Liouville Correspondences between Integrable Hierarchies

In this paper, we study explicit correspondences between the integrable Novikov and Sawada-Kotera hierarchies, and between the Degasperis-Procesi and Kaup-Kupershmidt hierarchies. We show how a pair of Liouville transformations between the isospectral problems of the Novikov and Sawada-Kotera equations, and the isospectral problems of the Degasperis-Procesi and Kaup-Kupershmidt equations relate the corresponding hierarchies, in both positive and negative directions, as well as their associated conservation laws. Combining these results with the Miura transformation relating the Sawada-Kotera and Kaup-Kupershmidt equations, we further construct an implicit relationship which associates the Novikov and Degasperis-Procesi equations.

(1.4) obtained directly from those of the KdV hierarchy. This argument does not work for the mCH and mKdV hierarchies, whose correspondence through a Liouville transformation is based upon a relationship between the corresponding recursion operators and some subtle identities relating the respective Hamiltonian operators [36]. It demonstrates that the positive flow and negative flow of the mCH hierarchy are generated by the negative flow and positive flow of the mKdV hierarchy, respectively. The correspondences between the Hamiltonian conservation laws for the CH (mCH) hierarchy and KdV (mKdV) hierarchy have also been derived [36,40]. The goal of this paper is to study the similar Liouville correspondences between the flows and Hamiltonian conservation laws in both the Novikov and SK hierarchies, as well as the DP and KK hierarchies. Furthermore, an underlying correspondence between the Novikov equation (1.1) and the DP equation (1.3) is also constructed. Our motivations are three-fold. First, it was shown that the Novikov equation is related to the first negative flow of the SK hierarchy [34], while the DP equation is related to the first negative flow of the KK hierarchy [17]. Second, the CH and mCH hierarchies are related, respectively, to the KdV and mKdV hierarchies through Liouville transformations relating their isospectral problems. Third, the SK equation is related to the KK equation by a Miura transformation [23], and there exists a transformation found in [36] which maps the mCH equation to the CH equation.
However, in the Novikov-SK and DP-KK settings, due to the non-standard bi-Hamiltonian structures [26], we neither have the dual relationship, as in both CH-KdV and mCH-mKdV settings, nor the subtle relationship between their Hamiltonian operators, as in the CH-KdV setting [40,50], nor between their recursion operators, as in the mCH-mKdV setting [36]. On the other hand, given that the Novikov and DP equations are both third-order nonlinear equations, while the SK and KK equations are of fifth order, it seems difficult to establish any relationship between the Novikov or DP equations with the flows in the negative direction of the SK hierarchy or the KK hierarchy. Nevertheless, based on the Liouville transformation between the isospectral problems of the Novikov and SK hierarchies, as well as the DP and KK hierarchies, we are able to establish certain nontrivial identities which reveal the underlying relationship between the recursion operator of the Novikov (DP) hierarchy and the adjoint operator of the recursion operator for the SK (KK) hierarchy. Using these operator identities, we are able to prescribe a Liouville correspondence between the flows involved in the Novikov-SK hierarchies and DP-KK hierarchies.
It is worth noting that, in the Novikov-SK setting, in order to establish the explicit relationship between the flows in the positive Novikov hierarchy and the flows in the negative SK hierarchy, we make use of a novel factorization of the recursion operator of the SK equation to identify the equations transformed from the positive flows in the Novikov hierarchy as the corresponding negative flows in the SK hierarchy exactly. The factorization is based on the following nontrivial operator identity for the recursion operator of the SK equation [8]: Since conservation laws play a key role in the study of well-posedness of solutions, stability of solitons, and wave-breaking phenomena, another topic of this paper is to establish relationships between the Hamiltonian conservation laws for the Novikov and SK hierarchies, and the DP and KK hierarchies. These rely on some new identities for Hamiltonian conservation laws related by the Liouville transformations and certain known results. This section is concluded by outlining the rest of the paper. In Section 2, we first present the Liouville transformation relating the isospectral problems of the Novikov hierarchy and the SK hierarchy in Section 2.1. Next in Section 2.2, several operator identities are combined with the Liouville transformation to establish the one-to-one correspondence between the flows in the Novikov and SK hierarchies. It is proved in Section 2.3 that the Liouville transformation establishes the correspondence between the series of Hamiltonian conservation laws of the Novikov equation and the SK equation. The Liouville correspondence between the DP hierarchy and the KK hierarchy, and the relationship of their conservation laws will be studied in Section 3. In Section 4, we obtain a nontrivial relationship between the Novikov equation ( In this section, we first obtain the Liouville transformation relating the Novikov and SK hierarchies. In accordance with standard terminology, a Liouville transformation is defined by a change of variables which maps one spectral problem to another [53,56]. If the transformation does not affect the independent variables, it is referred to as a Miura transformation. The Novikov equation can be expressed as the compatibility condition for the linear system [34] consisting of and Note that equation (2.2) is reduced to a scalar equation by setting Ψ = ψ 2 , namely It was proved in [34] that by the reciprocal transformation which is a third-order spectral problem for the SK equation. Note that the isospectral problems for the Novikov equation and the SK equation can also be written as the Zakharov-Shabat (ZS) formΨ is a diagonal sl(3) matrix. The functions here, g(y, t), h(y, t) satisfy the system in the case of the Novikov equation, and for the SK equation. Moreover, using (2.4), Notice that (2.7) is equivalent to after gauging Φ by a factor of e 2τ /(3µ) and setting W = −3V . Indeed, the linear system (2.5) and (2.8) provides the Lax pair for the first negative flow in the SK hierarchy [28] (see also [33,34]), and the compatibility condition Φ yyyτ = Φ τ yyy yields Therefore, we conclude that there exists a Liouville correspondence between the Novikov equation (2.1) and the first negative flow (2.9) of the SK hierarchy, where their corresponding Lax pairs are related by the transformations (2.4) and (2.6).
In light of this, we are led to generalize the Liouville correspondence between the Novikov equation and the first negative flow of the SK hierarchy to their entire hierarchies, establishing the correspondence between the flows of the Novikov and SK hierarchies. Motivated by (2.4) and (2.6), we pursue this study by utilizing the Liouville transformation Note that the first expression in (2.10) has the form of a reciprocal transformation [61].

The correspondence between the Novikov and SK hierarchies
Let us now study the correspondence between the Novikov hierarchy and the SK hierarchy. First of all, the Novikov equation (2.1) can be written in bi-Hamiltonian form [34] are the compatible Hamiltonian operators. The corresponding Hamiltonian functionals are given by Moreover, since using the Novikov equation (2.1), which implies that H 1 can be written in the following local form in terms of u and m: According to Magri's theorem [49,57,58], an integrable bi-Hamiltonian equation with two compatible Hamiltonian operators K and J belongs to an infinite hierarchy of higher-order bi-Hamiltonian systems, in both the positive and negative directions, where H n , n ∈ Z are all conserved functionals common to all members of the hierarchy. The Novikov equation (2.11) serves as the first member in the positive direction of (2.13). As for the negative direction, observe that and the Hamiltonian operator K admits the Casimir functional Therefore, we conclude that the negative flows of the Novikov hierarchy are generated from the Casimir equation 14) The SK equation exhibits a generalized bi-Hamiltonian system, whose corresponding integrable hierarchy is generated by a recursion operatorR =KJ , with As noted in [26],K maps the variational gradients of the conservation laws of the equation under consideration onto its symmetry groups, whileJ works in the opposite way.
is called a generalized bi-Hamiltonian system if there exist an implectic (Hamiltonian) operatorK and a functionalH 0 such that as well as a symplectic operatorJ and a corresponding functionalH 1 satisfyinḡ The term "generalized bi-Hamiltonian system" is taken from [26], and refers to the fact that we do not assume any nondegeneracy or invertibility conditions for the operatorsK andJ . These are particular instances of the general notion of compatible pairs of Dirac structures, whose properties are developed in Dorfman [19].
Therefore, defininḡ one finds that the SK equation (2.15) can be written as and the positive flows of the SK hierarchy are generated by applying successively the recursion operatorR =KJ toK 1 , namely On the other hand, in the negative direction, note that the trivial function f = 0 satisfies the equation Then the n-th negative flow is proposed to take the form Furthermore, it has been discovered in [8] that the recursion operatorR satisfies the following decomposition This factorization result demonstrates that the first negative flow (2.9) in the SK hierarchy derived in [34] satisfiesRQ τ = 0. We have thus confirmed the formulation (2.20) for the negative flows.
We are now in the position to establish the theorem which shows how the transformations (2.10) affect the underlying Liouville correspondence between the Novikov and SK hierarchies. In this theorem and hereafter, we denote, for a positive integer n, the n-th equation in the positive and negative directions of the Novikov hierarchy by (Novikov)  The proof of this theorem relies on the following two lemmas. Lemma 2.3. Let m(t, x) and Q(τ, y) be related by the transformation (2.10). Then the following operator identities hold: Proof . (i). In view of the transformation (2.10), one has ∂ x = m 2 3 ∂ y . It follows that where, by (2.10), a direct computation yields We thus arrive at (ii). Thanks to (2.22), we deduce that verifying (2.23).
(iii). Using the transformations (2.10) again, we find Hence, The relationship between the recursion operator for the Novikov hierarchy and the adjoint operator of recursion operator for the SK hierarchy is given by the following result.
holds for each integer n ≥ 1.
Proof . We prove (2.25) by induction on n. First, using the inverse operator K −1 along with (2.10), we deduce from (2.23) and (2.24) that which shows that (2.25) holds for n = 1. Next, we assume that (2.25) holds for n = k, say Then for n = k + 1, thanks to the result when n = 1, one has This completes the induction step, and thus proves the lemma.
Proof of Theorem 2.2. The proof of Theorem 2.2 contains two parts: (i). Let us begin with the (Novikov) −n equation for n ≥ 1. First, since the relation (2.10) can be rewritten as we deduce from (2.22) that the first negative flow (2.14) of the Novikov hierarchy satisfies Hence, the n-th member in the negative hierarchy takes the form Next, suppose that m(t, x) is the solution of the equation (2.27). We calculate the t-derivative of the corresponding function Q(τ, y) defined in (2.26). More precisely, we deduce that, on the one hand, and on the other hand, in view of (2.26), Hence, combining the preceding two equations, we arrive at This immediately implies that Q(τ, y) solves the (SK) n equation (2.19). Conversely, if Q(τ, y) is a solution of the (SK) n equation for n ≥ 1, since the transformation (2.10) is a bijection, tracing the previous steps backwards suffices to verify that the reverse argument is also true.
(ii). Now, we focus our attention on the (Novikov) n equation for n ≥ 1, which can be written as Plugging it into (2.28), we find As a consequence, the operator factorization identity (2.21), when combined with (2.25), allows us to deduce that, for each n ≥ 1, if m(t, x) solves the (Novikov) n equation (2.29), then for the operator B defined by the corresponding Q(τ, y) satisfies where we have made use of the operator identity (2.22). This immediately reveals that Q(τ, y) solves the (SK) −n equation (2.20). We thus prove that, for each n ≥ 1 the (Novikov) n equation is mapped into the (SK) −n equation under the transformation (2.10).
In analogy with the proof of part (i), the reverse argument follows from the fact that (2.10) is a bijection.

The correspondence between the Hamiltonian conservation laws of the Novikov and SK equations
According to Magri's theorem, one can also recursively construct an infinite hierarchy of Hamiltonian conservation laws of any bi-Hamiltonian structure. In particular, for the Novikov equation (2.1), at the n-th stage we determine the Hamiltonian conservation laws H n satisfying the recursive formula formally provides an infinite collection of Hamiltonian conservation laws for the SK equation (2.15), using the operator pairK andJ given in (2.16) and (2.17).
In this subsection we investigate the relationship between the two hierarchies of Hamiltonian conservation laws {H n } and {H n }. Let us begin with two preliminary lemmas.
Finally, to prove (2.32) holds for all n ≤ −1, we assume that (2.32) holds for n = k. Then for n = k − 1, using the recursive formulae (2.30) and (2.31) and Lemma 2.4 with n = 1 again, we infer that which establishes the induction step for n ≤ −1 and thus proves the lemma in general.
In order to establish the correspondence between Hamiltonian conservation laws admitted by the Novikov and SK equations, we require the formula for the change of variational derivatives.

Then the Fréchet derivative of F [m] is
On the other hand, where, by (2.10), Next, it follows from Finally, the assumption of the lemma implies that According to the usual definition of the variational derivative, we have, on the one hand, On the other hand, using the fact thatK is skew-symmetric, we infer that which, in comparison with (2.34) verifies (2.33), proving the lemma.
Finally, referring back to the form of the Hamiltonian operator K, one has It follows that the relation (2.32) can be written in an equivalent form, namely Therefore, subject to the hypothesis of Lemma 2.6, if we define the functional for some l ∈ Z, then Lemma 2.6 allows us to conclude that, for each n ∈ Z, This, when combined with (2.35), immediately leads to and then follows. We thus conclude that there exsits an one-to-one correspondence between the sequences of the Hamiltonian conservation laws admitted by the Novikov and SK equations. Indeed, we have proved the following theorem.
For instance, in the case of n = 2, which can be expressed in terms of Q according to (2.26), say As a consequence, 3 The correspondence between the DP and KK hierarchies

A Liouville transformation between the isospectral problems of the DP and KK hierarchies
The Lax pair for the DP equation

2)
and These can be rewritten in scalar form by setting Ψ = ψ 1 , namely Consider the KK equation P τ + P yyyyy + 20P P yyy + 50P y P yy + 80P 2 P y = 0. It has been shown in [34] that the Lax pair for the first negative flow of its associated hierarchy is Φ yyy + 4P Φ y + 2P y Φ = µΦ (3.4) and which is a reduction of a (2 + 1)-dimensional non-isospectral Lax pair given in [28]. The compatibility condition for (3.4) and (3.5) gives rise to where A is the fifth-order operator A = ∂ 5 y + 6 ∂ y P ∂ 2 y + ∂ 2 y P ∂ y + 4 ∂ 3 y P + P ∂ 3 y + 32 ∂ y P 2 + P 2 ∂ y .
In analogy with the Liouville correspondence between the Novikov equation and the first negative flow of the SK hierarchy, there exists a similar correspondence between the DP equation and the first negative flow of the KK hierarchy. In fact, it has been found [17] that the following coordinate transformations will convert the scalar form of the isospectral problem (3.2) into (3.4).
As before, in this section we investigate the Liouville correspondence between the DP and KK hierarchies. More precisely, the respective flows in the two hierarchies are related by the Liouville transformations and P = 1 4 In addition, the relationship between the Hamiltonian conservation laws for the DP hierarchy and those for the KK hierarchy is also clarified.

The correspondence between the DP and KK hierarchies
The DP equation (3.1) is also a bi-Hamiltonian system [17] are a pair of compatible Hamiltonian operators, and the corresponding Hamiltonian functionals are Applying the recursion operator R = LD −1 successively to the initial symmetry n t = G 1 gives rise to an infinite hierarchy of commuting bi-Hamiltonian flows and consequent conservation laws E l . As far as the associated negative flows are concerned, noting that and L admits the Casimir functional Therefore, we conclude that the first negative flow of the DP hierarchy is the Casimir equation and applying R −1 = DL −1 successively to it produces the hierarchy of negative flows, in which the l-th member takes the form Analogous to the SK hierarchy, the integrable hierarchy of the KK equation also arises from a generalized bi-Hamiltonian structure, the flow is governed by P τ =Ḡ l [P ], whereḠ l [P ] are determined by the relations y + 6(P ∂ y + ∂ y P ) + 4 ∂ 2 y P ∂ −1 y + ∂ −1 y P ∂ 2 y + 32 P 2 ∂ −1 y + ∂ −1 y P 2 , and R =LD is the consequent recursion operator. It is easy to see that the KK equation (3.3) in this hierarchy is exactly with the corresponding Hamiltonian functional Similarly, if we use the fact thatD · 0 = δĒ 0 /δP , we may conclude that the negative flows of the KK hierarchy take the form LD l P τ = 0, l = 1, 2, . . . .

(3.11)
It is worth noting that sinceD = ∂ −1 y A∂ −1 y , so the equation (3.6) arising from the compatibility condition of the Lax pair (3.4) and (3.5) is a reduction of the first negative flowLDP τ = 0.
As before, we hereafter denote, for a positive integer l, the l-th equation in the positive and negative directions of the DP hierarchy by (DP) l and (DP) −l , respectively, while the l-th positive and negative flows of the KK hierarchy by (KK) l and (KK) −l , respectively. With this notation, we state the main theorem on the Liouville correspondence between the DP and KK hierarchies as follows. The proof of this theorem is based on the following two preliminary lemmas, which clarify the relations between certain operators. Lemma 3.2. Let n(t, x) and P (τ, y) be related by the transformations (3.7) and (3.8). Then the following identities hold: (3.14) Proof . (i). Define χ = n 1 3 , so from (3.7) and (3.8), we have ∂ x = χ∂ y and And then, a direct calculation shows that where, by (3.15) We thus have which immediately leads to and verifies (3.12).
Proof . Due to the form of the inverse operator L −1 and the identities (3.13) and (3.14), we arrive at which verifies (3.17) for l = 1. Then an obvious induction procedure allows us to prove (3.17) in general. Hence the lemma is proved.
Proof of Theorem 3.1. To prove this theorem, we take the analogous steps as in the proof of Theorem 2.2. First of all, the derivative of P with respect to t is On the other hand, it follows from (3.8) that From the preceding equations and using the formula (3.12), we have Now, suppose n(t, x) is the solution of the (DP) −l equation and n(t, x) is related to P (τ, y) according to (3.7) and (3.8). Then plugging (3.19) into (3.18), one finds that the corresponding function P (τ, y) satisfies This shows that by the transformations (3.7) and (3.8), the (DP) −l equation is mapped into the (KK) l equation.
When it comes to the (DP) l equation for l ≥ 1, inserting the formula for the (DP) l equation into (3.18) yields Therefore, referring back to the form of the operator D and using the identity (3.17), we deduce that which shows that P (τ, y) is a solution for the (KK) −l equation (3.11). We thus have proved that for each l ≥ 1, the (DP) l equation is mapped, via the transformations (3.7) and (3.8), into the (KK) −l equation. As in Theorem 2.2, the converse argument is also valid.

The correspondence between the Hamiltonian conservation laws of the DP and KK equations
We now investigate the relationship between the Hamiltonian conservation laws of the DP and KK equations. For the DP equation (3.1), with the Hamiltonian pair L and D defined in (3.9) in hand, the corresponding recursive formula formally defines an infinite hierarchy of Hamiltonian conservation laws E l determined by For the KK equation, its Hamiltonian conservation lawsĒ l can be determined by the generalized bi-Hamiltonian (bi-Dirac) structurē withL andD given by (3.10). Before proving the main theorem for the correspondence between the two hierarchies of Hamiltonian conservation laws {E l } and {Ē l }, two lemmas regarding their variational derivatives are in order. Lemma 3.4. Let {E l } and {Ē l } be the hierarchies of Hamiltonian conservation laws of the DP and KK equations, respectively. Then, for each l ∈ Z, their corresponding variational derivatives are related according to the following identity Proof . We first consider the case of l ≤ −2. Since then the factD · 0 = δĒ 0 /δP reveals that (3.21) holds for l = −2.
We proceed by induction on l. Assume that (3.21) holds when l = k, namely From the recursive formula (3.20) and by the assumption, Then, thanks to Lemma 3.3 with l = 1, we conclude that (3.21) holds for l = k − 1. Furthermore, for the case of l = −1, we claim Note thatĒ −1 is a Casimir functional for Hamiltonian operatorL, it suffices to show that Indeed, from the definition of the operator L and the formula (3.13), we havē proving the claim and verifying that (3.21) holds for l = −1.
Finally, induction on l shows that if (3.21) holds for l = k, then for l = k + 1, from the recursive formula (3.20) and the identities (3.13) and (3.14), we infer that which completes the induction step, and thereby proves the lemma. Proof . As the first step, in view of (3.8), for convenience, we introduce the notatioñ Evaluating the Fréchet derivative ofF [n] produces On the other hand, we get Secondly, by the assumption, one has Furthermore, due to the fact thatL is skew-adjoint, the righ-hand side of the above expression yields which, combined with the definition of the variational derivative, produces (3.22).
As a consequence, the following theorem is thereby proved.
Theorem 3.6. Under the Liouville transformations (3.7) and (3.8), for each l ∈ Z, the Hamiltonian conservation lawĒ l (P ) of the KK equation is related to that E l (n) of the DP equation, according to the following identity E l (n) = 36Ē −(l+2) (P ), l ∈ Z.

The relationship between the Novikov equation and the DP equation
It has been shown in [23] that under the Miura transformations This, together with the fact that there exist the Liouville correspondences between the Novikov and SK hierarchies, as well as between the DP and KK hierarchies, inspires a natural question as to whether there exists some relationship between the Novikov equation (2.1) and the DP equation (3.1). We can regard (4.1) and (4.2) as Bäcklund transformations. According to Fokas and Fuchssteiner [21], all the positive flows in the SK hierarchy admit the same transformation (4.1). More precisely, set where B 1,V and B 1,Q are the Fréchet derivatives of (4.1) with respect to V and Q, respectively. Then, the recursion operatorR of the SK equation and the recursion operator R * of equation (4.3) satisfy Similarly, each member in the KK hierarchy admits the Miura transformation (4.2), and its corresponding recursion operator R is linked with the recursion operator R * according to the identity R * = T 2 RT −1 2 , where T 2 ≡ B −1 2,V B 2,P = (V + ∂ y ) −1 is the operator arising from the function (4.2).
In light of these relations, we claim that both the first negative flow of the SK hierarchy and the KK hierarchy are related to the same equation via the Miura transformations (4.1) and (4.2), respectively. Indeed, we have the following result. Proof . Thanks to (4.1), one has This, together with (4.4), implies Therefore, if R * V τ = 0, thenRQ τ = −(2V − ∂ y )R * V τ = 0, proving the SK part of the proposition. The KK part can be proved by in a similar manner.
Finally, using Proposition 4.1, combined with Theorems 2.2 and 3.1, we are able to establish a relationship between the Novikov equation (2.1) and the DP equation (3.1). This fact is summarized in the following proposition.
satisfies the Novikov equation (2.1), while the function n(t, x) determined by satisfies the DP equation (3.1).