Reduction operators of variable coefficient semilinear diffusion equations with an exponential source
aa r X i v : . [ n li n . S I] O c t Reduction operators of variable coefficientsemilinear diffusion equations withan exponential source
O.O. VANEEVA † , R.O. POPOVYCH †‡ and C. SOPHOCLEOUS §† Institute of Mathematics of NAS of Ukraine,3 Tereshchenkivska Str., 01601 Kyiv-4, Ukraine
E-mail: [email protected], [email protected] ‡ Fakult¨at f¨ur Mathematik, Universit¨at Wien,Nordbergstraße 15, A-1090 Wien, Austria § Department of Mathematics and Statistics,University of Cyprus, Nicosia CY 1678, Cyprus
E-mail: [email protected]
Reduction operators (called also nonclassical or Q -conditional symmetries)of variable coefficient semilinear reaction-diffusion equations with exponentialsource f ( x ) u t = ( g ( x ) u x ) x + h ( x ) e mu are investigated using the algorithm in-volving a mapping between classes of differential equations, which is generatedby a family of point transformations. A special attention is paid for checkingwhether reduction operators are inequivalent to Lie symmetry operators. Thederived reduction operators are applied to construction of exact solutions. Various processes in nature are successfully modeled by nonlinear systems of par-tial differential equations (PDEs). In order to study behavior of these processes,it is important to know exact solutions of corresponding model equations. Liesymmetries and the classical reduction method present a powerful and algorith-mic technique for the construction of exact solutions (of systems) of PDEs [14,16].In [2] Bluman and Cole introduced a new method for finding exact solutions ofPDEs. It was called “non-classical” to emphasize its difference from the classicalLie reduction method. A precise and rigorous definition of nonclassical invari-ance was firstly formulated in [6] as “a generalization of the Lie definition ofinvariance” (see also [26]). Later, operators satisfying the nonclassical invariancecriterion were called, by different authors, nonclassical symmetries, conditionalsymmetries and Q -conditional symmetries [4, 5, 7, 12]. Until now all names arein use. See, e.g., [11, 15, 20] for comprehensive reviews of the subject. Follow-ing Ref. [19] we call nonclassical symmetries reduction operators . The necessarydefinitions, including ones of equivalence of reduction operators, and statementsrelevant for this paper are collected in [24]. O.O. Vaneeva, R.O. Popovych and C. SophocleousThe problem of finding reduction operators for a PDE reduces to integration ofan overdetermined system of nonlinear PDEs. The complexity increases in timesin the case of classification problem of reduction operators for a class of PDEshaving nonconstant arbitrary elements.The experience of classification of Lie symmetries for classes of variable coeffi-cient PDEs shows that the usage of equivalence and gauging transformations canessentially simplify the group classification problem and even be a crucial pointin solving the problem [8, 22, 24]. The above transformations are of major impor-tance for studying reduction operators since under their classification one needs toovercome much more essential obstacles then those arising under the classificationof Lie symmetries.In [24] we propose an algorithm involving mapping between classes for findingreduction operators of the variable coefficient reaction-diffusion equations withpower nonlinearity f ( x ) u t = ( g ( x ) u x ) x + h ( x ) u m , (1)where f , g and h are arbitrary smooth functions of the variable x and m is anarbitrary constant such that f gh = 0 and m = 0 ,
1. In [23] reduction operators ofthe equations from class (1) with m = 2 were investigated using this algorithm.The case m = 2 was not systematically considered since it is singular from the Liesymmetry point of view and needs an additional mapping between classes (see [24]for more details). Nevertheless, all the reduction operators constructed in [23] forthe general case m = 0 , , m = 0 , , f ( x ) u t = ( g ( x ) u x ) x + h ( x ) e mu . (2)Here f , g and h are arbitrary smooth functions of the variable x , f gh = 0 and m is an arbitrary nonvanishing constant.The structure of this paper is as follows. For convenience of readers section 2contains a short review of results obtained in [21] and used here. Namely, inthis section the necessary information concerning equivalence transformations, themapping of class (2) to the so-called “imaged” class and the group classification ofequations from the imaged class is collected. Moreover, all additional equivalencetransformations connecting the cases of Lie symmetry extensions (cf. table 1) arefirst found and presented therein. As a result, the classifications of Lie symmetryextensions up to all admissible point transformations in the imaged and, therefore,initial classes are also obtained. Section 3 is devoted to the description of theoriginal algorithm for finding reduction operators of equations from class (2) usinga mapping between classes generated by a family of point transformations. Theresults of section 4 are completely new and concern the investigation of reductionoperators for equations from the imaged class. Reduction operators obtained inan explicit form is used for the construction of exact solutions of equations fromboth the imaged and initial classes.eduction operators of diffusion equations with an exponential source 3 Class (2) has complicated transformational properties. An indicator of this isthat it possesses the nontrivial generalized extended equivalence group, whichdoes not coincide with its usual equivalence group, cf. Theorem 1 below. Toproduce group classification of class (2), it is necessary to gauge arbitrary elementsof this class with equivalence transformations and a subsequent mapping of itonto a simpler class [21, 24]. It appears that the preimage set of each equationfrom the imaged class is a two-parametric family of equations from the initialclass (2). Moreover, preimages of the same equation belong to the same orbit ofthe equivalence group of the initial class. It allows one to look only for simplestrepresentative of the preimage to obtain its symmetries, exact solution, etc., andthen reproduce these results for two-parametric family of equations from the initialclass using equivalence transformations.
Theorem 1.
The generalized extended equivalence group ˆ G ∼ exp of class (2) consistsof the transformations ˜ t = δ t + δ , ˜ x = ϕ ( x ) , ˜ u = δ u + ψ ( x ) , ˜ f = δ δ ϕ x f, ˜ g = δ ϕ x g, ˜ h = δ δ ϕ x e − mψ ( x ) δ h, ˜ m = mδ , where ϕ is an arbitrary non-constant smooth function of x , ψ = δ R dxg ( x ) + δ ,and δ j , j = 0 , , . . . , , are arbitrary constants such that δ δ δ = 0 . Corollary 1.
The usual equivalence group of class (2) is the subgroup of ˆ G ∼ exp singled out by the condition δ = 0 . The transformations from ˆ G ∼ exp associated with varying the parameter δ in factdo not change equations from class (2) and hence form the gauge equivalence groupof this class. The values of arbitrary elements connected by a such transformationcorrespond to different representations of the same equation.Analogously to the power case we at first map class (2) onto its subclass f ( x ) u t = ( f ( x ) u x ) x + h ( x ) e u (3)(we omit tildes over the variables), using the family of equivalence transformationsparameterized by the arbitrary elements f , g and m ,˜ t = sign( f ( x ) g ( x )) t, ˜ x = Z (cid:12)(cid:12)(cid:12)(cid:12) f ( x ) g ( x ) (cid:12)(cid:12)(cid:12)(cid:12) dx, ˜ u = m u. (4)The new arbitrary elements are expressed via the old ones in the following way:˜ f (˜ x ) = ˜ g (˜ x ) = sign( g ( x )) | f ( x ) g ( x ) | , ˜ h (˜ x ) = m (cid:12)(cid:12)(cid:12)(cid:12) g ( x ) f ( x ) (cid:12)(cid:12)(cid:12)(cid:12) h ( x ) , ˜ m = 1 . The next step is to change the dependent variable in class (3): v ( t, x ) = u ( t, x ) + G ( x ) , where G ( x ) = ln | f ( x ) − h ( x ) | . (5) O.O. Vaneeva, R.O. Popovych and C. SophocleousFinally we obtain the class v t = v xx + F ( x ) v x + εe v + H ( x ) , (6)where ε = sign( f ( x ) h ( x )) and the new arbitrary elements F and H are expressedvia the arbitrary elements of class (6) according to the formulas F = f x f − , H = − G xx − G x F. (7)All results on Lie symmetries and exact solutions of class (6) can be extendedto class (3) by the inversion of transformation (5).The arbitrary elements f and h of class (6) are expressed via the functions F and H in the following way: f = c exp (cid:0)R F dx (cid:1) , h = εc exp (cid:0)R F dx + G (cid:1) , where G = R e − R F dx (cid:16) c − R He R F dx dx (cid:17) dx + c . (8)Here c , c and c are arbitrary constants, c = 0. The constant c is inessen-tial and can be set to the unity by an obvious gauge equivalence transformation.The equations from class (3), which have the same image in class (6) with re-spect to transformation (5), i.e., whose arbitrary elements are given by (8) anddiffer only by values of constants c and c , are ˆ G ∼ exp -equivalent. The equivalencetransformation˜ t = t, ˜ x = x, ˜ u = u + c R e − R F dx dx + c (9)maps an equation (6) having f and h of the form (8) with c + c = 0 to theone with c = c = 0. Hence up to ˆ G ∼ exp -equivalence we can consider, withoutloss of generality, only equations from class (3) that have the arbitrary elementsdetermined by (8) with c = c = 0. Theorem 2.
The generalized extended equivalence group G ∼ exp of class (6) coicideswith its usual equivalence group and is formed by the transformations ˜ t = δ t + δ , ˜ x = δ x + δ , ˜ v = v − ln δ , ˜ F = δ − F, ˜ H = δ − H, where δ j , j = 1 , , , are arbitrary constants, δ = 0 . The kernel of the maximal Lie invariance algebras of equations from class (6)is the one-dimensional algebra h ∂ t i . It means that any equation from class (6) isinvariant with respect to translations by t , and there are no more common Liesymmetries. Theorem 3. G ∼ exp -inequivalent cases of extension of the maximal Lie invariancealgebras in class (6) are exhausted by ones presented in table 1. eduction operators of diffusion equations with an exponential source 5 Table 1.
The group classification of the class v t = v xx + F ( x ) v x + εe v + H ( x ) . N F ( x ) H ( x ) Basis of A max ∀ ∀ ∂ t αx − + µx βx − + 2 µ ∂ t , e − µt ( ∂ t − µx∂ x + 2 µ∂ v )2 αx − βx − ∂ t , t∂ t + x∂ x − ∂ v µx γ ∂ t , e − µt ∂ x λ γ ∂ t , ∂ x µx µ ∂ t , e − µt ∂ x , e − µt ( ∂ t − µx∂ x + 2 µ∂ v )6 λ ∂ t , ∂ x , t∂ t + ( x − λt ) ∂ x − ∂ v Here λ ∈ { , } mod G ∼ exp , µ = ± G ∼ exp ; α, β, γ are arbitrary constants, α + β = 0. We also have γ = 2 µ and γ = 0 in cases 3 and 4, respectively.The corresponding results on group classification of class (2) up to ˆ G ∼ exp -equiv-alence is given in table 3 of [21].Additional equivalence transformations between G ∼ exp -inequivalent cases of Liesymmetry extension are also constructed. The pairs of point-equivalent cases fromtable 1 and the corresponding transformations are exhausted by the following:1 ˜2 , ˜6 | ˜ λ =0 : ˜ t = 12 µ e µt , ˜ x = e µt x, ˜ v = v − µt, ˜4 | ˜ λ =0 , ˜6 | ˜ λ =0 : ˜ t = t, ˜ x = x + λt, ˜ v = v. (10)The inequivalence of other different cases of table 1 can be proved using differencesin properties of the corresponding maximal Lie invariance algebras, which shouldcoincide for similar equations. Thus, the dimensions of the maximal Lie invariancealgebras are one, three and two in the general case (case 0), cases 5 and 6 andthe other cases, respectively. In contrast to cases 1–3, the algebra of case 4 iscommutative. The derivative of the algebra of case 3 has the zero projection onthe space of t , and this is not the case for cases 1 and 2. Possessing the zero(resp. non-zero) projection on the space of t is an invariant characteristic of Liealgebras of vector fields in the space of the variables t , x and v with respect topoint transformations connecting a pair of evolution equations since for any suchtransformation the expression of the transformed t is well known to depend onlyon t [9, 13].More difficult problem is to prove that there are no more additional equiv-alences within a parameterized case of table 1. (In fact, all the cases are pa-rameterized.) This needs at least a preliminary study of form-preserving [9] (or O.O. Vaneeva, R.O. Popovych and C. Sophocleousadmissible [17, 18]) transformations. In contrast to transformations from the cor-responding equivalence group, which transform each equation from the class L ofdifferential equations under consideration to an equation from the same class, aform-preserving transformation should transform at least a single equation from L to an equation from the same class. The notion of admissible transformations is aformalization of the notion of form-preserving transformations. The set of admis-sible transformations of the class L is formed by the triples each of which consistsof the tuples of arbitrary elements corresponding to the initial and target equa-tions and a point transformation connecting these equations. It is obvious thateach transformation from the equivalence group generates a family of admissibletransformations parameterized by arbitrary elements of the class L .A preliminary description of the set of admissible transformations of the class (6),which is sufficient for our purpose, is given by the following statements. Proposition 1.
Any admissible point transformation in the class (6) has the form ˜ t = T ( t ) , ˜ x = δ p T t x + X ( t ) , ˜ v = v − ln T t , where δ = ± and T and X are arbitrary smooth functions of t such that T t > .The corresponding values of the arbitrary elements are related via the formulas ˜ F = δ √ T t F − δ T tt √ T t x − X t T t , ˜ H = 1 T t H − T tt T t . Corollary 2.
Only equations from the class (6) whose arbitrary elements havethe form F = µx + λ + αx + κ , H = γ + β ( x + κ ) , (11) where α , β , γ , κ and µ are constants, possess admissible transformations that arenot generated by transformations from the equivalence group G ∼ exp . The subclassof the class (6) , singled out by the condition (11) , is closed under any admissibletransformation within the class (6) . The (constant) parameters of the representa-tion (11) are transformed by an admissible transformation in the following way: ˜ α = α, ˜ β = β, ˜ κ = δ p T t κ − X if ( α, β ) = (0 , , ˜ γ = γT t − T tt T t , ˜ µ = µT t − T tt T t , ˜ λ = − ˜ µX − X t T t + δλ √ T t . In particular, T tt = 0 if γ = 2 µ . Finally, we can formulate the assertion on group classification with respect tothe set of admissible transformations.
Theorem 4.
Up to point equivalence, cases of extension of the maximal Lieinvariance algebras in class (6) are exhausted by cases 0, 2, 3, λ =0 and | λ =0 oftable 1. eduction operators of diffusion equations with an exponential source 7 At first we adduce the definition of nonclassical symmetries [7, 19, 26], adopt-ing it for the case of one second-order PDE with two independent variables,relevant for this paper. Consider a second-order differential equation L of theform L ( t, x, u (2) ) = 0 for the unknown function u of the two independent vari-ables t and x , where u (2) = ( u, u t , u x , u tt , u tx , u xx ). Let Q be a first-order differ-ential operator of the general form Q = τ ( t, x, u ) ∂ t + ξ ( t, x, u ) ∂ x + η ( t, x, u ) ∂ u , ( τ, ξ ) = (0 , . Definition 1.
The differential equation L is called conditionally invariant withrespect to an operator Q if the relation Q (2) L ( t, x, u (2) ) (cid:12)(cid:12) L∩Q (2) = 0 (12)holds, which is called the conditional (or nonclassical) invariance criterion . Then Q is called conditional symmetry (or nonclassical symmetry, Q -conditional sym-metry or reduction operator) of the equation L .The symbol Q (2) stands for the standard second prolongation of Q (see e.g. [14,16]). Q (2) is the manifold determined in the second-order jet space by the dif-ferential consequences of the characteristic equation Q [ u ] := η − τ u t − ξu x = 0 , which have, as differential equations, orders not greater than two.It was proved in [26] that a differential equation L is conditionally invariantwith respect to the operator Q if and only if the ansatz constructed with thisoperator reduces the equation L . That is why it seems natural to call operatorsof conditional (nonclassical symmetries) reduction operators .Here we present the algorithm of application of equivalence transformations,gauging of arbitrary elements and mappings between classes of equations to clas-sification of reduction operators of class (2)1. At first we gauge class (2) to subclass (3) constrained by the condition f = g .Then class (3) is mapped to the imaged class (6) by transformation (5).2. Reduction operators should be classified up to the equivalence relations gen-erated by the corresponding equivalence groups or even by the whole sets ofadmissible transformations. As the singular case τ = 0 is “no-go” [10, 25],only the regular case τ = 0 (reduced to the case τ = 1) should be considered.Operators equivalent to Lie symmetry ones should be neglected.3. It is well-known (see e.g. [1, 3]) that the equations from the imaged class (6)with F = 0 and H = const and, therefore, all equations similar to themwith respect to point transformations possess no regular reduction operatorsthat are inequivalent to Lie symmetry operators. This is why all the aboveequations should be excluded from the consideration. O.O. Vaneeva, R.O. Popovych and C. Sophocleous4. Preimages of the nonclassical symmetries obtained and of equations admit-ting them should be found using the inverses of gauging transformationsand the push-forwards by these inverses on the sets of operators.Reduction operators of equations from class (3) are easily found from reductionoperators of corresponding equations from (6) using the formula˜ Q = τ ∂ t + ξ∂ x + ( η − ξG x ) ∂ u . (13)Here τ , ξ and η respectively are the coefficients of ∂ t , ∂ x and ∂ v in a reductionoperator of an equation from class (6). The function G is defined in (8).In [23, 24] we discussed two ways to use mappings between classes of equationsin the investigation of reduction operators and their usage to finding exact so-lutions. The preferable way is based on the implementation of reductions in theimaged class and preimaging of the obtained exact solutions instead of preimagingthe corresponding reduction operators. Following the above algorithm, we look for G ∼ exp -inequivalent reduction operatorswith nonvanishing coefficient of ∂ t for the equations from the imaged class (6).Up to the usual equivalence of reduction operators we need to consider only theoperators of the form Q = ∂ t + ξ ( t, x, v ) ∂ x + η ( t, x, v ) ∂ v . Applying conditional invariance criterion (12) to equation (6) we obtain a third-degree polynomial of v x with coefficients depending on t , x and v , which has toidentically equal zero. Splitting it with respect to different powers of v x results inthe following determining equations for the coefficients ξ and η : ξ vv = 0 , η vv = 2( ξ xv − ξξ v − F ξ v ) ,ξ t − ξ xx + 2 ξ x ξ + 3 ξ v ( H + εe v ) + 2 η vx − ξ v η + F ξ x + ξF x = 0 ,η t − η xx + 2 ξ x η = ξH x + F η x + (2 ξ x − η v ) H + εe v ( η + 2 ξ x − η v ) . (14)Integration of the first two equations of (14) gives us the expressions for ξ and η with an explicit dependence on v : ξ = av + b,η = − a v + ( a x − ab − aF ) v + cv + d, (15)where a = a ( t, x ) , b = b ( t, x ) , c = c ( t, x ) and d = d ( t, x ) are smooth functions of t and x .Substituting the expressions (15) for ξ and η into the third and forth equationsof (14) and collecting the coefficients of different powers of v in the resultingeduction operators of diffusion equations with an exponential source 9equations, we derive the conditions a = c = 0, d = − b x and two classifyingequations, which contain both the coefficient b = b ( t, x ) and the arbitrary elements F = F ( x ) and H = H ( x ). Summing up the above consideration, we have thefollowing assertion. Proposition 2.
Any regular reduction operator of an equation from the imagedclass (6) is equivalent to an operator of the form Q = ∂ t + b∂ x − b x ∂ v , (16) where the coefficient b = b ( t, x ) satisfies the overdetermined system of partialdifferential equations b t − b xx + 2 bb x + F b x + bF x = 0 ,bH x + 2 b x H − bb xx − F b ) xx − F b xx = 0 (17) with the corresponding values of the arbitrary elements F = F ( x ) and H = H ( x ) . The second equation of (17) can be written in the more compact form4( b + F ) b xx = 2 Kb x + K x b, where K = H − F x , which is more convenient for the study of compatibility.Analogously to the power case, we were not able to completely study all thecases of integration of system (17) depending on values of F and H . This is whywe try to solve this system under different additional constraints imposed eitheron b or on ( F, H ).The most interesting results are obtained for the constraint b t = 0. Then F and H are expressed, after a partial integration of (17), via the function b = b ( x )that leads to the following statement. Theorem 5.
For an arbitrary smooth function b = b ( x ) the equation from class (6) with the arbitrary elements F = 1 b (cid:0) b x + k − b (cid:1) ,H = 2 b (cid:0) k + b x ( k − b ) + bb xx (cid:1) , (18) where k and k are constants, admits the reduction operator (16) with the same b . An ansatz constructed by the reduction operator (16) with b t = 0 has the form v = z ( ω ) − | b | , where ω = t − Z dxb , The substitution of the ansatz into equation (6) leads to the reduced ODE z ωω − k z ω + εe z + 2 k = 0 . (19)0 O.O. Vaneeva, R.O. Popovych and C. SophocleousFor k = 0 the general solution of (19) is written in the implicit form Z ( c − k z − εe z ) − dz = ± ( ω + c ) . (20)Up to similarity of solutions of equation (6), the constant c is inessential and canbe set to equal zero by a translation of ω , which is always induced by a translationof t .Setting additionally k = 0 in (20), we are able to integrate in (20) in a closedform and to explicitly write down the general solution of (19). If ε = 1 then c > e z : e z = 2 s cosh ( s ω + s ) . Here and below s = p | c | / s = c s . If ε = −
1, the integration leads to e z = s sinh ( s ω + s ) , c > , s cos ( s ω + s ) , c < , ω + c ) , c = 0 . As a result, for the equation from class (6) of the form v t = v xx + 1 b (cid:0) b x − b (cid:1) v x + εe v + 2 b ( b xx − bb x ) (21)with ε = − v = − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ s b sinh (cid:18) s t − s Z dxb + s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,v = − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ s b cos (cid:18) s t − s Z dxb + s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (22) v = − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ b (cid:18) t − Z dxb + c (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where s , s and c are arbitrary constants, s = 0. Also we obtain a family ofexact solution v = − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ s b cosh (cid:18) s t − s Z dxb + s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (23)of the equation (21) with ε = 1.eduction operators of diffusion equations with an exponential source 11We continue the consideration by studying whether the equations from class (6)possessing nontrivial Lie symmetry properties, i.e. having the maximal Lie invari-ance algebras of dimension two or three, have nontrivial (i.e., inequivalent to Lieones) regular reduction operators. It has been already remarked that constantcoefficient equations from class (6) do not admit such reduction operators [1, 3].Hence it is needless to consider cases 4 and 6 of table 1 as well as case 5 connectedwith case 6 by point transformation (10). As case 1 reduces to case 2 with thesame transformation (10), we have to study only two cases, namely cases 2 and 3.We substitute the pairs of values of the parameter-functions F and H correspond-ing to cases 2 and 3 into system (17) in order to find relevant values for b . Weascertain that b t = 0 is a necessary condition for existing non-Lie regular reduc-tion operators for equations with the above values of ( F, H ). This is why we canuse equations (18) instead of (17) for further studying.The investigation of case 3 of table 1 leads to the conclusion that there are nonon-Lie regular reduction operators for this case.The functions F and H presented in case 2 of table 1 satisfy (18) if and onlyif β = 2(1 − α ), i.e. they have the form F = αx − , H = 2(1 − α ) x − , and k = k = 0. The corresponding value of b is b = − (1 + α ) x − . Hence α = − b = 0. Substituting the derived form of the function b into theformulas (22) and (23), we find that the equation v t = v xx + αx v x + εe v + 2(1 − α ) x (24)has the families of exact solutions v = − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ α )2 s x cosh (cid:18) s t + s x α ) + s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , if ε = 1 and v = − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ α )2 s x sinh (cid:18) s t + s x α ) + s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,v = − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ α )2 s x cos (cid:18) s t + s x α ) + s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,v = − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ α )2 x (cid:18) t + x α ) + c (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) if ε = −
1. Recall that s , s and c are arbitrary constants, s = 0.As a representative of the preimage of equation (24) with respect to the trans-formation (5), we can choose the equation x α u t = ( x α u x ) x + εx α +2 e u . (25)2 O.O. Vaneeva, R.O. Popovych and C. SophocleousSolutions of this equation can be easily constructed from the above solutions ofequation (24) using the transformation u = v − | x | . If α = 1, the chosenequation (25) can be replaced, e.g., by xu t = ( xu x ) x + εxe u which is just anotherrepresentation of equation (24).Non-Lie exact solutions of the equation v t = v xx + (cid:16) αx + µx (cid:17) v x + εe v + 2(1 − α ) x + 2 µ, where α = − b = − (1 + α ) x − − µx .We also prove the following assertions. Proposition 3.
Equations from class (6) with F = const or H = const mayadmit only nontrivial regular reduction operators that are equivalent to operatorsof the form (16) , where the function b does not depend on the variable t . Proposition 4.
Any reduction operator of an equation from class (6) , having theform (16) with b xx = 0 , is equivalent to a Lie symmetry operator of this equation. The proofs of these propositions are quite cumbersome and will be presentedelsewhere.
Acknowledgements
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