ∂u2 + λt ∂ ∂u1 , Ψ0(x) ∂ ∂u1 9 5. f1 = λeu2 , f2 = σeu2 λσ 6= 0 D − ∂ ∂u2 , Ψ0(x) ∂ ∂u1 9 6. f1 = λuν+1 1 e µ u2 u1 , f2 = e µ u2 u1 (λu2 + σu1)u ν 1 µλ 6= 0 µD − u1 ∂ ∂u2 , νD − u1 ∂ ∂u1 − u2 ∂ ∂u2 7. f1 = µ lnu2, f2 = ν lnu2 ν 6= 0 Ψ0(x) ∂ ∂u1 , D + u1 ∂ ∂u1 + u2 ∂ ∂u2 + ( µt− ν 2m x2 ) ∂ ∂u1 9 8. f1 = 0, f2 = ε lnu1 ε = ±1 D − t∂t + u1 ∂ ∂u1 +εt ∂ ∂u2 , t∂t + u2 ∂ ∂u2 , Φ(u1, x) ∂ ∂u2 3, 6, 17 9. f1 =ε (lnu2−κ lnu1)u1, f2 =ε (lnu2−κ lnu1)u2 m 6= 2, κ 6= m+2 m−2 (1 − κ)xa ∂ ∂xa +2κu2 ∂ ∂u2 + 2u1 ∂ ∂u1 , e(1−κ)εt ( u1 ∂ ∂u1 + u2 ∂ ∂u2 ) 1, ρ=ω if κ=1 10. f1 = εu1 ((m+ 2) lnu1 +(2 −m) lnu2) , m 6= 1, 2 α = 0 Q1, xaQ1 − x2∂xa, e4εt ( u1 ∂ ∂u1 + u2 ∂ ∂u2 ) f2 = εu2 ((m+ 2) lnu1 +(2 −m) lnu2) − αu1 m = 2, α = 0 Ha ∂ ∂xa −Ha xa u2 ∂ ∂u2 , e4εt ( u1 ∂ ∂u1 + u2 ∂ ∂u2 ) A. G. Nikitin 189 m = 1, α = 1, ε = 1 Q2, Q3, e4t ( u1 ∂ ∂u1 + u2 ∂ ∂u2 ) m = 1, α = 1, ε = −1 Q4, Q5, e−4t ( u1 ∂ ∂u1 + u2 ∂ ∂u2 ) 11. f1 = µu1 lnu1, f2 = µu2 lnu1 + νu2 µ 6= 0 eW ∂ ∂u2 , eµt(u1 ∂ ∂u1 + u2 ∂ ∂u2 ) 17 12. f1 = εu2, f2 = λ u2 2 u1 + 2νu2 +σu1 lnu1 λ = ±1, σ = ∓ν2 Q8 = eνt(λ(u1 ∂ ∂u1 + u2 ∂ ∂u2 ) +νu1 ∂ ∂u2 ), eνtu1 ∂ ∂u2 + tQ8 λ 6= 0, ν2 + λσ = 1 X± = eν±1(λ(u1 ∂ ∂u1 +u2 ∂ ∂u2 ) + (ν ± 1)u1 ∂ ∂u2 ) 1, ρ=ω if σ=0 λ 6= 0, ν2 + λσ = −1 eνt(λ cos t(u1 ∂ ∂u1 + u2 ∂ ∂u2 ) +(ν cos t− sin t)u1 ∂ ∂u2 ), eνt(λ sin t(u1 ∂ ∂u1 + u2 ∂ ∂u2 ) +(ν sin t+ cos t)u1 ∂ ∂u2 ) 12. Discussion In this paper we present the completed group classification of systems of two coupled reaction-diffusion equations with general diffusion matrix. In other words we specify essentially different equations of this type de- fined up to equivalence transformations and describe their symmetries. We consider only nonlinear equations, i.e., exclude the cases when f1 and f2 in the right hand side of (1.3) are linear in u1, u2. Such cases are presented in paper [23]. The analyzed class of equations includes six non-equivalent subclasses corresponding to different canonical forms of diffusion matrix A enumer- ated in (2.2). In the particular case when matrix A has the forms I and I∗ from (2.2) our results can be compared with those of [7] and also [3]–[5]. Paper [7] was apparently the first work were the problem of group classification of equations (1.3) with a diagonal diffusion matrix was for- mulated and partially solved. Unfortunately, the classification results presented in [7] are incomplete and in many points incorrect. Thus, all cases enumerated above in Table 7, Items 1,2 of Table 3, Items 1,2, 7-10, 15 of Table 4, Items 2, 12, 16 and 17 of Table 6, were overlooked, sym- metries of equations with non-linearities given in Items 1 and 2 of Table 5 were presented incompletely, etc. In papers [3]–[5] Lie symmetries of the same equations and also of sys- tems of diffusion equations with the unit diffusion matrix were classified. 190 Group Classification of Systems... The results obtained in [3]–[5] are much more advanced then the pioneer Davidov ones, nevertheless they are still incomplete. In particular, the cases indicated above in Items 5 and 6 of Table 3; Items 12–14 of Table 4; the last line of Item 1, Item 9 and Item 11 for a=1 of Table 5; Items 15 and 22 of Table 6 and Item I for σ = 0, µ 6= 0 of Table 7 were not indicated in [5], which is in conflict with the statement of Theorem 1 formulated here. Moreover, many of equations presented in [5] as non- equivalent ones, in fact are equivalent one to another even in frames of equivalence relations (7) of [3]. The related examples are not enumerated here in as much as we believe that all non-equivalent equations (1.3) with different symmetries are present in Tables 1–9. Except the points mentioned in the previous paragraph our results concerning equations with a diagonal diffusion matrix are in accordance with ones obtained in [3]–[5]. Consider examples of well known reaction diffusion equations which appear to be particular subjects of our analysis. • The Jackiw-Teitelboim model of two-dimension gravity with the non-relativistic gauge [19] ∂ ∂t u1 − ∂2u1 ∂x2 − 2ku1 + 2u2 1u2 = 0, ∂ ∂t u2 + ∂2u2 ∂x2 + 2ku2 − 2u1u 2 2 = 0 (12.1) admits the equivalence transformation 1 (2.10) for ρ = −ω. Choos- ing ρ = 2k we transform equation (12.1) to the form (1.2) where a = −1, f1 = −2u2 1u2 and f2 = 2u2 2u1. The symmetries corre- sponding to these non-linearities are given in the first line of Table 5. Symmetries of equations (12.1) were investigated in paper [16]. In accordance with our analysis, generalized equation (12.1) with two spatial variables admits additional conformal symmetry gener- ated by operator K (4.5). • The primitive predator-prey system can be defined by [20] u̇1 −D ∂2u1 ∂x2 = −u1u2, u̇2 − λD ∂2u2 ∂x2 = u1u2. and this is again a particular case of equation (1.2) with the non- linearities given in the first line of Table 3 where however µ=ν=1, F1 = −F2 = u1 u2 . In addition to the basic symmetries 〈 ∂∂t , ∂∂x〉 this equation admits the (main) symmetry: X = ( D − 2u1 ∂ ∂u1 − 2u2 ∂ ∂u2 ) . A. G. Nikitin 191 • The λ− ω reaction-diffusion system u̇1 = D∆u1 +λ(R)u1 −ω(R)u2, u̇2 = D∆u2 +ω(R)u1 +λ(R)u2, (12.2) where R2 = u2 1 + u2 2, has symmetries that were analyzed in paper [1]. Again we recognize that this system is a particular case of (1.2) with non-linearities given in Item 6 of Table 4 with µ = ν = 0. Hence it admits the five dimensional Lie algebra generated by main symmetries (2.2) with µ, ν = 1, 2 and: X = ( u1 ∂ ∂u2 − u2 ∂ ∂u1 ) (12.3) which is in accordance with results of paper [1] for arbitrary func- tions λ and ω. Moreover, using Table 5, Item 8 we find that for the cases when λ(R) = λ̃Rν , ω = σRν (12.4) equation (12.2) admits additional symmetry with respect to scaling transformations generated by the operator: X = ( u1 ∂ ∂u2 − u2 ∂ ∂u1 ) + νD. (12.5) The other extensions of the basic symmetries correspond to the case when λ(R) = µ ln(R), ω(R) = σ ln(R), the related additional symmetries are given in Table 8 where ν = λ = 0. • The nonlinear Schrödinger equation (NSE) in m-dimensional space: ( i ∂ ∂t − ∆ ) ψ = F (ψ,ψ∗) (12.6) also is a particular case of (1.2). If we denote ψ = u1 + iu2, F = f1+if2 then (12.6) reduces to the form (1.3) with A = ( 0 −1 1 0 ) . In other words, any solution given in Tables 3–6, 8 with matrices A belonging to Class II gives rise for the NSE (9.4) that admits a main or extended symmetry. Thus our analysis makes it possible to present the completed group classification of the NSE as a particular case of general study of systems of reaction-diffusion equations with arbitrary diffusion matrix. Our results are in complete accordance with ones obtained in paper [22] where symmetries of the general NSE were described. 192 Group Classification of Systems... Among the solutions present in Tables 3–6, 8 we recognize ones which correspond to the well-known non-linearities [11] F = F (ψ∗ψ)ψ, F = (ψ∗ψ)kψ, F = (ψ∗ψ) 2 mψ, F = ln(ψ∗ψ)ψ One more interesting particular case of the NSE with extended symmetry can be found using Table 6 Item 1 for ν = 2,m = 1: ( i ∂ ∂t − ∆ ) ψ = (ψ − ψ∗)2 which is a potential equation for the Boussinesq equation for func- tion V = ∂ ∂t(ψ − ψ∗). • Generalized complex Ginzburg-Landau (CGL) equation ∂W ∂τ − (1 + iβ)∆W = F (W,W ∗) (12.7) is a particular case of system (1.3) with matrix A belonging to Class II with a 6= 0, refer to (2.2). Indeed, representing W and F as W = (u1 + iu2), F = β(f1 + if2) and changing independent variable τ → t = βτ we transform (12.7) to the form (1.3) with A = ( β−1 −1 1 β−1 ) . All non-equivalent non-linearities f1, f2 and the corresponding symmetries are given in Table 3, Items 1, 3, Table 4, Items 5, 6, 15, Table 5, Items 8, 10, Table 6, Items 1, 4, 11 and Table 8. The ordinary CGL equation corresponds to the case F = W − (1+ iα)W |W |2,m = 2 and admits basic symmetries (5.1) only. • Non-autonomous dynamical systems in phase space [8] ∂u1 ∂t − ∂2u1 ∂x2 −A(u1, u2) = h1(t, x), ∂u2 ∂t + α ∂u1 ∂t − ∂2u2 ∂x2 − νu1 = h2(t, x) (12.8) also are equivalent to a system of type (1.3) at least in the case of constant h1 and h2. The related matrix A belongs to Type III. Using the results present in Tables 3–6 and 9 we can specify all cases when the considered system admits main or extended symmetries. We see that the class of equations which is classified in present paper includes a number of important particular systems. Moreover, we present a priori description of symmetries of all possible systems of two reaction- diffusion equations with general diffusion matrix. A. G. Nikitin 193 Appendix A.1. Algebras A for Equations (1.3) with Diagonal Diffusion Matrix Let us consider equation (1.3) with a diagonal matrix A (version I of (2.2) where a 6= 0) and find the related low-dimension algebras A. In this case matrix (9.5) and the equivalence transformation matrix (9.7) reduce to the forms g =   0 0 0 B1 C11 0 B2 0 C22   ∼   0 0 0 µ1 µ11 0 µ2 0 µ22   (A.1.1) and U =   1 0 0 b1 K1 0 b2 0 K2   . (A.1.2) Up to equivalence transformations (9.6), (A.1.2) there exist three non- equivalent matrices (A.1.1), namely g1 =   0 0 0 0 1 0 0 0 λ   , g2 =   0 0 0 1 0 0 0 0 1   , g3 =   0 0 0 λ 0 0 1 0 0   . (A.1.3) In accordance with (9.1)–(9.4) the related symmetry operator can be represented in one of the following forms X1 = µD − 2(ga)bcũc ∂ ∂ub , X2 = eλt(ga)bcũc ∂ ∂ub (A.1.4) or X3 = eλt+ω·x ( ∂ ∂u2 + µ ∂ ∂u1 ) . (A.1.5) Here (ga)bc are elements of matrices (A.1.3), b, c = 0, 1, 2, ũ = column (u0, u1, u2), u0 = 1. Formulae (A.1.4) and (A.1.5) give the principal description of one- dimension algebras A for equation (1.3), with matrix A of type I. To describe two-dimension algebras A we classify matrices g (A.1.1) forming two-dimension Lie algebras. Choosing one of the basis elements in the forms given in (A.1.3) and the other element in the general form (A.1.1) we find that up to equivalence transformations (9.6) there exist six algebras 〈e1, e2〉: A2,1 = {g̃1, g4}, A2,2 = {g̃1, g̃3}, A2,3 = {g5, g̃3}, (A.1.6) 194 Group Classification of Systems... A2,4 = {g1, g5}, A2,5 = {g′1, g3}, A2,6 = {g2, g̃3} (A.1.7) where g̃1 = g1|λ=0, g′1 = g1|λ=1, g̃3 = g3|λ=0, and g4 =   0 0 0 0 0 0 0 0 1   , g5 =   0 0 0 1 0 0 0 0 0   . (A.1.8) Algebras (A.1.6) are Abelian while algebras (A.1.7) are characterized by the following commutation relations: [e1, e2] = e2 (A.1.9) where e1 is the first element given in the brackets (A.1.7), i.e., for A2,4 e1 = g1, etc. Using (A.1.6), (A.1.7) and applying arguments analogous to those which follow equations (6.2) we easily find pairs of operators (5.2) forming Lie algebras. Denoting êα = (eα)abũb ∂ ∂ua , α = 1, 2 we represent them as follows: 〈µD + ê1 + νtê2, ê2〉, 〈µD + ê2 + νtê1, ê1〉, 〈µD − ê1, νD − ê2〉, 〈F1ê1 +G1ê2, F2ê1 +G2ê2〉 (A.1.10) for e1, e2 belonging to algebras (A.1.6), and 〈µD − ê1, ê2〉, 〈µD + ê1 + νtê2, ê2〉 (A.1.11) for e1, e2 belonging to algebras (A.1.7). Here µ and ν are parameters which can take on any (including zero) finite values, {F1, G1} and {F2, G2} are fundamental solutions of the following system Ft = λF + νG, Gt = σF + γG (A.1.12) with arbitrary parameters λ, ν, σ, γ. The list (A.1.10)–(A.1.11) does not includes algebras spanned on the vectors 〈F ê, Gê〉 (with F,G satisfying (A.1.12)) and 〈µD + λeνt+ω·xê, eνt+ω·xê〉 which are either incompatible with classifying equations (4.6) or reduce to one-dimension algebras. In the following we ignore algebras A which include such subalgebras. All the other two-dimension algebras A can be reduced to one of the form given in (A.1.10), (A.1.11) using equivalence transformations (2.4), (6.7). A. G. Nikitin 195 There exist one more type of (m+2)-dimensional algebras A generated by two-dimension algebras (A.1.6), namely: 〈µD + ê1 + (αt+ λσρxσxρ)ê2, xν ê2, ê2〉 where ν, σ, ρ run from 1 to m. The related classifying equations generated by all symmetries x1ê2, x2ê2, · · · , xmê2 and ê2 coincides and we have the same number of constrains for f1, f2 as in the case of two-dimension algebras A. Up to equivalence there exist three realizations of three-dimension algebras in terms of matrices (A.1.3), (A.1.8): A3,1 : e1 = g̃1, e2 = g4, e3 = g̃3, A3,2 : e1 = g5, e2 = g4, e3 = g̃3, (A.1.13) A3,3 : e1 = g′1, e2 = g5, e3 = g̃3. (A.1.14) Non-zero commutators for matrices (A.1.13) and (A.1.14) are [e2, e3] = e3 and [e1, eα] = eα(α = 2, 3). The algebras of operators (5.2) corre- sponding to realizations (A.1.13) and (A.1.14) are of the following general forms: 〈µD − 2ê1, νD − 2ê2 − 2λtê3, ê3〉 (A.1.15) and 〈µD − 2ê1 − 2νtê2 − 2σtê3, ê2, ê3〉, 〈ê1, F1ê2 +G1ê3, F2ê2 +G2ê3〉 (A.1.16) respectively. In addition, we have the only four-dimension algebra Â4,1 : e1 = g̃1, e2 = g5, e3 = g̃3, e4 = g4 (A.1.17) which generates the following algebras of operators (5.2): 〈µD − 2ê1 − 2νtê2, ê2, ê3, ê4〉, 〈µD − 2ê1 − 2νtê3, ê2, ê3, ê4〉, 〈µD − 2ê1, νD − ê4, ê2, ê3〉. (A.1.18) Thus we have specified all low dimension algebras A which can be admitted by equations (1.3) with a diagonal (but not unit) matrix A. 196 Group Classification of Systems... A.2. Algebras A for Equations (1.3) with A12 6= 0 Consider equation (1.3) with matrix A of type II (refer to (2.3)) and find the corresponding algebras A. The related matrices (9.5) and (9.7) are g =   0 0 0 µ1 µ2 µ3 µ4 −µ3 µ5   , U =   1 0 0 b1 k1 k2 b2 −k2 k3   . (A.2.1) Up to equivalence transformations (9.6), (A.2.1) there exist three ma- trices g, namely g′1 =   0 0 0 0 1 0 0 0 1   , g5 =   0 0 0 1 0 0 0 0 0   , g6 =   0 0 0 0 µ −1 0 1 µ   (A.2.2) and three two-dimension algebras of matrices g (A.2.1): A2,7 = {g′1, g6}, A2,8 = {g5, g̃3}, (A.2.3) A2,9 = {g′1, g5} (A.2.4) where g̃3 is matrix (A.1.3) with λ = 0. Algebras (A.2.3) are Abelian while the basis elements of A2,9 satisfy commutation relations (A.1.9). Like in previous subsection we easily find the related basis elements of one-dimension algebras A in the form (A.1.4) and (A.1.5) for µ = 0. The two-dimension algebras A generated by (A.2.3) and (A.2.4) again are given by relations (A.1.10) and (A.1.11) respectively, where e1 and e2 are the first and second elements of algebras A2,7 −A2.9. In addition, we have two three-dimension algebras A3,3 : e1 = g′1, e2 = g5, e3 = g̃3; A3,4 : e1 = g5, e2 = g6, e3 = g̃3 (A.2.5) and the only four-dimension algebra: A4,2 : e1 = g′1, e2 = g6, e3 = g̃3, e4 = g5. (A.2.6) Algebra A3,4 generates algebras (A.1.16) while A3,5 corresponds to (A.1.15) with ν = 0. Finally, A4,2 generates the following algebras A 〈µD − 2ê1, νD − 2ê2, ê3, ê4〉, 〈ê1, ê2, eµt+ν·xê3, eµt+ν·xê4〉. (A.2.7) A. G. Nikitin 197 A.3. Algebras A for Equations (1.3) with Triangular Matrix A If matrix A belongs to type III given in (2.3) the related matrices (9.5) and (9.7) take the form g =   0 0 0 µ1 µ2 0 µ3 µ4 µ5   , U =   1 0 0 b1 k1 0 b2 k2 k3   . (A.3.1) There exist six non-equivalent matrices g, i.e., matrices g′1, g3, g5 (A.1.3), (A.2.2), and the following ones g7 =   0 0 0 0 1 0 0 1 1   , g8 =   0 0 0 0 0 0 0 1 0   , g9 =   0 0 0 1 0 0 0 1 0   . (A.3.2) In addition, we have six two-dimension algebras, A2,3 = {g5, g̃3}, A2,10 = {g′1, g8}, A2,11 = {g8, g̃3}, A2,12 = {g9, g̃3}, (A.3.3) A2,5 = {g′1, g3}, A2,13 = {g′1, g5}, (A.3.4) four three-dimension algebras: A3,3 : e1 = g′1, e2 = g5, e3 = g̃3, A3,5 : e1 = g8, e2 = g′1, e3 = g̃3, A3,6 : e1 = g̃3, e2 = g8, e3 = g9, A3,7 : e1 = g̃3, e2 = g5, e3 = g7 (A.3.5) and the only four-dimension algebra: A4,3 : e1 = g̃3, e2 = g5, e3 = g′1, e4 = g8. (A.3.6) Algebras (A.3.3) are Abelian while (A.3.4) are characterized by com- mutation relations (A.1.9). The related two-dimension algebras A are given by formulae (A.1.10) and (A.1.11) respectively. Algebra A3,3 generates three-dimension algebras A enumerated in (A.1.16). Algebra A3,5 is isomorphic to A3,1 and so we come to the related algebras A given in (A.1.15). Algebras A3,6 and A3,7 are characterized by the following non-zero commutators [e2, e3] = e1 (A.3.7) and [e1, e2] = e2, [e1, e3] = e2 + e3 (A.3.8) 198 Group Classification of Systems... respectively. Using (A.3.7) and (A.3.8) we come to the following related three- dimension algebras A : 〈µD − 2ê2, νD − 2ê3, ê1〉, 〈ê1, D + 2eα + 2νtê1, êα′〉, 〈eνt+ω·xê1, eνt+ω·xêα, êα′〉 (A.3.9) where α, α′ = 2, 3, α′ 6= α, and 〈µD − 2ê1, ê2, ê3〉, 〈ê1, eνt+ω·xê2, eνt+ω·xê3〉. (A.3.10) Finally, four-dimension algebras A corresponding to A4,3 have the following general form 〈µD − 2ê1, νD − 2ê2, ê3, ê4〉, 〈eνt+ω·xê1, eνt+ω·xê2, ê3, ê4〉 (A.3.11) A.4. Algebras A for Equations (1.3) with the Unit Matrix A Group classification of these equations appears to be the most com- plicated. The related matrices g are of the most general form (9.5) and defined up to the general equivalence transformation (9.6), (9.7). In other words there are seven non-equivalent matrices (9.5), namely, g1, g2 (A.1.3), g5 , g6 (A.2.2) and g7 − g9 (A.3.2). In addition, we have fifteen two-dimension algebras of matrices (9.5), A2,1 = {g̃1, g4}, A2,2 = {g̃1, g̃3}, A2,3 = {g̃3, g5}, A2,10 = {g7, g8}, A2,11 = {g̃3, g8}, A2,12 = {g̃3, g9}, A2,13 = {g′1, g6}, (A.4.1) A2,4 = {g1, g5}, A2,5 = {g′1, g3}, A2,6 = {g2, g̃3}, A2,14 = {g1|λ6=1, g8}, A2,15 = {g11,−g8}, A2,16 = {g9, g′′1}, A2,17 = {g4, g8}, A2,18 = {g7, g̃3} (A.4.2) where g10 =   0 0 0 0 1 0 1 0 0  , g′′1 = g1|λ=2 =   0 0 0 0 1 0 0 0 2   . Algebras (A.4.1) are Abelian while algebras (A.4.2) are characterized by relations (A.1.9). Three-dimension algebras are A3,1 −A3,7 given by relations (A.1.13), (A.2.5) and (A.3.3) (where tildes should be omitted) and also A3,8−A3,11 given below: A3,8 : e1 = g1, e2 = g8, e3 = g̃3, A3,9 : e1 = g4, e2 = g8, e3 = g̃3, A3,10 : e1 = g2, e2 = g8, e3 = −g̃3, A3,11 : e1 = g̃1, e2 = −g8, e3 = g̃4. A. G. Nikitin 199 Algebras (A3,8, A3,11) and A3,9 and A3,10 are isomorphic to A3,1 and A3,3 and A3,6 respectively. The related algebras A are given by relations (A.1.15), (A.1.16) and (A.3.9) correspondingly. Finally, four-dimension algebras of matrices (9.6) are A4,1 , A4,2 and A4,3 given by equations (A.1.17), (A.2.6) and (A.3.6), and also A4,4, A4,5 given below: A4,4 : e1 = g1, e2 = g4, e3 = g8, e4 = g3; A4,5 : e1 = g4, e2 = g8, e3 = g5, e4 = g3. Using found algebras and solving the related equations (4.6) we easily make the group classification of equations (1.3). References [1] J. F. R. Archilla et al, Symmetry analysis of an integrable reaction-diffusion equation // J. Phys. 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