Symmetry structure of integrable hyperbolic third order equations

We explore the application of generating symmetries, i.e. symmetries that depend on a parameter, to integrable hyperbolic third order equations, and in particular to consistent pairs of such equations as introduced by Adler and Shabat in (2012 J. Phys. A: Math. Theor. 45 385207). Our main result is that different infinite hierarchies of symmetries for these equations can arise from a single generating symmetry by expansion about different values of the parameter. We illustrate this, and study in depth the symmetry structure, for two examples. The first is an equation related to the potential KdV equation taken from (Adler and Shabat 2012 J. Phys. A: Math. Theor. 45 385207). The second is a more general hyperbolic equation than the kind considered in (Adler and Shabat 2012 J. Phys. A: Math. Theor. 45 385207). Both equations depend on a parameter, and when this parameter vanishes they become part of a consistent pair. When this happens, the nature of the expansions of the generating symmetries needed to derive the hierarchies also changes.


Introduction
Although the original characterization of integrability for partial differential equations (PDE) was through the existence of an infinite number of conservation laws [24], it was soon realized that integrable PDE also exhibit infinitely many infinitesimal symmetries [28].The latter property is easier to investigate systematically, and has become a central pillar in the classification of integrable PDE.See [22,23] for surveys, and the papers [25,26,21] for some examples of applications to different classes of equations.
One of the important classes of equations to which the symmetry method has been applied [19,16] is the class of scalar, second order hyperbolic equations of the form u xy = h(x, y, u, u x , u y ) . ( (See also [39,20] for vector extensions.)For equations of this type, the conventional wisdom is that integrable cases are characterized by two infinite hierarchies of symmetries.In fact, a little more detail is needed in this discussion.Even for the archetypical example of an integrable PDE, the Korteweg-de Vries (KdV) equation, there are two infinite hierarchies of symmetries, the standard hierarchy of commuting symmetries, as discussed in [28], and the hierarchy of "additional" symmetries [14,15,29], which do not commute, either among themselves or with the standard symmetries.The latter are often ignored; this may be because with the exception of the two lowest order symmetries in this hierarchy, they are nonlocal.But these two lowest order symmetries are the Galilean and scaling symmetries, which certainly play a significant role in the theory of the KdV equation.In greater generality, for many (and maybe all) integrable PDE it is possible to identify one or more recursion operators that map symmetries to symmetries, and application of these recursion operators to simple point symmetries can generate many hierarchies (see [17] for a good example).However, it seems that integrable equations of the form (1) are characterized by the existence of at least two infinite hierarchies of local, commuting symmetries.
In our previous works [31,32,33,34,35,36,30] we showed that the infinite hierarchies of symmetries of integrable PDE can be conveniently encapsulated in generating symmetries, which are non-local symmetries depending on a parameter, that give rise to infinite hierarchies when expanded in suitable power series in the parameter.In the current paper we extend this work to some third order hyperbolic PDEs.As opposed to second order hyperbolic PDEs, third order hyperbolic PDEs have hardly been studied.An exception is the ground-breaking paper of Adler and Shabat [1] on equations of the form u xxy = f (x, y, u, u x , u y , u xy , u xx ) . ( In addition to giving various examples of integrable equations of this form, Adler and Shabat point out that there are examples of such equations which are consistent with another third order equation of the form but are irreducible, in the sense that (2) and ( 3) are not obtained simply by differentiation, with respect to x and y respectively, of a second order equation of the form (1). Adler and Shabat call such pairs of equations "consistent pairs" and point out that they "belong to an intermediate class between those of the second and third order equations".An interesting application of Adler and Shabat's idea can be found in [8].
From the point of view of symmetries, although full details are not given in [1], it seems that integrable equations of form (2) also have two infinite hierarchies of local, commuting symmetries, but some kind of degeneration takes place when the equation is part of a consistent pair.The contributions and structure of the current paper are as follows: in Section 2 we review the main definitions from [1], and show that the notion of consistent pair can be extended to more general third order hyperbolic equations.In section 3, we reconsider the example of Adler-Shabat that is related to the potential KdV equation, viz. the equation where c is a constant.We give the Lax pair for this equation, in several forms, write down generating symmetries, and show, remarkably, that both infinite hierarchies of local, commuting symmetries arise from a single generating symmetry, via expansions around two different values of the parameter.In the case c = 0, when the relevant equation is part of a consistent pair, we show that the dependence of the generating symmetry on the parameter changes, and thus different expansions are needed, but still the two hierarchies are obtained from the same generating symmetry, via two expansions around different values of the parameter.Thus the generating symmetry turns out to be a unifying object.In Section 4 we look at a more intricate example, viz. the equation that is not in the form (2). Once again, we give the Lax pair and write down generating symmetries, but in this case it is necessary to consider expansions around three different values of the parameter, and the resulting symmetries do not all commute.
Once again, the form of the expansions change when c = 0 and the equation is part of a consistent pair; in this case it is related by a differential substitution to the second order hyperbolic equation where f is a constant.But still the generating symmetry is a single unifying object that covers all the different cases and encodes numerous different hierarchies.For equation ( 5), we complete the theory by also looking at the generating symmetry that gives rise to additional symmetries and its expansions, and we show the origin of various different recursion operators from certain identities satisfied by the generating symmetries.Section 5 contains some concluding remarks and open questions.
We end this introduction by referring to the paper [11], which introduces a "generalized invariant manifold" for some integrable equations, including KdV.The authors demonstrate (Section 4) that the generalized invariant manifold W = φ 2 is a symmetry of pKdV, where φ is a solution of the Lax Pair.This symmetry was discovered in [9,18,27], and is characterized as a generating symmetry in [31].In another paper [10], the authors shows that for a second order hyperbolic equation, in the integrable case, two recursion operators (generating two distinct infinite hierarchies of symmetries) are produced by different parametrizations of the same generalized invariant manifold.This result coincides with the result that we obtain here.

Adler-Shabat Pairs and a Generalization
In this section we briefly recap the main definitions given by Adler and Shabat in [1] and show how the notion of a consistent pair can be generalized.The following is taken almost verbatim from [1]: • Second order and third order hyperbolic equations are, respectively, equations of the form (1) and (2).
• A pair of third order hyperbolic equations of the form (2) and (3). is called a consistent pair if D y (f ) = D x (g) on solutions of the pair of equations, where D x , D y are total derivative operators.
• A consistent pair of third order hyperbolic equations is called reducible if its general solution solves a 1-parameter family of second order hyperbolic equations u xy = h(α; x, y, u, u x , u y ).Otherwise it is called irreducible.
• A second order or third order hyperbolic equation is integrable if it is compatible with an infinite hierarchy of evolutionary symmetries.
• A representation of the consistent pair (2)-(3) in Bäcklund variables is a system from which the pair follows by elimination of v. (Cross differentiation and elimination of derivatives of v yields a relation of the form u xy = H(x, y, u, u x , u y , v) and further differentiation with respect to each of the variables and elimination of v yields the two equations of the pair.) • A representation of the single third order hyperbolic equation ( 2) in Bäcklund variables is a system from which the equation follows by elimination of v.
Quite clearly, equation ( 2) is not the most general form for a third order hyperbolic equation.For example, we might also consider equations of either of the forms or where in the second case we require b < 0 for hyperbolicity.We claim the following: if b = −a 2 then for suitable choices of a, f, g the pair of equations ( 9) and ( 10) can be consistent.
To show this we proceed as follows.Writing consistency requires the existence of an identity of the form where α 1 , α 2 depend on u and its first and second derivatives.The terms on the left hand side of this identity were chosen so that the only fourth derivative of u that appears is u xxxx .This appears linearly, with the coefficient −b − a 2 , and thus it is necessary to choose b = −a 2 .All the third derivatives of u also appear linearly (and u yyy does not appear at all).The functions α 1 , α 2 can be chosen so that the coefficients of u xxy and u xyy vanish, and for the coefficient of u xxx also to vanish we find that it is necessary to impose the condition where Once this is done, the consistency condition reduces to a condition involving only u and its first and second derivatives, viz.
We note the following: • In the case a = ∂g ∂uxx = 0 the condition ( 12) is satisfied automatically, and the condition (13) reduces to the requirement D y (f ) = D x (g) on solutions of the pair of equations ( 2)-( 3).Thus we recover the original consistent pairs from [1].
• Since, for given e 1 , the consistency condition (11) can be regarded as a first order differential equation for e 2 , there is in fact always the possibility to insert an extra term with an arbitrary function of y in the second equation of a pair of the form ( 9)-( 10).We will see this in examples later on.
• If a pair ( 9)-( 10) can be obtained by differentation, with respect to x and y, of a single second order differential equation of the form h(x, y, y, u x , u x , u y , u xy , u xx ) = 0, we call it reducible.Otherwise, the pair is irreducible.
• We leave for further investigation the question of how many solutions exist of the constraints ( 12)- (13).In this paper we focus on a single example (with a = 0 and g independent of u xx ) from [1], and a single example of the new type with A further solution of the constraints is The first equation of the corresponding pair is the Hunter-Saxton equation [13] and the other equation is where γ is an arbitrary constant.

Adler and Shabat's First Integrable Example
In this section we look at equation (4), the example that Adler and Shabat gave in [1], that is related to the potential KdV equation.We show that both infinite hierarchies of symmetries of this equation arise from different expansions of the same generating symmetry, but there is a critical difference between the cases c ̸ = 0 and the case c = 0, when the equation is part of a consistent pair.In (4), by a rescaling of y, the parameter c can be taken to be 0 or ±1, but it is convenient to keep it general.As noted in [1], equation ( 4) is a potential form of the associated Camassa-Holm equation [38,12], but multiplying by 2u y and differentiating with respect to x we obtain the fourth order equation This equation is the ∂ t = 0 reduction of the 3-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation [37,2,3,5,6,4,42,40,7] which is a 3-dimensional generalization of the potential KdV equation, which itself is obtained from the CBS equation by the ∂ x = ∂ y reduction.
Adler and Shabat state that for general c, equation ( 4) has two infinite hierarchies of commuting symmetries.The first is the potential KdV hierarchy, involving only x-derivatives of u.The second involves y-derivatives and mixed derivatives with a single x-derivative, the first two characteristics being These flows are related to the derivative nonlinear Schrödinger equation hierarchy.In the case c = 0, differentiating (4) with respect to y we get a total x-derivative, so Thus when c = 0, equations ( 4) and ( 14) form a consistent pair.Also when c = 0 equation ( 4) is invariant under reparametrizations of y.Using this freedom in (14) we can take f to be a constant, which we will assume from here on (the constant can be taken to be 0 or ±1 but we will keep it general).The pair ( 4)-( 14)) is irreducible, but if we write u y = e q in (14) it becomes q xy = e q +f e −q , the sine-Gordon or sinh-Gordon or Liouville equation, depending on the choice of f .The two hierarchies of symmetries of (4) are also symmetries of the pair (4)-( 14); the second hierarchy becomes the Schwarzian KdV hierarchy involving only y-derivatives of u.For example, the first two characteristics given above, after multiplication by a suitable constant and setting c = 0, become uy .Thus far all the results presented are the results of [1].
The generating symmetries for the Adler-Shabat equation ( 4) are nonlocal in u, and can be expressed in terms of the solutions of a number of related systems.We will use the z-system, to be defined below, but we could also work in terms of Bäcklund variables or solutions of the Lax pair.The Bäcklund transformation for the Adler-Shabat equation is that if u is a solution then so is u − 2v where v satisfies Here λ is a parameter.Note that we use the words "Bäcklund variables" here to refer to the auxiliary functions that appear in a Bäcklund transformation.This is distinct from Adler and Shabat's usage of the term (see section 2).However a representation of the consistent pair ( 4)-( 14) in terms of Bäcklund variables in the sense of Adler-Shabat can be found by settiing V = uxy 2uy , in which case The first equation of this system evidently has some commonality with the first equation for the Bäcklund variable v (in our sense).
The Bäcklund transformation for the Adler-Shabat equation ( 4) has the same superposition principle as the KdV equation.Specifically, if the solution u (1) is obtained from the solution u by Bäcklund transformation with the parameter λ 1 , and the solution u (2) is obtained from the solution u by Bäcklund transformation with the parameter λ 2 , Then the solution obtained by applying the two Bäcklund transformations to u, in either order, is − u (2) .The Lax pair for (4) can be obtained from the system for the Bäcklund variable by setting v = ϕx ϕ .This gives This is actually the Lax pair for (4) with c an arbitrary function of y, but we restrict c to be a constant as otherwise formulas become extremely lengthy.To write the generating symmetries in terms of solutions of the Lax pair requires the use of two linearly independent solutions of the Lax pair, and the formulas can be simplified by use of a single function z which is the ratio of two solutions.This satisfies what we call the z-system: Note that the z system is invariant under Möbius transformations of z.
We are now ready to present the generating symmetries of (4).However, we first mention that for general c the equation has 4 classical symmetries, with characteristic where c 1 , c 2 , c 3 , c 4 are constants.These correspond to translations of x, y and u and a scaling symmetry.In the case c = 0, as previously mentioned, the equation is also invariant under reparametrizations of y.The equation also has 4 generating symmetries with the characteristics In the case c = 0, then Q, S, T are also generating symmetries of ( 14), the second equation of the consistent pair.However R is only a symmetry of ( 14) if f = 0 (alternatively, when f ̸ = 0, it is necessary to define a non-trivial action of R on f for R to be a symmetry of the full pair).We clain the following: Both infinite hierarchies of symmetries of equation ( 4) in the case c ̸ = 0, and the pair ( 4)-( 14) in the case c = 0, are obatined by expansions of Q in powers of λ.To show this we consider the large |λ| expansion of Q, and then the small |λ| expansion, first in the case c ̸ = 0, and then in the case c = 0.

Large |λ| expansion. Writing the Schwarzian derivative of z in terms of
From this we see that Q has an expansion for large |λ| of the form Each coefficient is the characteristic of a symmetry of equation (4).These are the standard potential KdV flows, and constitute the first infinite hierarchy of symmetries of (4).
2. Small |λ| expansion, c ̸ = 0. Using the definition of Q and ( 17) it is straightforward to check that This allows us to express all x-derivatives of Q in terms of y derivatives.Using this to eliminate x-derivatives of Q from (20) gives Assuming c ̸ = 0 we write and find the following equation for Q: where Evidently for small |λ| we can find a solution of this for Q in the form of a power series in λ finding the coefficients Qn (x, y) recursively.This gives a power series solution for Q: with Each coefficient is the characteristic of a symmetry of equation ( 4).These constitue the second infinite hierarchy of symmetries of (4) in the case c ̸ = 0.
3. Small |λ| expansion, c = 0.In the case c = 0, making the substitution ( 24) in ( 23) is not appropriate.In this case, using (14), equation ( 23) reads Assuming f ̸ = 0, we make the substitution This has a solution as an expansion in the form (25), giving the following expansion for Q: We recognize that the components are the flows of the Schwarzian KdV (or UrKdV [41]) hierarchy.Furthermore, since there is no f dependence in any of the components, these flows are symmetries also in the case f = 0.
Comparison of the expansion (26) in the case c ̸ = 0 and the expansion (27) in the case c = 0 shows two unexpected features.First, while for c ̸ = 0 it seems that Q is analytic at λ = 0, for c = 0 it is not.But also, comparing the coefficients of the expansions, it seems that in some sense there are only half as many symmetries in the case c = 0 as there are in the case c ̸ = 0. We do not have any understanding of this.
The above discussion only shows how to obtain the two hierarchies of symmetries from the single generating symmetry Q.To show they all commute it is necessary to show that the different Q's associated with different values of the parameter λ commute.We refer the reader to [36] for a full discussion of how this is done for the KdV equation, and the necessary calculation for equation ( 4) is identical.In particular, one possible choice for the action of the symmetry Q(µ) (Q with parameter µ) on z(λ) (z with parameter λ) is given by We conclude our discussion of the equation ( 4) with a discussion of the recursion relation between the components of the two hierarchies of symmetries.Using the definition of Q, from equation (18) and equation ( 16), it is straightforward to show that Q satisfies the linear differential equation Substituting the expansion of Q for large |λ| into this, we derive the recursion Similarly, substituting the small |λ| expansion into the differential equation for Q we obtain the recursion which holds for n ≥ 0 if we take Q 0 = u y .In principle, these recursion relations can be used to find the two hierarchies of symmetries, but it is highly nontrival to prove they are local, and the generating symmetry approach is preferable.We note that with these defintions of Q 0 and Q 0 , the two hierachies include translations of x, y and u, but do not include the scaling symmetry, which belongs to the hierarchy associated with the generating symmetry R that we will not discuss here.
Note that Q, S, T are not invariant under Mobius transformations of z.In fact, they transform into each other.So in writing the above, we have assumed that we fix z, and are writing different symmetries associated with this z.Alternatively, it is possible to consider just Q for different choices of z (and later on we will do this).In the case c = 0, equation ( 5) has an additional generating symmetry If, in addition, f = 0, then this is also a symmetry of (28).In greater generality, R is a symmetry of (5) for arbitrary c, and, if c = 0, of (28) for arbitary f , if we allow changes of c and f given by We consider three different expansions of the generating symmetries: For small λ − 1, for small λ and for large λ.
Expansions for small λ − 1.To obtain an expansion of Q(λ) in powers of λ − 1 we rewrite equation (32) in terms of Q to get We can find a solution of this in the form with Each component in this series is a symmetry of (5).This is the first hierarchy of local symmetries of (5), and coincides with the hierarchy of symmetries of the Hunter-Saxton equation [13] given in [32].To obtain a corresponding expansion of R(λ) we observe that since 1 Q = (log z) x , we have with qi = k 2i+1 q i .The symmetries q i do not have a limit as k → 0, but the symmetries qi do.The coefficients q0 , q1 , q2 , which are valid in the k → 0 limit, can be obtained from the formulas given above for q 0 , q 1 , q 2 in the k ̸ = 0 case, simply by setting k = 1.
To summarize: although the expansions are different for the cases c ̸ = 0 (equation 42), c = 0 and f ̸ = 0 (equation 43), and c = f = 0 (equation 44), in each case there is an expansion of Q for small λ, giving a second hierarchy of local symmetries of (5).In each case the lowest order symmetry in the hierarchy is simply u y (which, we recall, is one of the point symmetries of (5), see ( 29)).
To find the corresponding expansions of R we use (35), for which we first need to find an expression for z.We know Q = z zx , from which where C(y, λ) is a currently undetermined function of y and λ.To satisfy (33) we need In the case c ̸ = 0, from (42), we have Q = uy √ c + O(λ) for small λ, and the first term on the right hand side is regular at λ = 0.It follows that C y cannot be regular at λ = 0, and must have a singular part − √ c λ .In fact, without loss of generality, we can take as the regular part of C y at λ = 0 can be absorbed into the choice of limits in the integral, and adding a function of λ alone to C corresponds to adding some (λ-dependent) multiple of Q to R. Thus, using (35), we have where Apart from the first two, these symmetries are nonlocal, involving a single integral.
Moving to the analysis of R in the case c = 0, the appropriate form of C to take in equation (45) in order to satisfy (32) is Using this we obtain Expansions for large λ.It is in fact more convenient to consider this expansion as an expansion in powers of λ − 1.We start with the z-system (32)- (33).For large λ the left-hand sides are both zero, so to leading order we have where α, β, γ, δ are constants, with αδ − βγ = 1 without loss of generality.(The solution of ( 32)- (33) with vanishing left hand side takes this form with α, β, γ, δ all functions of λ.We assume these functions tend to constants in the large λ limit, and these are the constants indicated above.Further constants of integration appear at each stage of constructing the large λ expansion; these can all be handled by treating α, β, γ, δ as polynomials in λ − 1, to the relevant order.)It is then convenient to take the expansion of z in the form Substituting in ( 32)-( 33) we obtain Substituting the expansion for z into the formula in (34) for Q we obtain an expansion for Q, depending on the parameters α, β, γ, δ.Linearizing in β, γ we obtain where We see in the first few terms 6 of the 7 generalized point symmetries given in (29).The higher terms, involving z2 , z3 , . . .are nonlocal, though the simplest one (associated with the third term in Q 2 ), u t = u 2 x − (z 2 ) x , can be differentiated twice to give u txx = u x u xxx + 1 2 u 2 xx which is the Hunter-Saxton equation [13].What we have done in the previous paragraph is to compute Q for the most general solution of the z-system.As mentioned previously, this is equivalent to computing the 3 generating symmetries Q, S, T for a single solution of the z-symmetry.The symmetries that appear as the coefficients of different powers of 1 λ−1 in a single expansion commute with each other.However symmetries appearing in different expansions do not necessarilly commute with each other (see [36]).
Moving now to the corresponding expansion for R, we note that the formula (35) for R is invariant under Mobius transformations of z, and thus there is only a single expansion.This takes the form Summary.The following picture emerges.The generating symmetry Q has distinct expansions for small λ − 1, small λ and large λ.For small λ − 1 there is a single expansion (39), valid for all values of the constants c and f , giving an infinite hierarchy of local symmetries.For small λ, there is a single expansion, but this takes different forms in the three cases c ̸ = 0 (equation ( 42)), c = 0, f ̸ = 0 (equation ( 43)), and c = f = 0 (equation ( 44)).In each case there is an infinite hierarchy of local symmetries, with the lowest order symmetry corresponding to y-translation, one of the classical symmetries.When c = 0 there appear, in some sense, to only be half as many of these symmetries as when c ̸ = 0. Finally, for large λ, there are three distinct expansions, (47), ( 48) and (49), valid for all values of the constants c and f .In each of these expansions only the two lowest order symmetries are local, and these give the remaining six classical symmetries given in (29).Before summarizing the results for the generating symmetry R, we recall that this is only a symmetry in the standard sense of ( 5) if c = 0, and of (28) if in addition f = 0.However, if we allow R to also act on the constants c and f then it is a symmetry for arbitrary values of c and f .For small λ − 1, R has the expansion (40), in which all terms are nonlocal, involving a single integral.For small λ, R has an expansion of the form (46), in which the coefficients are different depending on whether c ̸ = 0 or c = 0; in the latter case the only local symmetry in the hierarchy is y-rescaling, but in the former there is a second local symmetry.All other symmetries in the hierarchy involve a single integral.Finally, for large λ there is also only a single expansion of R, equation ( 50), with all terms nonlocal.

Recursion operators
We mention in conclusion of our symmetry analysis for equation (5) that recursion formulae between the different components of the different hierarchies of symmetries can be obtained from the identities We give just two examples.Substituting the small λ − 1 expansion (39) into the Q-identity, we obtain the recursion (2u xxx ∂ x + u xxxx )q i+1 = q i,xxx , i = 0, 1, . . .

Concluding remarks
The research presented in this paper very much reflects the problem that although we know that a necessary property for integrability of a PDE is the existence of an infinite number of symmetries, and this can be used to search for integrable equations, it is much harder to know when we have found all the symmetries of a given equation or whether a given infinite set is sufficient.
We have shown that different infinite hierarchies of symmetries for a given equation can be obtained by different expansions, around different values of the parameter, of a single generating symmetry.This suggests to us that the existence of a generating symmetry (or symmetries) for an equation is fundamental to integrability -but we still have no idea how many generating symmetries there should be, nor (since they are written in terms of some nonlocal variables) in what form to search for them.
Nevertheless, a routine effort should be made to identify generating symmetries, and we think this may well be easier than, for example, looking for recursion operators.
Also in this paper we have extended Adler and Shabat's notion of a consistent pair of third order equations.Although the only application we know so far of this kind of system is [8], it shows how even one of the most basic notions in differential equations -that of the order of a differential equation -is not as simple as it might seem.In both examples that we have studied in depth we have seen how the symmetry structure changes when the value of the parameter c is set to 0 and the equation becomes part of a consistent pair (and an extra scaling symmetry appears).In particular, the type of expansions needed to derive hierarchies from the generating symmetry changes, and we have no understanding of why this happens.A more detailed study of the limit c → 0 would certainly be interesting.