Abelian Lie symmetry algebras of two-dimensional quasilinear evolution equations
aa r X i v : . [ n li n . S I] D ec Abelian Lie symmetry algebras of two-dimensional quasi-linear evolution equations
Rohollah Bakhshandeh-Chamazkoti †† Department of Mathematics, Faculty of Basic Sciences, Babol Noshirvani University of Tech-nology, Babol, Iran.
E-mail: r [email protected]
We carry out the classification of abelian Lie symmetry algebras of two-dimensional second-order nondegenerate quasilinear evolution equations. It is shown that such an equation islinearizable if it admits an abelian Lie symmetry algebra that is of dimension greater thanor equal to five or of dimension greater than or equal to three with rank one.
Transformation properties of evolution equations especially Lie group of point symmetries hasbeen widely studied because of many practical benefits that such knowledge gives and alsobecause of the variety of physical applications that are modeled by these equations.The purpose of this paper is to classify all admissible abelian symmetry algebras of the class L of two-dimensional second-order non-degenerate quasilinear evolution equations, which are ofthe general form u t = F ( t, x, y, u, u x , u y ) u xx + G ( t, x, y, u, u x , u y ) u yy + H ( t, x, y, u, u x , u y ) u xy + K ( t, x, y, u, u x , u y ) , (1)where F, G, H and K are some non-zero smooth functions in terms of t, x, y, u, u x , u y variableswith F G − H / = 0. The present paper is the first part of a project to classify Lie symmetryalgebras of differential equations of the class L and develops the methods which were applied in[3].In [3], the authors gave a complete Lie point symmetry classification of all third-order evolu-tion equations u t = F ( t, x, u, u x , u xx ) u xxx + G ( t, x, u, u x , u xx ) , (2)where F = 0 and G are two functions in terms of t, x, u, u x , u xx which (2) admit semi-simple sym-metry algebras. Before in [9], this method was applied for equation (2) for F = 1. This methodfor finding symmetries and integrability properties for (1+1)-dimension evolution equations werecarried out in [1, 7, 8, 9, 13, 18, 19], and recently this method is developed in [2, 4, 10, 11, 17].In [11], the authors carried out enhanced symmetry analysis of the two-dimensional Burgerssystem (cid:26) u t + uu x + vu y − u xx − u yy = 0 ,v t + uv x + vv y − v xx − v yy = 0 , (3)Under the constraint v = 0, the second equation of the system (3) is equaled to zero and its firstequation reduces to a (1 + 2)-dimensional generalization of the Burgers equation u t + uu x − u xx − u yy = 0 , (4)which was deduced in [15] as an equation for the wave phase of two-dimensional sound simplewaves in weakly dissipative flows. Therein, symmetry analysis of this equation was carried out,1hich included the first exhaustive study of its Lie reductions in an optimized way and theconstruction of several new families of its exact solutions. The group classification of (1 + 2)-dimensional diffusion-convection equation (4) was also investigated in [6]. Recently, Bihlo andPopovych in [5], classified zeroth-order conservation laws of systems from the class of two-dimensional shallow water equations with variable bottom topography.The present paper is organized as follows: Section 2 is presented to some basic definitionsof equivalence groupoid and then we show the infinitesimal Lie point symmetry of differentialequations of the class L , has the form Q = a ( t ) ∂ t + b ( t, x, y, u ) ∂ x + c ( t, x, y, u ) ∂ y + ϕ ( t, x, y, u ) ∂ u , where a ( t ) , b ( t, x, y, u ) , c ( t, x, y, u ) and ϕ ( t, x, y, u ) are smooth functions of their argumentswhich Q is equal to ∂ t or ∂ u as a canonical form under the equivalence group (7) and also the ∂ x , ∂ y and ∂ u are equivalent as canonical forms under this equivalence group (7).In section 3, we compute the determining equations by applying Lie symmetry conditions.In section 4, we classify the admissible abelian symmetry algebras and we prove that anyequation of the the class L equipped with an five-dimensional abelian symmetry algebra or withdimension bigger than five, or three-dimensional abelain Lie algebra or with dimension biggerthan three with rank-one realization, is linearizable. Finally, a table included some invariantsolutions corresponding to abelian Lie symmetries for non-linear cases of the evolution equationis listed. Let L θ denote a system of differential equations of the form L ( x, u ( r ) , θ ( q ) ( x, u ( r ) )) = 0, where x, u , and u ( r ) are the tuples of independent variables, of dependent variables and of derivativesof u with respect to x up to order r . The tuple of functions L = ( L , . . . , L l ) of ( x, u ( r ) , θ ) isfixed whereas the tuple of functions θ = ( θ , . . . , θ k ) of ( x, u ( r ) ) runs through the solution set S of an auxiliary system of differential equations and inequalities in θ , where x and u ( r ) jointlyplay the role of independent variables. Thus, the class of (systems of) differential equations L is the parameterized family of systems L θ with θ running through the set S . The componentsof θ are called the arbitrary elements of the class L . The equivalence groupoid G ∼ of the class L consists of admissible transformations of this class, G ∼ = G ∼ ( L ) = (cid:8) ( θ, ξ, ˜ θ ) | θ, ˜ θ ∈ S , ξ ∈ T( θ, ˜ θ ) (cid:9) . Here L θ and L ˜ θ are the source and the target systems, belonging to the class L and correspondingto the values θ and ˜ θ of the arbitrary-element tuple, respectively, which in turn runs through thesolution set S of the auxiliary system S of equations and inequalities for arbitrary elements, and ξ is a point transformation relating the equations L θ and L ˜ θ (the set of all such transformationsis denoted by T( θ, ˜ θ )) [16]. Definition 1.
The (usual) equivalence group G ∼ = G ∼ ( L ) of the class L is the (pseudo)groupof point transformations in the space of ( x, u ( r ) , θ ) which are projectable to the space of ( x, u ( r ′ ) )for any 0 ≤ r ′ ≤ r , are consistent with the contact structure on the space of ( x, u ( r ) ) and preservethe class L . Definition 2.
The class of differential equations L is called normalized if its equivalencegroupoid G ∼ is induced by its equivalence group G ∼ , meaning that for any triple ( θ, ξ, ˜ θ ) from G ∼ ,there exists a transformation Ξ from G ∼ such that ˜ θ = Ξ ∗ θ and ξ = Ξ | ( x,u ) .For the class L of equations of the form (1), the tuple θ of arbitrary elements is constituted byfour arbitrary smooth functions F , G , H and K of ( t, x, y, u, u t , u x , u y ) that satisfy the auxiliary2ystem S consisting of the single inequality F G − H / = 0, that is, θ = ( F, G, H, K ) and S = { ( F, G, H, K ) | F G − H / = 0 } . The usual equivalence group G ∼ of the class L can beassumed to act in the space with coordinates ( t, x, y, u, u t , u x , u y , F, G, H, K ) and to consist ofthe point transformations of the form t ′ = T ( t, x, y, u ) , x ′ = X ( t, x, y, u ) , y ′ = Y ( t, x, y, u ) , u ′ = U ( t, x, y, u ) ,u ′ t ′ = U t ( t, x, y, u, u x , u y ) , u ′ x ′ = U x ( t, x, y, u, u x , u y ) , u ′ y ′ = U y ( t, x, y, u, u x , u y ) ,F ′ = F ( t, x, y, u, u x , u y , F, G, H, K ) , G ′ = G ( t, x, y, u, u x , u y , F, G, H, K ) ,H ′ = H ( t, x, y, u, u x , u y , F, G, H, K ) , K ′ = K ( t, x, y, u, u x , u y , F, G, H, K ) , (5)that preserve the contact structure of the space with coordinates ( t, x, y, u, u x , u y ) and map eachequation L θ from the class L to an equation L θ ′ from the same class, u ′ t ′ = F ′ ( t ′ , x ′ , y ′ , u ′ , u ′ x ′ , u ′ y ′ ) u ′ x ′ x ′ + G ′ ( t ′ , x ′ , y ′ , u ′ , u ′ x ′ , u ′ y ′ ) u ′ y ′ y ′ + H ′ ( t ′ , x ′ , y ′ , u ′ , u ′ x ′ , u ′ y ′ ) u ′ x ′ y ′ + K ′ ( t ′ , x ′ , y ′ , u ′ , u ′ x ′ , u ′ y ′ ) , (6) Theorem 3.
A point transformation ξ in the space with coordinates ( t, x, y, u ) transforms anequation (1) from the class L to an equation (6) from the same class if and only if the componentsof ξ are of the form t ′ = T ( t ) , x ′ = X ( t, x, y, u ) , y ′ = Y ( t, x, y, u ) , u ′ = U ( t, x, y, u ) . (7) with the condition T t (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( X, Y, U ) ∂ ( x, y, u ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 . The transformed arbitrary elements F ′ , G ′ , H ′ and K ′ are given by F ′ = − T t ζ u h ( X x ζ u − X u ζ x ) F + ( X y ζ u − X u ζ y ) G + ( X x ζ u − X u ζ x )( X y ζ u − X u ζ y ) H i (8) G ′ = − T t ζ u h ( Y x ζ u − Y u ζ x ) F + ( Y y ζ u − Y u ζ y ) G + ( Y x ζ u − Y u ζ x )( Y y ζ u − Y u ζ y ) H i (9) H ′ = − T t ζ u h ( X x ζ u − X u ζ x )( Y x ζ u − Y u ζ x ) F − ( X y ζ u − X u ζ y )( Y y ζ u − Y u ζ y ) G − ( X x ζ u − X u ζ x )( Y y ζ u − Y u ζ y ) H i (10) K ′ = 1 T t h(cid:16) − ζ xx + ζ x ζ u − ζ x ζ uu ζ u (cid:17) F + (cid:16) − ζ yy + ζ y ζ u − ζ y ζ uu ζ u (cid:17) G + (cid:16) − ζ xy + ζ x ζ y ζ u − ζ x ζ y ζ uu ζ u (cid:17) H + Kζ u + ζ t ζζ u i (11) where ζ = ∂u = U − u t T − u x X − u y Y .Proof. We employ the direct method for finding point transformations relating differential equa-tions. Suppose that a point transformation ξ of the general form t ′ = T ( t, x, y, u ) , x ′ = X ( t, x, y, u ) , y ′ = Y ( t, x, y, u ) , u ′ = U ( t, x, y, u ) (12)with J := (cid:12)(cid:12) ∂ ( T, X, Y, U ) /∂ ( t, x, y, u ) (cid:12)(cid:12) = 0 maps the equation (1) to the equation (6).Computing u x , u y and u t , we get u µ = − ∂ µ u ′ ∂ u u ′ = − U µ − u ′ t ′ T µ − u ′ x ′ X µ − u ′ y ′ Y µ U u − u ′ t ′ T u − u ′ x ′ X u − u ′ y ′ Y u (13)3here µ runs through { t, x, y } . Now, making the change of variables (5), we get the derivatives u xx , u yy , u xy by the following relations u µν = − ( ∂ µν u ′ )( ∂ u u ′ ) − ( ∂ uν u ′ )( ∂ µ u ′ ) ∂ u u ′ (14)where µ, ν could be equal to t, x and y . Then, we insert u t , u xx , u yy , u xy from (13) and (14) into(1), and then in the transformed equation, the coefficients u ′ t ′ x ′ , u ′ t ′ y ′ and u ′ t ′ t ′ are set equal tozero because by comparing the transformed equation with equation (6), the right-hand side ofthe transformed equation must not contain u ′ t ′ x ′ , u ′ t ′ y ′ and u ′ t ′ t ′ . Therefore F ( T x ∂ u u ′ − T u ∂ x u ′ ) + G ( T y ∂ u u ′ − T u ∂ y u ′ ) + H ( T x ∂ u u ′ − T u ∂ x u ′ )( T y ∂ u u ′ − T u ∂ y u ′ ) = 0 , (15) F ( T x ∂ u u ′ − T u ∂ x u ′ )( X x ∂ u u ′ − X u ∂ x u ′ ) + G ( T y ∂ u u ′ − T u ∂ y u ′ )( X y ∂ u u ′ − X u ∂ y u ′ )+ H ( T x ∂ u u ′ − T x ∂ x u ′ )( X y ∂ u u ′ − X u ∂ y u ′ ) = 0 , (16) F ( T x ∂ u u ′ − T u ∂ x u ′ )( Y x ∂ u u ′ − Y u ∂ x u ′ ) + G ( T y ∂ u u ′ − T u ∂ y u ′ )( Y y ∂ u u ′ − Y u ∂ y u ′ )+ H ( T x ∂ u u ′ − T x ∂ x u ′ )( Y y ∂ u u ′ − Y u ∂ y u ′ ) = 0 , (17)According to the above system can be writtenrank T x ∂ u u ′ − T u ∂ x u ′ T y ∂ u u ′ − T u ∂ y u ′ T y ∂ u u ′ − T u ∂ y u ′ X x ∂ u u ′ − X u ∂ x u ′ X y ∂ u u ′ − X u ∂ y u ′ X y ∂ u u ′ − X u ∂ y u ′ Y x ∂ u u ′ − Y u ∂ x u ′ Y y ∂ u u ′ − Y u ∂ y u ′ Y y ∂ u u ′ − Y u ∂ y u ′ = rank T x T y T u X x X y X u Y x Y y Y u ∂ x u ′ ∂ y u ′ ∂ u u ′ − T x T y T u X x X y X u Y x Y y Y u U x U y U u − , since J = 0. Now since the matrix (cid:20) F H/ H/ G (cid:21) is nondegenerate then we conclude T x ∂ u u ′ − T u ∂ x u ′ = 0 ,T y ∂ u u ′ − T u ∂ y u ′ = 0 ,T y ∂ u u ′ − T u ∂ y u ′ = 0 . (18)Splitting the variables in (18) we find T µ U u − T u U µ = 0 and T µ T u = 0 , T µ X u − T u X µ = 0 , T µ Y u − T u Y µ = 0 . (19)where µ = x, y . If T u = 0 then each row ( X x , X y , X u ), ( Y x , Y y , Y u ) and ( U x , U y , U u ) are parallelto the ( T x , T y , T u ) which contradicts with J = 0, and therefore T u = 0. Using J = 0 condition weconclude ( X u , Y u , U u ) = (0 , ,
0) and T µ U u = T µ X u = T µ Y u = 0 here µ are x, y . So T x = T y = 0.Putting ζ = ∂u = U − u t T − u x X − u y Y , and using the following formula u t = − ∂ t u ′ ∂ u u ′ = − ζ t ζ u + T t ζ u u ′ t ′ (20)4e have F h − ζ xx ζ u + ζ x ζ u − ζ x ζ uu ζ u − ζ u (cid:2) ( X x ζ u − X u ζ x ) u ′ x ′ x ′ + ( Y x ζ u − Y u ζ x ) u ′ y ′ y ′ + 2( X x ζ u − X u ζ x )( Y x ζ u − Y u ζ x ) u ′ x ′ y ′ (cid:3)i + G h − ζ yy ζ u + ζ y ζ u − ζ y ζ uu ζ u − ζ u (cid:2) ( X y ζ u − X u ζ y ) u ′ x ′ x ′ + ( Y y ζ u − Y u ζ y ) u ′ y ′ y ′ + 2( X y ζ u − X u ζ y )( Y y ζ u − Y u ζ y ) u ′ x ′ y ′ (cid:3)i + H h − ζ xy ζ u + ζ x ζ y ζ u − ζ x ζ y ζ uu ζ u − ζ u (cid:2) ( X x ζ u − X u ζ x )( X y ζ u − X u ζ y ) u ′ x ′ x ′ + ( Y x ζ u − Y u ζ x )( Y y ζ u − Y u ζ y ) u ′ y ′ y ′ + 2( X x ζ u − X u ζ x )( Y y ζ u − Y u ζ y ) u ′ x ′ y ′ (cid:3)i + K = − ζ t ζ + T t ζ u (cid:0) F ′ u ′ x ′ x ′ + G ′ u ′ y ′ y ′ + H ′ u ′ x ′ y ′ + K ′ (cid:1) and then, splitting the last equation with respect to u ′ x ′ x ′ , u ′ x ′ y ′ and u ′ y ′ y ′ , we obtain the equa-tions (8), (9), (10) and (11).In other words, Theorem 3 means that the equivalence groupoid G ∼ of the class L consistsof of the triples ( θ, ξ, ˜ θ ), where the point transformations ξ is of the form (7), the tuples θ runsthrough the entire set S , and the tuples ˜ θ are defined by the equations (8)–(11). Thus, theequivalence groupoid G ∼ can be partitioned into families of admissible transformations each ofwhich corresponds to a fixed point transformations ξ and parameterized by arbitrary valuesof θ ∈ S . This means that the each point transformation of the form (7) is the projectionof an equivalence transformation to the space with coordinates ( t, x, y, u ), the ( F, G, H, K )-components of this equivalence transformation are defined by the equations (8), (9), (10) and(11), respectively, and the action groupoid of the equivalence group G ∼ of the class L coincideswith the entire equivalence groupoid G ∼ of this class. As a result, we have the following theorem. Theorem 4.
The class L of equations of the form (1) with F G − H / = 0 is normalized inthe usual sense. Its usual equivalence group G ∼ is constituted by the point transformations inthe space with coordinates ( t, x, y, u, u t , u x , u y , F, G, H, K ) , where the ( t, x, y, u ) -components aregiven by (7) , the ( u t , u x , u y ) -components are computed using the chain rule, and the ( F, G, H, K ) -components are defined by the equations (8) , (9) , (10) and (11) , respectively. Corollary 5.
A smooth vector field Q which defines an infinitesimal point symmetry of equation (1) , has the form Q = a ( t ) ∂ t + b ( t, x, y, u ) ∂ x + c ( t, x, y, u ) ∂ y + ϕ ( t, x, y, u ) ∂ u . (21) where a ( t ) , b ( t, x, y, u ) , c ( t, x, y, u ) , ϕ ( t, x, y, u ) are smooth functions of their arguments. Theorem 6.
A vector field Q = a ( t ) ∂ t + b ( t, x, y, u ) ∂ x + c ( t, x, y, u ) ∂ y + ϕ ( t, x, y, u ) ∂ u can be transformed by transformations of the form (7) into one of the following canonical forms: Q = ∂ t , Q = ∂ u . (22) Proof.
Any operator Q = a ( t ) ∂ t + b ( t, x, y, u ) ∂ x + c ( t, x, y, u ) ∂ y + ϕ ( t, x, y, u ) ∂ u , is transformed by our allowed transformations into Q ′ = Q ( T ) ∂ t ′ + Q ( X ) ∂ x ′ + Q ( Y ) ∂ y ′ + Q ( U ) ∂ u ′ .If a ( t ) = 0 then we choose T ( t ) so that Q ( T ) = a ( t ) ˙ T ( t ) = 1 (at least locally) and also we canchoose X, Y and U to be each of any three independent integrals of the PDE Q ( V ) = 0. Thisgives the canonical form Q = ∂ t in some coordinate system. If we now have a ( t ) = 0 then Q is transformed into Q ′ = Q ( X ) ∂ x ′ + Q ( Y ) ∂ y ′ + Q ( U ) ∂ u ′ . We then choose X, Y and U so that Q ( X ) = Q ( Y ) = 0 , Q ( U ) = 1. This gives us the canonical form Q = ∂ u in some coordinatesystem. 5 emark 7. It is sometimes useful to use the fact that t ′ = t, x ′ = y ′ = u, u ′ = x = y isan equivalence transformation preserving the form of equation (1), and it gives us the followingtransformations of derivatives: ∂ t ↔ ∂ t , ∂ x ↔ ∂ y , ∂ x ↔ ∂ u , ∂ y ↔ ∂ u . The second prolongation of Q in (21) is the vector fieldpr (2) Q = Q + ϕ x ∂ u x + ϕ y ∂ u y + ϕ t ∂ u t + ϕ xx ∂ u xx + ϕ xy ∂ u xy + ϕ xt ∂ u xt + ϕ yy ∂ u yy + ϕ yt ∂ u yt + ϕ tt ∂ u tt , (23)with coefficients ϕ i = D i ( ϕ − bu x − cu y − au t ) + bu xi + cu yi + au ti , (24) ϕ ij = D i ( D j ( ϕ − bu x − cu y − au t )) + bu xij + cu yij + au tij , (25)where D i represents total derivative and subscripts of u are derivative in terms of the respectivecoordinates i and j in above could be x, y or t coordinates. Then the vector field Q is a symmetryof (1) precisely when its second-order prolongation (23) annihilates equation (1) on its solutionmanifold, that is we havepr (2) Q (∆) (cid:12)(cid:12)(cid:12) ∆=0 = 0 , ∆ = u t − F u xx − Gu yy − Hu xy − K, (26)namely h ϕ t − [ Q ( F ) + ϕ x F u x + ϕ y F u y ] u xx − [ Q ( G ) + ϕ x G u x + ϕ y G u y ] u yy − [ Q ( H ) + ϕ x H u x + ϕ y H u y ] u xy − [ Q ( K ) + ϕ x K u x + ϕ y K u y ] − ϕ xx F − ϕ yy G − ϕ xy H (cid:12)(cid:12)(cid:12) ∆=0 = 0 . (27) Proposition 8.
The symmetry group of the nonlinear equation (1) for functions
F, G, H and K is generated by the vector field Q = a ( t ) ∂ t + b ( t, x, y, u ) ∂ x + c ( t, x, y, u ) ∂ y + ϕ ( t, x, y, u ) ∂ u (28) where the functions a , b , c and ϕ satisfy the determining equations aF t + bF x + cF y + ϕF u + [ ϕ x + ( ϕ u − b x ) u x − c x u y − b u u x − c u u x u y ] F u x + [ ϕ y + ( ϕ u − c y ) u y − b y u x − c u u y − b u u x u y ] F u y + [ a t − b x − b u u x ] F − [ b y + b u u y ] H = 0 , (29a) aG t + bG x + cG y + ϕG u + [ ϕ x + ( ϕ u − b x ) u x − c x u y − b u u x − c u u x u y ] G u x + [ ϕ y + ( ϕ u − c y ) u y − b y u x − c u u y − b u u x u y ] G u y + [ a t − c y − c u u y ] G − [ c x + c u u x ] H = 0 , (29b) aH t + bH x + cH y + ϕH u + [ ϕ x + ( ϕ u − b x ) u x − c x u y − b u u x − c u u x u y ] H u x + [ ϕ y + ( ϕ u − c y ) u y − b y u x − c u u y − b u u x u y ] H u y + [ a t − b x − c y − c u u y − b u u x ] H − c x + c u u x ) F − b y + b u u y ) G = 0 , (29c)6 K t + bK x + cK y + ϕK u + [ ϕ x + ( ϕ u − b x ) u x − c x u y − b u u x − c u u x u y ] K u x + [ ϕ y + ( ϕ u − c y ) u y − b y u x − c u u y − b u u x u y ] K u y + [ a t − ϕ u + b u u x + c u u y ] K + [ ϕ xy + ( ϕ yu − b xy ) u x + ( ϕ xu − c xy ) u y + ( ϕ uu − b xu − c yu ) u x u y − b yu u x − b uu u x u y − c xu u y − c uu u x u y ] H + [ ϕ yy + (2 ϕ yu − c yy ) u y + ( ϕ uu − c yu ) u y − c uu u y − b yy u x − b yu u x u y − b uu u x u y ] G + [ ϕ xx + (2 ϕ xu − b xx ) u x − c xx u y + ( ϕ uu − b xu ) u x − b uu u x − c xu u x u y − c uu u x u y ] F − ϕ t + b t u x + c t u y = 0 . (29d) Theorem 9.
There are only two inequivalent forms for one-dimensional abelian Lie symmetryalgebras A = h ∂ t i and A = h ∂ u i . Theorem 10.
All inequivalent, admissible two-dimensional abelian Lie symmetry algebras are h ∂ t , ∂ u i , h ∂ x , ∂ u i and h ∂ u , x∂ u i .Proof. Now we take dim A = 2. We have A = h Q , Q i . For rank A = 1 the case Q = ∂ t isimpossible because it leads to Q = a ( t ) ∂ t and [ Q , Q ] = 0 gives ˙ a ( t ) = 0, so dim A = 1. Wetake Q = ∂ u and Q = ϕ ( t, x, y ) ∂ u . The residual equivalence group G ∼ ( ∂ u ) is given by t ′ = T ( t ) , x ′ = X ( t, x, y ) , y ′ = Y ( t, x, y ) , u ′ = u + U ( t, x, y )where ˙ T ( t ) ( X x Y y − X y Y x ) = 0. Under such a transformation Q is mapped to Q ′ = ϕ ( t, x, y ) ∂ u ′ .If ϕ x = 0 (or ϕ y = 0) we may take X ( t, x, y ) = ϕ ( t, x, y ) so that the Q ′ = x ′ ∂ u ′ and then wehave h ∂ u , x∂ u i in canonical form. If ϕ x = ϕ y = 0 then Q = ϕ ( t ) ∂ u . Substituting ∂ u and ϕ ( t ) ∂ u into the equation (29b) we find G u = 0 , ˙ ϕ ( t ) = 0 so ϕ ( t ) is a constant and thus dim A = 1which is a contradiction, so we have no admissible canonical form in this case.If dim A = 2 , rank A = 2 then we may take Q = ∂ t or Q = ∂ u . If Q = ∂ t then, bycommutativity and the fact that A = h Q , Q i , we may take Q = b ( x, y, u ) ∂ x + c ( x, y, u ) ∂ y + ϕ ( x, y, u ) ∂ u . The residual equivalence group G ∼ ( ∂ t ) consists of invertible transformations of theform t ′ = t + l, x ′ = X ( x, , y, u ) , y ′ = Y ( x, , y, u ) , u ′ = U ( x, y, u )where l = constant. Under such a transformation Q is mapped to Q ′ = Q ( X ) ∂ x ′ + Q ( Y ) ∂ y ′ + Q ( U ) ∂ u ′ . We can always choose X, Y and U so that Q ( U ) = 1 , Q ( X ) = Q ( Y ) = 0. Thuswe have the canonical form h ∂ t , ∂ u i . If we take Q = ∂ u , then we have Q = a ( t ) ∂ t + b ( t, x, y ) ∂ x + c ( t, x, y ) ∂ y + ϕ ( t, x, y ) ∂ u . The residual equivalence group G ∼ ( ∂ u ) is the group of transformations t ′ = T ( t ) , x ′ = X ( t, x, y ) , y ′ = Y ( t, x, y ) , u ′ = u + U ( t, x, y )with ˙ T ( t ) ( X x Y y − X y Y x ) = 0. Under such a transformation, Q is mapped to Q ′ = a ( t ) ˙ T ( t ) ∂ t ′ + Q ( X ) ∂ x ′ + Q ( Y ) ∂ y ′ + [ ϕ + Q ( U )] ∂ u ′ . If a ( t ) = 0, we may choose T ( t ) , X ( t, x, y ) , Y ( t, x, y ) and U ( t, x, y ) so that a ( t ) ˙ T ( t ) = 1 , Q ( X ) =0 , Q ( Y ) = 0 , and ϕ + Q ( U ) = 0. Then we may take Q = ∂ u , Q = ∂ t . If, however, a ( t ) = 0,then Q = b ( t, x, y ) ∂ x + c ( t, x, y ) ∂ y + ϕ ( t, x, y ) ∂ u and we then have b + c = 0 since we haverank A = 2. Then Q ′ = [ b ( t, x, y ) X x + c ( t, x, y ) X y ] ∂ x ′ +[ b ( t, x, y ) Y x + c ( t, x, y ) Y y ] ∂ y ′ +[ ϕ + b ( t, x, y ) U x + c ( t, x, y ) U y ] ∂ u ′ and because b + c = 0 we may choose X ( t, x, y ) , Y ( t, x, y ) and U ( t, x, y ) so that ϕ + b ( t, x, y ) U x + c ( t, x, y ) U y = 0 , b ( t, x, y ) Y x + c ( t, x, y ) Y y = 0 , b ( t, x, y ) X x + c ( t, x, y ) X y = 17nd this gives Q ′ = ∂ x ′ , so we have the canonical form A = h ∂ u , ∂ x i . Hence there are thefollowing admissible, canonical forms for dim A = 2: h ∂ u , x∂ u i , h ∂ t , ∂ u i , h ∂ x , ∂ u i . Theorem 11.
Inequivalent three-dimensional abelian admissible Lie symmetry algebras are h ∂ t , ∂ x , ∂ u i , h ∂ t , ∂ u , x∂ u i , h ∂ u , x∂ u , y∂ u i , h ∂ x , ∂ y , ∂ u i , h ∂ u , x∂ u , ϕ ( t, x ) ∂ u i , with ϕ xx = 0 , Proof.
For dim A = 3 we have A = h Q , Q , Q i . If rank A = 1 then we may take Q = ∂ u , Q = x∂ u by the previous argument for dim A = 2. We then have Q = ϕ ( t, x, y ) ∂ u . If ϕ x = ϕ y = 0then we find Q = ϕ ( t ) ∂ u , which dim A = 3 gives ˙ ϕ ( t ) = 0. But substituting ∂ u , ϕ ( t ) ∂ u in(29d) as symmetries lead to ˙ ϕ ( t ) = 0 that is a contradiction. If ϕ x = 0 , ϕ y = 0 then ϕ xx = 0.Because ϕ xx = 0 gives ϕ ( t, x ) = α ( t ) + β ( t ) x . Then substituting ∂ u , x∂ u , [ α ( t ) + β ( t ) x ] ∂ u into(29d) we obtain ˙ α ( t ) + ˙ β ( t ) x = 0, which is possible only for constant α ( t ) , β ( t ). But thismeans that Q is a constant linear combination of Q , and Q , contradicting dim A = 3. So, ϕ xx ( t, x ) = 0. For A = h ∂ u , x∂ u , ϕ ( t, x ) ∂ u i as a symmetry algebra we find the following form forthe two–dimensional evolution equation: u t = ϕ t ( t, x ) ϕ xx ( t, x ) u xx + G ( t, x, y ) u yy + H ( t, x, y ) u xy + K ( t, x, y ) , (30)with ϕ t ( t, x ) = 0 , ϕ xx ( t, x ) = 0. Thus, the evolution equation is semi-linear. Now, if ϕ y = 0,since the residual equivalence group G ∼ ( ∂ u , x∂ u ) is t ′ = T ( t ) , x ′ = x, y ′ = Y ( t, x, y ) , u ′ = u + U ( t, x, y )with Y y = 0, then we can choose ϕ ( t, x, y ) = Y ( t, x, y ) and so Q = y∂ u , thus A = h ∂ u , x∂ u , y∂ u i .We now come to dim A = 3 , rank A = 2. If we take Q = ∂ t then we may take Q = ∂ u since h Q , Q i is a two-dimensional abelian subalgebra of A , and there is only one type of two-dimensional abelian algebra containing ∂ t as we have seen. Then we have Q = a∂ t + b ( x, y ) ∂ x + c ( x, y ) ∂ y + ϕ ( x, y ) ∂ u with a = constant and so we may take Q = b ( x, y ) ∂ x + c ( x, y ) ∂ y + ϕ ( x, y ) ∂ u .The requirement that rank A = 2 then gives b ( x, y ) = c ( x, y ) = 0 and we find that Q = ϕ ( x, y ) ∂ u . Obviously, ϕ x + ϕ y = 0 since we must have dim A = 3. Now, the residual equivalencegroup G ∼ ( ∂ t , ∂ u ) is given by transformations of the form t ′ = t + k, x ′ = X ( x, y ) , y ′ = Y ( x, y ) , u ′ = u + U ( x, y )with X x .Y y = 0. We see that under such a transformation, Q = ϕ ( x, y ) ∂ u is transformed to Q ′ = ϕ ( x, y ) ∂ u ′ and we may take ϕ ( x, y ) = X ( x, y ) (or ϕ ( x, y ) = Y ( x, y )) giving Q ′ = x ′ ∂ u ′ (or Q ′ = y ′ ∂ u ′ ), so we find the canonical form A = h ∂ t , ∂ u , x∂ u i . If Q = ∂ u , then we may take Q = ∂ t or Q = ∂ x if Q ∧ Q = 0. If Q = ∂ t then Q = x∂ u as before. If Q = ∂ x then Q = b ( t ) ∂ x + ϕ ( t ) ∂ u because rank A = 2. However, putting Q = ∂ u , Q = ∂ x , Q = b ( t ) ∂ x + ϕ ( t ) ∂ u into the (29d) equation, we obtain ˙ ϕ ( t ) − ˙ b ( t ) u x = 0, which means that ˙ b ( t ) = ˙ ϕ ( t ) = 0 and hence Q is a linear combination of Q and Q , which is a contradiction. Thus we cannot have Q = ∂ x in this case. If Q ∧ Q = 0 then Q = ϕ ( t, x, y ) ∂ u , and since h Q , Q i is a two-dimensionalsubalgebra of A = h Q , Q , Q i then we know from the above that we may take Q = x∂ u . Thisthen gives us Q = ∂ t in canonical form. Hence we have only the case A = h ∂ t , ∂ u , x∂ u i whendim A = 3 , rank A = 2.Now consider the case dim A = 3 , rank A = 3. If we take Q = ∂ t and Q = ∂ u then we maytake Q = b ( x, y ) ∂ x + c ( x, y ) ∂ x + ϕ ( x, y ) ∂ u with b + c = 0 since we have ∂ t ∧ ∂ u ∧ Q = 0.Under G ∼ ( ∂ t , ∂ u ) transformation Q is mapped to Q ′ = [ b ( x, y ) X x + c ( x, y ) X y ] ∂ x ′ + [ b ( x, y ) Y x + c ( x, y ) Y y ] ∂ y ′ + [ ϕ + b ( x, y ) U x + c ( x, y ) U y ] ∂ u ′ b + c = 0 we can choose X ( x, y ) , Y ( x, y ) and U ( x, y ) so that ϕ + b ( x, y ) U x + c ( x, y ) U y = 0 , b ( x, y ) Y x + c ( x, y ) Y y = 0 , b ( x, y ) X x + c ( x, y ) X y = 1 and this gives Q ′ = ∂ x ′ , sowe have the canonical form A = h ∂ t , ∂ y , ∂ x i . Now if we take Q = ∂ x and Q = ∂ u then we cantake Q = a ( t ) ∂ t + b ( t, y ) ∂ x + c ( t, y ) ∂ x + ϕ ( t, y ) ∂ u with a + c = 0 since we have ∂ x ∧ ∂ u ∧ Q = 0.The residual equivalence group G ∼ ( ∂ x , ∂ u ) is t ′ = T ( t ) , x ′ = x + ξ ( t, y ) , y ′ = Y ( t, y ) , u ′ = u + U ( t, y )with ˙ T ( t ) = 0 and Y y = 0. Under this transformation Q maps to Q ′ = a ( t ) ˙ T ( t ) ∂ t ′ + [ a ( t ) ξ t + cξ y + ϕ ] ∂ x ′ + [ a ( t ) Y t + cY y ] ∂ y ′ + [ a ( t ) U t + cU y + ϕ ] ∂ u ′ . If a ( t ) = 0 then c = 0 and we may choose ξ, Y and U so that cξ y + ϕ = 0, cY y = 1 and cU y + ϕ = 0. Thus we find A = h ∂ x , ∂ u , ∂ y i . The a ( t ) = 0 doesn’t lead to a new case. Theorem 12.
Inequivalent abelian admissible Lie symmetry algebras A with dimension A ≥ are h ∂ t , ∂ u , x∂ u , y∂ u i , h ∂ t , ∂ x , ∂ u , y∂ u i , h ∂ t , ∂ x , ∂ u , y∂ x i , h ∂ t , ∂ x , ∂ y , ∂ u i , h ∂ u , x∂ u , y∂ u , ϕ ( t, x, y ) ∂ u , q ( t, x, y ) ∂ u , . . . , q k ( t, x, y ) ∂ u i , with ϕ xx = 0 , ( q i ) xx = 0 , h ∂ t , ∂ u , x∂ u , y∂ u , ϕ ( x, y ) ∂ u , q ( x, y ) ∂ u , . . . , q k ( x, y ) ∂ u i , with ϕ xy = 0 , ( q i ) xy = 0 . Proof.
Now consider the case dim A = 4 , rank A = 4. We have A = h Q , Q , Q , Q i and Q i = a i ( t ) ∂ t + b i ( t, x, y, u ) ∂ x + c i ( t, x, y, u ) ∂ y + ϕ i ( t, x, y, u ) ∂ u for i = 1 , ,
3. Since Q ∧ Q ∧ Q ∧ Q = 0 at least one of the coefficients a i ( t ) must be differentfrom zero. So, assume a ( t ) = 0. In this case, we may take Q = ∂ t in canonical form, accordingto Theorem 6. Then we must have Q i = a i ∂ t + b i ( x, y, u ) ∂ x + c i ( x, y, u ) ∂ y + ϕ i ( x, y, u ) ∂ u for i = 2 , ,
4. We may take a i = 0 for i = 2 , , Q , Q , Q , Q .Hence Q = b ( x, y, u ) ∂ x + c ( x, y, u ) ∂ y + ϕ ( x, y, u ) ∂ u , Q = b ( x, y, u ) ∂ x + c ( x, y, u ) ∂ y + ϕ ( x, y, u ) ∂ u and Q = b ( x, y, u ) ∂ x + c ( x, y, u ) ∂ y + ϕ ( x, y, u ) ∂ u . Consequently, we may, by Theorem 6take Q = ∂ x and Q = ∂ u . Finally, this means that Q = b ( y ) ∂ x + c ( y ) ∂ y + ϕ ( y ) ∂ u bycommutativity, and c ( y ) = 0 because ∂ t ∧ ∂ x ∧ ∂ u ∧ Q = 0. The residual equivalence group G ∼ ( ∂ t , ∂ x , ∂ u ) is given by transformations of the form t ′ = t + k, x ′ = x + Y ( y ) , y ′ = Y ( y ) , u ′ = u + U ( y )with Y y = 0. Under such a transformation, Q is transformed to Q ′ = [ b ( y ) + c ( y ) X y ] ∂ x ′ + c ( y ) Y y ∂ y ′ + [ ϕ ( y ) + c ( y ) U y ] ∂ y ′ . Since c ( y ) = 0 then we may choose X ( y ) , Y ( y ) and U ( y ) so that b ( y )+ c ( y ) X y = 0 , c ( y ) Y y = 1and ϕ ( y ) + c ( y ) U y = 0, so we may take Q = ∂ y in canonical form. Consequently, we haveonly one canonical form A = h ∂ t , ∂ x , ∂ u , ∂ y i .For dim A ≥ , rank A = 4, we may choose Q = ∂ t , Q = ∂ x , Q = ∂ u , Q = ∂ y . For anyother symmetry Q = a ( t ) ∂ t + b ( t, x, y, u ) ∂ x + c ( t, x, y, u ) ∂ y + ϕ ( t, x, y, u ) ∂ u , the coefficients mustbe constants, so that Q will be a constant linear combination of Q , Q , Q , Q contradictingwith dim A ≥ A = 4 , rank A = 3, we may take Q = ∂ t , Q = ∂ x , Q = ∂ u and for any othersymmetry we take Q = b ( y ) ∂ x + ϕ ( y ) ∂ u . The residual equivalence group G ∼ ( ∂ t , ∂ x , ∂ u ) is t ′ = t + k, x ′ = x + ξ ( y ) , y ′ = Y ( y ) , u ′ = u + U ( y )9ith Y y = 0. Since dim A = 4 then we have b y + ϕ y = 0. If b y = 0 , ϕ = 0, since Y y = 0 thus wemay choose ϕ ( y ) = Y ( y ) and then we obtain Q = y∂ u . Therefore we find A = h ∂ t , ∂ x , ∂ u , y∂ u i .Now if b y = 0 , ϕ = 0, we may choose b ( y ) = Y ( y ) and then we find Q = y∂ x and thus wehave A = h ∂ t , ∂ x , ∂ u , y∂ x i . We may Q = ∂ x , Q = ∂ y , Q = ∂ u and any other symmetry is Q = b ( t ) ∂ x + c ( t ) ∂ y + ϕ ( t ) ∂ u , because A is a abelian Lie algebra with dim A = 4. Putting Q , Q , Q , Q in determining equations (29) leads to ˙ ϕ ( t ) − ˙ b ( t ) u x − ˙ c ( t ) u y = 0. Therefore wefind ˙ ϕ ( t ) = ˙ b ( t ) = ˙ c ( t ) = 0, that means Q is a Linear combination of Q , Q , Q contradictingwith dim A = 4. Consequently we have only two canonical forms A = h ∂ t , ∂ x , ∂ u , y∂ u i and A = h ∂ t , ∂ x , ∂ u , y∂ x i for an abelian Lie algebra with dim A = 4 , rank A = 3.Now we come to dim A ≥ , rank A = 3. We may take Q = ∂ t , Q = ∂ x , Q = ∂ u . Forany other symmetry vector field Q = a ( t ) ∂ t + b ( t, x, u ) ∂ x + c ( t, x, u ) ∂ u the coefficients must beconstants, so that Q will be a constant linear combination of Q , Q , Q , contradicting dim A ≥ A with dim A ≥ , rank A = 3.For dim A = 4 , rank A = 2 we may take Q = ∂ t , Q = ∂ u , Q = x∂ u , and for Q wetake Q = ϕ ( x, y ) ∂ u because A is abelian and rank A = 2. Further ϕ y = 0, for otherwise Q would be a constant linear combination of Q and Q , contradicting dim A = 4. Thus we have A = h ∂ t , ∂ u , x∂ u , ϕ ( x, y ) ∂ u i with ϕ y = 0. The residual equivalence group G ∼ ( ∂ t , ∂ u , x∂ u ) is givenby transformations of the form t ′ = t + k, x ′ = x, y ′ = Y ( x, y ) , u ′ = u + U ( x, y )with Y y = 0. Under this transformation Q maps to Q ′ = ϕ ( x, y ) ∂ ′ u and since ϕ y = 0 we maychoose ϕ ( x, y ) = Y ( x, y ). Thus Q = y∂ u and A = h ∂ t , ∂ u , x∂ u , y∂ u i .For dim A = 5 , rank A = 2 we find that any additional symmetry operator Q must be ofthe form Q = ϕ ( x, y ) ∂ u with ϕ y = 0 and, substituting the vector fields a symmetries in the(29) equations gives ϕ xy K + ϕ yy G + ϕ xx F = 0 with ϕ xy + ϕ yy + ϕ xx = 0. From this it followsthat ϕ x = 0 , ϕ y = 0 and then ϕ xy = 0. Hence we find A = h ∂ t , ∂ u , x∂ u , y∂ u , ϕ ( x, y ) ∂ u i with ϕ xy = 0. For dim A ≥ , rank A = 2, arguments similar to those for dim A = 5 , rank A = 2 giveus abelian Lie algebras A = h ∂ t , ∂ u , x∂ u , y∂ u , ϕ ( x, y ) ∂ u , q ( x, y ) ∂ u , . . . , q k ( x, y ) ∂ u i (31)with the sole proviso that each of the q : s satisfy q xy = 0 and ϕ xx F + ϕ yy Gϕ xy = q xx F + q yy Gq xy , as well as linear independence of the vector fields (31). We obtain equations of the form u t = F ( x, y ) u xx + G ( x, y ) u yy + H ( x, y ) u xy − ϕ yy G + ϕ xx Fϕ xy , (32)with ϕ xy = 0 , ϕ xx + ϕ yy = 0.For dim A ≥ , rank A = 1, arguments similar to those for dim A = 3 , rank A = 1 give usabelian Lie algebras A of the form A = h ∂ u , x∂ u , y∂ u , ϕ ( t, x, y ) ∂ u , q ( t, x, y ) ∂ u , . . . , q k ( t, x, y ) ∂ u i (33)with the sole proviso that each of the q : s satisfy q xx ( t, x, y ) = 0 , ϕ ( t, x, y ) = 0 and ϕ t − ϕ yy G − ϕ xy Kϕ xx = q t − q yy G − q xy Kq xx , as well as linear independence of the vector fields (33). We obtain equations of the form u t = (cid:20) ϕ t − ϕ yy G − ϕ xy Kϕ xx (cid:21) u xx + G ( t, x, y ) u yy + H ( t, x, y ) u xy + K ( t, x, y ) . (34)10e have the following corollary of the above results: Corollary 13.
The evolution equations of the form (1) admitting abelian Lie algebras A with dim A ≥ or if dim A ≥ , rank A = 1 as symmetries are linearizable.Proof. The proof is just by calculation: for dim A ≥ , rank A = 1 we find the evolutionequations (30) and (34) which is linear in u and its derivatives. Similarly, the rank-two algebra A = h ∂ t , ∂ u , x∂ u , y∂ u , ϕ ( x, y ) ∂ u , q ( x, y ) ∂ u , . . . , q k ( x, y ) ∂ u i , with ϕ xy = 0 , ( q i ) xy = 0 , gives the (32) equation.We have the following table of non-linear equations form of (1) admitting abelian Lie algebrasas symmetries:The invariant solutions of symmetries admitting abelian Lie algebras for non-linear forms of (1)Realizations invariant solutions h ∂ t i u t = e F ( x, y, u, u x , u y ) u xx + e G ( x, y, u, u x , u y ) u yy + e H ( x, y, u, u x , u y ) u xy + e K ( x, y, u, u x , u y ) h ∂ u i u t = e F ( t, x, y, u x , u y ) u xx + e G ( t, x, y, u x , u y ) u yy + e H ( t, x, y, u x , u y ) u xy + e K ( t, x, y, u x , u y ) h ∂ t , ∂ u i u t = e F ( x, y, u x , u y ) u xx + e G ( x, y, u x , u y ) u yy + e H ( x, y, u x , u y ) u xy + e K ( x, y, u x , u y ) h ∂ x , ∂ u i u t = e F ( t, y, u x , u y ) u xx + e G ( t, y, u x , u y ) u yy + e H ( t, y, u x , u y ) u xy + e K ( t, y, u x , u y ) h ∂ u , x∂ u i u t = e F ( t, x, y, u y ) u xx + e G ( t, x, y, u y ) u yy + e H ( t, x, y, u y ) u xy + e K ( t, x, y, u y ) h ∂ t , ∂ x , ∂ u i u t = e F ( y, u x , u y ) u xx + e G ( y, u x , u y ) u yy + e H ( y, u x , u y ) u xy + e K ( y, u x , u y ) h ∂ t , ∂ u , x∂ u i u t = e F ( x, y, u y ) u xx + e G ( x, y, u y ) u yy + e H ( x, y, u y ) u xy + e K ( x, y, u y ) h ∂ x , ∂ y , ∂ u i u t = e F ( t, u x , u y ) u xx + e G ( t, u x , u y ) u yy + e H ( t, u x , u y ) u xy + e K ( t, u x , u y ) h ∂ t , ∂ x , ∂ u , y∂ u i u t = e F ( y, u x ) u xx + e G ( y, u x ) u yy + e H ( y, u x ) u xy + e K ( y, u x ) h ∂ t , ∂ x , ∂ u , y∂ x i u t = (cid:20) u y u x e G ( y, u x ) − u y u x ee G ( y, u x ) + eee G ( y, u x ) (cid:21) u xx + e G ( x, y, u y ) u yy + e H ( x, y, u y ) u xy + u y u x e G ( y, u x ) + ee G ( y, u x ) h ∂ t , ∂ x , ∂ y , ∂ u i u t = e F ( u x , u y ) u xx + e G ( u x , u y ) u yy + e H ( u x , u y ) u xy + e K ( u x , u y ) We have carried out the abelain Lie symmetry classification of evolution equation (1). There arejust two inequivalent forms for one–dimensional abelian Lie symmetry algebras h ∂ t i , h ∂ u i andthree inequivalent forms for two–dimensional h ∂ t , ∂ u i , h ∂ x , ∂ u i , h ∂ u , x∂ u i . For three–dimensionalwe have h ∂ t , ∂ x , ∂ u i , h ∂ t , ∂ u , x∂ u i , h ∂ u , x∂ u , y∂ u i , h ∂ x , ∂ y , ∂ u i , h ∂ u , x∂ u , ϕ ( t, x ) ∂ u i with ϕ xx = 0abelian Lie symmetry algebras. We have four inequivalent four–dimensional admissible abelianLie symmetry algebras h ∂ t , ∂ u , x∂ u , y∂ u i , h ∂ t , ∂ x , ∂ u , y∂ u i , h ∂ t , ∂ x , ∂ u , y∂ x i , h ∂ t , ∂ x , ∂ y , ∂ u i and foradmissible abelian Lie symmetry algebras with dimension bigger than four are h ∂ u , x∂ u , y∂ u , ϕ ( t, x, y ) ∂ u , q ( t, x, y ) ∂ u , . . . , q k ( t, x, y ) ∂ u i , with ϕ xx = 0 , ( q i ) xx = 0 , h ∂ t , ∂ u , x∂ u , y∂ u , ϕ ( x, y ) ∂ u , q ( x, y ) ∂ u , . . . , q k ( x, y ) ∂ u i , with ϕ xy = 0 , ( q i ) xy = 0 . The evolution equations of the form (1) admitting five-dimensional abelian Lie algebras or with adimension higher than five or if with three-dimensional abelian Lie algebras or with a dimensionhigher than three with rank-one as symmetries are linearizable.In future work, we show the evolution equations (1) admit sl (2 , R ) as semi-simple symmetryalgebras we classify these realizations. 11 Acknowledgement
I thank the Mathematics Department, Link¨oping University and professor Peter Basarab-Horwathfor their hospitality during my stay in Link¨oping, where this work was started. The author alsowishs to thank Professor R. O. Popovych for useful comments.This work was supported by the Babol Noshirvani University of Technology under Grant No.BNUT/391024/99.
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