Point and contact equivalence groupoids of two-dimensional quasilinear hyperbolic equations
aa r X i v : . [ m a t h . A P ] S e p Point and contact equivalence groupoidsof two-dimensional quasilinear hyperbolic equations
Roman O. Popovych
Fakult¨at f¨ur Mathematik, Universit¨at Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, AustriaInstitute of Mathematics of NAS of Ukraine, 3 Tereshchenkivska Str., 01024 Kyiv, Ukraine
E-mail: [email protected]
We describe the point and contact equivalence groupoids of an important class of two-dimensional quasilinear hyperbolic equations. In particular, we prove that this class is nor-malized in the usual sense with respect to point transformations, and its contact equivalencegroupoid is generated by the first-order prolongation of its point equivalence groupoid, thecontact vertex group of the wave equation and a family of contact admissible transformationsbetween trivially Darboux-integrable equations.
Genuine (first-order) contact transformations between differential equations [7, 10], are rarerobjects than their point counterparts. These transformations were introduced by Sophus Lie.In particular, it was known to him [3] that the class of Monge–Amp´ere equations is closed underthe action of its contact equivalence (pseudo)group, which consists of all contact transformationsof two independent and one dependent variables. In the modern terminology [8, 9, 11], this meansthat this class is normalized in the usual sense with respect to contact transformations. Thewell-known B¨acklund theorem [7, Theorem 4.32] states that genuine contact transformations,which are not the first-order prolongations of point transformations, exist only in the case ofone dependent variable.In the present paper, we compute the point and contact equivalence groupoids of the class H gen of two-dimensional quasilinear hyperbolic equations of the form u xy = f ( x, y, u, u x , u y ) , (1)which is properly contained in the class of Monge–Amp´ere equations. Here x and y are theindependent variables, u is is the unknown function of ( x, y ), which is treated as the dependentvariable, and the arbitrary element f of the class runs through the set A of differential func-tions for these variables that depend at most on ( x, y, u, u x , u y ), i.e., whose order is less thanor equal to one. Subscripts of functions denote derivatives with respect to the correspondingvariables, e.g., u xy := ∂ u/∂x ∂y and f u x := ∂f /∂u x , and we use the same notation for thederivatives u x , u y , u xx , u xy and u yy and for the corresponding coordinates of the second-orderjet space J ( R x,y , R u ) with the independent variables ( x, y ) and the dependent variable u . Theconsideration is within the local framework. We employ the terms “group” and “groupoid” forpseudogroups and pseudogroupoids, respectively. For a fixed value of the arbitrary element f ,let E f , G c f and G c f denote the equation from the class H gen with this value of f , its contactsymmetry group and its vertex group, G c f := { ( f, Φ , f ) | Φ ∈ G c f } . See [11] for necessary no-tions of the theory of admissible transformations within classes of differential equations. Thespace with coordinates ( x, y, u, u x , u y ) is the minimal underlying space for both the point andthe contact equivalence groups of the class H gen , cf. [2, footnote 1] and [8, footnote 1]. ByD x and D y we denote the operators of total derivatives with respect to x and y , respectively,D x = ∂ x + u x ∂ u + u xx ∂ u x + u xy ∂ u y + · · · and D y = ∂ y + u y ∂ u + u xy ∂ u x + u yy ∂ u y + · · · .The form (1) is the most general form for equations that can be interpreted as (1+1)-dimensional generalized nonlinear Klein–Gordon equations in the light-cone (characteristic) co-ordinates. It is also the canonical form for single second-order two-dimensional partial differential1quations each of whose two Monge systems admits a first integral of the first order [10, Theorem12.1.2]. The class H gen contains a number of famous equations, including the wave equation,the Klein–Gordon equation, the Liouville equation, the Tzitzeica equation and the sine-Gordon(or Bonnet) equation. The problem of studying the subclass of equations of the form (1) with f u x = f u y = 0 within the framework of group analysis of differential equations was posed bySophus Lie [5]. The group classification problem for this subclass was completely solved in [1];see also a review of results on this subclass therein. The equations from the entire class H gen forwhich each of the two Monge systems admits at least two functionally independent first integralsare called hyperbolic Goutsat equations. Such equations were classified by Goutsat with respectto the equivalence generated by the point equivalence group G ∼ gen of the class H gen (this groupis given Theorem 2 below). See [10, Chapter 13] for the presentation of the enhanced classifi-cation proof by Vessiot and [4] for references on modern results related to Darboux-integrableequations.We describe contact admissible transformations within the class H gen , which essentially gen-eralizes Lie’s study in [5]. After deriving determining equations for such transformations inSection 2, we compute point and contact equivalence groupoids of the class H gen in Section 3.In particular, we prove that both the sources and targets of genuine contact admissible trans-formations are trivially Darboux-integrable equations. Applying the direct method of finding admissible transformations, we fix a contact admissibletransformation T = ( f, Φ , ˜ f ) in the class H gen . Here the equations E f : u xy = f ( x, y, u, u x , u y )and E ˜ f : ˜ u ˜ x ˜ y = ˜ f (˜ x, ˜ y, ˜ u, ˜ u ˜ x , ˜ u ˜ y ) belong to the class H gen , where f ∈ C ∞ (Ω , R ) and ˜ f ∈ C ∞ ( ˜Ω , R )with connected open domains Ω and ˜Ω contained in R . Φ is a contact transformation with theindependent variables ( x, y ) and the dependent variable u , Φ: (˜ x, ˜ y, ˜ u, ˜ u ˜ t , ˜ u ˜ y ) = ( X, Y, U, U x , U y ),that is defined by a diffeomorphism from Ω onto ˜Ω and maps E f to E ˜ f , Φ ∗ E f = E ˜ f . Thus, thefunctions X , Y , U , U x and U y in the components of the transformation Φ are smooth functionsof z := ( x, y, u, u x , u y ) ∈ Ω with J := (cid:12)(cid:12) ∂ ( X, Y, U, U x , U y ) /∂ ( x, y, u, u x , u y ) (cid:12)(cid:12) = 0 that satisfy thecontact condition U x D x X + U y D x Y = D x U,U x D y X + U y D y Y = D y U. (2)Collecting coefficients of the second derivatives of u in this condition leads to the system U x X u x + U y Y u x = U u x , U x ˆD x X + U y ˆD x Y = ˆD x U,U x X u y + U y Y u y = U u y , U x ˆD y X + U y ˆD y Y = ˆD y U, (3)where ˆD x = ∂ x + u x ∂ u and ˆD y = ∂ y + u y ∂ u are the truncated operators of total derivatives withrespect to x and y , respectively. Lemma 1.
Up to the discrete equivalence transformation I : ˜ x = y , ˜ y = x , ˜ u = u , ˜ u ˜ x = u y , ˜ u ˜ y = u x , ˜ f = f of the class H gen , the components of the transformational part Φ of any contactadmissible transformation T in this class satisfy, on the entire corresponding domain Ω , thejoint system of (3) and ˆD y X + f X u x = 0 , X u y = 0 , U xu y = (Φ ∗ ˜ f ) Y u y , ˆD y U x + f U xu x = (Φ ∗ ˜ f ) ˆD y Y, (4a)ˆD x Y + f Y u y = 0 , Y u x = 0 , U yu x = (Φ ∗ ˜ f ) X u x , ˆD x U y + f U yu y = (Φ ∗ ˜ f ) ˆD x X. (4b) Proof.
We expand the condition Φ ∗ E f = E ˜ f via substituting the expression for the involved jetvariables with tildes in terms of the jet variables without tildes, (cid:12)(cid:12)(cid:12)(cid:12) D x U y D y U y D x Y D y Y (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) D x X D y X D x U x D y U x (cid:12)(cid:12)(cid:12)(cid:12) = (Φ ∗ ˜ f ) (cid:12)(cid:12)(cid:12)(cid:12) D x X D y X D x Y D y Y (cid:12)(cid:12)(cid:12)(cid:12) on solutions of E f . (5)2ere Φ ∗ denotes the pullback by Φ, Φ ∗ ˜ f := ˜ f ( X, Y, U, U x , U y ). The first equality in (5) is adifferential consequence of the system (2). Define two 2 × A := (cid:18) Q X Q U x − (Φ ∗ ˜ f ) Q Y (cid:19) Q ∈M and A := (cid:18) Q Y Q U y − (Φ ∗ ˜ f ) Q X (cid:19) Q ∈M , where M := ( ∂ u x , ∂ u y , ˆD x + f ∂ u y , ˆD y + f ∂ u x ).It is obvious that the set Ω := { z ∈ Ω | rank A ( z ) = rank A ( z ) = 2 } is open. Moreover, itis dense in Ω. Indeed, Ω = Ω \ (Ω ∪ Ω ∪ Ω ∪ Ω ), whereΩ := (cid:8) z ∈ Ω | Q X ( z ) = 0 , Q ∈ M (cid:9) , Ω := (cid:8) z ∈ Ω \ Ω | rank A = 1 (cid:9) , Ω := (cid:8) z ∈ Ω | Q Y ( z ) = 0 , Q ∈ M (cid:9) , Ω := (cid:8) z ∈ Ω \ Ω | rank A = 1 (cid:9) . If the set Ω (resp. Ω ) contained an open subset U , then the row of J that is associated with X (resp. Y ) and thus the Jacobian J itself would vanish on U , which is a contradiction. Therefore,the subsets Ω and Ω are nowhere dense. Suppose that the set Ω contains an open subset U .Then there exists Λ ∈ C ∞ ( U , R ) such that Q U x − (Φ ∗ ˜ f ) Q Y = Λ Q X on U for any Q ∈ M ,which implies the equations D x U x = ΛD x X + (Φ ∗ ˜ f )D x Y and D y U x = ΛD y X + (Φ ∗ ˜ f )D y Y on U .We respectively subtract these equation from the equalities D x U x = U xx D x X + U xy D x Y andD y U x = U xx D y X + U xy D y Y , where U xx := Φ (2) ∗ ˜ u ˜ x ˜ x , U xy := Φ (2) ∗ ˜ u ˜ x ˜ y , and Φ (2) denotes theprolongation of the contact transformation Φ on the jet space J ( R x,y , R u ). As a result, wederive the equations( U xx − Λ)D x X + ( U xy − Φ ∗ ˜ f )D x Y = 0 , ( U xx − Λ)D y X + ( U xy − Φ ∗ ˜ f )D y Y = 0 (6)on U . Restricting the equation (6) on the manifold defined by E f in J ( R x,y , R u ), where u xy = f and U xy = Φ ∗ ˜ f , leads to the equalities (cid:0) U xx − Λ (cid:1) ( ˆD x X + f X u y + X u x u xx ) = 0 , (cid:0) U xx − Λ (cid:1) ( ˆD y X + f X u x + X u y u yy ) = 0 . Since the derivative ˜ u ˜ x ˜ x is not constrained on the solution set of E ˜ f , this implies the equationsˆD x X + f X u y + X u x u xx = 0 and ˆD y X + f X u x + X u y u yy = 0, which can be split with respectto u xx and u yy into the condition Q X ( z ) = 0, Q ∈ M , on U , contradicting the definition of Ω .Therefore, the subset Ω (and, similarly, Ω ) is nowhere dense.The splitting of the equation (5) with respect to u xx and u yy leads to the condition that the2 × A and A associated with the pairs ( Q , Q ) from the set M × M , where M := (cid:8) ˆD x + f ∂ u y , ∂ u x (cid:9) , M := (cid:8) ˆD y + f ∂ u x , ∂ u y (cid:9) , vanish. This condition implies, in view of the condition rank A ( z ) = rank A ( z ) = 2 on Ω , thesystem( Q X )( Q X ) = 0 , ( Q X ) (cid:0) Q U x − (Φ ∗ ˜ f ) Q Y (cid:1) = 0 , (cid:0) Q U x − (Φ ∗ ˜ f ) Q Y (cid:1) ( Q X ) = 0 , ( Q Y )( Q Y ) = 0 , ( Q Y ) (cid:0) Q U y − (Φ ∗ ˜ f ) Q X (cid:1) = 0 , (cid:0) Q U y − (Φ ∗ ˜ f ) Q X (cid:1) ( Q Y ) = 0 , where the operator Q i runs through the set M i , i = 1 ,
2; which is satisfied on Ω and thus,by continuity, on Ω. Therefore, in each point of Ω , we have either ( Q X ) Q ∈M = (0 ,
0) or( Q X ) Q ∈M = (0 ,
0) and either ( Q Y ) Q ∈M = (0 ,
0) or ( Q Y ) Q ∈M = (0 , Q X ) Q ∈M = (0 ,
0) at a point z ∈ Ω , then ( Q Y ) Q ∈M = (0 ,
0) at z . Indeed,otherwise ( Q X ) Q ∈M = (0 ,
0) and ( Q Y ) Q ∈M = (0 ,
0) simultaneously at z and thus onsome neighborhood U of z in Ω . This implies the equations Q X = Q Y = 0 and then Q U x = Q U y = 0 with Q ∈ M . In particular, X u y = Y u y = U xu y = U yu y = 0 and, in viewof the third equation in (3), U u y = 0, which contradicts the nondegeneracy of Φ. It is obviousthat the inverse implication, ( Q X ) Q ∈M = (0 ,
0) at a point of Ω if ( Q Y ) Q ∈M = (0 ,
0) at3he same point, also holds true. Therefore, we also have that ( Q X ) Q ∈M = (0 ,
0) at a point z ∈ Ω if and only if ( Q Y ) Q ∈M = (0 ,
0) at z . DenotingΞ := (cid:8) z ∈ Ω | ( Q X ) Q ∈M = (0 , , ( Q Y ) Q ∈M = (0 , (cid:9) , Ξ := (cid:8) z ∈ Ω | ( Q X ) Q ∈M = (0 , , ( Q Y ) Q ∈M = (0 , (cid:9) , we get Ω = Ξ ⊔ Ξ . On the closure cl(Ξ ) of Ξ in Ω, the components of Φ satisfy the system (4).Note that on the subset, where X u x = 0 or Y u y = 0, the last equations in (4a) and in (4b) aredifferential consequences of the other equations in (4) supplemented with the equations (3). Onthe closure cl(Ξ ) of Ξ in Ω, the components of Φ satisfy the analogous system obtained fromthe system (4) by permutation of ( x, u x ) and ( y, u y ). The sets cl(Ξ ) and cl(Ξ ) are disjoint sinceotherwise J = 0 on cl(Ξ ) ∩ cl(Ξ ). Therefore, Ω = cl(Ω ) = cl(Ξ ⊔ Ξ ) = cl(Ξ ) ⊔ cl(Ξ ), whichimplies that both the sets cl(Ξ ) and cl(Ξ ) are open and closed in Ω, and hence one of the setsis empty and the other coincides with Ω. Up to the discrete equivalence transformation I , wecan assume that cl(Ξ ) = Ω and cl(Ξ ) = ∅ , i.e., the system (4) holds on the entire set Ω. We first compute the point equivalence groupoid of the class H gen and then single out its threedisjoint subclasses, H xy := H x ∩ H y , H ′ x := H x \ H y and H ′ y := H y \ H x , where H x := (cid:8) E f ∈ H gen | f = F ( x, y, u, u y ) + F ( x, y, u, u y ) u x , F x + F F u y = F u + F F u y (cid:9) , H y := (cid:8) E f ∈ H gen | f = F ( x, y, u, u x ) + F ( x, y, u, u x ) u y , F y + F F u x = F u + F F u x (cid:9) , and hence H xy = (cid:8) E f ∈ H gen | f = f ( x, y, u ) + f ( x, y, u ) u x + f ( x, y, u ) u y + f ( x, y, u ) u x u y ,f y = f u , f x = f u , f y = f x = f u + f f − f f (cid:9) . We will prove that any contact admissible transformation in H gen that is not the first-orderprolongation of a point admissible transformation in H gen belongs to the contact equivalencegroupoid of H x ∪ H y . It is obvious that the subclasses H x and H y (resp. the subclasses H ′ x and H ′ y ) are mapped onto each other by the permutation I . This is why any assertion that istrue for one of these two subclasses holds true for the other after the modification using I , andeach of them can substituted by the other in the assertions below. Any equation E f from thesubclass H x (resp. H y or H xy ) can be (locally) represented in the form D x g = 0 (resp. D y h = 0or D x D y θ = 0). Here g = g ( x, y, u, u y ), h = h ( x, y, u, u x ) and θ = θ ( x, y, u ) are arbitrary smoothfunctions of their arguments with g u y = 0, h u x = 0 and θ u = 0. (The representation for equationsfrom the subclass H xy follows from those for the subclasses H x and H y in view of the theorem onnull divergences [6, Theorem 4.24].) This is why all the equations from the subclasses H x and H y ,not to mention their intersection H xy , are Darboux integrable in a trivial way. The reparameter-ized class H y , where h is assumed as the arbitrary element instead of f , possesses gauge equiv-alence transformations (see the definition of this notion in [9]) (˜ x, ˜ y, ˜ u, ˜ u ˜ x , ˜ u ˜ y ) = ( x, y, u, u x , u y ),˜ h = H ( x, h ), where H is an arbitrary smooth function of ( x, h ) with H h = 0. Theorem 2. (i) The class H gen is normalized in the usual sense with respect to point trans-formations. Its point equivalence group G ∼ gen coincides with its contact equivalence group and isgenerated by the discrete equivalence transformation I : ˜ x = y , ˜ y = x , ˜ u = u , ˜ u ˜ x = u y , ˜ u ˜ y = u x , ˜ f = f and the transformations of the form ˜ x = X ( x ) , ˜ y = Y ( y ) , ˜ u = U ( x, y, u ) , ˜ u ˜ x = U x + U u u x X x , ˜ u ˜ y = U y + U u u y Y y , (7a)˜ f = 1 X x Y y ( U u f + U xy + U xu u y + U yu u x + U uu u x u y ) , (7b) where X , Y and U are arbitrary smooth functions of their arguments with X x Y y U u = 0 .(ii) The contact equivalence groupoid G ∼ cgen of the class H gen is generated by the first-order prolongation of the point equivalence groupoid G ∼ pgen of this class, • the contact vertex group G c0 of the wave equation E : u xy = 0 and • the subgroupoid G ∼ q y of the equivalence groupoid G ∼ c y of the subclass H ′ y whose elements areinduced, modulo gauge equivalence within the reparameterized class H ′ y , by the pullbacksof the transformations of the form ˜ τ = η , ˜ ξ = ξ , ˜ υ = Υ( τ, ξ, υ, η ) , ˜ η = τ with Υ υ = 0 in the space with the coordinates ( τ, ξ, υ, η ) by the mapping Ψ : ( τ, ξ, υ ) = ( x, y, u ) , η = h ( x, y, u, u x ) , where ˜ u ˜ x = Ψ ∗ Υ η , ˜ u ˜ y = Ψ ∗ Υ ξ + u y Ψ ∗ Υ υ , and h is an arbitrary element ofthe reparameterized class H ′ y .The subclasses H xy , C := H x △ H y = H ′ x ∪ H ′ y and C := H gen \ ( H x ∪ H y ) are invariant underthe action of G ∼ cgen . In other words, the partition of the class H gen into these three subclasses, H gen = H xy ⊔ C ⊔ C , induces the corresponding partitions of its contact equivalence groupoid, G ∼ cgen = G ∼ c H xy ⊔ G ∼ c C ⊔ G ∼ c C .(iii) The G ∼ cgen -orbit of the wave equation E coincides with the subclass H xy .(iv) The contact symmetry group G c0 of E is generated by the discrete permutation transfor-mation ˜ x = y , ˜ y = x , ˜ u = u , ˜ u ˜ x = u y , ˜ u ˜ y = u x and contact transformations of the form ˜ x = X ( x, u x ) , ˜ y = Y ( y, u y ) , ˜ u = cu + U ( x, u x ) + U ( y, u y ) , (8a)˜ u ˜ x = U x ( x, u x ) , ˜ u ˜ y = U y ( y, u y ) , (8b) where X , Y , U and U are arbitrary smooth functions of their arguments and c is an arbitraryconstant with ( X x , X u x ) = (0 , , ( Y y , Y u y ) = (0 , , c = 0 and U u x = U x X u x , cu x + U x = U x X x , U u y = U y Y u y , cu y + U y = U y Y y . (9) Proof.
We sketch the proof using the statement, the notation and the proof of Lemma 1 and theequivalence generated by the transformation I . Looking for contact equivalence transformationsof the class H gen , we split, with respect to the source and target values of the arbitrary element, f and ˜ f , the equations of the system (4) that do not involve these values simultaneously. Takinginto account the contact condition (3), we obtain the system X u x = X u y = X u = X y = 0 , Y u x = Y u y = Y u = Y x = 0 , U u x = U u y = 0 , (10a) U x = ˆD x UX x , U y = ˆD y UY y , (10b)where X x Y y U u = 0 since J = 0. In addition, we derive the equation (7b) defining the f -components of equivalence transformations. Any point admissible transformation in the class H gen relates its source and target according to (7b), its transformational part satisfies the equa-tions (10a) and thus it is induced by a point equivalence transformation of the class H gen . Thisproves point (i) of the theorem.Under the constraint X u x = Y u y = 0, the systems (3) and (4) imply the system (10), i.e., anycontact admissible transformation whose x - and y -components do not depend on u x and on u y is the first-order prolongation of a point admissible transformation.If both f = 0 and ˜ f = 0, then the joint system (3)–(4) reduces to X u = X y = 0, Y u = Y x = 0, U xu y = U xu = U xy = 0, U yu x = U yu = U yx = 0, U x X u x = U u x , U x ˆD x X = ˆD x U , U y Y u y = U u y , U y ˆD y Y = ˆD y U , where ( X x , X u x ) = (0 , Y y , Y u y ) = (0 ,
0) and U u = 0 in view of J = 0, whichimplies U u x u y = U u x u = U u x y = U u y u = U u y x = 0, and hence point (iv) holds true.If just f = 0, then similarly we derive X u = X y = 0 and Y u = Y x = 0, i.e., ˜ x = X ( x, u x ) and˜ y = Y ( y, u y ) with ( X x , X u x ) = (0 ,
0) and ( Y y , Y u y ) = (0 , G c0 with the same x - and y -components as T and with the identity u -component, ˜ u = u , from T , wecan set X = x and Y = y , and then U = U ( x, y, u ). This makes point (iii) obvious. Moreover,it becomes clear that the subclass H xy is invariant under the action of G ∼ cgen .5f both X u x = 0 and Y u y = 0, then the first equations in (4a) and (4b) immediately imply f u x u x = f u y u y = 0 and, after splitting them with respect to u x and u y and testing the obtainedequations on the compatibility with respect to X and Y , give the other constraints on f , jointlymeaning that E f ∈ H xy . In other words, the ( x, y )-components of T ∈ G ∼ cgen simultaneouslydepend on the first derivatives of u only if T ∈ G ∼ c H xy .Suppose that only one of the derivatives X u x and Y u y does not vanish. Up to the equivalencegenerated by I , we can assume that X u x = 0 and Y u y = 0. Then we derive from the jointsystem (3)–(4) that X = X ( x, y, u, u x ), Y = Y ( y ), U = U ( x, y, u, u x ), U x = U u x /X u x , U y =( ˆD y U − U x ˆD y X ) /Y y , ˜ f = U yu x /X u x , f = ( ˆD y X ) /X u x , and U u x ˆD x X = X u x ˆD x U . The componentsof the inverse T − of T are of the same form as the respective components of T . In particular,the derivative of the x -component of T − with respect to u x does not vanish as well sinceotherwise the admissible transformation T − and thus the admissible transformation T are thefirst-order prolongations of point admissible transformations in the class H gen , which contradictsthe condition X u x = 0. This is why the first equation in (4a) implies that both the equations E f and E ˜ f belong to the class H y . Together with point (i) and the proven G ∼ cgen -invariance of H xy ,this implies the G ∼ cgen -invariance of H x ∪ H y and, hence, of C and of C . We can set Y = y upto the s- G ∼ gen -equivalence on G ∼ cgen . Then, taking X as a value of the arbitrary element h for thereparameterized form D y h = 0 of the equation E f , we obtain that T ∈ G ∼ q y . Remark 3.
We explain, reformulate and expand the points of Theorem 2.(i) The normalization of the class H gen in the usual sense with respect to point transformationsmeans that its point equivalence groupoid G ∼ pgen coincides with the action groupoid G G ∼ gen of thegroup G ∼ gen , i.e., any point admissible transformation within the class H gen is induced by itspoint equivalence transformation; see definitions in [8, 9, 11].(ii) Two equations from the class C are related by a contact transformation if and only if thistransformation is the first-order prolongation of a point transformation that is the projection ofan element of the group G ∼ gen to the space with coordinates ( x, y, u ). In other words, the class C is normalized in the usual sense with respect to point transformations, its point equivalencegroup coincides with G ∼ gen and with its contact equivalence group, and its contact equivalencegroupoid is the first-order prolongation of its point equivalence groupoid.(iii) Let G ∼ u gen denote the subgroup of G ∼ gen that is constituted by the transformations of theform (7) with X ( x ) = x and Y ( y ) = y . The G ∼ cgen -orbit H xy of the equation E coincides with its G ∼ gen -orbit and, more specifically, with its G ∼ u gen -orbit.An equation E f from the class H gen reduces to E by a contact transformation if and only ifit reduces to E by an element of G ∼ u gen . Then the corresponding value of the arbitrary element f is necessarily of the form f = ( θ xy + θ xu u y + θ yu u x + θ uu u x u y ) /θ u , where θ is a smooth functionof ( x, y, u ) with θ u = 0.The contact equivalence groupoid of the class H xy is generated by the contact vertex group G c0 of the wave equation E and the restriction of the action groupoid of the group G ∼ u gen to H xy .(iv) In view of (9), the u x -component (resp. the u y -component) of a contact symmetry trans-formation of E can be locally expressed in terms of the x - and the u -components (resp. the y -and the u -components). More specifically, U x = U u x /X u x , U x = ( cu x + U x ) /X x , U y = U u y /Y u y and U y = ( cu y + U y ) /Y y , whenever X u x = 0, X x = 0, Y u y = 0 and Y y = 0, respectively.Excluding U x and U y from (9), we derive U u x X x = ( cu x + U x ) X u x , U u y Y y = ( cu y + U y ) Y u y . Whenever X u x = 0, the former equation gives the following expression for U : U ( x, u x ) = − c Z xt Θ (cid:0) x ′ , X ( x, u x ) (cid:1) d x ′ + ϕ (cid:0) X ( x, u x ) (cid:1) , is the inverse of the function X with respect to u x , Θ (cid:0) x, X ( x, u x ) (cid:1) = u x , and ϕ isan arbitrary smooth function of its single argument. Similarly, the later equation integrates toan expression for U whenever Y u y = 0, U ( y, u y ) = − c Z yy Θ (cid:0) y ′ , Y ( y, u y ) (cid:1) d y ′ + ϕ (cid:0) Y ( y, u y ) (cid:1) , where Θ is the inverse of the function Y with respect to u y , Θ (cid:0) y, Y ( y, u y ) (cid:1) = u x , and ϕ is anarbitrary smooth function of its single argument.For contact symmetry transformation (8) of E , the equation (9) implies that U u x = 0 if X u x = 0 and U u y = 0 if Y u y = 0. Therefore, such a transformation with X u x = Y u y = 0 is thefirst-order prolongation of a point symmetry transformation of E .The point symmetry group G of the wave equation E is generated by the discrete permu-tation transformation ˜ x = y , ˜ y = x , ˜ u = u and point transformations of the form ˜ x = X ( x ),˜ y = Y ( y ), ˜ u = cu + U ( x ) + U ( y ), where X , Y , U and U are arbitrary smooth functions oftheir arguments with X x = 0 and Y y = 0, and c is an arbitrary nonzero constant. Therefore,a complete list of discrete point symmetry transformations of the equation E that are inde-pendent up to combining with each other and with continuous point symmetry transformationsof this equation is exhausted by the ( x, y )-permutation and the three transformations each ofwhich alternates the sign of one of the variables x , y and u . The quotient group of the pointsymmetry group G of E . with respect to its identity component is isomorphic to the group Z × Z × Z × Z . Acknowledgments
The author thanks Vyacheslav Boyko, Michael Kunzinger and Dmytro Popovych for helpfuldiscussions. The research was supported by the Austrian Science Fund (FWF), projects P25064and P28770.
References [1] Boyko V.M., Lokaziuk O.V. and Popovych R.O., Realizations of Lie algebras on the line and the new groupclassification of (1+1)-dimensional generalized nonlinear Klein–Gordon equations, arXiv:2008.05460, 30 pp.[2] Kurujyibwami C. and Popovych R.O., Equivalence groupoids and group classification of multidimensionalnonlinear Schr¨odinger equations,
J. Math. Anal. Appl. (2020), 124271, arXiv:2003.02781.[3] Kushner A.G., On contact equivalence of Monge–Amp`ere equations to linear equations with constant coef-ficients,
Acta Appl. Math. (2010), 197–210.[4] Kuznetsova M.N., Pekcan A., Zhiber A.V., The Klein–Gordon equation and differential substitutions of theform v = φ ( u, u x , u y ), SIGMA d z/dx dy = F ( z ), Arch. fur Math. (1881), 112–124. (Reprintedin: Lie S., Gesammelte Abhandlungen , Vol. 3, B.G. Teubner, Leipzig and H. Aschehoug & Co, Kristiania,469–478.)[6] Olver P.J.,
Application of Lie groups to differential equations , Springer, New York, 1993.[7] Olver P.J.,
Equivalence, invariants, and symmetry , Cambridge University Press, Cambridge, 1995.[8] Opanasenko S., Bihlo A. and Popovych R.O., Equivalence groupoid and group classification of a class ofvariable-coefficient Burgers equations,
J. Math. Anal. Appl. (2020), 124215, arXiv:1910.13500.[9] Popovych R.O., Kunzinger M. and Eshraghi H., Admissible transformations and normalized classes of non-linear Schr¨odinger equations,
Acta Appl. Math. (2010), 315–359, arXiv:math-ph/0611061.[10] Stormark O.,
Lie’s structural approach to PDE systems , Cambridge University Press, Cambridge, 2000.[11] Vaneeva O.O., Bihlo A. and Popovych R.O., Generalization of the algebraic method of group classificationwith application to nonlinear wave and elliptic equations,
Commun. Nonlinear Sci. Numer. Simul. (2020),105419, arXiv:2002.08939.(2020),105419, arXiv:2002.08939.