Pre-Hamiltonian operators related to hyperbolic equations of Liouville type
aa r X i v : . [ n li n . S I] F e b Pre-Hamiltonian operators related tohyperbolic equations of Liouville type
S. Ya. Startsev
Institute of Mathematics, Ufa Federal Research Centre, Russian Academy of Sciences
Abstract.
This text is devoted to hyperbolic equations admitting differential operators thatmap any function of one independent variable into a symmetry of the corresponding equation.We use the term ‘symmetry driver’ for such operators and prove that any symmetry driverof the smallest order is pre-Hamiltonian (i.e., the image of the driver is closed with respectto the standard bracket). This allows us to prove that the composition of a symmetrydriver with the Fr´echet derivative of an integral is also pre-Hamiltonian (in a new set of thevariables) if both the symmetry driver and the integral have the smallest orders.1
Introduction: Pre-Hamiltonian operators
Suppose that u is a function of x and y . Let F denote the set of all functions depending on a finitenumber of the variables y, x, u = u, u = u x , u = u xx , . . . . (1.1)Notice that (1.1) does not contain derivatives of u with respect to y . Moreover, the objects definedin this Section do not employ the variable y , and this variable serves here as a parameter (which isabsent in the standard definitions). The origin of the parameter y will be explained in Section 2.By D we denote the total derivative with respect to x . On functions from F , it is defined by theformula D = ∂∂x + + ∞ X i =0 u i +1 ∂∂u i . (1.2)For any function a ∈ F , the differential operator a ∗ def = + ∞ X i =0 ∂a∂u i D i is called the Fr´echet derivative of a .For any a ∈ F we can also consider the total derivative ∂ a with respect to t in virtue (i.e. onsolutions) of the evolution equation u t = a . It easy to check that ∂ a ( h ) = h ∗ ( a ) if h ∈ F . Thecommutator [ ∂ f , ∂ g ] of such derivatives corresponds to the Lie bracket h f, g i def = g ∗ ( f ) − f ∗ ( g ) (1.3)on F . Namely, it is well-known (see, for example, [1]) that[ ∂ f , ∂ g ] = ∂ [ f,g ] . (1.4)Let M be a differential operator of the form M = k X i =0 ξ i D i , ξ i ∈ F , k ≥ . (1.5)The direct calculation shows that D ( a ) ∗ = D ◦ a ∗ for any a ∈ F , where the symbol ◦ denotes thecomposition of differential operators. Taking this into account, we see that the relation h M ( a ) , M ( b ) i = M ( b ∗ ( M ( a )) − a ∗ ( M ( b ))) + k X i =0 (cid:0) D i ( b )( ξ i ) ∗ ( M ( a )) − D i ( a )( ξ i ) ∗ ( M ( b )) (cid:1) holds for any differential operator (1.5) and any functions a, b ∈ F . Denoting the highest of theorders of the operators ( ξ i ) ∗ ◦ M by m , we can rewrite this relation in the form h M ( a ) , M ( b ) i − M ( b ∗ ( M ( a )) − a ∗ ( M ( b ))) = m X i =0 m X j =0 c ij D i ( a ) D j ( b ) , (1.6) In the formal sense, just as a way to define new differentiations on F . c ij ∈ F do not depend on a and b .Generally speaking, the right-hand side of (1.6) does not belong to the image of M for all a and b . But, for example, if M = D + u , then h M ( a ) , M ( b ) i = M (cid:16) b ∗ ( M ( a )) − a ∗ ( M ( b )) + D ( a ) b − D ( b ) a (cid:17) (1.7)for any a, b ∈ F . Definition 1.1.
A differential operator M of the form (1.5) is called pre-Hamiltonian if for any a, b ∈ F there exists ϑ ∈ F such that h M ( a ) , M ( b ) i = M ( ϑ ).It is known (see [1] for example) that the image Im H of an operator H satisfies the relation h Im H , Im H i ⊆ Im H (i.e., Im H forms a Lie subalgebra in F ) if the operator H is Hamiltonian.Thus, we can consider pre-Hamiltonian operators as a generalization of Hamiltonian operators. (Notethat the pre-Hamiltonian operator D + u in the above example is not Hamiltonian because it is notskew-symmetric.)Pre-Hamiltonian operators (under different names or without a name) were studied in severalworks of this century. As far as the author knows, the definition of pre-Hamiltonian operatorsactually arose for the first time in subsection 7.1 of [2] as an ‘experimentally observed’ remarkableproperty of operators associated with hyperbolic equations of the Liouville type (i.e., with Darbouxintegrable equations).The present text can be considered as an addition to these ‘experimental observations’ of [2] andprovides them with the proof. Namely, we prove in Section 3 that any Darboux integrable equationgenerates four pre-Hamiltonian operators (a couple of such operators per each of the characteristics). From now on, we deal with hyperbolic equations of the form u xy = F ( x, y, u, u x , u y ) . (2.1)If u ( x, y ) is a solution of equation (2.1), then we can express all mixed derivatives of u in terms ofthe variables x, y, u = ¯ u = u, u = u x , ¯ u = u y , u = u xx , ¯ u = u yy , . . . . (2.2)These variables will be considered as independent. We use notation f h u i for a function f dependingon a finite number of the variables (2.2). We also use the notation of Section 1 and, in particular,continue to write f ∈ F if f h u i does not depend on ¯ u j for all j > References to these works will be added in the next version of this text; a part of them are mentioned in subsec-tion 3.5.2 of https://arxiv.org/abs/1711.10624 x and y -characteristics.If we denote an x -object by a symbol, then the corresponding y -object is denoted by the same symbolwith a dash above. For example, by D we denote the total derivative with respect to x defined onsolutions of equation (2.1), while the total derivative with respect to y is denoted by ¯ D . Thesederivatives are defined by formulas D = ∂∂x + ∞ X i =0 u i +1 ∂∂u i + ∞ X i =1 ¯ D i − ( F ) ∂∂ ¯ u i , ¯ D = ∂∂y + ∞ X i =0 ¯ u i +1 ∂∂ ¯ u i + ∞ X i =1 D i − ( F ) ∂∂u i . Note that the restriction of D onto F coincides with (1.2). Definition 2.1.
A function W h u i is called an x - integral of equation (2.1) if ¯ D ( W ) = 0. Any function W ( x ) is called a trivial x -integral.If we replace x with y (and ¯ D with D ) in the above definition, then we obtain the definitionof y -integrals . Using the symmetry of formula (2.1) with respect to the interchange x ↔ y , wehereafter give only one of two ‘symmetric’ definitions and statements.Differentiating the defining relation ¯ D ( W ) = 0 with respect to the highest variable of the form¯ u i , we obtain that any x -integral W does not depend on ¯ u i for all i > W = W ( x, y, u, u , u , . . . , u p ) . The number p is called order of the x -integral W . Example . The function w = u − u is an x -integral of the Liouville equation u xy = exp u. (2.3)Obviously, if w is an x -integral of equation (2.1), then the expression W = S (cid:0) x, w, D ( w ) , . . . , D j ( w ) (cid:1) (2.4)is also an x -integral of (2.1) for any function S and any j ≥ Proposition 2.3. [3]
Any x -integral W can be represented in the form (2.4) , where w is an x -integralof the smallest order. In particular, w = φ ( x, ˜ w ) if ˜ w is another x -integral of the smallest order. Definition 2.4.
An equation of the form (2.1) is said to be
Darboux integrable if this equationpossesses both nontrivial x and y -integrals.The Liouville equation (2.3) is the most known example of nonlinear Darboux integrable equation. Notice that in different papers (even in articles of the same authors) the names for the integrals are varied: in apart of the papers, the term ‘ x -integral’ is used to denote y -integrals and vice versa. .3 Hyperbolic equations of Liouville type Let us introduce the functions H def = − D (cid:18) ∂F∂u (cid:19) + ∂F∂u ∂F∂ ¯ u + ∂F∂u , H def = − ¯ D (cid:18) ∂F∂ ¯ u (cid:19) + ∂F∂u ∂F∂ ¯ u + ∂F∂u . Then we can define the functions H i for i > i < D ¯ D (log H i ) = − H i +1 − H i − + 2 H i , i ∈ Z . (2.5)The functions H i are called Laplace invariants of equation (2.1).The direct calculation shows that H = H = exp u and H = H − = 0 for the Liouville equation(2.3). Definition 2.5.
Equation (2.1) is called a
Liouville-type equation if H r = H − s ≡ r ≥ s ≥ Theorem 2.6. [4, 5]
An equation of the form (2.1) is a Liouville-type equation if and only if thisequation admits both a non-trivial x -integrals W ( x, y, u, u , u , . . . , u p ) and a non-trivial y -integral ¯ W ( x, y, u, ¯ u , ¯ u , . . . , ¯ u ¯ p ) . In addition, s < p and r ≤ ¯ p . Thus, Definitions 2.5 and 2.4 are equivalent for scalar hyperbolic equations.Differentiating the defining relations D ( W ) = 0 and ¯ D ( ¯ W ) = 0 with respect to u p and ¯ u ¯ p ,respectively, we obtain the following statement. Corollary 2.7. If (2.1) is a Liouville-type equation, then the relations ∂F∂u = ¯ D log ψ ( x, y, u, u , . . . , u p ) , ∂F∂ ¯ u = D log ¯ ψ ( x, y, u, ¯ u , . . . , ¯ u ¯ p ) (2.6) hold for some functions ψ and ¯ ψ . An inherent feature of Darboux integrable equations is the existence of higher symmetries in both x and y -directions. Definition 3.1.
An equation of the form u t = g h u i (3.1)is a symmetry of equation (2.1) if g satisfies the following relation (cid:16) D ¯ D − ∂F∂u D − ∂F∂ ¯ u ¯ D + ∂F∂u (cid:17) ( g ) = 0 . If the right-hand side of a symmetry belongs to F , then we call it an x - symmetry . The right-hand side of y -symmetries depends on x, y, u, ¯ u , ¯ u , ...
5e often identify symmetry (3.1) with its right-hand side g . The set of x -symmetries is a Liealgebra with respect to the bracket (1.3).The following formula generates symmetries for Liouville-type equations (see, for example, The-orem 5 in [2], where this formula was proved but was given in the slightly different form (3.3)). Theorem 3.2.
Suppose that for equation (2.1) there exists a non-zero function ψ h u i ∈ ker( ¯ D − F u ) and H r = 0 for some r > . Let us define the operator M by the formula M = ( H ( D − F ¯ u ) ◦ H ( D − F ¯ u ) ◦ · · · ◦ H r − ( D − F ¯ u ) ◦ H r − . . . H ψ if r > , ψ if r = 1 . (3.2) Then u t = M ( W ) is a symmetry of the equation (2.1) for any x -integral W . The above Theorem is applicable to any Liouville-type equation because H r = 0 by Definition 2.5and the function ψ exists by Corollary 2.7. In the case of Liouville-type equations (as well as in othercases when both relations (2.6) hold), we can rewrite (3.2) for r > M = ¯ ψH D ◦ H D ◦ · · · ◦ H r − D ◦ ψH . . . H r − ¯ ψ (3.3)found in [2].Notice that Theorem 3.2 can be applied not only to Liouville-type equations but also to someequations (2.1) without nontrivial x -integrals. In the latter case, W is an arbitrary function of x . Example . We have H = 0 and ψ = u for any equation of the form u xy = η ( y, u, u y ) u x . Inthis case, M is the operator of multiplication by u . This operator maps arbitrary function of x to a symmetry of this equation, while the equation generically is not Darboux integrable (admitsnontrivial x -integrals not for all η in accordance with [3, 6]). Definition 3.4.
An operator M = P ki =0 ξ i h u i D i , ξ k = 0 is called an x -symmetry driver if k ≥ u t = M ( W ) is a symmetry of equation (2.1) for any x -integral W . The integer k is called order ofthe driver M .The above definition remains applicable even if (2.1) admits only trivial x -integrals (see Exam-ple 3.3). It is clear that for a driver M the operator M ◦ D i ◦ W is a driver for any i ≥ x -integral W (in particular, W can be equal to 1). Lemma 3.5.
The coefficients of any x -symmetry driver do not depend on ¯ u j for all j > . Theleading coefficient belongs to the kernel of the operator ¯ D − F u .Proof. Let M = P ki =0 ξ i h u i D i be an x -symmetry driver. Then( D ¯ D − F u D − F ¯ u ¯ D − F u )( M ( g )) = 0for arbitrary function g ( x ). Collecting the coefficients of g ( i ) in this identity, we arrive at the followingrelations: ( ¯ D − F u )( ξ k ) = 0 , ( ¯ D − F u )( ξ i − ) = ( F u + F u D + F ¯ u ¯ D − D ¯ D )( ξ i ) , ≤ i ≤ k. (3.4)It follows from the first relation that ξ k does not depend on derivatives of u with respect to y , whilethe second one implies that ( ξ i − ) ¯ u j = 0 for all j > ξ i has the same property.6 orollary 3.6. Let M and ˜ M be x -symmetry drivers of the smallest possible order for an equationof the form (2.1) . Then ˜ M = M ◦ W , where W is an x -integral of (2.1) .Proof. Let M and ˜ M have order q and their leading coefficients be denoted by ξ q and ˜ ξ q , respectively.Both ¯ D (log ξ q ) and ¯ D (log ˜ ξ q ) are equal to F u by Lemma 3.5. Therefore, ¯ D (log ˜ ξ q − log ξ q ) = 0 and˜ ξ q = W ξ q , where W ∈ ker ¯ D . This means that the operator ˜ M − M ◦ W has order less than q .On the other hand, the last operator maps x -integrals into symmetries. But (2.1) admits no x -symmetry drivers of order less than q by the assumption of Corollary. Hence, all the coefficientsof ˜ M − M ◦ W must be equal to zero.According to Theorem 3.2, the operator M is an x -symmetry driver of order r −
1. As it isdemonstrated in [6], H j = 0 for some positive j ≤ r if (2.1) admits an x -symmetry driver of order r −
1. This means that (2.1) admits no driver of order less than r − H r − = 0. Therefore, M is adriver of the smallest order. In addition, relation (2.6) defines the function ψ up to multiplication by x -integrals. Hence, the operator M is defined up to transformations M → M ◦ W , where W is an x -integral. Comparing this with Corollary 3.6 and taking Lemma 3.5 and Theorem 3.2 into account,we obtain the following statement. Proposition 3.7.
Let equation (2.1) admit x -symmetry drivers. Then H r = 0 for some r > , thekernel of ¯ D − F u contains non-zero elements and any x -symmetry driver of the smallest order isdefined by the formula (3.2) with some ψ ∈ ker( ¯ D − F u ) . The equations of Liouville type also possess differential operators that map symmetries to inte-grals.
Lemma 3.8.
Let W ( x, y, u, u , u , . . . , u p ) be an x -integral of equation (2.1) . Then the differentialoperator W ∗ maps any symmetry to some x -integral. The above Lemma directly follows from Lemma 1 in [7]. Not so formal, Lemma 3.8 is validbecause the total derivative with respect to t in virtue of any symmetry u t = g and the operator ¯ D commute: [ ∂ g , ¯ D ] = 0. Corollary 3.9.
For any x -symmetry driver M and any x -integral W the operator L = W ∗ ◦ M maps x -integrals into x -integrals again. After rewriting L in the form P µ i D i , the coefficients µ i ofthis operator are x -integrals. The most significant operator of this kind is the composition L = w ∗ ◦ M , (3.5)where w is a nontrivial x -integral of the smallest order and the operator M is given by (3.2) (i.e., M is an x -symmetry driver of the smallest order by Proposition 3.7).7 xample . Recall that H = H = exp( u ) and H = 0 in the case of the Liouville equation (2.3).In this case, we can set ψ = 1 and the formula (3.2) gives us M = exp( − u ) D ◦ exp( u ) = D + u . (3.6)Since H does not vanish, w = u − u is an x -integral of the smallest order (see the last sentenceof Theorem 2.6). We have w ∗ = D − u D and L = ( D − u D ) ◦ ( D + u ) = D + 2 wD + D ( w ) . (3.7)In Section 1 we show that the operator (3.6) is pre-Hamiltonian (see (1.7)). As it is noted in [2],the operator (3.7) is also pre-Hamiltonian if we change the notation and rewrite this operator as D + 2 uD + u . Theorem 3.11.
Let M be an x-symmetry driver of the smallest order for equation (2.1) . Then theoperator M is pre-Hamiltonian and, moreover, there exist functions γ ij ∈ ker ¯ D such that (cid:2) M ( a ) , M ( b ) (cid:3) = M b ∗ ( M ( a )) − a ∗ ( M ( b )) + n X i =0 n X j =0 γ ij D i ( a ) D j ( b ) ! , (3.8) for any a, b ∈ F . In particular, for any x -integrals f and g there exists φ ∈ ker ¯ D such that (cid:2) M ( f ) , M ( g ) (cid:3) = M ( φ ) . Note that M is always defined by formula (3.2) in accordance with Proposition 3.7. It shouldalso be emphasized that we do not assume the existence of nontrivial x -integrals in Theorem 3.11.If nontrivial x -integrals are absent for equation (2.1), then this theorem means that γ ij (as well as f , g and φ ) are functions of x only. Remark . We can rewrite equation (3.8) in the form m X i =0 m X j =0 c ij D i ( a ) D j ( b ) = M n X i =0 n X j =0 γ ij D i ( a ) D j ( b ) ! , (3.9)where the functions c ij are defined by (1.6) for M . The functions a and b are arbitrary here.Therefore, D i ( a ) and D j ( b ) can be considered as independent variables . Collecting the coefficientsat these variables in the right-hand side of (3.9) and equating them to c ij , we see that formula (3.8)is equivalent to relations between the coefficients of the operator M (which, in particular, determine c ij ) and the functions γ ij . These relations do not contain a and b in any way and therefore remainunchanged if we prove (3.9) for a and b belonging to an appropriate subset of F (such that D i ( a )and D j ( b ) continue to play the role of independent variables when a and b are arbitrary elements ofthis subset). Thus, it is enough for the proof of Theorem 3.11 to demonstrate that (3.8) holds forthe functions a and b of a special form. We take arbitrary x -integrals as such a special form for a and b . I.e. a polynomial in these variables is equal to zero if and only if all the coefficients of this polynomial are zero.
Lemma 3.13.
Let M be an x -symmetry driver of order k for equation (2.1) and an expression N = q X i =0 ℓ X j =0 c ij D i ( f ) D j ( g ) , c ij ∈ F , q ≥ k, be a symmetry of (2.1) for arbitrary x -integrals f and g . Then there exist x -integrals θ j such thatthe expression ˜ N = N − M D q − k ( f ℓ X j =0 θ j D j ( g )) ! has the form P q − i =0 P ˜ ℓj =0 ˜ c ij D i ( f ) D j ( g ) and is a symmetry of (2.1) for any x -integrals f and g . In simpler words, the driver M allows us to reduce the order q of N without loosing the otherproperty of N . Therefore, Lemma 3.13 remains applicable to the reduced expression ˜ N if its order ˜ q is greater than k − Proof of Lemma.
The assumptions of the lemma imply that the operator R = P qi =0 C i ( g ) D i , where C i ( g ) = P ℓj =0 c ij D j ( g ), is an x -symmetry driver for any g ∈ ker ¯ D . According to Lemma 3.5, we have( ¯ D − F u )( C q ( g )) = ℓ X j =0 ( ¯ D − F u )( c qj ) D j ( g ) = 0 . Since the integral g is arbitrary, we get ( ¯ D − F u )( c qj ) = 0. The leading coefficient ξ k of the driver M also belongs to the kernel of the operator ¯ D − F u . Therefore c qj = θ j ξ k , where θ j ∈ ker ¯ D (seethe proof of Corollary 3.6 if a more detailed explanation is needed). The last equalities imply that M D q − k f ℓ X j =0 θ j D j ( g ) !! = C q ( g ) D q ( f ) + . . . , where the dots denote terms with D i ( f ), i < q . Since the M ◦ D q − k is a driver, the above expressionis a symmetry for any f, g ∈ ker ¯ D . Subtracting this symmetry from N , we complete the proof. Proof of Theorem 3.11. As M is an x -symmetry driver, the functions M ( g ) and M ( f ) are sym-metries of (2.1) for any x -integrals f and g . Lemma 3.8 implies that f ∗ ( M ( g )) and g ∗ ( M ( f )) are x -integrals (in particular, are zero if (2.1) admits trivial x -integrals only). Therefore, M (cid:16) g ∗ ( M ( f )) − f ∗ ( M ( g )) (cid:17) is a symmetry. In addition, (cid:2) M ( f ) , M ( g ) (cid:3) is also a symmetry since x -symmetries forms aLie algebra with respect to the bracket (1.3). Thus, substituting M , f , g for M , a , b in formula (1.6),we obtain that m X i =0 m X j =0 c ij D i ( f ) D j ( g ) (3.10)is an x -symmetry for any x -integrals f and g . 9y the assumption of Theorem 3.11, M is a driver of smallest order. Let us denote this orderby k (recall that k = r − M ( f ) , M ( g )] − M g ∗ ( M ( f )) − f ∗ ( M ( g )) + n X i =0 n X j =0 γ ij D i ( f ) D j ( g ) ! = ˆ m X i =0 ˆ ℓ X j =0 ς ij D i ( f ) D j ( g ) , where γ ij ∈ ker ¯ D and the non-negative integer ˆ m < k . If k = 0, then the right-hand side of the aboverelation is zero because we can completely absorb (3.10) into the image of M by using Lemma 3.13.If k > ς ij , then the right-hand side defines a driver for arbitrary g ∈ ker ¯ D . But the order of this driver is less that the smallest order k and, therefore, all thecoefficients ς ij must be equal to zero.We have proved that the operator M is pre-Hamiltoninan with respect to the bracket (1.3) definedon functions from F .It turns out that there exists a pre-Hamiltonian operator in a set of variables other than (1.1) ifequation (2.1) admits nontrivial x -integrals in addition to x -symmetry drivers. Let w be an x -integral of smallest order (see Proposition 2.3). Consider the operator L definedby (3.5). According to Corollary 3.9, all coefficients of L are function of x , w , w i def = D i ( w ). Theorem 3.14.
The operator L is pre-Hamiltonian with respect to the bracket { a, b } = + ∞ X i =0 (cid:18) ∂b∂w i D i ( a ) − ∂a∂w i D i ( b ) (cid:19) defined on functions of the variables x , w , w , . . . . It was already noted in [2] that the operator L is pre-Hamiltonian for many (all checked) examplesof Liouville-type equations. In other words, the work [2], in fact, contains the above theorem as aconjecture. By using Theorem 3.11, we can now prove Theorem 3.14. The proof below belongs toV. V. Sokolov. Proof.
Consider the evolution equations u t = M ( f ) and u t = M ( g ), where f and g are arbitraryfunction of the form (2.4) (i.e. f and g are x -integrals). Recall (see Section 1) that differentiationswith respect to t and t in virtue of these equation are denoted by ∂ M ( f ) and ∂ M ( g ) . Let us applythe commutator [ ∂ M ( f ) , ∂ M ( g ) ] to the x -integral w of smallest order. Taking (1.4) and Theorem 3.11into account, we obtain[ ∂ M ( f ) , ∂ M ( g ) ]( w ) = w ∗ (cid:0)(cid:2) M ( f ) , M ( g ) (cid:3)(cid:1) = w ∗ ( M ( φ )) = L ( φ )for some x -integral φ (which is a function of x , w , w i by Proposition 2.3).On the other hand, we have w t = ∂ M ( f ) ( w ) = L ( f ) , w t = ∂ M ( g ) ( w ) = L ( g ) . As it is demonstrated in [6], equation (2.1) is a Liouville-type equation if it admits both x -symmetry drivers andnontrivial x -integrals. ∂ M ( f ) and ∂ M ( g ) respectively coincide with ∂ L ( f ) and ∂ L ( g ) onfunctions of x , w , w i . Since L ( f ) and L ( g ) are function of x , w , w i by Corollary 3.9, we have[ ∂ M ( f ) , ∂ M ( g ) ]( w ) = ∂ M ( f ) ( L ( g )) − ∂ M ( g ) ( L ( f )) = ∂ L ( f ) ( L ( g )) − ∂ L ( g ) ( L ( f )) = {L ( f ) , L ( g ) } . Thus, we arrive at the relation {L ( f ) , L ( g ) } = L ( φ ) (3.11)that holds when w i are functions from F . Since the functions x , w , w , . . . ∈ F are functionallyindependent, the relation (3.11) holds identically (i.e. without substituting w i with their values from F ). Acknowledgements
The author thanks V. V. Sokolov for many useful discussions as well as for suggestions that madethe text more readable even in this very preliminary version.
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