Extended symmetry analysis of generalized Burgers equations
aa r X i v : . [ m a t h - ph ] A ug Extended symmetry analysisof generalized Burgers equations
Oleksandr A. Pocheketa † and Roman O. Popovych ‡† Center for evaluation of activity of research institutions and scientific support of regionaldevelopment of Ukraine, NAS of Ukraine, 54 Volodymyrska Str., 01030 Kyiv, Ukraine ‡ Wolfgang Pauli Institute, Oskar-Morgenstern-Platz 1, A-1090 Vienna, AustriaInstitute of Mathematics of NAS of Ukraine, 3 Tereshchenkivska Str., 01004 Kyiv, Ukraine
E-mail: † [email protected], ‡ [email protected] Using enhanced classification techniques, we carry out the extended symmetry analysis of theclass of generalized Burgers equations of the form u t + uu x + f ( t, x ) u xx = 0. This enhancesall the previous results on symmetries of these equations and includes the description of ad-missible transformations, Lie symmetries, Lie and nonclassical reductions, hidden symmetries,conservation laws, potential admissible transformations and potential symmetries. The studyis based on the fact that the class is normalized, and its equivalence group is finite-dimensional. The methods of group analysis have been comprehensively developed and applied to wide andcomplicated classes of differential equations. Nevertheless, there are some simple famous modelequations and classes of differential equations that have not been properly investigated from thesymmetry point of view yet.In this paper we exhaustively classify Lie symmetries and Lie reductions, reduction operators,conservation laws and potential symmetries of equations from the delightedly simple class u t + uu x + f ( t, x ) u xx = 0 with f = 0 , (1)which are called generalized Burgers equations. This enhances and essentially extends known re-sults on these equations. We combine modern methods of group analysis of differential equationswith original techniques, which include the algebraic method for group classification of normal-ized classes of differential equations [7, 49], the special technique of classifying appropriate sub-algebras [7, 30], the classification of Lie reductions for normalized classes of differential equationsup to equivalence transformations, the selection of optimal ansatzes for reduction [18, 19, 44],mappings between classes generated by families of point transformations [55] and the classifica-tion of reduction operators up to admissible transformations [52].In (1) and in what follows subscripts of functions denote derivatives with respect to thecorresponding variables. The consideration is within the local framework. The equation of theform (1) with a fixed f , which is assumed to be nonvanishing for all ( t, x )’s from the relateddomain, is denoted by L f .The classical Burgers equation L − µ with positive constant µ was suggested in the 1930sas a model for one-dimensional turbulence [11]. Among its most famous solutions there areso-called traveling waves. In [32] it is shown that, under proper interpretation, the same equa-tion describes the propagation of one-dimensional weak planar waves. A formal multiple-scalesmethod was employed to extend this model to weak cylindrical and spherical waves in [31].Thus, generalized Burgers equations are used to model a wide variety of phenomena in physics,chemistry, mathematical biology, etc.; see, e.g., [60, Chapter 4].As important but simple model, the Burgers equation was intensively studied and was usedas illustrative and toy example or standard test benchmark in the course of developing various1athematical concepts and methods, in particular, in the field of group analysis of differentialequations; see [3, 35] and references therein. This tradition spread to generalized Burgers equa-tions. The study of admissible transformations for the class (1) in [26] was a pioneer work onsuch transformations in the literature. It appeared that the equivalence groupoid constitutedby admissible transformations can be described in terms of normalization. More specifically, theclass (1) is normalized with respect to the usual equivalence group, which is six-dimensional,and this fact is of principal value for the entire consideration in this paper.Equations from the class (1) and similar classes were subjects of many papers. In particular,Lie symmetries and similarity solutions of equations of the form (1) with f x = 0 were consideredin [14, 59]. It was also shown in [59] that among these equations only the classical Burgersequation, for which f = const, admits nontrivial potential symmetries and regular reductionoperators inequivalent to Lie symmetries. More references and detail comments on studiesrelated to group analysis of generalized Burgers equations are given in the corresponding sectionsof this paper. Since an essential part of such results presented in the literature are not exhaustiveor completely correct, it is in fact necessary to carry out extended symmetry analysis of equationsfrom the class (1) from the very beginning.The structure of the paper is as follows. Basic notions of group analysis of differentialequations are briefly reviewed in Section 2. The equivalence groupoid, the equivalence group andthe equivalence algebra of the class (1) are computed in Section 3. In Section 4 we exhaustivelyclassify Lie symmetries of equations from the class (1) using the algebraic method of groupclassification. This method is especially effective for the class (1) due to this class is normalizedand its equivalence group is finite-dimensional. Section 5 is devoted to Lie reductions, hiddensymmetries and similarity solutions of generalized Burgers equations that admit nonzero Lieinvariance algebras. Section 6 deals with reduction operators and nonclassical reductions ofequations from the class (1). Conservation laws and potential symmetries of these equations aswell as potential admissible transformations between them are studied in Section 7. Implicationsof paper’s results are discussed in the last section. Recall the concepts of a class of differential equations, the equivalence groupoid and the equiv-alence group of a class, and others involved in the present study. As in this paper we considerthe class of equations of the simple form (1), we give definitions for the specific case of a class ofsingle second-order partial differential equations with the two independent variables ( t, x ) andthe single dependent variable u that are merely parameterized by the single arbitrary element f = f ( t, x, u ) without any derivatives of f . For the general definition of a class of systemsof differential equations involving a tuple of arbitrary elements with their derivatives see, e.g.,[7, 49]. We deal here with usual equivalence groups, unless another type of an equivalence groupis stated.Consider a family of differential equations L f : L f [ u ] := L ( t, x, u, u t , u x , u tt , u tx , u xx , f ) = 0parameterized by a parameter-function f = f ( t, x, u ) running through the set S of solutionsof an auxiliary system of differential equations and differential inequalities on f , where all thevariables t , x and u are assumed to be independent. Definition 1.
The set {L f | f ∈ S} denoted by L| S is called a class of differential equations ,that is defined by the parameterized form of equations L ( t, x, u, u t , u x , u tt , u tx , u xx , f ) = 0 andthe set S of values of the arbitrary element f .For the class (1), the form of equations and the set of arbitrary elements are L f [ u ] := u t + uu x + f u xx = 0 and S = (cid:8) f = f ( t, x, u ) | f u = 0 , f = 0 (cid:9) , f u t = f u x = f u tt = f u tx = f u xx = 0. These constraintswill be used implicitly, e.g. while solving determining equations.A point transformation in the space of ( t, x, u ) has the form ϕ : ˜ t = T ( t, x, u ) , ˜ x = X ( t, x, u ) , ˜ u = U ( t, x, u ) , (2)where T , X and U are smooth functions of t , x and u with | ∂ ( T, X, U ) /∂ ( t, x, u ) | 6 = 0. Giventwo fixed equations L f and L ˜ f from the class L| S with arbitrary elements f and ˜ f , by T ( f, ˜ f ) wedenote the set of point transformations in the space ( t, x, u ) that map L f to L ˜ f . An admissibletransformation [45, 49] in the class L| S is a triple consisting of two arbitrary elements f, ˜ f ∈ S (or, in other words, the corresponding two equations, which are called the initial one and thetarget one) and a point transformation ϕ ∈ T( f, ˜ f ).The notion of admissible transformation is a formalization of the earlier notions of form-preserving [26, 27], or allowed [22] transformations. Definition 2.
The equivalence groupoid G ∼ = G ∼ ( L| S ) of the class L| S is the set of admissibletransformations of this class, { ( f, ϕ, ˜ f ) | f, ˜ f ∈ S , ϕ ∈ T( f, ˜ f ) } , equipped with the operation“ ◦ ” of composition of admissible transformations.The composition “ ◦ ” of admissible transformations ( f , ϕ , ˜ f ) and ( f , ϕ , ˜ f ) is defined onlyif ˜ f = f , and its result is ( f , ϕ ϕ , ˜ f ). It is obvious that the axioms of groupoid hold for G ∼ :1. (( f , ϕ , f ) ◦ ( f , ϕ , f )) ◦ ( f , ϕ , f ) = ( f , ϕ , f ) ◦ (( f , ϕ , f ) ◦ ( f , ϕ , f )), which meansthe associativity of the composition.2. For each f the role of the neutral element is played by the triple ( f, id , f ), where id is theidentical transformation, ˜ t = t , ˜ x = x , ˜ u = u .3. Any admissible transformation ( f, ϕ, ˜ f ) is invertible, and the inverse is ( ˜ f , ϕ − , f ). Definition 3.
The usual equivalence group G ∼ = G ∼ ( L| S ) of the class L| S is the (pseudo)groupof point transformations in the extended space of ( t, x, u, f ), T : ˜ t = T ( t, x, u ) , ˜ x = X ( t, x, u ) , ˜ u = U ( t, x, u ) , ˜ f = Φ( t, x, u, f ) , that are projectable to the variable space ( t, x, u ) and map each equation from the class L| S toan equation from the same class.Each equivalence transformation T ∈ G ∼ generates a family of admissible transformations { ( f, T | ( t,x,u ) , T f ) | f ∈ S } ⊂ G ∼ , where T | ( t,x,u ) denotes the restriction of T to the space ( t, x, u ). Definition 4.
A class of differential equations L| S is called normalized if its equivalencegroupoid G ∼ is generated by its equivalence group G ∼ , meaning that for each triple ( f, ϕ, ˜ f )from G ∼ there exists a transformation T from G ∼ such that ˜ f = T f and ϕ = T | ( t,x,u ) .In the case of single dependent variable we might also consider contact transformations, see,e.g., [12]. They are of the form ˜ t = T ( t, x, u, u t , u x ), ˜ x = X ( t, x, u, u t , u x ), ˜ u = U ( t, x, u, u t , u x ),˜ u ˜ t = U t ( t, x, u, u t , u x ), ˜ u ˜ x = U x ( t, x, u, u t , u x ), where the transformation components satisfy thenondegenerate assumption (i.e., the corresponding Jacobi matrix is nondegenerate) and the con-tact condition (meaning the consistence with the contact structure), and thus the components U t and U x are defined via the chain rule. At the same time, any contact transformation betweenany two second-order evolution equations that are linear with respect to the second derivativeis induced by a specific point transformation, cf. [51, Proposition 2] and [56, Section 2]. Thisis why all contact transformations between equations from the class (1) are exhausted by theprolongations of point admissible transformations, and thus the problem of describing contacttransformations in the class (1) is reduced to that for point transformations.3 Equivalence groupoid and equivalence group
It is possible to find not only the equivalence group and the equivalence algebra of the class (1)but also its entire equivalence groupoid. In an implicit form, this groupoid was described in thepioneer paper [26], which is, actually, the first study of the set of admissible transformations ina class of differential equations. Earlier, in [13], a conformal transformation was found betweenequations of the form (1) with f x = 0. Theorem 5.
The class (1) is normalized in the usual sense. The usual equivalence group G ∼ of the class (1) consists of the transformations ˜ t = αt + βγt + δ , ˜ x = κx + µ t + µ γt + δ , ˜ u = κ ( γt + δ ) u − κγx + µ δ − µ γαδ − βγ , (3)˜ f = κ αδ − βγ f, (4) where α , β , γ , δ , µ , µ and κ are arbitrary constants that are defined up to a nonzero multiplier, αδ − βγ = 0 and κ = 0 .Proof. We fix any two equations from the class (1), L f : u t + uu x + f ( t, x ) u xx = 0 and L ˜ f :˜ u ˜ t + ˜ u ˜ u ˜ x + ˜ f (˜ t, ˜ x )˜ u ˜ x ˜ x = 0, and find all point transformations, which are of the form (2), betweenthese two equations. For this purpose, we substitute all the tilded variables and derivatives withtheir expressions in terms of untilded values in L ˜ f , including˜ u ˜ t = 1D t T (cid:18) D t U − D x U D t X D x X (cid:19) , ˜ u ˜ x = D x U D x X , ˜ u ˜ x ˜ x = D x (cid:18) D x U D x X (cid:19) , where D t = ∂ t + u t ∂ u + u tt ∂ u t + u tx ∂ u x + . . . and D x = ∂ x + u x ∂ u + u tx ∂ u t + u xx ∂ u x + . . . are theoperators of total derivatives with respect to t and x . The obtained equation should be satisfiedby all solutions of L f . This is why the equality derived by substituting u t with − uu x − f u xx in view of L f can be split with respect to u x and u xx , which gives the system of determiningequations for the transformation components T , X and U .The computation can be simplified by taking into account the specific structure of generalizedBurgers equations, which are second-order quasilinear evolution equations, i.e., they are linearwith respect to the derivative u xx . In view of [23, Lemma 1], each point transformation betweenthe equations L f and L ˜ f is projectable both on the space of t and on the space of ( t, x ), ˜ t = T ( t ) , ˜ x = X ( t, x ) , ˜ u = U ( t, x, u ) , where T t X x U u = 0. The determining equations can be reduced to U uu = 0 , and hence U = U ( t, x ) u + U ( t, x ) , ˜ f = X x T t f, U = X x T t , U = X t T t , X xx = 0 , U t + U x = 0 , U t = 0 . The equivalence groupoid G ∼ of the class (1) is established after solving these determiningequations. The elements of G ∼ are defined by (3), where the initial and the resulting valuesof the arbitrary element are connected by (4). Each transformation of the form (3) mapsany equation from the class (1) to an equation from the same class, and its prolongation to thearbitrary element f , which is given by (4), is a point transformation in the joint space ( t, x, u, f ).Hence such prolongations of the transformations of the form (3) according to (4) constitute theequivalence group G ∼ of the class (1). Since any element of G ∼ is induced by an equivalencetransformation, this class is normalized. Throughout the paper, a linear function means a polynomial of degree one or zero. We can also use results of [39] for the narrower superclass u t + F ( t, x, u ) u xx + H ( t, x, u ) u x + H ( t, x, u ) = 0,which gives more constraints for the transformation components, but the property of double projectability itselfsufficiently simplifies the computation. G ∼ is singled out by the inequal-ities αδ − βγ > κ >
0. Up to composing with each other and with continuous equivalencetransformations, discrete equivalence transformations are exhausted by alternating signs in thetuples ( t, u, f ) and ( x, u ).The Lie algebra g ∼ = h ˜ P t , ˜ P x , ˜ D t , ˜ D x , ˜ G, ˜Π i that corresponds to the equivalence group G ∼ is called the equivalence algebra of the class (1). Its basis elements may be chosen as˜ P t = ∂ t , ˜ P x = ∂ x , ˜ D t = t∂ t − u∂ u − f ∂ f , ˜ D x = x∂ x + u∂ u + 2 f ∂ f , ˜ G = t∂ x + ∂ u , ˜Π = t ∂ t + tx∂ x + ( x − tu ) ∂ u . The elementary one-parameter transformations from the group G ∼ that correspond to thesebasis elements areˆ P t ( β ) : ˜ t = t + β, ˜ x = x, ˜ u = u, ˜ f = f, ˆ P x ( µ ) : ˜ t = t, ˜ x = x + µ , ˜ u = u, ˜ f = f, ˆ D t ( α ) : ˜ t = αt, ˜ x = x, ˜ u = 1 α u, ˜ f = 1 α f, ˆ D x ( κ ) : ˜ t = t, ˜ x = κx, ˜ u = κu, ˜ f = κ f, ˆ G ( µ ) : ˜ t = t, ˜ x = x + µ t, ˜ u = u + µ , ˜ f = f, ˆΠ( γ ) : ˜ t = tγt + 1 , ˜ x = xγt + 1 , ˜ u = ( γt + 1) u − γx, ˜ f = f. (5)These transformations are deduced from (3)–(4) by setting all the constants (except one, whichis present in a transformation) with the values that correspond to the identity transformation, α = 1 , β = 0 , γ = 0 , δ = 1 , κ = 1 , µ = 0 , µ = 0 . The equivalence algebra g ∼ can be found directly using the infinitesimal Lie method, ina similar way as finding Lie symmetries of single systems of differential equations [2, 37]. Infact, this is not needed since, knowing the complete equivalence group G ∼ , we can construct thealgebra g ∼ as the set of infinitesimal generators of one-parameter subgroups of G ∼ , cf. [30].The projection of g ∼ to the space ( t, x, u ) is the algebra g = h P t , P x , D t , D x , G, Π i ≃ g ∼ with P t = ∂ t , P x = ∂ x , D t = t∂ t − u∂ u , D x = x∂ x + u∂ u ,G = t∂ x + ∂ u , Π = t ∂ t + tx∂ x + ( x − tu ) ∂ u . Both the algebras g ∼ and g are realizations of the so-called reduced (i.e., centerless) full Galileialgebra [15] with space dimension one, which is isomorphic to the affine Lie algebra aff(2 , R ) [22].The nonzero commutation relations of g are[ P t , D t ] = P t , [ D t , Π] = Π , [ P t , Π] = 2 D t + D x , [ P x , D x ] = P x , [ P x , Π] = G, [ P t , G ] = P x , [ D t , G ] = G, [ G, D x ] = G. The Levi decomposition of the algebra g is g = h P t , D t + D x , Π i ∈ h D x , P x , G i . Here thesubalgebra f = h P t , D t + D x , Π i is a Levi factor of g , which is a realization of the algebra sl(2 , R ).The radical r = h D x , P x , G i of g is a realization of the algebra A , from the Mubarakzyanov’slist of low-dimensional real algebras [34] (see also [47]), which is the almost abelian algebraassociated with the 2 × r = c ∈ n , where n = h P x , G i isthe nilradical (as well as the maximal abelian ideal) of both r and g , and the span c = h D x i isa Cartan subalgebra of r . By pr f and pr c we denote the projectors defined by the decomposition g = f ∈ ( c ∈ n ).As the class (1) is normalized, the algebra g contains the union of Lie invariance algebras ofall equations L f from the class (1). Moreover, the algebra g appears to coincide with this union,cf. Section 4, which displays the strong normalization of the class (1) in the infinitesimal sense.5 Lie symmetries
Classical symmetry analysis of the classical Burgers equation and its generalizations related tothe class (1) has been carried out since the 1960s. The maximal Lie invariance group of theBurgers equation was computed by Katkov [25] in the course of group classification of equationsof the general form u t + uu x = ( f ( u ) u x ) x . The maximal Lie invariance algebra of the Burgersequation is five-dimensional and is spanned by the vector fields P x , G , P t , D t + D x and Π,cf. the algebra g in Table 1 below.The group classification problem for the subclass of (1) singled out by the constraint f x = 0was considered in [14, 59] without proper use of equivalence transformations. As a result, theclassification lists presented there contain needless cases, which was already remarked in [26]concerning [14]; see also footnote 4 in [43]. The same remark is true for the group classificationof equations of the general form u t + g ( t, x ) uu x + f ( t, x ) u xx = 0 with f g = 0 in [54]. Moreover,classification cases were unnecessarily split into several subcases depending on the structure ofsymmetry algebras therein, and, as far as it can be analyzed, some classification cases weremissed, in particular, due to over-gauging parameters, which is not allowed. A vector field that generates a one-parameter Lie symmetry group of an equation L f : L f [ u ] = 0from the class (1) is of the form Q = τ ( t, x, u ) ∂ t + ξ ( t, x, u ) ∂ x + η ( t, x, u ) ∂ u and satisfies theinfinitesimal invariance criterion Q (2) L f [ u ] (cid:12)(cid:12) L f ≡ (cid:0) η t + ηu x + η x u + ξf x u xx + τ f t u xx + η xx f (cid:1)(cid:12)(cid:12) L f = 0 , (6)where Q (2) is the usual second-order prolongation of Q [35, 37], and η t , η x , η xx are prolongationcomponents, which are computed by η t = D t η − u t D t τ − u x D t ξ , η x = D x η − u t D x τ − u x D x ξ and η xx = D x η − u t D x τ − u tx D x τ − u x D x ξ − u xx D x ξ . Using the restriction L f [ u ] = 0, wesubstitute u t = − uu x − f u xx for u t and then split the result with respect to u tx , u xx , u x and u .After simplifying we obtain a system of determining equations on the components τ , ξ and η , τ x = 0 , τ u = 0 , ξ u = 0 , η uu = 0 , η x = 0 , (7) η − ξ t = 0 , η + τ t − ξ x = 0 , η t = − η x , η t = 0 , (8) τ f t + ξf x + ( τ t − ξ x ) f = 0 . (9)Equations (7) imply τ = τ ( t ), ξ = ξ ( t, x ) and η = η ( t ) u + η ( t, x ). Then making use ofequations (8) we specify the form of the components of Q , τ = c t + c t + c , ξ = ( c t + c ) x + c t + c , η = ( − c t + c − c ) u + c x + c , (10)where c , . . . , c are arbitrary constants. In view of results of Section 3, we could postulate theform (10) for the components of Lie symmetry vector fields of equations from the class (1) fromthe very beginning. Indeed, the normalization of the class (1) proved in Theorem 5 implies thatthe maximal Lie invariance algebra g f of any equation L f from the class (1) is contained by thealgebra g , and the components of any vector field from g are of the form (10). Moreover, for anyconstant tuple ( c , . . . , c ) the equation (9) has a nonzero solution for f . This means that eachelement of the algebra g is a Lie symmetry of an equation from the class (1), i.e. g = S f ∈S g f .The equation (9) is the only classifying condition for Lie symmetries of equations from theclass (1). Depending on values of the arbitrary element f , the classifying condition imposesadditional constraints for the constants c , . . . , c . Varying f and splitting (9) with respectto f and its derivatives, we get c = · · · = c = 0. Therefore, the kernel invariance algebra ofthe class (1), i.e. the intersection of the maximal Lie invariance algebras of equations from thisclass, is { } . 6 .2 Appropriate subalgebras As the class (1) is normalized (see Theorem 5) and its equivalence algebra g ∼ is finite-dimensional, it is convenient to carry out its group classification using the algebraic method.Although one could solve the group classification problem for the class (1) using the directmethod, the algebraic method is much more effective on both the steps of computing and ar-ranging classification cases, in particular, checking their inequivalence; cf. [55, p. 3]. Recall thatthe normalization of the class (1) has two consequences: • The maximal Lie invariance algebra of every equation from this class is contained in theprojection g of the equivalence algebra g ∼ to the space ( t, x, u ). • Equations L f and L ˜ f from the class (1) are similar with respect to point transformationsif and only if they are G ∼ -equivalent.Therefore, in order to obtain the exhaustive group classification of the class (1), it suffices toconstruct a list of inequivalent appropriate subalgebras of g and then to find the correspondingvalues of the arbitrary element f for each subalgebra from the list. We call a subalgebra s ⊂ g appropriate if s is the maximal Lie invariance algebra of an equation L f from the class (1).In other words, a subalgebra s of g is appropriate if there exists a value f of the arbitraryelement f such that the following conditions hold:1. The components of every Q ∈ s satisfy the classifying condition (9) with f = f or,equivalently, f is an invariant of a subalgebra ˜ s ⊂ g ∼ whose projection to the space of( t, x, u ) coincides with s .2. The algebra s is maximal among the Lie invariance algebras of the equation L f .We classify appropriate subalgebras of the algebra g up to the equivalence relation generatedby the adjoint action of the group G ∼ on g . See, e.g., [35, Chapter 3.3], [37, Section 14.7] or[7, 8] for relevant elementary techniques and [38] for more sophisticated methods. The radical r and the nilradical n are megaideals (i.e., fully characteristic ideals) of g and hence they are G ∼ -invariant. To characterize classification cases, with any subalgebra s of g we associate the G ∼ -invariant values dim s ∩ r , dim s ∩ n , dim pr f s and dim pr c s . The adjoint actions of theelementary equivalence transformations (5) on the basis vector fields of g are as follows: Ad P t P x D t D x G Πˆ P t ( β ) P t P x D t − βP t D x G − βP x Π − β (2 D t + D x )+ β P t ˆ P x ( µ ) P t P x D t D x − µ P x G Π − µ G ˆ D t ( α ) αP t P x D t D x α − G α − Πˆ D x ( κ ) P t κP x D t D x κG Πˆ G ( µ ) P t + µ P x P x D t + µ G D x − µ G G
ΠˆΠ( γ ) P t − γ (2 D t + D x )+ γ Π P x − γG D t − γ Π D x G Π The subalgebras of an algebra isomorphic to aff(2 , R ) were first classified in [22] with respect to the groupof internal automorphisms. Parameterized families of inequivalent subalgebras were additionally partitioneddepending on their algebraic structure, which is unnecessary for the group classification of the class (1). Moreover,the list of subalgebras obtained therein is large. It consists of 44 families of subalgebras, and most of themare not appropriate as maximal Lie invariance algebras of equations from the class (1). Since the proof of theclassification was not presented, it is impossible to check its correctness at a glance. This is why we classifyappropriate subalgebras of g independently, without using the results of [22], which is much easier than theclassification of all subalgebras of g . The list of inequivalent (nonzero) appropriate subalgebras presented inTable 1 below includes only 19 families of subalgebras.
7o efficiently recognize inequivalent appropriate subalgebras of g , we consider their projec-tions on the Levi factor f . These projections are necessarily subalgebras of f , and, moreover, theprojections of equivalent subalgebras of g are equivalent as subalgebras of f . A complete list ofinequivalent subalgebras of the algebra sl(2 , R ) is well known. In terms of the realization f , it isexhausted by { } , h P t i , h D t + D x i , h P t + Π i , h P t , D t + D x i and f itself. Considering each ofthe listed subalgebras of f as a projection of an appropriate subalgebra, we try to add elementsof the radical r to the basis elements of this subalgebra, and to additionally extend the basis byelements from the radical r .Some properties of appropriate subalgebras of g directly follow from the classifying condi-tion (9). Below by s we denote an appropriate subalgebra of g . Lemma 6. If s ∩ n = { } , then s ∩ r = n .Proof. The condition s ∩ n = { } implies that the algebra s contains a vector field aG + bP x with constants ( a, b ) = (0 , τ = 0 and ξ = at + b corresponding to aG + bP x into (9), we get f x = 0. For such f , both the pairs ( τ, ξ ) = (0 ,
1) and ( τ, ξ ) = (0 , t )solve (9), and the pair ( τ, ξ ) = (0 , x ) is not a solution since f = 0. Therefore, the algebra s contains both P x and G and does not contain D x . Corollary 7. s ∩ r ∈ (cid:8) { } , c , n (cid:9) . Lemma 8. If s ∩ r = c , then s ⊂ f ⊕ c and dim s .Proof. Suppose that s * f ⊕ c and thus s \ ( f ⊕ c ) = ∅ . Each element of this set differenceis of the form Q = a P t + a D t + a Π + a D x + a G + a P x with ( a , a ) = (0 , D x , Q ∈ s , we have [ Q, D x ] = a G + a P x ∈ s . Then Lemma 6 implies that D x s , which givesa contradiction. Therefore, s ⊂ f ⊕ c .If dim s >
2, then dim s ∩ f > G ∼ -equivalence, s ⊇ h P t , D t + D x , D x i .Therefore, we also get s ⊇ h P t , D t i . The substitution of the values ( τ, ξ ) = (1 ,
0) and ( τ, ξ ) =( t,
0) corresponding to P t and D t , respectively, into (9) leads to a system on f that is inconsistentwith the constraint f = 0 for the arbitrary element f . The obtained contradiction means thatdim s s ∩ f Corollary 9. If s ∩ r = c , then s ∈ (cid:8) h D x i , h D x , P t i , h D x , D t + D x i , h D x , P t + Π i (cid:9) mod G ∼ . This gives the subalgebras g . , g . – g . of Table 1, respectively. Lemma 10. If s ∩ r = n , then either dim s and s ∩ f = { } or s = f ∈ n .Proof. Suppose that dim s >
3. Then dim pr f s >
2, i.e., modulo G ∼ , the algebra s containsthe vector fields Q = P t + aD x and Q = D t + bD x with some constants a and b . Therefore,the commutator [ Q , Q ] = P t also belongs to s , and the classifying condition (9) in view ofinvolving P x and P t implies that f = const.In the same way, the condition f = const is derived in the case s ∩ f = { } since aftersubstituting the components of P x and of any nonzero element of f into (9) we obtain theequations f x = 0 and f t = 0.The maximal Lie invariance algebra of the equation L f with f = const, which is the classicalBurgers equation, is the five-dimensional algebra s = f ∈ n . Corollary 11. If s ∩ r = n , then s = s f ∈ n , where s f ∈ (cid:8) { } , h P t + D x i , h D t + aD x i , h P t + Π + aD x i , f (cid:9) mod G ∼ , the parameter a runs through R \ { } , and a > G ∼ . g . , g . – g . and g of Table 1,respectively. Corollary 12.
The dimension of any appropriate subalgebra of g is not greater than . Below we consider the last case for s ∩ r , s ∩ r = { } . Then we obviously have dim s
3. Infact, the upper bound for dim s can be lowered. Lemma 13. If s ∩ r = { } , then dim s . Moreover, if additionally dim s = 2 , then pr c s = c and s = h P t , D t i mod G ∼ .Proof. Suppose that dim s > c s = { } . Modulo G ∼ -equivalence, we can assume thatpr f s ⊇ h P t , D t + D x i . In view of the classifying condition (9), the invariance of L f with respectto s then implies f = const. Recall that the maximal Lie invariance algebra of the equation L f for any (nonzero) constant f contains n , which contradicts the condition s ∩ r = { } . Therefore,pr c s = c if dim s > s = 3. Therefore, pr f s = f and hence s ≃ sl(2 , R ), i.e. s is a Levi factorof g . Then the Levi–Malcev theorem (or the direct computation of commutation relations of s )implies pr c s = { } . This contradicts the above conclusion that pr c s = c if dim s > s ⊇ h P t , D t i implies f = 0, which contradicts the originalinequality f = 0 for the arbitrary element f .If dim s = 1, then s = h Q f + a D x + a G + a P x i , where Q f ∈ (cid:8) P t , D t , P t + Π (cid:9) mod G ∼ and a , a and a are constants. Consider Q f = P t . First suppose that pr c s = { } and thus a = 0.The coefficient a is gauged to zero by ˆ G ( − a ), and a ∈ { , } up to scaling ˆ D t ( a − ) for a = 0.If pr c s = c , i.e. a = 0, then we scale a to 1 by ˆ D t ( a ) and by a subsequent rescaling of the basisvector field. Then we set the modified parameters a and a to zero by ˆ G ( a ) and ˆ P x ( a + a ).The other two G ∼ -inequivalent values of Q f are studied in a similar way. For the coefficient a ,we can then only alternate the sign of a − or a , respectively. The coefficients a and a canbe set to zero by ˆ G ( µ ) and ˆ P x ( µ ) for some µ and µ , except for Q f = D t and a ∈ { , } where the nonzero value a (resp. a ) can be only scaled to one if a = 0 (resp. if a = 1). Intotal, this results in the subalgebras g . – g . a .In the case dim s = 2, up to G ∼ -equivalence we have that the subalgebra s is spanned bytwo vector fields of the form Q = P t + b D x + b G + b P x and Q = D t + a D x + a G + a P x with some constants a , a , a , b , b and b . The commutation relation for s is [ Q , Q ] = Q .Expanding it and collecting the coefficients of D x , we derive b = 0 and hence a = sincepr c s = c . Then collecting the coefficients of G and P x leads to the equations ( a − b = 0and ( a − b + a = 0, respectively. We set b to zero by ˆ G ( − b ), and hence also a = 0.The coefficient b is zero if a = 2, and its nonzero value is scaled to one. For a = 0 wecan set a = 0 by ˆ P x ( − a /a ), and for a = 0 the nonzero value of a is scaled to one. Thesimultaneous vanishing a = a = 0 is not possible in view of Lemma 13. Therefore, in thiscase we obtain the subalgebras g . – g . , which completes the computation of G ∼ -inequivalentappropriate subalgebras of g . In order to compute the associated values of the arbitrary element f , for each of the listedinequivalent subalgebras of g we substitute the components τ and ξ of its basis vector fields intothe classifying condition (9) and then solve the obtained system of differential equations on f .This system does have solutions and, moreover, at least for a subset of its solutions the involvedsubalgebra of g is the maximal Lie invariance algebra for the corresponding equations from theclass (1). This means that the collection of properties of appropriate subalgebras derived inSection 4.2 gives a necessary and sufficient condition for a subalgebra of g to be appropriate.9 complete list of inequivalent appropriate subalgebras of g and the corresponding valuesfor f is presented in Table 1. Since the class (1) is normalized, the table provides its exhaustivegroup classification. Table 1: The group classification of the class (1) s ⊂ g Basis of g f f ( t, x ) ω Constraints g . D x x h ( ω ) t (cid:0) ( αω + βω + γ ) h (cid:1) ω = 0 g . P t h ( ω ) x ( αω + β ) h ω = γh g . P t + G h ( ω ) x − t αω + β ) h ω = γh g . P t + D x e t h ( ω ) e − t x h ω = 0 , ωh ω = 2 h g . D t + P x t h ( ω ) x − ln | t | h ω = 0 , h ω = − h g . a D t + aD x | t | a t h ( ω ) | t | − a x see table notes; a > mod G ∼ g . D t + D x + G t h ( ω ) xt − ln | t | h ω = 0 , h ω = − h g . a P t + Π + aD x e a arctan t h ( ω ) e − a arctan t √ t + 1 x ωh ω = 2 h ; a > G ∼ g . P x , G h ( ω ) t ( αω + βω + γ ) h ω = δh g . D x , P t x g . D x , D t κ x t κ = 0, κ > G ∼ g . D x , P t + Π κ x t + 1 κ = 0, κ > G ∼ g . P t , D t + P x e − x g . a P t , D t + aD x | x | − /a a = 0 , g . P t + G, D t + 2 D x κ (cid:12)(cid:12)(cid:12)(cid:12) x − t (cid:12)(cid:12)(cid:12)(cid:12) / κ = 0, κ > G ∼ g . P x , G, P t + D x εe t g . a P x , G, D t + aD x ε | t | a − a = , a > mod G ∼ g . a P x , G, P t + Π + aD x εe a arctan t a = 0 , a > G ∼ g P x , G, P t , D t + D x , Π 1Here ε = ± G ∼ . The constants a and κ and the (nonvanishing) function h should satisfy constraints inthe last column for the expression of f to be well defined and for the corresponding Lie invariance algebra to bemaximal. If possible, we gauge the constants a and κ by equivalence transformations, which is also indicated inthe last column. For the algebras g . , g . , g . and g . , the constants α , β , γ and δ involved in the correspondinginequalities for h are arbitrary but are not simultaneously zeros. The constraints for g . a are h ω = 0 and a ( ω + α ) h ω = (2 a − h if ( a − a − α = 0 , ( ω + β ) h ω = 2 h if ( a − aβ = 0 , ( a − ω + γ ) h ω = (2 a − h if a ( a + 1) γ = 0 . Classical similarity solutions
The solution of the group classification problem for a class of differential equations can beused for finding exact solutions of equations from the class. The standard procedure for thispurpose starts with classifying subalgebras of the maximal Lie invariance algebra of each equationlisted in the course of group classification. Then, using invariants of obtained inequivalentsubalgebras, one constructs ansatzes for the unknown function and derives the correspondingreduced equations. In general, reduced equations are simpler for solving than their originalcounterparts since they have less number of independent variables. The last step of the procedureis to construct at least particular solutions of reduced equations, which gives exact solutions ofthe corresponding original equations.In order to optimize the reduction process for equations from the class (1), we exploit twospecial reduction techniques.The first technique is available due to the class (1) is normalized. Roughly speaking, thistechnique can be characterized as the classification of Lie reductions with respect to the equiv-alence group G ∼ of the whole class, rather than with respect to the Lie symmetry group of theequation to be reduced. Thus, this technique is related to the algebraic method of group classifi-cation. Since the class (1) is normalized, the projection of G ∼ to the space ( t, x, u ) contains thepoint symmetry groups of all equations from the class (1), and hence the maximal Lie invariancealgebras of these equations are subalgebras of the projection g of the equivalence algebra g ∼ .Recall that equations L f and L ˜ f from the class (1) are similar with respect to a point trans-formation if and only if they are G ∼ -equivalent. The similarity of equations L f and L ˜ f impliesthe equivalence of their maximal Lie invariance algebras g f and g ˜ f and establishes a one-to-onecorrespondence between the sets of subalgebras of these algebras. Subalgebras of g f and g ˜ f areobviously subalgebras of the algebra g . So, it suffices to classify inequivalent subalgebras of thealgebra g (cf. Section 4.2), that are appropriate for Lie reduction of equations from the class (1).This approach allows us to avoid the separate implementation of the Lie reduction procedurefor each of the nineteen classification cases listed in Table 1.The second technique, which was systematically used in [18, 19] and discussed in [44], isto construct ansatzes in such a way that reduced equations are of a simple and similar form.Thus, the algebras g . and g . give trivial first-order ordinary differential equations. Reducedequations constructed using the algebras g . – g . a are of order two. For all these algebras, wechoose the invariant independent variable ω linear in x with coefficients dependent at most on t ,and the general form of ansatzes is u = F ( t ) ϕ ( ω )+ G ( t, x ) with G xx = 0. Here the intention is tomake reduced equations of the same general form (11). After constructing intermediate ansatzesand the correspondent reduced equations, in some cases it is necessary to change the invariantdependent variable ϕ , e.g. ϕ = φ + 1, in order to push all second-order reduced equations intothe class (11).Note that classical Lie reductions of equations from the class (1) were carried out earlier onlyfor the subclass with f x = 0 [14, 59] with some weaknesses and were later enhanced in [43]. Up to More specifically, in [14] optimal systems of subalgebras were constructed for the corresponding maximal Lieinvariance algebras. These subalgebras were used for finding ansatzes for u and reduced ordinary differentialequations. At the same time, the consideration was needlessly overcomplicated since the cases of Lie symmetryextensions were not simplified by point equivalence transformations, and two cases are equivalent to others withrespect to point transformations. Some of the optimal systems are incorrect, cf. [43, footnote 7]. Moreover,no reduced equations were integrated. In [59], Lie reductions were performed only with respect to the one-dimensional subalgebras spanned by single basis elements, not to mention the presence of equivalent cases andneedless parameters in the classification list. The reduced equation (95) in [59] contains two misprints and shouldin fact read as F f ′′ + ff ′ + mz λf ′ − mz f + z λ = 0, cf. [43, footnote 8]. The further integration procedure is notapplicable to the correct version of the reduced equation, and the functions (99)–(101) in [59] do not satisfy thecorresponding generalized Burgers equation. The only nontrivial solutions (91)–(93) presented in [59] look, up toan equivalence transformation, like particular cases of the solution (15) for the first value of ˜ ω in (16), ˜ ω = ω/ν . ∼ -equivalence, Lie symmetry extensions in this subclass are exhausted by the algebras g . , g . , g . , g . and g of Table 1.Reduced equations for all possible G ∼ -inequivalent one-dimensional subalgebras of g arepresented in Table 2. The case with g . in this table corresponds to the case with g . inTable 1; cf. Lemma 6.Table 2: Lie reductions with respect to one-dimensional subalgebras of g ⊂ g Basis Ansatz, ϕ = ϕ ( ω ) ω Reduced equation g . P x u = ϕ t ϕ ω = 0 g . D x u = xϕ t ϕ ω + ϕ = 0 g . P t u = ϕ x h ( ω ) ϕ ωω + ϕϕ ω = 0 g . P t + G u = ϕ + t x − t h ( ω ) ϕ ωω + ϕϕ ω + 1 = 0 g . P t + D x u = e t ϕ e − t x h ( ω ) ϕ ωω + ϕϕ ω − ωϕ ω + ϕ = 0 g . D t + P x u = t − ϕ x − ln | t | h ( ω ) ϕ ωω + ϕϕ ω − ϕ ω − ϕ = 0 g . a D t + aD x u = | t | a t − ϕ | t | − a x h ( ω ) ϕ ωω + ϕϕ ω − ωϕ ω + ( a − ϕ = 0 g . D t + D x + G u = ϕ + ln | t | xt − ln | t | h ( ω ) ϕ ωω + ϕϕ ω − ( ω + 1) ϕ ω + 1 = 0 g . a P t + Π + aD x u = e a arctan t √ t + 1 ϕ + t + at + 1 x e − a arctan t √ t + 1 x h ( ω ) ϕ ωω + ϕϕ ω + 2 aϕ + ( a + 1) ω = 0 In order to solve the second-order reduced equations listed in Table 2, we consider the super-class of ordinary differential equations of the form h ( ω ) φ ωω + φφ ω + αφ + βω + γ = 0 with h ( ω ) = 0 , (11)which contains all the reduced equations (except those for g . and g . ). The change of thevariable ϕ (if needed) and the values of the constants α , β and γ for them are as follows: g . : α = 0 , β = 0 , γ = 0 , ϕ = φ ; g . : α = 0 , β = 0 , γ = 1 , ϕ = φ ; g . : α = 2 , β = 1 , γ = 0 , after the change ϕ = φ + ω ; g . : α = − , β = 0 , γ = − , after the change ϕ = φ + 1; g . a : α = a, β = a − , γ = 0 , after the change ϕ = φ + ω ; g . : α = 1 , β = 0 , γ = 1 , after the change ϕ = φ + ω + 1; g . a : α = 2 a, β = a + 1 , γ = 0 , ϕ = φ. Linear solutions of reduced equations of the form (11), as well as all solutions of the reducedequations for g . and g . , lead to solutions of equations from the class (1) that are linear withrespect to x . The solutions being linear with respect to x are only common for all equationsfrom the class (1) and are exhausted by the two families, u = c and u = ( x + c ) / ( t + c ), where c , c and c are arbitrary constants. They also arise in Section 6.1 within the framework ofreduction operators.We find Lie symmetries of ordinary differential equations from the class (11) and use them forsolving reduced equations presented for the subalgebras g . – g . a from Table 2. For these sym-metries to be well interpreted as symmetries of reduced equations, equivalence transformationsbetween equations from the class (11) are not involved in the consideration.12 roposition 14. The values of the arbitrary elements h , α , β and γ that correspond to equationsfrom the class (11) with nonzero maximal Lie invariance algebras, h , are exhausted by1. h = h (cid:0) ω + γβ (cid:1) , β = 0 : h = h (cid:0) ω + γβ (cid:1) ∂ ω + φ∂ φ i ;2. h = h , β = 0 , γ = 0 : h = h ∂ ω i ;3. h = h | ω + µ | / , α = β = 0 , γ = 0 : h = h ω + µ ) ∂ ω + φ∂ φ i ;4. h = − α ω + µω + ν , β = γ = 0 : h = h h∂ ω − αh∂ φ , − (cid:0) h R d ωh (cid:1) ∂ ω + (cid:0) φ + αh R d ωh + αω − µ (cid:1) ∂ φ i ;5. h ωω = κh − α , β = γ = 0 : h = h h∂ ω + ( κ − αh ) ∂ φ i ;6. h ωω + α ( h ω + αω + µ ) = κh , β = γ = 0 : h = h ξ∂ ω + ( φ − αξ + αω + µ ) ∂ φ i .Here h , µ , ν and κ are arbitrary constants with h = 0 and κ = 0 , and ξ = h ω + αω + µh ωω + α .Proof. Any Lie symmetry operator of an equation from the class (11) has the general form ξ ( ω ) ∂ ω + [ c φ − αξ ( ω ) + αc ω + c ] ∂ φ , where c and c are arbitrary constants and the component ξ = ξ ( ω ) satisfies the classifying equations( ξ ω − c )( βω + γ ) + βξ = 0 , (12) ξ ω − h ω h ξ = − c , (13) hξ ωω = − αξ + c αω + c , and hence ( h ωω + α ) ξ = c h ω + c αω + c . (14)Both the forms of the equation (14) are equivalent when (13) holds, and are useful for the furtherclassification. The simplest way to solve the system (12)–(14) is to start its integration fromequation (12) considering the cases β = 0 and β = 0 separately.For β = 0, the equation (12) implies ξ = c ( βω + γ ) β + bβω + γ . Here and below b denotes an integration constant. The assumption b = 0 leads to a contradiction.As a result, b = 0, i.e. ξ = c ( ω + γβ ), and thus ξ ωω = 0. Hence from the first form of (14) weget c = c αγβ . Since the algebra h is supposed to be nonzero, it should contain a vector fieldwith a nonzero value of c . Therefore, from (13) we have ( ω + γβ ) h ω = 2 h , which gives item 1 ofthe proposition.If β = 0 and γ = 0, then from (12) we obtain ξ = 2 c ω + b and split the first form of (14)with respect to ω to derive αc = 0 and c = αb . Then we exploit (13) and obtain items 2 and 3depending on whether c vanishes for all elements of h or not, respectively.The equation (12) with β = γ = 0 is an identity. Consider the subcases h ωω + α = 0 and h ωω + α = 0 separately.If h ωω + α = 0, integrating (13) we derive ξ = bh − c h R d ωh . This gives item 4.For h ωω + α = 0, the second form of the equation (14) gives ξ = h ω + αωh ωω + α c + 1 h ωω + α c . Then the equation (13) can be represented as K c + K c = 0, where K = (cid:20) h ( h ωω + α ) (cid:21) ω h and K = 2 + ( h ω + αω ) K . K = 0, then K = 2, c = 0 and κ := h ( h ωω + α ) = const. That is why we have item 5.Now suppose that K = 0. K and K are linearly dependent, otherwise c = c = 0, whichcorresponds to the trivial algebra. Therefore, µ := − K /K = const, and hence c = µc . Thenthe equation (13) reduces to hh ω + αω + µ (cid:20) ( h ω + αω + µ ) h ( h ωω + α ) (cid:21) ω = 0 , or, equivalently, h ωω + α ( h ω + αω + µ ) = κh when once integrated. Here κ is an integration constant. Thus we obtain item 6. In the lasttwo items we have κ = 0 since h ωω + α = 0.The equation (11) with h = − α ω + µω + ν and β = γ = 0 (item 4 of Proposition 14) admitsthe widest (two-dimensional) symmetry algebra. In this case in the variables˜ ω = Z d ω − α ω + µω + ν , ˜ φ = φ + αω − µ the equation (11) can be once integrated, which leads to 2 ˜ φ ˜ ω = c − ˜ φ , where c is an integrationconstant. The solution of the integrated equation depends on the sign of c ,˜ φ = − κ tan (cid:16) κ ω + c (cid:17) if c < , κ := √− c , ˜ φ = 2˜ ω + c or ˜ φ = 0 if c = 0 , (15)˜ φ = κ c e κ ˜ ω − c e κ ˜ ω + 1 or ˜ φ = κ if c > , κ := √ c , where c is another integration constant. The form of ˜ ω depends on the sign of ∆ := µ + 2 αν and on α , µ and ν , namely˜ ω = ων , α = 0 , µ = 0 , ν = 0 , µ ln (cid:12)(cid:12)(cid:12)(cid:12) ω + νµ (cid:12)(cid:12)(cid:12)(cid:12) , α = 0 , µ = 0 , − √− ∆ arctan αω − µ √− ∆ , α = 0 , ∆ < , αω − µ , α = 0 , ∆ = 0 , √ ∆ ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) αω − µ + √ ∆ αω − µ − √ ∆ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , α = 0 , ∆ > . (16)Therefore, substituting the expressions for ˜ φ and ˜ ω into φ = ˜ φ (˜ ω ) − αω + µ we have fifteendifferent expressions for solutions of (11) with h quadratic in ω and β = γ = 0.We can apply this result to the reduced equations that correspond to the subalgebras g . and g . (cf. Table 2), where α = 0 and α = 1, respectively.The differential equation for h in item 5 being multiplied by 2 h ω and once integrated takesthe form h ω = 2 ν ln | h |− αh + c , where c is an arbitrary constant. After the second integrationwe have the implicit general solution ± Z d h p ν ln | h | − αh + c = ω + c . φ = c p | ω + µ | . The constant c is a solutionof the reduced algebraic equation 2 εc − h c + 4 γ = 0, where ε = sgn( ω + µ ), and hence c = 14 ε (cid:0) h ± q h − εγ (cid:1) . Hereby for γ = 1 we can use this solution φ of the reduced equation associated with the subal-gebra g . , see Table 2, to find the solution u ( t, x ) = c s(cid:12)(cid:12)(cid:12)(cid:12) x − t µ (cid:12)(cid:12)(cid:12)(cid:12) + t (17)of the generalized Burgers equation L f with f = h (cid:12)(cid:12) x − t + µ (cid:12)(cid:12) / , and µ = 0 mod G ∼ .If we suppose α = µ = 0 in item 6, then the corresponding equation h ωω /h ω = κh ω /h for h can be integrated, h = h | ω + λ | − κ if κ = 1 and h = h e λω if κ = 1, and the associated Liesymmetry algebra leads to the ansatz φ = c ( h/h ) κ . Here h and λ are integration constants.This ansatz reduces the equation (11) to a quadratic equation in c whose two solutions are c = 0 and c = − κ − κ h sgn( ω + λ ) if κ = 1 , − λh if κ = 1 . Using this result for the reduced equation that corresponds to the algebra g . , see Table 2, weobtain the stationary solutions u ( t, x ) = c | x + λ | κ − κ and u ( t, x ) = − λh e λx for the equations ofthe form (1) with f = h | x + λ | − κ and f = h e λx , respectively. Remark 15.
If a reduced equation admits Lie symmetries that are not induced by Lie sym-metries of the initial equation, then the initial equation is said to have additional [35] (or hidden [1]) symmetries with respect to the corresponding reduction. The first example of suchsymmetries was presented in [24]; see also the discussion of this example in [35, Example 3.5].A comprehensive study of such symmetries for the Navier–Stokes equations was carried outin [18, 19]. The Lie reductions of relevant equations from the class (1) with respect to alge-bras g . and g . lead to first-order reduced equations. Therefore, the corresponding initialequations admit infinite-dimensional families of hidden symmetries with respect to the abovereductions, but these symmetries are not essential for consideration because they provide no newexact solutions. The other algebras from Table 2 give second-order reduced equations of thegeneral form (11). Among Lie symmetries of such reduced equations there are both induced andhidden symmetries. Namely, in items 1–3 of Proposition 14, all symmetries of related reducedequations (which are constructed using the algebras g . , g . a with a = 1, g . a ; g . , g . , g . ; g . , respectively) are induced by Lie symmetries of the corresponding initial equations from theclass (1). The condition β = γ = 0 of items 4–6 can be satisfied only by reduced equationsobtained using the algebras g . and g . . In item 4, the first basis vector field is induced only if f = x for the reduction with respect to g . , and the second basis vector field is induced only if f = t ( ω + ¯ ν ) with ω = x/t for the reduction with respect to g . . All the other Lie symmetriesof reduced equations presented in items 4–6 are hidden symmetries of the corresponding initialequations from the class (1).Consider possible reductions with respect to the two-dimensional inequivalent subalgebrasof the algebra g . For this purpose for each basis vector field of a subalgebra we write the15haracteristic equation and thus obtain a system of two differential equations. If the systemis consistent, then its solution gives an ansatz reducing the corresponding equations from theclass (1) to algebraic equations. The system obtained for g . is inconsistent, i.e. there is noansatz associated with this subalgebra. An ansatz constructed with g . is u = ( t + c ) x/ ( t + 1)but the corresponding reduced equation c + 1 = 0 has no solutions. The other subalgebrasallow us to construct some simple solutions, namely, u = 0 from g . , u = 0 and u = x/t from g . , u = 0 and u = e − x from g . , u = 0 and u = a − x | x | − /a from g . a , and from g . were-obtain solution (17) with µ = 0 and h = κ . Up to G ∼ -equivalence, the other two-dimensionalsubalgebras of g can be assumed to contain the vector field P x [22] and, therefore, to lead atmost to constant solutions of equations from the class (1). Achieving new possible reductions opens a way for finding more exact solutions, which may beof interest for modeling physical phenomena and verifying approximate methods of solving dif-ferential equations. Nonclassical reductions were first considered in [9] as a generalization of theclassical Lie reduction method. An attempt of formalizing them was made in [17]. Vector fieldsassociated with nonclassical reductions are called nonclassical, or conditional, or Q -conditionalsymmetries [16, 36]. Another, more proper, name for such a vector field Q is a reduction opera-tor [29], which relates Q to reducing the number of independent variables in a partial differentialequation with an ansatz constructed by Q [64].After the linear heat equation [9], the Burgers equation was the second one that was con-sidered from the point of view of reduction operators [61, 62]; see also a review of these resultsin [3]. Later, reduction operators of the Burgers equation were objects of study and discussionin a number of papers [4, 5, 33, 36, 53]. These studies were summed up in [42]. Attemptsto describe nonclassical symmetries for equations from the class (1) with nonconstant f ’s werestarted in [59] for the subclass of equations with f x = 0. It was also shown that in this subclassreduction operators inequivalent to Lie symmetries exist only for f = const, i.e. for the classicalBurgers equation. Preliminary results on reduction operators of equations from the class (1)with general f ’s were first outlined in [41].Here we arrange the consideration of reduction operators for generalized Burgers equationspresented in [41, 42] and extend it with complete proofs of the corresponding assertions.Roughly speaking, a reduction operator of an equation L f from the class (1) is a vector fieldof the form Q = τ ( t, x, u ) ∂ t + ξ ( t, x, u ) ∂ x + η ( t, x, u ) ∂ u with ( τ, ξ ) = (0 ,
0) that leads to an ansatzreducing the initial equation to an ordinary differential equation (see [28, 46, 64] for precisedefinitions). Due to the equivalence relation of reduction operators, one can multiply Q bya nonzero function of ( t, x, u ) in order to gauge a component of Q to one. The set of reductionoperators for any (1+1)-dimensional evolution equation can be naturally partitioned into twosubsets depending on whether τ = 0 or τ = 0. Moreover, reduction operators with τ = 0 are singular [10, 28], and the problem of finding them is equivalent to solving a single determiningequation, which reduces to the original equation [20, 28, 63] (so-called “no-go” problem). Inparticular, the determining equation for singular reduction operators of the equation L f is η t + uη x + η + f x ( η x + ηη u ) + f ( η xx + 2 ηη xu + η η uu ) = 0 , where the component ξ is already set to 1 using the equivalence relation of reduction operators;cf. [42, Section 2] for the Burger equation L . Sometimes such a determining equation may be useful, when it is possible to guess some ad hoc forms of itsparticular solutions, although there is no algorithmic procedure to do this. See, e.g., [21]. Singular reduction oper-ators corresponding to these particular solutions can be used to construct exact solutions for the original equation. regular reduction operators of equationsfrom the class (1), which have, up to the equivalence relation of reduction operators, the generalform Q = ∂ t + ξ ( t, x, u ) ∂ x + η ( t, x, u ) ∂ u . The conditional invariance criterion [16, 52, 64] impliesthe condition Q (2) L f [ u ] (cid:12)(cid:12) L f ∩Q (2) = 0 (18)for the vector field Q to be a reduction operator of the equation L f : L f [ u ] = 0. Here, again, Q (2) is the standard second-order prolongation of the vector field Q , the manifold defined by theequation L f in the second-order jet space with the variables ( t, x, u, u t , u x , u tt , u tx , u xx ) is denotedby the same symbol L f , and Q (2) is the manifold defined in the same jet space by the invariantsurface condition Q [ u ] := η − u t − ξu x = 0 and its differential consequences D t Q [ u ] = 0 andD x Q [ u ] = 0. These consequences are not needed in the course of expanding the condition (18)since the expression Q (2) L f [ u ] does not contain the derivatives u tt and u tx due to the gaugingof the component τ of Q to 1. Thus, the expanded condition (18) is η t + ηu x + uη x + ( f t + ξf x ) u xx + f η xx = 0 if u t + uu x + f u xx = 0 , u t + ξu x = η. Substituting u t = η − ξu x and u xx = ( ξu x − uu x − η ) /f and splitting the result with respectto u x , we obtain the system of determining equations ξ uu = 0 , η uu = 2 f ξ u ( ξ − u ) + 2 ξ xu , (2 ξ u + 1) η + (cid:18) f t f + f x f ξ (cid:19) ( ξ − u ) + 2 f η xu − ξ t − ξ x ξ + uξ x − f ξ xx = 0 ,η t + uη x + f η xx − (cid:18) f t f + f x f ξ (cid:19) η + 2 ξ x η = 0 . (19)Integrating the first two equations, we can represent ξ and η as polynomials of u with coefficientsdepending on t and x , ξ = ξ ( t, x ) u + ξ ( t, x ) , η = ξ (cid:0) ξ − (cid:1) f u + (cid:18) ξ x + ξ ξ f (cid:19) u + η ( t, x ) u + η ( t, x ) , where the coefficients ξ , ξ , η and η are assumed as new unknown functions. Substituting theexpressions for ξ and η into the third determining equation and splitting the result with respectto u , we derive the system ξ ( ξ − ξ + 1) = 0 ,ξ (2 ξ + 1) ξ − ξ ( ξ − f x = 0 , ( ξ − f t + (2 ξ + 1)( f η + f ξ x − f x ξ ) = 0 , (2 ξ + 1) η + (cid:18) f t f + f x f ξ (cid:19) ξ + 2 f η x − ξ t − ξ ξ x − f ξ xx = 0 . (20)In the second and third equations of (20) we immediately take into account the implication ξ = const of the first equation of (20).The further consideration depends on the choice of solution of the first equation of (20). Wedevote the next subsections to the cases ξ = 1 and ξ = − , and the case ξ = 0 is partitionedinto two subcases, ξ xx = 0 and ξ xx = 0. Note that the last determining equation (19) will berepresented in terms of ξ , ξ , η and η and split with respect to u in every particular case.The partition of reduction operators of equations from the class (1) into the singular andregular reduction operators and the further partition of the regular case into the above subcases17re invariant under the action of G ∼ on the pairs (‘equation’, ‘its reduction operator’). See, e.g.,[46, Section 3] or [28, Definition 3]. Therefore, these reduction operators can be classified up to G ∼ -equivalence, which coincides with G ∼ -equivalence due to the normalization of the class (1).The following theorem holds. Theorem 16.
Up to G ∼ -equivalence, all regular reduction operators of equations from theclass (1) are exhausted by1. Q = ∂ t + u∂ x for any equation L f from the class (1) ,2. Lie symmetry operators with nonzero coefficients of ∂ t ,3. Q θ = ∂ t − ( θ t /θ x ) ∂ x for each equation L f θ with f θ = − /θ x , where θ = θ ( t, x ) is an arbi-trary nonconstant solution of the equation θ t = θ xx θ x + h ( θ ) θ x , and h is an arbitrary smooth function of θ ,4. Q ξ η η = ∂ t + (cid:0) − u + ξ (cid:1) ∂ x + (cid:0) u − ξ u + η u + η (cid:1) ∂ u only for the classical Burgersequation L (modulo G ∼ -equivalence, any constant f can be set to one), where ξ = 12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u z u z u z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u y u y u y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , η = 14 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y z y z y z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u y u y u y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , η = − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u y z u y z u y z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u y u y u y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,u i are solutions of L such that the determinant in the denominators does not vanish, y i = 2 u ix + ( u i ) and z i = 4 u ixx + 6 u i u ix + ( u i ) , i = 1 , , . The proof of the theorem is split into the four parts corresponding to theorem’s cases andaccompanied with brief discussions of associated nonclassical reductions. ξ = 1 The case ξ = 1 was considered in [4, 53] for the classical Burgers equation and in [41] for allequations from the class (1). The determining equations (20) imply ξ = 0, η = 0, η = 0,which gives item 1 of Theorem 16. The vector field Q = ∂ t + u∂ x is a unique common reductionoperator for all equations of the class (1). The set of Q -invariant solutions of every equation L f from the class (1) is exhausted by two families, u ( t, x ) = ( x + c ) / ( t + c ) and u = c , where c , c and c are arbitrary constants, and these are only common solutions for all equations fromthe class (1), cf. Section 5. See [42, Section 3.1] for more details on related reductions. Notethat any transformation from the restriction of G ∼ to the space ( t, x, u ) pushes forward Q toan equivalent vector field differing from Q by a nonvanishing multiplier and preserves both thefamilies of Q -invariant solutions. ξ = − This value for ξ is possible only if f = const, hence the subproblem in question is equivalentto the problem of finding reduction operators with ξ = − for the classical Burgers equation.Since the usual equivalence group contains scale transformations impacting the constant f , wecan set f = 1 modulo G ∼ , and thus obtain the Burgers equation L : L [ u ] := u t + uu x + u xx = 0 . ξ = ξ ( t, x ), η = η ( t, x ) and η = η ( t, x )become ξ t + 2 ξ x ξ + ξ xx − η x = 0 ,η t + 2 ξ x η + η xx + η x = 0 ,η t + 2 ξ x η + η xx = 0 . (21)Reduction operators in this case take the form Q = ∂ t + (cid:18) − u + ξ (cid:19) ∂ x + (cid:18) u − ξ u + η u + η (cid:19) ∂ u . (22)As it was established in [5, 33], solving the system (21) is equivalent to solving the system ofthree copies of the linear heat equation. Therefore, it has been referred to as a “no-go” problem.Using a technique developed in [46], we showed in [42] that this equivalence immediately followsfrom the fact that Q is a reduction operator of the Burgers equation. Moreover, we derivedthe representation of solutions of the system (21) via solutions of the uncoupled system of threecopies of the initial Burgers equation. Below we briefly review results of [42]. Note that theproofs given in [5, 33] did not use the relation of the system (21) with reduction operators ofthe Burgers equation. Lemma 17.
Any solution of the determining system (21) on the coefficients of reduction oper-ators of the form (22) is represented as ξ = ( W (¯ v )) x W (¯ v ) , η = | ¯ v, ¯ v xx , ¯ v xxx | W (¯ v ) , η = − W (¯ v x ) W (¯ v ) , (23) where ¯ v = ( v , v , v ) is a triple of linearly independent solutions of the heat equation v t + v xx = 0 , W (¯ v ) = | ¯ v, ¯ v x , ¯ v xx | and W (¯ v x ) = | ¯ v x , ¯ v xx , ¯ v xxx | are the Wronskians of this triple and the triple ofthe corresponding derivatives with respect to x , respectively, and | ¯ p, ¯ q, ¯ r | denotes the determinantof the matrix constructed with ternary columns ¯ p , ¯ q and ¯ r . Conversely, any triple ( ξ , η , η ) admitting the representation (23) satisfies the system (21) . The proof of the lemma is based on properties of reduction operators. Considering Q -invariantsolutions for an operator Q of the form (22), we solve the system of two equations L [ u ] = 0and Q [ u ] = 0, recombined for convenience as L [ u ] = 0, L [ u ] + Q [ u ] = 0. The Hopf–Coletransformation u = 2 v x /v maps this system to the linear system v t + v xx = 0 , (24) v xxx − ξ v xx + η v x + 12 η v = 0 . (25)Since the family of Q -invariant solutions of L is two-parameter, the space of solutions of the sys-tem (24)–(25) with respect to v is three-dimensional. Let the functions v i = v i ( t, x ), i = 1 , , Q -invariant solutions of L is represented as u = 2 c v x + c v x + c v x c v + c v + c v , (26)where only two of the constants c , c , c are essential.Now take three arbitrary linearly independent solutions v i of the heat equation (24) andsubstitute each of them into (25). Solving the obtained system of three copies of (25) as a systemof linear algebraic equations on ξ , η and η , we derive the representation (23).19onversely, if the coefficients ξ , η and η are of the form (23), then the equation L admitsa two-parametric family (26) of Q -invariant solutions. Hence the vector field Q is a reductionoperator of L , which completes the proof of Lemma 17.Lemma 17 and the Hopf–Cole transformation allow us to represent the coefficients ξ , η and η in terms of three solutions of the classical Burgers equation, ξ = 12 | ¯ e, ¯ u, ¯ z || ¯ e, ¯ u, ¯ y | , η = 14 | ¯ e, ¯ y, ¯ z || ¯ e, ¯ u, ¯ y | , η = − | ¯ u, ¯ y, ¯ z || ¯ e, ¯ u, ¯ y | , (27)where the columns ¯ e , ¯ y and ¯ z consist of e i = 1, y i = 2 u ix + ( u i ) and z i = 4 u ixx + 6 u i u ix + ( u i ) ,respectively, i = 1 , ,
3, and ¯ u is a column of three solutions of the Burgers equation with | ¯ e, ¯ u, ¯ y | 6 = 0. This results in item 4 of Theorem 16. The associated nonclassical reductions werediscussed in [42, Section 3.3]. ξ = 0 with ξ xx = 0 The condition ξ = 0 implies that ξ = ξ ( t, x ) and η = η ( t, x ) u + η ( t, x ). Substituting theseexpression into the system (19)–(20) and splitting (19) with respect to u , we obtain η x = 0, i.e. η = η ( t ), and η x + η ξ x = ( η ) − η t , (28) f t + f x ξ − f ( η + ξ x ) = 0 , (29) ξ t + ξ ξ x + f ξ xx = η + η ξ , (30) η t + η ξ x + f η xx = η η . (31) Lemma 18.
For the case ξ = 0 , ξ xx = 0 the component η vanishes modulo G ∼ .Proof. For convenience we denote α := ( η ) − η t . The equation (28) can be easily integratedwith respect to x , η + η ξ = αx + β, (32)where an arbitrary function β = β ( t ) arises in the course of the integration. After eliminating η in view of (32) and recombining, the equations (30) and (31) take the form ξ t + ξ ξ x + f ξ xx = αx + β, (33) (cid:0) ( αx + β ) ξ (cid:1) x = 2 η ( αx + β ) − ( α t x + β t ) . (34)In fact, α = 0 and β = 0 for any solution of the system (28)–(31). Indeed, suppose that α = 0. Integrating (34) with respect to x , we get ξ = (cid:0) αη − α t (cid:1) x + (2 η β − β t ) x + δαx + β = γ x + γ + µαx + β , where δ is an arbitrary function of t that arises in the course of integration, and γ = η − α t α , γ = η β − β t α + α t β α , µ = δ + βα ( β t − η β ) − α t β α . Then the equation (33) gives an expression for f , which is a polynomial in ( αx + β ), and thecoefficient of ( αx + β ) is µ/ (2 α ). Substituting the expressions for ξ and f into (29), wederive the condition that a polynomial in ( αx + β ) and ( αx + β ) − with coefficients dependingon t identically equals zero. This means that all the coefficients of this polynomial vanish.20n particular, the coefficient of the lowest power ( αx + β ) − is µ /
2. Hence µ = 0 and then wehave ξ = γ x + γ , which contradicts the assumption ξ xx = 0.Knowing that α = 0, we differentiate the equation (34) with respect to x and obtain βξ xx = 0.Therefore, β = 0 in view of the assumption ξ xx = 0.In view of the notation of α and the equation (32), the condition α = β = 0 implies that η t = ( η ) and η = − η ξ . The solutions of the equation η t = ( η ) are η = 0 and η = − ( t + c ) − , where c is an integration constant. The second value of η can be set to zero usingthe equivalence transformation ˜ t = − ( t + c ) − , ˜ x = x ( t + c ) − , ˜ u = ( t + c ) u − x , ˜ f = f .As η = 0, the system of determining equations (28)–(31) reduces to the system f t + ξf x − ξ x f = 0 , (35) ξ t + ξξ x + f ξ xx = 0 , (36)which is well determined as it consists of two differential equations in two unknown functions, f = f ( t, x ) and ξ = ξ ( t, x ), and definitely has no nontrivial differential consequences.We write the equation (35) in conserved form, (1 /f ) t +( ξ/f ) x = 0, and use it to introduce thepotential θ = θ ( t, x ) which is defined by the system θ x = − /f , θ t = ξ/f , and thus f = − /θ x , ξ = − θ t /θ x . Substituting these expressions for f and ξ into (36) and integrating, we obtain θ t = θ xx θ x + h ( θ ) θ x , (37)where h is an arbitrary smooth function of θ [41]. As a result, the system (35)–(36) reduces tothe equation (37). For any h and for an arbitrary solution θ of the equation (37), the vectorfield Q θ = ∂ t − θ t θ x ∂ x is a reduction operator of the equation L f θ with f θ = − /θ x , which proves item 3 of Theorem 16.The impossibility of complete explicit description of reduction operators in this case, whichis equivalent to the problem of finding the general solution of the system (35)–(36), was signedout in [41]. Nevertheless, new solutions of equations from the class (1) were constructed ibidby nonclassical reduction via establishing the connection between the system (35)–(36) and thepotential fast diffusion equation (37) with h = 0 and using a number of already known exactsolutions of the latter equation [52]. ξ = 0 with ξ xx = 0 We prove that in this case reduction operators are equivalent to Lie symmetry operators. Thisassertion was first stated in [41, Section 2]. Under the condition ξ xx = 0, the equation (28)implies η xx = 0. Hence ξ ( t, x ) = ξ ( t, x ) = ξ ( t ) x + ξ ( t ) and η ( t, x ) = η ( t ) u + η ( t ) x + η ( t ).In terms of the new parameter-functions, the determining equations (28)–(31) take the form ξ t = ( η − ξ ) ξ + η , (38) ξ t = ( η − ξ ) ξ + η , (39) η t = ( η − ξ ) η − η , (40) η t = ( η − ξ ) η , (41) η t = ( η − ξ ) η , (42) η = f t f + f x f (cid:0) ξ x + ξ (cid:1) − ξ . (43)21he next step is to modify the determining equations by the substitution ξ = ϕ t /ϕ and η = − ψ t /ψ , where ϕ and ψ are smooth functions of t with ϕψ = 0. Specifically, from (41) weget the equation η t η + ϕ t ϕ + ψ t ψ = 0 , which integrates to the condition η ϕψ = a = const and thus implies η = a ϕψ . (44)Substituting the expression (44) for η into the modified equations (38) and (40), we derive ϕ tt ψ + ϕ t ψ t = a and ϕψ tt + ϕ t ψ t = a , or, after integration, ϕ t ψ = a t + a , ϕψ t = a t + a , (45)respectively. The sum of the equations (45) is directly integrated to ϕψ = a t + ( a + a ) t + a , (46)where a , a and a are arbitrary constants with ( a , a + a , a ) = (0 , ,
0) since ϕψ = 0. Thenthe equation (44) leads to η = a a t + ( a + a ) t + a . Dividing each of the equations (45) by (46) we obtain ξ = ϕ t ϕ = a t + a a t + ( a + a ) t + a , η = − ψ t ψ = − a t + a a t + ( a + a ) t + a . We sequentially integrate the two still unused equations (42) and (39) to derive η = a a t + ( a + a ) t + a , ξ = a t + a a t + ( a + a ) t + a with two more integration constants a and a .Finally, in view of the obtained representations of ξ and η we have the family of vector fields Q a = ∂ t + ( a t + a ) x + a t + a a t + ( a + a ) t + a ∂ x + − ( a t + a ) u + a x + a a t + ( a + a ) t + a ∂ u , where the arbitrary constant tuple a = ( a , . . . , a ) is defined up to a nonzero multiplier, and( a , a + a , a ) = (0 , , Q a is equivalent (up to multiplication by nonvanishing function)to the vector field˜ Q a = (cid:0) a t + ( a + a ) t + a (cid:1) ∂ t + (cid:0) ( a t + a ) x + a t + a (cid:1) ∂ x + (cid:0) − ( a t + a ) u + a x + a (cid:1) ∂ u . Note that ˜ Q a is of the general form obtained for Lie symmetry operators in Section 4.1. For Q a to be a reduction operator of L f with certain f , its coefficients should additionally satisfy theequation (43). This is equivalent to the fact that the components of ˜ Q a satisfy the classifyingcondition (9) with the same f . In other words, the vector field Q a is a reduction operator of anequation L f if and only if the equivalent vector field ˜ Q a is the Lie symmetry generator of thesame equation, which results in item 2 of Theorem 16.Lie reductions and Lie solutions of equations from the class (1) were considered in Section 5.22 Conservation laws and potential admissible transformations
Given a generalized Burgers equation L f of the form (1), in order to study its (local) conservationlaws we use the common technique based on the notion of characteristic [35] which was developedby Vinogradov [58]. As this equation is a second-order quasilinear evolution equation, it issufficient to consider the characteristics of conservation laws that depend only on t , x and u , λ = λ ( t, x, u ); see [51, Corollary 2].The necessary condition for a conservation-law characteristic λ of L f to be its co-symmetryleads to the determining equation − D t λ + D x ( f λ ) + u x λ − D x ( uλ ) = 0 holding on solutionsof L f . After the substitution u t = − uu x − f u xx we split this determining equation with respectto u xx , which gives λ u = 0 and then simplifies it to − λ t + ( f λ ) xx − uλ x = 0. Further splittingwith respect to u results in λ x = 0 and f xx λ = λ t . Hence f xxx λ = 0. This means that theequation L f possesses nontrivial conservation laws if and only if f xxx = 0, i.e. f = f ( t ) x + f ( t ) x + f ( t ) , (47)where the coefficients f , f and f are smooth functions of t not vanishing simultaneously.Then λ = λ ( t ) = e R f xx d t is a unique linearly independent characteristic of the equation L f . Hereinafter, an integral withrespect to t means a fixed antiderivative. In other words, the space of conservation laws of L f isone-dimensional. After multiplying by λ the equation L f can be rewritten in the conserved formD t ( λu ) + λ D x (cid:18) u + f u x − f x u (cid:19) = 0 . Using this form of L f , we introduce the potential v = v ( t, x ) defined by the system v x = λu, v t = − λ (cid:18) u + f u x − f x u (cid:19) . (48)which is called a potential system for L f and is denoted by R f,λ . After excluding the dependentvariable u from R f,λ , we obtain the associated potential equation P f,λ for L f , v t + 12 λ v x + f v xx − f x v x = 0 with f xxx = 0 , λ t = f xx λ, f λ = 0 . (49)Since it is impossible to choose a canonical representative in a one-dimensional space ofconservation-law characteristics of the equation L f with general f quadratic in x , in the courseof the consideration of the class (48) of potential systems, R f,λ , and of the class (49) of potentialequations, P f,λ , it is necessary to extend the arbitrary element with λ . The set ˘ S run by theextended arbitrary element ( f, λ ) is defined by the auxiliary system f xxx = 0, λ t = f xx λ , f λ = 0.Potential systems R f,λ and R f, ˜ λ (resp. potential equations P f,λ and P f, ˜ λ ) with linearly depen-dent λ and ˜ λ are assumed to be gauge-equivalent in the sense of potential systems (resp. potentialequations). Therefore, we associate the single equation L f with the set of gauge-equivalent po-tential systems {R f,λ } and with the set of gauge-equivalent potential equations {P f,λ } .We can also study potential conservation laws of the equation L f , which are local conservationlaws of the corresponding potential system R f,λ , or, equivalently, the potential equation P f,λ . Let µ be the reduced characteristic of a local conservation law for the potential equation P f,λ . There exists an isomorphism between the spaces of conservation laws of the equation (49) and of the sys-tem (48). µ = µ ( t, x, v ) [51, Corollary 2]. We write the equation − D t µ +D x ( f µ ) − D x (( λ v x − f x ) µ ) = 0holding on solutions of P f,λ and solve it with respect to µ , which gives only zero solutions if f = const. As a result, we conclude that the equation L f with f = const has no nonzeropotential conservation laws, and thus R f,λ is the only canonical potential system for L f . Theclassical Burgers equation (for which f = const) possesses potential conservation laws, which arelocal conservation laws of the corresponding potential equation P f,λ with reduced characteristicsof the form µ = ψ ( t, x ) e v f , where ψ ( t, x ) is an arbitrary solution of the linear heat equation ψ t = f ψ xx . Since the potential Burgers equation P f,λ is similar to the linear heat equation, theclassical Burgers equation admits no higher-level potential conservation laws [50, Theorem 5].Therefore, the class (1) f xxx =0 of equations of the form (1) with f quadratic in x is naturallypartitioned into the subclasses L = {L f | f = const } and L = {L f | f xxx = 0 , f = const } . Both these subclasses are closed under the action of the equivalence group G ∼ and, therefore,are normalized as well as the class (1) in view of [49, Proposition 4].Admissible transformations and the equivalence groupoid of the class of potential systems (48)can be called potential admissible transformations and the potential equivalence groupoid of theclass (1) f xxx =0 , respectively. Lemma 19.
The equivalence groupoids of the class of potential systems (48) and of the class ofpotential equations (49) are isomorphic.Proof.
We fix any two values of the arbitrary element from the set ˘ S , ( f, λ ) and ( ˜ f , ˜ λ ). Everypoint transformation between the equations P f,λ and P ˜ f, ˜ λ is prolonged to u according to theequation u = v x /λ , which gives a point transformation between the systems R f,λ and R ˜ f, ˜ λ .Conversely, let ϕ be a point transformation between the systems R f,λ and R ˜ f, ˜ λ , ϕ : ˜ t = T ( t, x, u, v ) , ˜ x = X ( t, x, u, v ) , ˜ u = U ( t, x, u, v ) , ˜ v = V ( t, x, u, v ) . Substituting u = v x /λ into T , X and V and neglecting the transformation component of ϕ for u , we obtain a contact transformation between P f,λ and P ˜ f, ˜ λ , which is necessarily inducedby a point transformation since both the equations P f,λ and P ˜ f, ˜ λ are second-order evolutionequations that are linear with respect to the second derivative u xx [51, Proposition 2].In view of Lemma 19, we can deal with the class of potential equations (49) instead ofthe class of potential systems (48) when studying potential admissible transformations of theclass (1) f xxx =0 . Corollary 20.
For each fixed value f with f xxx = 0 , there exists an isomorphism between theLie invariance algebras g P f,λ and g R f,λ of the equation P f,λ and the system R f,λ . This isomorphism is provided by the projection to the space of ( t, x, v ) when mapping g R f,λ onto g P f,λ and by the prolongation according to the equation u = v x /λ for the inverse mapping.In order to find the equivalence groupoid of the class (49), it is convenient to reparameterizethis class by assuming the coefficients f , f and f of f in (47) to be new arbitrary elements.We use both the parameterizations simultaneously.The above partition of the class (1) f xxx =0 hints at the partition of the class (49) into thesubclasses P = {P f,λ | f = const } and P = {P f,λ | f xxx = 0 , f = const } , which have essentially different transformational properties and are potential counterparts ofthe subclasses L and L , respectively. 24 roposition 21. The equivalence groupoid of the subclass P consists of the triples of the form ( f, ϕ, ˜ f ) where the components of the transformation ϕ are ˜ t = αt + βγt + δ , ˜ x = κx + µ t + µ γt + δ , ˜ v = c (cid:18) v − γλ ( t )2( γt + δ ) x + δµ − γµ κ ( γt + δ ) λ ( t ) x + V ( t ) (cid:19) , the relation between the source and target arbitrary elements is given by ˜ f = ( γt + δ ) ∆ f , ˜ f = γt + δ ∆ (cid:0) κf − µ t + µ ) f (cid:1) , ˜ f = 1∆ (cid:0) κ f − κ ( µ t + µ ) f + ( µ t + µ ) f (cid:1) , i.e. ˜ f = κ ∆ f, ˜ λ = c ∆ κ λ, (50) α , β , γ , δ , µ , µ and κ are arbitrary constants defined up to a nonzero multiplier with ∆ := αδ − βγ = 0 and κ = 0 ; c and σ are arbitrary constants with c = 0 , and V = Z (cid:18) ( δµ − γµ ) κ ( γt + δ ) + δµ − γµ κ ( γt + δ ) f ( t ) + γγt + δ f ( t ) (cid:19) λ ( t ) d t + σ, (51) Proof.
Each equation P f,λ from the class (49) is mapped by the point transformation ψ f,λ : ˆ t = 12 Z λ d t, ˆ x = λx + Z f λ d t, ˆ v = v, (52)parameterized by f (more precisely, by f and f ) and λ to a shorter equation ˆ P ˆ f from the classˆ v ˆ t + ˆ v x + ˆ f ˆ v ˆ x ˆ x = 0 with ˆ f = 0 , ˆ f ˆ x ˆ x ˆ x = 0 . (53)The image of f under the mapping is computed by ˆ f (ˆ t, ˆ x ) = 2 λ ( t ) f ( t, x ). Equivalently, we cancompute the images of the coefficients of f . The subclasses P and P of the class (49) arethen mapped to the subclasses ˆ P = { ˆ P ˆ f | ˆ f = const } and ˆ P = { ˆ P ˆ f | ˆ f ˆ x ˆ x ˆ x = 0 , ˆ f = const } ofthe class (53), respectively. The transformational properties of the subclass ˆ P essentially differfrom those of the subclass ˆ P , and there are no point transformations between equations fromthe different subclasses; see [40] (note that the subclasses notation therein is slightly different).The transformational part of each admissible transformation of the subclass ˆ P has the followingproperties: • the transformational component for ˆ t depends only on ˆ t , • the transformational component for ˆ x depends only on (ˆ t, ˆ x ) and is linear in ˆ x , • the transformational component for ˆ v is linear in ˆ v with a constant coefficient of ˆ v and isquadratic in ˆ x .Consider an arbitrary admissible transformation ( f, ϕ, ˜ f ) of the subclass P . Denoteˆ f := ψ f,λ f, ˆ˜ f := ψ f,λ ˜ f , ˆ ϕ := ψ ˜ f, ˜ λ ϕψ − f,λ , and hence ϕ = ψ − f, ˜ λ ˆ ϕψ f,λ . Then the triple ( ˆ f , ˆ ϕ, ˆ˜ f ) is an admissible transformation of the subclass ˆ P , and hence itstransformational part ˆ ϕ has the properties listed above. In view of (52) and the relation of ϕ with ˆ ϕ , the point transformation ϕ has the same properties, i.e. ϕ : ˜ t = T ( t ) , ˜ x = X ( t ) x + X ( t ) , ˜ v = c (cid:0) v + V ( t ) x + V ( t ) x + V ( t ) (cid:1) , T t X c = 0. We substitute the expressions of tilded entities in terms of untilded ones,including the derivatives˜ v ˜ t = c T t (cid:18) v t + V t x + V t x + V t − X t x + X t X ( v x + 2 V x + V ) (cid:19) , ˜ v ˜ x = c v x + 2 V x + V X , ˜ v ˜ x ˜ x = c v xx + 2 V ( X ) , and also v t = − λ v x − f v xx + f x v x into P ˜ f , and split the obtained equation with respect to v xx , v x and x . After simplifying, this gives˜ f = ( X ) T t f, ˜ λ = c T t ( X ) λ, V = c X t X λ, V = c X t X λ, (54) (cid:18) X t ( X ) (cid:19) t = 0 , (cid:18) X t ( X ) (cid:19) t = 0 , V t = c λ (cid:18) X t X (cid:19) + X t X f − X t X f ! . (55)The first equation of (54) implies the relation between coefficients of f and ˜ f ,˜ f = 1 T t f , ˜ f = X T t f − X T t f , ˜ f = ( X ) T t f − X X T t f + ( X ) T t f . (56)Since λ = e R f d t , ˜ λ = e R ˜ f d˜ t and T t ˜ f = f , we obtain ˜ λλ ! t = 2 (cid:16) T t ˜ f − f (cid:17) ˜ λλ = 0 , i.e. ˜ λλ = const . Hence the second equation of (54) gives T t / ( X ) = ˜ λ/ ( c λ ) = const. The first equation of (55)can be represented as (1 /X ) tt = 0, which means that 1 /X is a linear function of t . Then therelation T t = const · ( X ) implies that T is fractional linear in t . As a result, we derive T = αt + βγt + δ , X = κγt + δ , X = µ t + µ γt + δ , where the expression for X is obtained from the second equation of (55). The last equationof (55) leads, after integration, to the expression (51). Finally, from (56) we have (50). Corollary 22.
There are no point transformations relating equations from the different sub-classes P and P . Since admissible transformations of the subclass P are uniformly parameterized by theextended arbitrary element in a nonlocal way, we may say that this subclass is normalized inthe extended generalized sense although there is no point transformation group underlying thisnormalization in a natural way.Comparing the transformations in Theorem 5 and Proposition 21 and knowing the connec-tion (48) between u and v , we conclude that the equivalence groupoids of the classes L and P are isomorphic up to gauge shifts ˜ v = v + σ and gauge scalings (˜ v, ˜ λ ) = ( c v, c λ ). The same istrue for the equivalence groups of the classes L and P although these equivalence groups areof different kinds. This also implies that for each nonconstant f with f xxx = 0 the Lie symmetrygroups of the equations L f and P f,λ are isomorphic up to shifts of v , which belong to the kernelgroup of P , and so the group classifications for the classes L and P are equivalent. Since theclass L is closed under the action of G ∼ , its group classification can be easily separated outfrom the group classification of the entire class (1), which is given in Table 1.26 roposition 23. Potential admissible transformations of the class L are induced by its usualadmissible transformations. In particular, equations from this class have no nontrivial potentialsymmetries. Consider the potential counterpart P of the class L . For constant values of f , the trans-formation (52) degenerates to a simple rescaling of t . Hence the class P in fact coincides, upto the scaling ˆ t = t , with the class ˆ P , see equation (53). The equivalence groupoid of theclass ˆ P is described in [40, Proposition 2]. We modify it up to the multipliers and κ for betterconsistence with the present paper. Proposition 24.
The subclass P = {P f,λ | f = const } of the class (49) is normalized in thegeneralized sense. Its generalized equivalence group is constituted by the transformations ˜ t = αt + βγt + δ , ˜ x = κx + µ t + µ γt + δ , ˜ v = 2 κ f ∆ ln (cid:12)(cid:12)(cid:12)(cid:12) F (cid:16) e v f + F (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) , ˜ f = κ ∆ f, where α , β , γ , δ , κ , µ , µ are arbitrary constants with ∆ := αδ − βγ = 0 and κ = 0 , the tuple ( α, β, γ, δ, κ ) is defined up to a nonzero multiplier, F = k p | γt + δ | exp (cid:18) − ( γκx − µ δ + µ γ ) f κ γ ( γt + δ ) (cid:19) , γ = 0 ,k exp 2 κµ x + µ t κ f , γ = 0 ,k is a nonzero constant, and F is a solution of the linear equation F t + f F xx = 0 . Proposition 24 is quite obvious since the class P is the orbit of any its single equationunder the action of a scaling group. The complicated form of admissible transformations in theclass P is a consequence of the fact that any equation P f,λ with f = const can be linearizedto the heat equation w t + f w xx = 0 by the variable change w = e v/ (2 f ) , and thus generalsymmetry transformations of P f,λ are of complicated form and involve an arbitrary solution ofthe heat equation. The potential symmetries of the classical Burgers equation are commonlyknown, see, e.g., [35, Example 2.42] and [48]. Up to linear combining, the nontrivial potentialsymmetries of L f that are associated with local conservation laws are exhausted by the imagesof linear superposition symmetries of the corresponding linear heat equation w t + f w xx = 0under push-forwarding with v = 2 f ln w and prolonging to u according to u = v x .Propositions 21 and 24 jointly with Corollary 22 give the complete description of the equiv-alence groupoid of the class (49). Extended symmetry analysis of the class (1) of generalized Burgers equations was merely one ofthe purposes of this paper. Although the class (1) looks quite simple, it has several interestingspecific properties that are related to the field of group analysis of differential equations. This iswhy the class (1) is a convenient object for testing various methods that were recently developedin this field. Moreover, the study of the class (1) in the present paper can serve as a source ofideas for improving and modifying known techniques.Although the arbitrary element f of the class (1) depends on two arguments and the usualequivalence group G ∼ of this class is finite-dimensional, the class (1) is normalized in the usualsense, and thus it is a rare bird among classes of differential equations considered in the litera-ture on symmetries. Similar subclasses of variable-coefficient Korteweg–de Vries equations weresingled out in [22] in the course of symmetry analysis of a wider class of KdV-like equations,27 t + f ( t, x ) uu x + g ( t, x ) u xxx = 0 with f g = 0. These subclasses are associated with the con-straints f = x , f = x − and f = 1, and the last subclass is the most relevant to the class (1).Therein, Lie symmetries of equations from these three subclasses were studied using an early ver-sion of the algebraic method of group classification. See also a modern treatment of these resultsin [57]. At the same time, the algebraic method of group classification was (implicitly or explic-itly) used mostly for normalized classes whose equivalence algebras are infinite-dimensional; see[6, 7, 30, 49, 50] and references therein. The attempt of extending results of [22] to variable-coefficient Burgers equations of the form u t + f ( t, x ) uu x + g ( t, x ) u xx = 0 with f g = 0 in [54] wasnot completely successful, in particular, due to weaknesses of the used sets of subalgebras of therelated (finite-dimensional) algebras of vector fields. This is why it was instructive to carry outthe group classification of the class (1) accurately using various notions and techniques within theframework of the algebraic method, including the ascertainment of normalization of the class (1)and the selection of appropriate subalgebras of (the projection of) its equivalence algebra.For optimizing the procedure of Lie reductions of equations from the class (1), we have appliedtwo special techniques. The technique of classifying of Lie reductions of partial differentialequations from a normalized class with respect to its equivalence group is completely original.This technique allowed us to study Lie reductions for the whole class (1) at once but notseparately for each case of Lie symmetry extensions listed in Table 1. Moreover, the classificationof appropriate subalgebras of the projection of the equivalence algebra can then be adapted forthe classification of Lie reductions. Each subalgebra leads to a set of equivalent ansatzes, andthe second technique is aimed to optimize the form of reduced equations via selecting specificansatzes among equivalent ones. In other words, the selected ansatzes are not of the simplestform but they are complicated as much as it is necessary for simplifying the further study ofreduced equations.There are two kinds of Lie reductions of generalized Burgers equations from the class (1),singular and regular ones. The algebras g . and g . lead to singular reductions since the orderof associated reduced equations, which equals one, is less than the order of original equations,which equals two. These reduced equations can easily be integrated although this gives onlytrivial solutions of original equations. These solutions linearly depend on x and are thus commonfor all equations from the class (1). Lie reductions with the algebras g . – g . a are regular. Due tothe selection of ansatzes, the associated reduced equations are included in the single class (11),which allowed us to unify symmetry analysis of reduced equations and to completely describehidden symmetries of equations from the class (1).Theorem 16 together with the preceding discussion of singular reduction operators presentsone of a few examples existing in the literature, where reduction operators are completely de-scribed for a nontrivial class of differential equations as well as nonclassical reductions result ineffectively finding new exact solutions. The technique of classifying reduction operators withrespect to the equivalence group of the class under consideration plays an important but notcrucial role here.Three “no-go” cases of different nature are singled out in the course of the study. For everyequation L f from the class (1), the problem of describing its singular reduction operators is “no-go” since the corresponding ansatzes reduce L f to first-order ordinary differential equations.This is a general property of partial differential equations with two independent variables thatpossess first co-order singular sets of vector fields parameterized by single arbitrary functionsof both independent and dependent variables [28, Section 8]. Another “no-go” case is relatedto regular reduction operators of the classical Burgers equation. It was studied in [5, 33] andcan be explained by the linearization of the Burgers equation to the heat equation with theHopf–Cole transformation and the existence of the similar “no-go” case for regular reductionoperators of linear evolution equations [20, 46]. The last “no-go” case arises due to the study ofreduction operators for a class of differential equations, and this phenomenon was not discussed28n the literature before [41]. Involving the arbitrary element f to the determining equations forthe components of regular reduction operators leads to the necessity of solving the well-definedsystem (35)–(36). At the same time, the last case is of most interest in spite of its “no-go”essence. The system (35)–(36) reduces to the single equation (37). Many exact solutions of theequation (37) are known for the value h = 0, which immediately results in exact solutions ofvarious equations from the class (1).We studied both local and potential conservation laws of these equations. Only equationswith f xxx = 0 admit nonzero conservation laws. Using the subclass (1) f xxx =0 of such equations asillustrating example, we introduced the notion of potential equivalence groupoid of a class of dif-ferential equations and then computed the potential equivalence groupoid of the above subclass.All assertions on potential symmetries of equations from the class (1) are direct consequencesof the comparison of the usual and potential equivalence groupoids of the subclass (1) f xxx =0 . Acknowledgements
The authors thank Vyacheslav Boyko for useful discussions. The research of ROP was supportedby project P25064 of FWF.
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