Liouville Correspondences between Integrable Hierarchies

Liouville Correspondences
between Integrable Hierarchies

Jing KANG †, Xiaochuan LIU †, Peter J. OLVER ‡ and Changzheng QU §

J. Kang, X.C. Liu, P.J. Olver and C.Z. Qu

† Center for Nonlinear Studies and School of Mathematics, Northwest
University,
† Xi’an 710069, P.R. China<EMAIL_ADDRESS><EMAIL_ADDRESS>
‡ School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
<EMAIL_ADDRESS>http://www.math.umn.edu/~olver/

§ Center for Nonlinear Studies and Department of Mathematics, Ningbo
University,
§ Ningbo 315211, P.R. China<EMAIL_ADDRESS>
Received February 07, 2017, in final form May 22, 2017; Published online May
28, 2017

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.

Liouville transformation; Miura transformation; bi-Hamiltonian structure;
conservation law; Novikov equation; Degasperis–Procesi equation; Sawada–Kotera
equation; Kaup–Kupershmidt equation

37K05; 37K10

##  1 Introduction

This paper is devoted to studying Liouville correspondences between certain
integrable hierarchies and their interrelationships. One pair consists of the
Novikov and Sawada–Kotera (SK) hierarchies, which are initiated respectively
from the Novikov equation [34, 55]

$\displaystyle m_{t}=3uu_{x}m+u^{2}m_{x},\qquad m=u-u_{xx},$ (1.1)

and the SK equation [7, 62]

$\displaystyle Q_{\tau}+Q_{yyyyy}+5QQ_{yyy}+5Q_{y}Q_{yy}+5Q^{2}Q_{y}=0.$ (1.2)

The second pair of integrable hierarchies is initiated by the
Degasperis–Procesi (DP) equation [17, 18]

$\displaystyle m_{t}=3u_{x}m+um_{x},\qquad m=u-u_{xx},$ (1.3)

and the Kaup–Kupershmidt (KK) equation [37, 39]

$\displaystyle P_{\tau}+P_{yyyyy}+10PP_{yyy}+25P_{y}P_{yy}+80P^{2}P_{y}=0.$
(1.4)

Furthermore, combining these explicit correspondences with the known Miura
transformation linking the SK and KK equations [23], we derive a nontrivial
underlying correspondence between the Novikov equation (1.1) and the DP
equation (1.3).

The DP equation (1.3) was derived by Degasperis and Procesi [18] as a result
of the asymptotic integrability method for classifying (a class of) third-
order nonlinear dispersive evolution equations. It was subsequently shown that
the DP equation is integrable with a Lax pair involving a $3\times 3$
isospectral problem as well as a bi-Hamiltonian structure [17]. Furthermore,
the Lax representation can be written in the form of a matrix spectral problem
of Zakharov–Shabat (ZS) form. The associated inverse scattering transform and
the dressing method can be applied to construct smooth soliton solutions for
the DP equation [12, 13, 52]. Physically, the DP equation provides a model
describing the propagation of shallow water waves [14, 20]. It admits peaked
solitons (peakons) [17] as well as multi-peakon solutions which recover the
soliton interaction dynamics [17, 48] and exhibits a particular shock peakon
structure [47]. The integrability, well-posedness, wave breaking phenomenon
and stability of peakons for the DP equation have been studied extensively;
see [9, 33, 43, 46] and references therein. The Novikov equation (1.1) with
cubic nonlinear terms was discovered as a consequence of the symmetry
classification of nonlocal partial differential equations involving both cubic
and quadratic nonlinearities [55]. A Lax pair formulation based on a $3\times
3$ isospectral problem and the associated bi-Hamiltonian structure were
established in [34]. It was also shown that the Novikov equation possesses
peaked solitons and multi-peakon solutions [32, 34]. The well-posedness, wave
breaking and blow-up phenomena, as well as stability of peakons for the
Novikov equation have been studied in a number of papers, including [30, 45,
63].

The SK equation (1.2) and the KK equation (1.4) are two typical fifth-order
integrable equations [7, 37, 39, 62]. Their integrability can be verified from
several different standpoints: for instance, they both possess $3\times 3$
isospectral problems and bi-Hamiltonian structures, enjoy the Painlevé
property, admit multi-soliton solutions, etc. Like the DP equation, the KK
equation also admits a Lax operator of ZS form, which can be applied to
construct solitons of the KK hierarchy, [2, 27, 52, 64]. In Section 2, we will
see that both the Novikov and SK equations support Lax operators of ZS form.
Geometrically, the SK equation arises naturally from an integrable planar
curve flow in affine geometry [8, 59], while the KK equation comes from an
integrable planar curve flow in projective geometry [42, 54]. Interestingly,
both equations are related to the so-called Fordy–Gibbons–Jimbo–Miwa equation
via certain Miura transformations [23].

Like the Camassa–Holm (CH) equation [4, 5, 25] and the modified Camassa–Holm
(mCH) equation [24, 60], the Novikov and DP equations exhibit nonlinear
dispersion. Recent years have seen a proliferation of papers for the CH and
mCH equations studying their integrable properties, geometric formulations,
well-posedness for solutions of the Cauchy problem, and the stability of
peaked solitary waves solutions; see [1, 3, 6, 10, 11, 14, 15, 16, 29, 31, 35,
38, 41, 44, 51]. Interestingly, these equations support a notable variety of
non-smooth soliton-like solutions and can model the phenomenon of wave
breaking.

The method of tri-Hamiltonian duality was developed [22, 24, 60] to
systematically derive additional nonlinear dispersive integrable systems. This
method begins with the basic observation that most classical integrable
soliton equations that possess a bi-Hamiltonian structure, actually support a
compatible triple of Hamiltonian structures using a particular scaling
argument, leading to a systematic algorithm [60], to construct their dual
nonlinear dispersive integrable systems. In particular, the CH and mCH
equations appear as the duals to, respectively, the KdV and mKdV equations.

In view of this duality, it is of interest to establish relationships between
the full integrable hierarchy and the corresponding dual integrable hierarchy.
In [40] and [50], the correspondence between the CH hierarchy and the KdV
hierarchy is established through a Liouville transformation, the key point
being that the two Hamiltonian operators of the CH hierarchy can be 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]:

$\displaystyle\bar{\mathcal{R}}=-\big{(}\partial_{y}^{3}+2Q\partial_{y}+2\partial_{y}Q\big{)}\big{(}2\partial_{y}^{3}+2\partial_{y}^{2}Q\partial_{y}^{-1}+2\partial_{y}^{-1}Q\partial_{y}^{2}+Q^{2}\partial_{y}^{-1}+\partial_{y}^{-1}Q^{2}\big{)}$
$\displaystyle\hphantom{\bar{\mathcal{R}}}{}=-2\big{(}\partial_{y}^{4}+5Q\partial_{y}^{2}+4Q_{y}\partial_{y}+Q_{yy}+4Q^{2}+2Q_{y}\partial_{y}^{-1}Q\big{)}\big{(}\partial_{y}^{2}+Q+Q_{y}\partial_{y}^{-1}\big{)}.$

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
(1.1) and the DP equation (1.3) by exploiting the Miura transformations
relating the SK equation (1.2) and the KK equation (1.4) and the results in
previous Sections.

##  2 The correspondence between the Novikov and SK hierarchies

###  2.1 A Liouville transformation between the isospectral problems
of the Novikov and SK hierarchies

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

$\displaystyle m_{t}=u^{2}m_{x}+3uu_{x}m,\qquad m=u-u_{xx},$ (2.1)

can be expressed as the compatibility condition for the linear system [34]
consisting of

$\displaystyle\mathbf{\Psi}_{x}=\begin{pmatrix}0&\lambda m&1\\\ 0&0&\lambda
m\\\
1&0&0\end{pmatrix}\mathbf{\Psi},\qquad\mathbf{\Psi}=\begin{pmatrix}\psi_{1}\\\
\psi_{2}\\\ \psi_{3}\end{pmatrix},$ (2.2)

and

$\displaystyle\mathbf{\Psi}_{t}=\begin{pmatrix}\frac{1}{3}\lambda^{-2}-uu_{x}&\lambda^{-1}u_{x}-\lambda
u^{2}m&u_{x}^{2}\\\
\lambda^{-1}u&-\frac{2}{3}\lambda^{-2}&-\lambda^{-1}u_{x}-\lambda u^{2}m\\\
-u^{2}&\lambda^{-1}u&\frac{1}{3}\lambda^{-2}+uu_{x}\end{pmatrix}\mathbf{\Psi}.$

Note that equation (2.2) is reduced to a scalar equation by setting
$\Psi=\psi_{2}$, namely

$\displaystyle\Psi_{xxx}=2m^{-1}m_{x}\Psi_{xx}+\big{(}m^{-1}m_{xx}-2m^{-2}m_{x}^{2}+1\big{)}\Psi_{x}+\lambda^{2}m^{2}\Psi.$
(2.3)

It was proved in [34] that by the reciprocal transformation

$\displaystyle\mathrm{d}y=m^{\frac{2}{3}}\mathrm{d}x+m^{\frac{2}{3}}u^{2}\mathrm{d}t,\qquad\mathrm{d}\tau=\mathrm{d}t,$
(2.4)

the isospectral problem (2.3) is converted into

$\displaystyle\Phi_{yyy}+Q\Phi_{y}=\mu\Phi,$ (2.5)

with

$\displaystyle\Phi=\Psi,\qquad\mu=\lambda^{2},\qquad
Q=\frac{4}{9}m^{-\frac{10}{3}}m_{x}^{2}-\frac{1}{3}m^{-\frac{7}{3}}m_{xx}-m^{-\frac{4}{3}},$
(2.6)

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) form

$\displaystyle\tilde{\Psi}_{y}+\big{(}Q_{l}-\tilde{\lambda}J\big{)}\tilde{\Psi}=0,$

where $\tilde{\lambda}=\lambda^{\frac{2}{3}}$, while

$\displaystyle Q_{l}=\begin{pmatrix}-g^{-1}g_{y}&0&0\\\
0&g^{-1}g_{y}+h^{-1}h_{y}&0\\\ 0&0&-h^{-1}h_{y}\end{pmatrix},$

is a diagonal $\mathfrak{sl}(3)$ matrix. The functions here, $g(y,t)$,
$h(y,t)$ satisfy the system

$\displaystyle g_{yy}+h^{-1}h_{y}g_{y}+hg=0,\qquad
h^{-1}h_{yy}-2h^{-1}h_{y}^{2}+h=-\frac{1}{3}m^{-1}m_{yy}+\frac{2}{9}m^{-2}m_{y}^{2}-m^{-\frac{4}{3}},$

in the case of the Novikov equation, and

$\displaystyle g_{yy}+h^{-1}h_{y}g_{y}+hg=0,\qquad
h^{-1}h_{yy}-2h^{-1}h_{y}^{2}+h=Q,$

for the SK equation.

Moreover, using (2.4),

$\displaystyle\partial_{t}=\partial_{\tau}+m^{\frac{2}{3}}u^{2}\partial_{y},$

the $t$ evolution of $\Psi=\psi_{2}$ in (2.2) is transformed into

$\displaystyle\Phi_{\tau}-\frac{1}{\mu}(V\Phi_{yy}-V_{y}\Phi_{y})+\frac{2}{3\mu}\Phi=0,\qquad\mathrm{with}\qquad
V=um^{\frac{1}{3}}.$ (2.7)

Notice that (2.7) is equivalent to

$\displaystyle\Phi_{\tau}+\frac{1}{3\mu}(W\Phi_{yy}-W_{y}\Phi_{y})=0$ (2.8)

after gauging $\Phi$ by a factor of $e^{2\tau/(3\mu)}$ 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 $\Phi_{yyy\tau}=\Phi_{\tau yyy}$ yields

$\displaystyle Q_{\tau}=W_{y},\qquad W_{yy}+QW=T,\qquad T_{y}=0.$ (2.9)

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

$\displaystyle
y=\int^{x}m^{\frac{2}{3}}(t,\xi)\mathrm{d}\xi,\qquad\tau=t,\qquad
Q=\frac{4}{9}m^{-\frac{10}{3}}m_{x}^{2}-\frac{1}{3}m^{-\frac{7}{3}}m_{xx}-m^{-\frac{4}{3}}.$
(2.10)

Note that the first expression in (2.10) has the form of a reciprocal
transformation [61].

###  2.2 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]

$\displaystyle m_{t}=K_{1}=\mathcal{K}\frac{\delta\mathcal{H}_{0}}{\delta
m}=\mathcal{J}\frac{\delta\mathcal{H}_{1}}{\delta m},\qquad m=u-u_{xx},$
(2.11)

where

$\displaystyle\mathcal{K}=\frac{1}{2}m^{\frac{1}{3}}\partial_{x}m^{\frac{2}{3}}\big{(}4\partial_{x}-\partial_{x}^{3}\big{)}^{-1}m^{\frac{2}{3}}\partial_{x}m^{\frac{1}{3}},\qquad\mathcal{J}=\big{(}1-\partial_{x}^{2}\big{)}m^{-1}\partial_{x}m^{-1}\big{(}1-\partial_{x}^{2}\big{)}$
(2.12)

are the compatible Hamiltonian operators. The corresponding Hamiltonian
functionals are given by

$\displaystyle\mathcal{H}_{0}=9\int\big{(}u^{2}+u_{x}^{2}\big{)}\mathrm{d}x$

and

$\displaystyle\mathcal{H}_{1}=\frac{1}{6}\int
um\partial_{x}^{-1}m\big{(}1-\partial_{x}^{2}\big{)}^{-1}\big{(}u^{2}m_{x}+3uu_{x}m\big{)}\mathrm{d}x.$

Moreover, since

$\displaystyle\partial_{x}^{-1}(mu_{t})=\int_{-\infty}^{x}(u-u_{xx})u_{t}\mathrm{d}x=-(u_{x}u_{t}-uu_{xt})(t,x)+\int_{-\infty}^{x}u(u_{t}-u_{xxt})\mathrm{d}x$
$\displaystyle\hphantom{\partial_{x}^{-1}(mu_{t})}{}=-(u_{x}u_{t}-uu_{xt})(t,x)+\int_{-\infty}^{x}u\big{(}u^{2}m_{x}+3uu_{x}m\big{)}\mathrm{d}x$
$\displaystyle\hphantom{\partial_{x}^{-1}(mu_{t})}{}=\big{(}uu_{xt}-u_{x}u_{t}+u^{3}m\big{)}(t,x),$

using the Novikov equation (2.1),

$\displaystyle\mathcal{H}_{1}=\frac{1}{6}\int_{\mathbb{R}}um\partial_{x}^{-1}(mu_{t})\mathrm{d}x=\frac{1}{6}\int_{\mathbb{R}}um\big{(}uu_{xt}-u_{x}u_{t}+u^{3}m\big{)}\mathrm{d}x$
$\displaystyle\hphantom{\mathcal{H}_{1}}{}=\frac{1}{6}\int_{\mathbb{R}}u^{2}mu_{xt}\mathrm{d}x-\frac{1}{6}\int_{\mathbb{R}}uu_{x}mu_{t}dx+\frac{1}{6}\int_{\mathbb{R}}u^{4}m^{2}\mathrm{d}x$
$\displaystyle\hphantom{\mathcal{H}_{1}}{}=\frac{1}{6}\int_{\mathbb{R}}\big{(}{-}3uu_{x}m-u^{2}m_{x}\big{)}u_{t}\mathrm{d}x+\frac{1}{6}\int_{\mathbb{R}}u^{4}m^{2}\mathrm{d}x,$

which implies that $\mathcal{H}_{1}$ can be written in the following local
form in terms of $u$ and $m$:

$\displaystyle\mathcal{H}_{1}=\frac{1}{6}\int\big{(}u^{4}m^{2}-m_{t}u_{t}\big{)}\mathrm{d}x.$

According to Magri’s theorem [49, 57, 58], an integrable bi-Hamiltonian
equation with two compatible Hamiltonian operators $\mathcal{K}$ and
$\mathcal{J}$ belongs to an infinite hierarchy

$\displaystyle m_{t}=K_{n}=\mathcal{K}\frac{\delta\mathcal{H}_{n-1}}{\delta
m}=\mathcal{J}\frac{\delta\mathcal{H}_{n}}{\delta m},\qquad n\in\mathbb{Z}$
(2.13)

of higher-order bi-Hamiltonian systems, in both the positive and negative
directions, where $\mathcal{H}_{n}$, $n\in\mathbb{Z}$ 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

$\displaystyle K_{0}=\mathcal{J}\frac{\delta\mathcal{H}_{0}}{\delta m}=0,$

and the Hamiltonian operator $\mathcal{K}$ admits the Casimir functional

$\displaystyle\mathcal{H}_{C}=\frac{9}{2}\int
m^{\frac{2}{3}}\mathrm{d}x\qquad\hbox{with
}\qquad\frac{\delta\mathcal{H}_{C}}{\delta m}=3m^{-\frac{1}{3}}.$

Therefore, we conclude that the negative flows of the Novikov hierarchy are
generated from the Casimir equation

$\displaystyle m_{t}=K_{-1}=\mathcal{J}\frac{\delta\mathcal{H}_{-1}}{\delta
m}=\mathcal{J}\frac{\delta\mathcal{H}_{C}}{\delta
m}=3\mathcal{J}m^{-\frac{1}{3}}.$ (2.14)

The SK equation

$\displaystyle Q_{\tau}+Q_{yyyyy}+5QQ_{yyy}+5Q_{y}Q_{yy}+5Q^{2}Q_{y}=0$ (2.15)

exhibits a generalized bi-Hamiltonian system, whose corresponding integrable
hierarchy is generated by a recursion operator
$\bar{\mathcal{R}}=\bar{\mathcal{K}}\bar{\mathcal{J}}$, with

$\displaystyle\bar{\mathcal{K}}=-\big{(}\partial_{y}^{3}+2Q\partial_{y}+2\partial_{y}Q\big{)}$
(2.16)

and

$\displaystyle\bar{\mathcal{J}}=2\partial_{y}^{3}+2\partial_{y}^{2}Q\partial_{y}^{-1}+2\partial_{y}^{-1}Q\partial_{y}^{2}+Q^{2}\partial_{y}^{-1}+\partial_{y}^{-1}Q^{2}.$
(2.17)

As noted in [26], $\bar{\mathcal{K}}$ maps the variational gradients of the
conservation laws of the equation under consideration onto its symmetry
groups, while $\bar{\mathcal{J}}$ works in the opposite way.

######  Definition 2.1.

The equation $Q_{\tau}=\bar{K}[Q]$ is called a generalized bi-Hamiltonian
system if there exist an implectic (Hamiltonian) operator $\bar{\mathcal{K}}$
and a functional $\bar{\mathcal{H}}_{0}$ such that

$\displaystyle\bar{K}[Q]=\bar{\mathcal{K}}\frac{\delta\bar{\mathcal{H}}_{0}}{\delta
Q},$

as well as a symplectic operator $\bar{\mathcal{J}}$ and a corresponding
functional $\bar{\mathcal{H}}_{1}$ satisfying

$\displaystyle\bar{\mathcal{J}}\bar{K}[Q]=\frac{\delta\bar{\mathcal{H}}_{1}}{\delta
Q}.$

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 operators $\bar{\mathcal{K}}$ and $\bar{\mathcal{J}}$. These are
particular instances of the general notion of _compatible pairs of Dirac
structures_ , whose properties are developed in Dorfman [19].

Therefore, defining

$\displaystyle\bar{\mathcal{H}}_{0}=\frac{1}{6}\int\big{(}Q^{3}-3Q_{y}^{2}\big{)}\mathrm{d}y\qquad\hbox{with
}\qquad\frac{\delta\bar{\mathcal{H}}_{0}}{\delta Q}=\frac{1}{2}Q^{2}+Q_{yy},$
(2.18)

one finds that the SK equation (2.15) can be written as

$\displaystyle
Q_{\tau}=\bar{K}_{1}=\bar{\mathcal{K}}\frac{\delta\bar{\mathcal{H}}_{0}}{\delta
Q},$

and the positive flows of the SK hierarchy are generated by applying
successively the recursion operator
$\bar{\mathcal{R}}=\bar{\mathcal{K}}\bar{\mathcal{J}}$ to $\bar{K}_{1}$,
namely

$\displaystyle
Q_{\tau}=\bar{K}_{n}=\big{(}\bar{\mathcal{K}}\bar{\mathcal{J}}\big{)}^{n-1}\bar{K}_{1},\qquad
n=1,2,\ldots.$ (2.19)

On the other hand, in the negative direction, note that the trivial function
$f=0$ satisfies the equation

$\displaystyle\bar{\mathcal{J}}\cdot
f=\frac{\delta\bar{\mathcal{H}}_{0}}{\delta Q}.$

Then the $n$-th negative flow is proposed to take the form

$\displaystyle\bar{\mathcal{R}}^{n}Q_{\tau}=0,\qquad n=1,2,\ldots.$ (2.20)

Furthermore, it has been discovered in [8] that the recursion operator
$\bar{\mathcal{R}}$ satisfies the following decomposition

$\displaystyle\bar{\mathcal{R}}=\bar{\mathcal{K}}\bar{\mathcal{J}}=-\big{(}\partial_{y}^{3}+2Q\partial_{y}+2\partial_{y}Q\big{)}\big{(}2\partial_{y}^{3}+2\partial_{y}^{2}Q\partial_{y}^{-1}+2\partial_{y}^{-1}Q\partial_{y}^{2}+Q^{2}\partial_{y}^{-1}+\partial_{y}^{-1}Q^{2}\big{)}$
$\displaystyle\hphantom{\bar{\mathcal{R}}=\bar{\mathcal{K}}\bar{\mathcal{J}}}{}=-2\big{(}\partial_{y}^{4}+5Q\partial_{y}^{2}+4Q_{y}\partial_{y}+Q_{yy}+4Q^{2}+2Q_{y}\partial_{y}^{-1}Q\big{)}\big{(}\partial_{y}^{2}+Q+Q_{y}\partial_{y}^{-1}\big{)}.$
(2.21)

This factorization result demonstrates that the first negative flow (2.9) in
the SK hierarchy derived in [34] satisfies $\bar{\mathcal{R}}Q_{\tau}=0$. 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)n and (Novikov)-n,
respectively, while the $n$-th positive and negative flows of the SK hierarchy
are denoted by (SK)n and (SK)-n, respectively.

######  Theorem 2.2.

Under the Liouville transformation (2.10), for each nonzero integer
$n\in\mathbb{Z}$, the (Novikov)n equation is mapped into the (SK)-n equation,
and conversely.

The proof of this theorem relies on the following two lemmas.

######  Lemma 2.3.

Let $m(t,x)$ and $Q(\tau,y)$ be related by the transformation (2.10). Then the
following operator identities hold:

$\displaystyle
m^{-1}\big{(}1-\partial_{x}^{2}\big{)}m^{-\frac{1}{3}}=-\big{(}Q+\partial_{y}^{2}\big{)},$
(2.22) $\displaystyle
m^{-1}\mathcal{J}m^{-\frac{1}{3}}=\frac{1}{2}\partial_{y}\bar{\mathcal{J}}\partial_{y},$
(2.23) $\displaystyle
m^{-\frac{4}{3}}\big{(}4\partial_{x}-\partial_{x}^{3}\big{)}m^{-\frac{2}{3}}=\bar{\mathcal{K}}.$
(2.24)

###### Proof.

(i). In view of the transformation (2.10), one has
$\partial_{x}=m^{\frac{2}{3}}\partial_{y}$. It follows that

$\displaystyle\partial_{x}^{2}m^{-\frac{1}{3}}=m^{\frac{2}{3}}\partial_{y}m^{\frac{2}{3}}\partial_{y}=m\partial_{y}^{2}+\big{(}m^{-\frac{1}{3}}\big{)}_{xx},$

where, by (2.10), a direct computation yields

$\displaystyle\big{(}m^{-\frac{1}{3}}\big{)}_{xx}=-\frac{1}{3}\big{(}m^{-\frac{4}{3}}m_{x}\big{)}_{x}=\frac{1}{3}\left(\frac{4}{3}m^{-\frac{7}{3}}m_{x}^{2}-m^{-\frac{4}{3}}\left(\frac{4}{3}m^{-1}m_{x}^{2}-3m-3m^{\frac{7}{3}}Q\right)\right)$
$\displaystyle\hphantom{\big{(}m^{-\frac{1}{3}}\big{)}_{xx}}{}=m\big{(}m^{-\frac{4}{3}}+Q\big{)}.$

We thus arrive at

$\displaystyle
m^{-1}\big{(}1-\partial_{x}^{2}\big{)}m^{-\frac{1}{3}}=m^{-1}\big{(}m^{-\frac{1}{3}}-\big{(}m\partial_{y}^{2}+m^{-\frac{1}{3}}+mQ\big{)}\big{)}=-\big{(}Q+\partial_{y}^{2}\big{)}.$

(ii). Thanks to (2.22), we deduce that

$\displaystyle
m^{-1}\big{(}1-\partial_{x}^{2}\big{)}m^{-1}\partial_{x}m^{-1}\big{(}1-\partial_{x}^{2}\big{)}m^{-\frac{1}{3}}\partial_{y}^{-1}=\big{(}Q+\partial_{y}^{2}\big{)}\partial_{y}\big{(}Q+\partial_{y}^{2}\big{)}\partial_{y}^{-1}=\frac{1}{2}\partial_{y}\bar{\mathcal{J}},$

verifying (2.23).

(iii). Using the transformations (2.10) again, we find

$\displaystyle\partial_{x}^{2}m^{-\frac{2}{3}}=m^{\frac{2}{3}}\partial_{y}\left(\partial_{y}-\frac{2}{3}m^{-\frac{5}{3}}m_{x}\right)=m^{\frac{2}{3}}\left(\partial_{y}^{2}-\frac{2}{3}m^{-\frac{5}{3}}m_{x}\partial_{y}-\frac{2}{3}m^{-\frac{2}{3}}\big{(}m^{-\frac{5}{3}}m_{x}\big{)}_{x}\right)$
$\displaystyle\hphantom{\partial_{x}^{2}m^{-\frac{2}{3}}}{}=\frac{2}{9}m^{-\frac{8}{3}}m_{x}^{2}+2m^{-\frac{2}{3}}+2m^{\frac{2}{3}}Q-\frac{2}{3}m^{-1}m_{x}\partial_{y}+m^{\frac{2}{3}}\partial_{y}^{2}.$

Hence,

$\displaystyle
m^{-\frac{4}{3}}\big{(}4\partial_{x}-\partial_{x}^{3}\big{)}m^{-\frac{2}{3}}=m^{-\frac{2}{3}}\partial_{y}(4-\partial_{x}^{2})m^{-\frac{2}{3}}$
$\displaystyle\hphantom{m^{-\frac{4}{3}}\big{(}4\partial_{x}-\partial_{x}^{3}\big{)}m^{-\frac{2}{3}}}{}=m^{-\frac{2}{3}}\partial_{y}\left(2m^{-\frac{2}{3}}-\frac{2}{9}m^{-\frac{8}{3}}m_{x}^{2}-2m^{\frac{2}{3}}Q+\frac{2}{3}m^{-1}m_{x}\partial_{y}-m^{\frac{2}{3}}\partial_{y}^{2}\right)$
$\displaystyle\hphantom{m^{-\frac{4}{3}}\big{(}4\partial_{x}-\partial_{x}^{3}\big{)}m^{-\frac{2}{3}}}{}=2m^{-\frac{4}{3}}\big{(}m^{-\frac{2}{3}}\big{)}_{x}-\frac{2}{9}m^{-\frac{4}{3}}\big{(}m^{-\frac{8}{3}}m_{x}^{2}\big{)}_{x}-2m^{-\frac{4}{3}}\big{(}m^{\frac{2}{3}}\big{)}_{x}Q-2Q_{y}$
$\displaystyle\hphantom{m^{-\frac{4}{3}}\big{(}4\partial_{x}-\partial_{x}^{3}\big{)}m^{-\frac{2}{3}}=}{}+\left(2m^{-\frac{4}{3}}-\frac{2}{9}m^{-\frac{10}{3}}m_{x}^{2}+\frac{2}{3}m^{-\frac{4}{3}}\big{(}m^{-1}m_{x}\big{)}_{x}-2Q\right)\partial_{y}$
$\displaystyle\hphantom{m^{-\frac{4}{3}}\big{(}4\partial_{x}-\partial_{x}^{3}\big{)}m^{-\frac{2}{3}}=}{}+\left(m^{-\frac{2}{3}}\big{(}m^{\frac{2}{3}}\big{)}_{x}-\frac{2}{3}m^{-1}m_{x}\right)\partial_{y}^{2}-\partial_{y}^{3},$

then (2.24) follows. ∎

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.

######  Lemma 2.4.

Under the transformation (2.10), the relation

$\displaystyle
m^{-1}\big{(}\mathcal{J}\mathcal{K}^{-1}\big{)}^{n}m=\partial_{y}\big{(}\bar{\mathcal{J}}\bar{\mathcal{K}}\big{)}^{n}\partial_{y}^{-1}$
(2.25)

holds for each integer $n\geq 1$.

###### Proof.

We prove (2.25) by induction on $n$. First, using the inverse operator
$\mathcal{K}^{-1}$ along with (2.10), we deduce from (2.23) and (2.24) that

$\displaystyle
m^{-1}\mathcal{J}\mathcal{K}^{-1}m=2m^{-1}\mathcal{J}m^{-\frac{1}{3}}\partial_{x}^{-1}m^{-\frac{2}{3}}\big{(}4\partial_{x}-\partial_{x}^{3}\big{)}m^{-\frac{2}{3}}\partial_{x}^{-1}m^{\frac{2}{3}}=\partial_{y}\bar{\mathcal{J}}\bar{\mathcal{K}}\partial_{y}^{-1},$

which shows that (2.25) holds for $n=1$. Next, we assume that (2.25) holds for
$n=k$, say

$\displaystyle
m^{-1}\big{(}\mathcal{J}\mathcal{K}^{-1}\big{)}^{k}m=\partial_{y}\big{(}\bar{\mathcal{J}}\bar{\mathcal{K}}\big{)}^{k}\partial_{y}^{-1}.$

Then for $n=k+1$, thanks to the result when $n=1$, one has

$\displaystyle
m^{-1}\big{(}\mathcal{J}\mathcal{K}^{-1}\big{)}^{k+1}m=m^{-1}\mathcal{J}\mathcal{K}^{-1}\big{(}\mathcal{J}\mathcal{K}^{-1}\big{)}^{k}m=\partial_{y}\big{(}\bar{\mathcal{J}}\bar{\mathcal{K}}\big{)}^{k+1}\partial_{y}^{-1}.$

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 $n\geq 1$. First, since
the relation (2.10) can be rewritten as

$\displaystyle Q=-m^{-1}\big{(}1-\partial_{x}^{2}\big{)}m^{-\frac{1}{3}},$
(2.26)

we deduce from (2.22) that the first negative flow (2.14) of the Novikov
hierarchy satisfies

$\displaystyle
m_{t}=K_{-1}=-3\big{(}1-\partial_{x}^{2}\big{)}m^{-1}Q_{x}=3m\big{(}Q+\partial_{y}^{2}\big{)}Q_{y}=3m\partial_{y}\left(\frac{1}{2}Q^{2}+Q_{yy}\right).$

Hence, the $n$-th member in the negative hierarchy takes the form

$\displaystyle
m_{t}=K_{-n}=\big{(}\mathcal{J}\mathcal{K}^{-1}\big{)}^{n-1}K_{-1}=3\big{(}\mathcal{J}\mathcal{K}^{-1}\big{)}^{n-1}m\partial_{y}\left(\frac{1}{2}Q^{2}+Q_{yy}\right),\qquad
n=1,2,\ldots.\\!\\!\\!$ (2.27)

Next, suppose that $m(t,x)$ is the solution of the equation (2.27). We
calculate the $t$-derivative of the corresponding function $Q(\tau,y)$ defined
in (2.26). More precisely, we deduce that, on the one hand,

$\displaystyle
Q_{t}=Q_{\tau}+Q_{y}\int^{x}\big{(}m^{\frac{2}{3}}(t,\xi)\big{)}_{t}\mathrm{d}\xi=Q_{\tau}+\frac{2}{3}Q_{y}\partial_{x}^{-1}m^{-\frac{1}{3}}m_{t}=Q_{\tau}+\frac{2}{3}Q_{y}\partial_{y}^{-1}m^{-1}m_{t},$

and on the other hand, in view of (2.26),

$\displaystyle
Q_{t}=m^{-2}m_{t}\big{(}1-\partial_{x}^{2}\big{)}m^{-\frac{1}{3}}+\frac{1}{3}m^{-1}\big{(}1-\partial_{x}^{2}\big{)}m^{-\frac{4}{3}}m_{t}=-m^{-1}\left(Q-\frac{1}{3}\big{(}1-\partial_{x}^{2}\big{)}m^{-\frac{4}{3}}\right)m_{t}.$

Hence, combining the preceding two equations, we arrive at

$\displaystyle
Q_{\tau}=\left(-\frac{2}{3}Q_{y}\partial_{y}^{-1}m^{-1}-m^{-1}Q+\frac{1}{3}m^{-1}\big{(}1-\partial_{x}^{2}\big{)}m^{-\frac{4}{3}}\right)m_{t}=\frac{1}{3}\bar{\mathcal{K}}\partial_{y}^{-1}m^{-1}m_{t},$
(2.28)

where we have made use of the formula (2.22).

Finally, according to Lemma 2.4, we deduce that if $m(t,x)$ is the solution of
the (Novikov)-n equation (2.27), the corresponding $Q(\tau,y)$ satisfies

$\displaystyle
Q_{\tau}=\bar{\mathcal{K}}\partial_{y}^{-1}m^{-1}\big{(}\mathcal{J}\mathcal{K}^{-1}\big{)}^{n-1}m\partial_{y}\left(\frac{1}{2}Q^{2}+Q_{yy}\right)$
$\displaystyle\hphantom{Q_{\tau}}{}=\bar{\mathcal{K}}\big{(}\bar{\mathcal{J}}\bar{\mathcal{K}}\big{)}^{n-1}\left(\frac{1}{2}Q^{2}+Q_{yy}\right)=\bar{\mathcal{R}}^{n-1}\bar{K}_{1}=\bar{K}_{n}.$

This immediately implies that $Q(\tau,y)$ solves the (SK)n equation (2.19).

Conversely, if $Q(\tau,y)$ is a solution of the (SK)n equation for $n\geq 1$,
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 $n\geq 1$,
which can be written as

$\displaystyle
m_{t}=K_{n}=\big{(}\mathcal{K}\mathcal{J}^{-1}\big{)}^{n-1}\mathcal{K}\frac{\delta\mathcal{H}_{0}}{\delta
m}=9\big{(}\mathcal{K}\mathcal{J}^{-1}\big{)}^{n-1}m^{\frac{1}{3}}\partial_{x}m^{\frac{2}{3}}\big{(}4\partial_{x}-\partial_{x}^{3}\big{)}^{-1}m^{\frac{2}{3}}\partial_{x}m^{\frac{1}{3}}u.$
(2.29)

Plugging it into (2.28), we find

$\displaystyle
Q_{\tau}=3\bar{\mathcal{K}}\partial_{y}^{-1}m^{-1}\big{(}\mathcal{K}\mathcal{J}^{-1}\big{)}^{n-1}m^{\frac{1}{3}}\partial_{x}m^{\frac{2}{3}}\big{(}4\partial_{x}-\partial_{x}^{3}\big{)}^{-1}m^{\frac{2}{3}}\partial_{x}m^{\frac{1}{3}}u.$

As a consequence, the operator factorization identity (2.21), when combined
with (2.25), allows us to deduce that, for each $n\geq 1$, if $m(t,x)$ solves
the (Novikov)n equation (2.29), then for the operator $\mathcal{B}$ defined by

$\displaystyle\mathcal{B}=-2\big{(}\partial_{y}^{4}+5Q\partial_{y}^{2}+4Q_{y}\partial_{y}+Q_{yy}+4Q^{2}+2Q_{y}\partial_{y}^{-1}Q\big{)},$

the corresponding $Q(\tau,y)$ satisfies

$\displaystyle\bar{\mathcal{R}}^{n}Q_{\tau}=\mathcal{B}\big{(}\partial_{y}^{2}+Q+Q_{y}\partial_{y}^{-1}\big{)}\big{(}\bar{\mathcal{K}}\bar{\mathcal{J}}\big{)}^{n-1}Q_{\tau}$
$\displaystyle\hphantom{\bar{\mathcal{R}}^{n}Q_{\tau}}{}=3\mathcal{B}\partial_{y}\big{(}\partial_{y}^{2}\\!+Q\big{)}\partial_{y}^{{-}1}\big{(}\bar{\mathcal{K}}\bar{\mathcal{J}}\big{)}^{n{-}1}\bar{\mathcal{K}}\partial_{y}^{{-}1}m^{{-}1}\big{(}\mathcal{K}\mathcal{J}^{{-}1}\big{)}^{n{-}1}m^{\frac{1}{3}}\partial_{x}m^{\frac{2}{3}}\big{(}4\partial_{x}\\!-\partial_{x}^{3}\big{)}^{{-}1}m^{\frac{2}{3}}\partial_{x}m^{\frac{1}{3}}u$
$\displaystyle\hphantom{\bar{\mathcal{R}}^{n}Q_{\tau}}{}=3\mathcal{B}\partial_{y}\big{(}\partial_{y}^{2}+Q\big{)}\partial_{y}^{-1}\bar{\mathcal{K}}m^{\frac{2}{3}}\big{(}4\partial_{x}-\partial_{x}^{3}\big{)}^{-1}m^{\frac{2}{3}}\partial_{x}m^{\frac{1}{3}}u$
$\displaystyle\hphantom{\bar{\mathcal{R}}^{n}Q_{\tau}}{}=3\mathcal{B}\partial_{y}\big{(}\partial_{y}^{2}+Q\big{)}m^{\frac{1}{3}}u=-3\mathcal{B}\partial_{y}\cdot
1=0,$

where we have made use of the operator identity (2.22). This immediately
reveals that $Q(\tau,y)$ solves the (SK)-n equation (2.20). We thus prove
that, for each $n\geq 1$ 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. ∎

###  2.3 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 $\mathcal{H}_{n}$ satisfying the recursive
formula

$\displaystyle\mathcal{K}\frac{\delta\mathcal{H}_{n-1}}{\delta
m}=\mathcal{J}\frac{\delta\mathcal{H}_{n}}{\delta m},\qquad n\in\mathbb{Z},$
(2.30)

where $\mathcal{K}$ and $\mathcal{J}$ are the two compatible Hamiltonian
operators (2.12) admitted by the Novikov equation. On the other hand, the
recursive formula

$\displaystyle\bar{\mathcal{J}}\bar{\mathcal{K}}\frac{\delta\bar{\mathcal{H}}_{n-1}}{\delta
Q}=\frac{\delta\bar{\mathcal{H}}_{n}}{\delta Q},\qquad n\in\mathbb{Z},$ (2.31)

formally provides an infinite collection of Hamiltonian conservation laws for
the SK equation (2.15), using the operator pair $\bar{\mathcal{K}}$ and
$\bar{\mathcal{J}}$ given in (2.16) and (2.17).

In this subsection we investigate the relationship between the two hierarchies
of Hamiltonian conservation laws $\\{\mathcal{H}_{n}\\}$ and
$\\{\bar{\mathcal{H}}_{n}\\}$. Let us begin with two preliminary lemmas.

######  Lemma 2.5.

Let $\\{\mathcal{H}_{n}\\}$ and $\\{\bar{\mathcal{H}}_{n}\\}$ be the
hierarchies of Hamiltonian conservation laws of the Novikov and SK equations,
respectively. Then, for each $n\in\mathbb{Z}$, their corresponding variational
derivatives satisfy the relation

$\displaystyle\frac{\delta\bar{\mathcal{H}}_{n}}{\delta
Q}=\frac{1}{3}\partial_{x}^{-1}m^{-\frac{1}{3}}\mathcal{K}\frac{\delta\mathcal{H}_{-(n+2)}}{\delta
m}.$ (2.32)

###### Proof.

The proof relies on an induction argument. First of all, since

$\displaystyle\frac{\delta\mathcal{H}_{-2}}{\delta
m}=\mathcal{K}^{-1}\mathcal{J}\frac{\delta\mathcal{H}_{-1}}{\delta
m}=3\mathcal{K}^{-1}\big{(}1-\partial_{x}^{2}\big{)}m^{-1}\partial_{x}m^{-1}\big{(}1-\partial_{x}^{2}\big{)}m^{-\frac{1}{3}}$
$\displaystyle\hphantom{\frac{\delta\mathcal{H}_{-2}}{\delta
m}}{}=-3\mathcal{K}^{-1}\big{(}1-\partial_{x}^{2}\big{)}m^{-1}Q_{x}=3\mathcal{K}^{-1}m\big{(}Q+\partial_{y}^{2}\big{)}Q_{y}=3\mathcal{K}^{-1}m\partial_{y}\frac{\delta\bar{\mathcal{H}}_{0}}{\delta
Q},$

by (2.10), (2.22), and (2.26), and so, clearly, (2.32) holds for $n=0$.

Suppose by induction, that (2.32) holds for $n=k$ with $k\geq 0$, say

$\displaystyle\frac{\delta\bar{\mathcal{H}}_{k}}{\delta
Q}=\frac{1}{3}\partial_{x}^{-1}m^{-\frac{1}{3}}\mathcal{K}\frac{\delta\mathcal{H}_{-(k+2)}}{\delta
m}.$

Then, for $n=k+1$, by (2.30) and (2.31), we deduce that

$\displaystyle\begin{split}&\frac{\delta\bar{\mathcal{H}}_{k+1}}{\delta
Q}=\bar{\mathcal{J}}\bar{\mathcal{K}}\frac{\delta\bar{\mathcal{H}}_{k}}{\delta
Q}=\frac{1}{3}\bar{\mathcal{J}}\bar{\mathcal{K}}\partial_{x}^{-1}m^{-\frac{1}{3}}\mathcal{K}\frac{\delta\mathcal{H}_{-(k+2)}}{\delta
m}\\\ &\hphantom{\frac{\delta\bar{\mathcal{H}}_{k+1}}{\delta
Q}}{}=\frac{1}{3}\bar{\mathcal{J}}\bar{\mathcal{K}}\partial_{y}^{-1}m^{-1}\mathcal{K}\mathcal{J}^{-1}\mathcal{K}\frac{\delta\mathcal{H}_{-(k+3)}}{\delta
m}=\frac{1}{3}\partial_{x}^{-1}m^{-\frac{1}{3}}\mathcal{K}\frac{\delta\mathcal{H}_{-(k+3)}}{\delta
m},\end{split}$

where we have made use of Lemma 2.4 with $n=1$. This verifies (2.32) for
$n\geq 0$, completing the first step.

Next, in the case of $n=-1$, it follows from $\delta\mathcal{H}_{-1}/\delta
m=3m^{-1/3}$ that

$\displaystyle\frac{1}{3}\bar{\mathcal{K}}\partial_{x}^{-1}m^{-\frac{1}{3}}\mathcal{K}\frac{\delta\mathcal{H}_{-1}}{\delta
m}=\frac{1}{2}m^{-\frac{2}{3}}\partial_{x}\cdot 1=0,$

which, together with the fact that
$\bar{\mathcal{K}}\cdot(\delta\bar{\mathcal{H}}_{-1}/\delta Q)=0$ shows that
(2.32) holds for $n=-1$.

Finally, to prove (2.32) holds for all $n\leq-1$, 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

$\displaystyle\frac{\delta\mathcal{H}_{-(k-1)}}{\delta
m}=\mathcal{J}^{-1}\mathcal{K}\frac{\delta\mathcal{H}_{-k}}{\delta
m}=3\mathcal{J}^{-1}m\partial_{y}\frac{\delta\bar{\mathcal{H}}_{k-2}}{\delta
Q}=3\mathcal{J}^{-1}m\partial_{y}\bar{\mathcal{J}}\bar{\mathcal{K}}\frac{\delta\bar{\mathcal{H}}_{k-3}}{\delta
Q}=3\mathcal{K}^{-1}m^{\frac{1}{3}}\partial_{x}\frac{\delta\bar{\mathcal{H}}_{k-3}}{\delta
Q},\\!$

which establishes the induction step for $n\leq-1$ 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.

######  Lemma 2.6.

Let $m(t,x)$ and $Q(\tau,y)$ be related by the transformations (2.10). If
$\mathcal{H}(m)=\bar{\mathcal{H}}(Q)$, then

$\displaystyle\frac{\delta\mathcal{H}}{\delta
m}=\frac{1}{3}m^{-\frac{1}{3}}\partial_{y}^{-1}\bar{\mathcal{K}}\frac{\delta\bar{\mathcal{H}}}{\delta
Q},$ (2.33)

where $\bar{\mathcal{K}}$ is the Hamiltonian operator (2.16) admitted by the
SK equation (2.15).

###### Proof.

First of all, motivated by (2.10) and (2.26), we introduce

$\displaystyle
F[m(t,x)]\equiv-m^{-1}\big{(}1-\partial_{x}^{2}\big{)}m^{-\frac{1}{3}}=Q(\tau,y).$

Then the Fréchet derivative of $F[m]$ is

$\displaystyle\mathcal{D}_{F[m]}=\frac{4}{3}m^{-\frac{7}{3}}-m^{-2}\big{(}m^{-\frac{1}{3}}\big{)}_{xx}-\frac{1}{3}m^{-1}\partial_{x}^{2}m^{-\frac{4}{3}}$
$\displaystyle\hphantom{\mathcal{D}_{F[m]}}{}=-m^{-1}Q-\frac{1}{3}\big{(}Q+\partial_{y}^{2}\big{)}m^{-1}=-\frac{1}{3}\big{(}4Q+\partial_{y}^{2}\big{)}m^{-1}.$

On the other hand,

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\epsilon}\Big{|}_{\epsilon=0}F[m+\epsilon\rho]=Q_{y}\frac{\mathrm{d}}{\mathrm{d}\epsilon}\Big{|}_{\epsilon=0}y(m+\epsilon\rho)+\frac{\mathrm{d}}{\mathrm{d}\epsilon}\Big{|}_{\epsilon=0}^{y\,{\rm
f\/ixed}}F[m+\epsilon\rho],$

where, by (2.10),

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\epsilon}\Big{|}_{\epsilon=0}y(m+\epsilon\rho)=\frac{2}{3}\partial_{x}^{-1}m^{-\frac{1}{3}}\rho.$

Next, it follows from

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\epsilon}\Big{|}_{\epsilon=0}F[m+\epsilon\rho]=\mathcal{D}_{F[m]}(\rho)=-\frac{1}{3}\big{(}4Q+\partial_{y}^{2}\big{)}m^{-1}\rho$

that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\epsilon}\Big{|}_{\epsilon=0}^{y\,{\rm
f\/ixed}}F[m+\epsilon\rho]=-\frac{1}{3}\big{(}4Q+\partial_{y}^{2}\big{)}m^{-1}\rho-\frac{2}{3}Q_{y}\partial_{y}^{-1}m^{-1}\rho=\frac{1}{3}\bar{\mathcal{K}}\partial_{y}^{-1}m^{-1}\rho.$

Finally, the assumption of the lemma implies that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\epsilon}\Big{|}_{\epsilon=0}\mathcal{H}(m+\epsilon\rho)=\frac{\mathrm{d}}{\mathrm{d}\epsilon}\Big{|}_{\epsilon=0}\bar{\mathcal{H}}\left(F[m+\epsilon\rho]\right).$

According to the usual definition of the variational derivative, we have, on
the one hand,

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\epsilon}\Big{|}_{\epsilon=0}\mathcal{H}(m+\epsilon\rho)=\int\frac{\delta\mathcal{H}}{\delta
m}\cdot\rho\mathrm{d}x.$ (2.34)

On the other hand, using the fact that $\bar{\mathcal{K}}$ is skew-symmetric,
we infer that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\epsilon}\Big{|}_{\epsilon=0}\bar{\mathcal{H}}(F[m+\epsilon\rho])=\int\frac{\delta\bar{\mathcal{H}}}{\delta
Q}\cdot\frac{\mathrm{d}}{\mathrm{d}\epsilon}\Big{|}_{\epsilon=0}^{y\,{\rm
f\/ixed}}F[m+\epsilon\rho]\mathrm{d}y=\frac{1}{3}\int\frac{\delta\bar{\mathcal{H}}}{\delta
Q}\cdot\bar{\mathcal{K}}\partial_{y}^{-1}m^{-1}\rho\mathrm{d}y$
$\displaystyle\hphantom{\frac{\mathrm{d}}{\mathrm{d}\epsilon}\Big{|}_{\epsilon=0}\bar{\mathcal{H}}(F[m+\epsilon\rho])}{}=\frac{1}{3}\int
m^{\frac{2}{3}}\big{(}\bar{\mathcal{K}}\partial_{y}^{-1}m^{-1}\big{)}^{*}\frac{\delta\bar{\mathcal{H}}}{\delta
Q}\cdot\rho\mathrm{d}x=\frac{1}{3}\int
m^{-\frac{1}{3}}\partial_{y}^{-1}\bar{\mathcal{K}}\frac{\delta\bar{\mathcal{H}}}{\delta
Q}\cdot\rho\mathrm{d}x,$

which, in comparison with (2.34) verifies (2.33), proving the lemma. ∎

Finally, referring back to the form of the Hamiltonian operator $\mathcal{K}$,
one has

$\displaystyle\mathcal{K}^{-1}m^{\frac{1}{3}}\partial_{x}=2m^{-\frac{1}{3}}\partial_{x}^{-1}m^{-\frac{2}{3}}\big{(}4\partial_{x}-\partial_{x}^{3}\big{)}m^{-\frac{2}{3}}=2m^{-\frac{1}{3}}\partial_{y}^{-1}\bar{\mathcal{K}}.$

It follows that the relation (2.32) can be written in an equivalent form,
namely

$\displaystyle\frac{\delta\mathcal{H}_{n}}{\delta
m}=3\mathcal{K}^{-1}m^{\frac{1}{3}}\partial_{x}\frac{\delta\bar{\mathcal{H}}_{-(n+2)}}{\delta
Q}=6m^{-\frac{1}{3}}\partial_{y}^{-1}\bar{\mathcal{K}}\frac{\delta\bar{\mathcal{H}}_{-(n+2)}}{\delta
Q}.$ (2.35)

Therefore, subject to the hypothesis of Lemma 2.6, if we define the functional

$\displaystyle\mathcal{G}_{l}(Q)\equiv\mathcal{H}_{n}(m),$

for some $l\in\mathbb{Z}$, then Lemma 2.6 allows us to conclude that, for each
$n\in\mathbb{Z}$,

$\displaystyle\frac{\delta\mathcal{H}_{n}}{\delta
m}=\frac{1}{3}m^{-\frac{1}{3}}\partial_{y}^{-1}\bar{\mathcal{K}}\frac{\delta\mathcal{G}_{l}}{\delta
Q}.$

This, when combined with (2.35), immediately leads to

$\displaystyle\mathcal{G}_{l}(Q)=18\bar{\mathcal{H}}_{-(n+2)}(Q),$

and then

$\displaystyle\mathcal{H}_{n}(m)=18\bar{\mathcal{H}}_{-(n+2)}(Q)$

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.

######  Theorem 2.7.

Under the Liouville transformation (2.10), for each $n\in\mathbb{Z}$, the
Hamiltonian conservation law $\bar{\mathcal{H}}_{n}(Q)$ of the SK equation is
related to the Hamiltonian conservation law $\mathcal{H}_{-n}(m)$ of the
Novikov equation, according to the following identity

$\displaystyle\mathcal{H}_{n}(m)=18\bar{\mathcal{H}}_{-(n+2)}(Q),\qquad
n\in\mathbb{Z}.$

For instance, in the case of $n=2$,

$\displaystyle\frac{\delta\mathcal{H}_{-2}}{\delta
m}=\mathcal{K}^{-1}\mathcal{J}\frac{\delta\mathcal{H}_{-1}}{\delta m},$

which can be expressed in terms of $Q$ according to (2.26), say

$\displaystyle\frac{\delta\mathcal{H}_{-2}}{\delta
m}=6m^{-\frac{1}{3}}\partial_{y}^{-1}\bar{\mathcal{K}}\left(\frac{1}{2}Q^{2}+Q_{yy}\right)=-6m^{-\frac{1}{3}}\left(Q_{yyyy}+5QQ_{yy}+\frac{5}{3}Q^{3}\right).$

As a consequence,

$\displaystyle\mathcal{H}_{-2}(m)=9\int
m^{\frac{2}{3}}\left(\frac{1}{5}Q_{yyyy}+QQ_{yy}+\frac{1}{3}Q^{3}\right)\mathrm{d}x=3\int\big{(}Q^{3}-3Q_{y}^{2}\big{)}\mathrm{d}y,$

with $Q$ being determined by (2.10) and (2.26), which, when compared with
(2.18), shows that $\mathcal{H}_{-2}(m)=18\bar{\mathcal{H}}_{0}(Q)$, in
accordance with Theorem 2.7.

##  3 The correspondence between the DP and KK hierarchies

###  3.1 A Liouville transformation between the isospectral problems
of the DP and KK hierarchies

The Lax pair for the DP equation

$\displaystyle n_{t}=vn_{x}+3v_{x}n,\qquad n=v-v_{xx},$ (3.1)

takes the form [34]

$\displaystyle\mathbf{\Psi}_{x}=\begin{pmatrix}0&1&0\\\ 0&0&1\\\ -\lambda
n&1&0\end{pmatrix}\mathbf{\Psi},\qquad\mathbf{\Psi}=\begin{pmatrix}\psi_{1}\\\
\psi_{2}\\\ \psi_{3}\end{pmatrix},$ (3.2)

and

$\displaystyle\mathbf{\Psi}_{t}=\begin{pmatrix}v_{x}&-v&-\lambda^{-1}\\\
v&-\lambda^{-1}&-v\\\ \lambda
vn+v_{x}&0&-\lambda^{-1}-v_{x}\end{pmatrix}\mathbf{\Psi}.$

These can be rewritten in scalar form by setting $\Psi=\psi_{1}$, namely

$\displaystyle\Psi_{xxx}-\Psi_{x}+\lambda
n\Psi=0,\qquad\Psi_{t}+\lambda^{-1}\Psi_{xx}+v\Psi_{x}-v_{x}\Psi=0.$

Consider the KK equation

$\displaystyle P_{\tau}+P_{yyyyy}+20PP_{yyy}+50P_{y}P_{yy}+80P^{2}P_{y}=0.$
(3.3)

It has been shown in [34] that the Lax pair for the first negative flow of its
associated hierarchy is

$\displaystyle\Phi_{yyy}+4P\Phi_{y}+2P_{y}\Phi=\mu\Phi$ (3.4)

and

$\displaystyle\Phi_{\tau}+\mu^{-1}\left(U\Phi_{yy}-\frac{1}{2}U_{y}\Phi_{y}+\frac{1}{6}(U_{yy}+16PU)\Phi\right)=0,$
(3.5)

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

$\displaystyle P_{\tau}=\frac{3}{4}U_{y},\qquad\mathcal{A}U=0,$ (3.6)

where $\mathcal{A}$ is the fifth-order operator

$\displaystyle\mathcal{A}=\partial_{y}^{5}+6\big{(}\partial_{y}P\partial_{y}^{2}+\partial_{y}^{2}P\partial_{y}\big{)}+4\big{(}\partial_{y}^{3}P+P\partial_{y}^{3}\big{)}+32\big{(}\partial_{y}P^{2}+P^{2}\partial_{y}\big{)}.$

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

$\displaystyle\mathrm{d}y=n^{\frac{1}{3}}\mathrm{d}x+n^{\frac{1}{3}}v^{2}\mathrm{d}t,\qquad\mathrm{d}\tau=\mathrm{d}t,$

together with

$\displaystyle\Psi=n^{-\frac{1}{3}}\Phi,\qquad\lambda=-\mu,\qquad
P=\frac{1}{4}\left(\frac{7}{9}n^{-\frac{8}{3}}n_{x}^{2}-\frac{2}{3}n^{-\frac{5}{3}}n_{xx}-n^{-\frac{2}{3}}\right)$

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

$\displaystyle y=\int^{x}n^{\frac{1}{3}}(t,\xi)\mathrm{d}\xi,\qquad\tau=t,$
(3.7)

and

$\displaystyle
P=\frac{1}{4}\left(\frac{7}{9}n^{-\frac{8}{3}}n_{x}^{2}-\frac{2}{3}n^{-\frac{5}{3}}n_{xx}-n^{-\frac{2}{3}}\right)=\frac{1}{4}n^{-\frac{1}{2}}\big{(}4\partial_{x}^{2}-1\big{)}n^{-\frac{1}{6}}.$
(3.8)

In addition, the relationship between the Hamiltonian conservation laws for
the DP hierarchy and those for the KK hierarchy is also clarified.

###  3.2 The correspondence between the DP and KK hierarchies

The DP equation (3.1) is also a bi-Hamiltonian system [17]

$\displaystyle n_{t}=G_{1}=\mathcal{L}\frac{\delta\mathcal{E}_{0}}{\delta
n}=\mathcal{D}\frac{\delta\mathcal{E}_{1}}{\delta n},\qquad n=v-v_{xx},$

where

$\displaystyle\mathcal{L}=n^{\frac{2}{3}}\partial_{x}n^{\frac{1}{3}}\big{(}\partial_{x}-\partial_{x}^{3}\big{)}^{-1}n^{\frac{1}{3}}\partial_{x}n^{\frac{2}{3}},\qquad\mathcal{D}=\partial_{x}\big{(}1-\partial_{x}^{2}\big{)}\big{(}4-\partial_{x}^{2}\big{)}$
(3.9)

are a pair of compatible Hamiltonian operators, and the corresponding
Hamiltonian functionals are

$\displaystyle\mathcal{E}_{0}=\frac{9}{2}\int
n\mathrm{d}x,\qquad\mathcal{E}_{1}=\frac{1}{6}\int u^{3}\mathrm{d}x.$

Applying the recursion operator
$\widetilde{\mathcal{R}}=\mathcal{L}\mathcal{D}^{-1}$ successively to the
initial symmetry $n_{t}=G_{1}$ gives rise to an infinite hierarchy

$\displaystyle n_{t}=G_{l}=\mathcal{L}\frac{\delta\mathcal{E}_{l-1}}{\delta
n}=\mathcal{D}\frac{\delta\mathcal{E}_{l}}{\delta n},\qquad l\in\mathbb{Z},$

of commuting bi-Hamiltonian flows and consequent conservation laws
$\mathcal{E}_{l}$. As far as the associated negative flows are concerned,
noting that

$\displaystyle G_{0}=\mathcal{D}\frac{\delta\mathcal{E}_{0}}{\delta n}=0,$

and $\mathcal{L}$ admits the Casimir functional

$\displaystyle\mathcal{E}_{C}=18\int
n^{\frac{1}{3}}\mathrm{d}x\qquad\hbox{with}\qquad\frac{\delta\mathcal{E}_{C}}{\delta
n}=6n^{-\frac{2}{3}}.$

Therefore, we conclude that the first negative flow of the DP hierarchy is the
Casimir equation

$\displaystyle n_{t}=G_{-1}=\mathcal{D}\frac{\delta\mathcal{E}_{C}}{\delta
n}=6\mathcal{D}n^{-\frac{2}{3}}$

and applying $\widetilde{\mathcal{R}}^{-1}=\mathcal{D}\mathcal{L}^{-1}$
successively to it produces the hierarchy of negative flows, in which the
$l$-th member takes the form

$\displaystyle
n_{t}=G_{-l}=6\big{(}\mathcal{D}\mathcal{L}^{-1}\big{)}^{l-1}\mathcal{D}n^{-\frac{2}{3}},\qquad
l=1,2,\ldots.$

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 $P_{\tau}=\bar{G}_{l}[P]$, where $\bar{G}_{l}[P]$ are determined by the
relations

$\displaystyle\bar{G}_{l}[P]=\bar{\mathcal{L}}\frac{\delta\bar{\mathcal{E}}_{l-1}}{\delta
P}\qquad\hbox{and}\qquad\bar{\mathcal{D}}\bar{G}_{l}[P]=\frac{\delta\bar{\mathcal{E}}_{l}}{\delta
P},\qquad l\in\mathbb{Z},$

with

$\displaystyle\begin{split}&\bar{\mathcal{L}}=-\big{(}\partial_{y}^{3}+2P\partial_{y}+2\partial_{y}P\big{)},\\\
&\bar{\mathcal{D}}=\partial_{y}^{3}+6(P\partial_{y}+\partial_{y}P)+4\big{(}\partial_{y}^{2}P\partial_{y}^{-1}+\partial_{y}^{-1}P\partial_{y}^{2}\big{)}+32\big{(}P^{2}\partial_{y}^{-1}+\partial_{y}^{-1}P^{2}\big{)},\end{split}$
(3.10)

and $\widehat{\mathcal{R}}=\bar{\mathcal{L}}\bar{\mathcal{D}}$ is the
consequent recursion operator. It is easy to see that the KK equation (3.3) in
this hierarchy is exactly

$\displaystyle
P_{\tau}=\bar{G}_{1}[P]=\bar{\mathcal{L}}\frac{\delta\bar{\mathcal{E}}_{0}}{\delta
P}=\bar{\mathcal{L}}\big{(}P_{yy}+8P^{2}\big{)},$

with the corresponding Hamiltonian functional

$\displaystyle\bar{\mathcal{E}}_{0}=\int\left(\frac{8}{3}P^{3}-\frac{1}{2}P_{y}^{2}\right)\mathrm{d}y.$

Similarly, if we use the fact that $\bar{\mathcal{D}}\cdot
0=\delta\bar{\mathcal{E}}_{0}/\delta P$, we may conclude that the negative
flows of the KK hierarchy take the form

$\displaystyle\big{(}\bar{\mathcal{L}}\bar{\mathcal{D}}\big{)}^{l}P_{\tau}=0,\qquad
l=1,2,\ldots.$ (3.11)

It is worth noting that since
$\bar{\mathcal{D}}=\partial_{y}^{-1}\mathcal{A}\partial_{y}^{-1}$, 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 flow
$\bar{\mathcal{L}}\bar{\mathcal{D}}P_{\tau}=0$.

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.

######  Theorem 3.1.

Under the Liouville transformations (3.7) and (3.8), for each nonzero integer
$0\neq l\in\mathbb{Z}$, the (DP)l equation is mapped into the (KK)-l equation,
and conversely.

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 $P(\tau,y)$ be related by the transformations (3.7) and
(3.8). Then the following identities hold:

$\displaystyle
n^{-\frac{1}{2}}\left(\frac{1}{4}-\partial_{x}^{2}\right)n^{-\frac{1}{6}}=-\big{(}P+\partial_{y}^{2}\big{)},$
(3.12) $\displaystyle
n^{-\frac{2}{3}}\big{(}\partial_{x}-\partial_{x}^{3}\big{)}n^{-\frac{1}{3}}=\bar{\mathcal{L}},$
(3.13) $\displaystyle
n^{-1}\mathcal{D}n^{-\frac{2}{3}}=\partial_{y}\bar{\mathcal{D}}\partial_{y}.$
(3.14)

###### Proof.

(i). Define $\chi=n^{\frac{1}{3}}$, so from (3.7) and (3.8), we have
$\partial_{x}=\chi\partial_{y}$ and

$\displaystyle
P=\frac{1}{4}\chi^{-2}\chi_{y}^{2}-\frac{1}{2}\chi^{-1}\chi_{yy}-\frac{1}{4}\chi^{-2}.$
(3.15)

And then, a direct calculation shows that

$\displaystyle\partial_{x}^{2}\chi^{-\frac{1}{2}}=\chi\partial_{y}\chi\partial_{y}\chi^{-\frac{1}{2}}=\chi^{\frac{3}{2}}\partial_{y}^{2}-\frac{1}{2}\chi\big{(}\chi^{-\frac{1}{2}}\chi_{y}\big{)}_{y},$

where, by (3.15)

$\displaystyle\chi\big{(}\chi^{-\frac{1}{2}}\chi_{y}\big{)}_{y}=-\frac{1}{2}\chi^{-\frac{1}{2}}-2\chi^{\frac{3}{2}}P.$

We thus have

$\displaystyle\partial_{x}^{2}\chi^{-\frac{1}{2}}=\frac{1}{4}\chi^{-\frac{1}{2}}+\chi^{\frac{3}{2}}\big{(}P+\partial_{y}^{2}\big{)},$

which immediately leads to

$\displaystyle\left(\frac{1}{4}-\partial_{x}^{2}\right)\chi^{-\frac{1}{2}}=-\chi^{\frac{3}{2}}\big{(}P+\partial_{y}^{2}\big{)},$

and verifies (3.12).

(ii). To prove (3.13), according to (3.7) and (3.8), we only need to verify

$\displaystyle\bar{\mathcal{L}}=\chi^{-1}\partial_{y}\big{(}1-\partial_{x}^{2}\big{)}\chi^{-1}.$
(3.16)

In fact, since

$\displaystyle\big{(}1-\partial_{x}^{2}\big{)}\chi^{-1}=\chi^{-1}+\chi\big{(}\chi^{-1}\chi_{y}\big{)}_{y}+\chi_{y}\partial_{y}-\chi\partial_{y}^{2},$

one has

$\displaystyle\chi^{-1}\partial_{y}\big{(}1-\partial_{x}^{2}\big{)}\chi^{-1}=\chi^{-1}\big{(}\chi^{-1}+\chi\big{(}\chi^{-1}\chi_{y}\big{)}_{y}\big{)}_{y}+\chi^{-1}\big{(}\chi^{-1}+\chi\big{(}\chi^{-1}\chi_{y}\big{)}_{y}+\chi_{yy}\big{)}\partial_{y}-\partial_{y}^{3}.$

This, when combined with (3.15), proves (3.16).

(iii). In view of the explicit form of $\mathcal{D}$, we deduce that

$\displaystyle\chi^{-3}\mathcal{D}\chi^{-2}=\chi^{-1}\bar{\mathcal{L}}\chi\big{(}4-\partial_{x}^{2}\big{)}\chi^{-2},$

where we have made used of (3.13). In the right-hand side of the preceding
equation

$\displaystyle\chi\big{(}4-\partial_{x}^{2}\big{)}\chi^{-2}=\chi\big{(}4\chi^{-2}+2\chi\big{(}\chi^{-2}\chi_{y}\big{)}_{y}+3\chi^{-1}\chi_{y}\partial_{y}-\partial_{y}^{2}\big{)}$
$\displaystyle\hphantom{\chi\big{(}4-\partial_{x}^{2}\big{)}\chi^{-2}}{}=-16\chi
P-6\chi_{yy}+3\chi_{y}\partial_{y}-\chi\partial_{y}^{2}.$

Hence,

$\displaystyle\chi^{-1}\bar{\mathcal{L}}\chi\big{(}4-\partial_{x}^{2}\big{)}\chi^{-2}$
$\displaystyle\qquad{}=\chi^{-1}\big{(}32\chi PP_{y}+64(\chi
P)_{y}P+12\chi_{yy}P_{y}+24\chi_{yyy}P+16(\chi P)_{yyy}+6\chi_{yyyyy}\big{)}$
$\displaystyle\qquad\quad{}+\chi^{-1}\big{(}64\chi
P^{2}-6\chi_{y}P_{y}+12\chi_{yy}P+48(\chi
P)_{yy}+15\chi_{yyyy}\big{)}\partial_{y}$
$\displaystyle\qquad\quad{}+\chi^{-1}\big{(}2\chi P_{y}-8\chi_{y}P+48(\chi
P)_{y}+10\chi_{yyy}\big{)}\partial_{y}^{2}+20P\partial_{y}^{3}+\partial_{y}^{5}$
$\displaystyle\qquad{}=4(16PP_{y}+P_{yyy})+2\big{(}32P^{2}+9P_{yy}\big{)}\partial_{y}+30P_{y}\partial_{y}^{2}+20P\partial_{y}^{3}+\partial_{y}^{5},$

which implies

$\displaystyle\chi^{-3}\mathcal{D}\chi^{-2}=\partial_{y}\bar{\mathcal{D}}\partial_{y},$

and then (3.14) follows. ∎

######  Lemma 3.3.

Under the transformations (3.7) and (3.8), the relation

$\displaystyle
n^{-1}\big{(}\mathcal{D}\mathcal{L}^{-1}\big{)}^{l}n=\partial_{y}\big{(}\bar{\mathcal{D}}\bar{\mathcal{L}}\big{)}^{l}\partial_{y}^{-1}$
(3.17)

holds for each integer $l\geq 1$.

###### Proof.

Due to the form of the inverse operator $\mathcal{L}^{-1}$ and the identities
(3.13) and (3.14), we arrive at

$\displaystyle
n^{-1}\mathcal{D}\mathcal{L}^{-1}n=\partial_{y}\bar{\mathcal{D}}\partial_{y}\partial_{x}^{-1}n^{-\frac{1}{3}}\big{(}\partial_{x}-\partial_{x}^{3}\big{)}n^{-\frac{1}{3}}\partial_{x}^{-1}n^{\frac{1}{3}}=\partial_{y}\bar{\mathcal{D}}\bar{\mathcal{L}}\partial_{y}^{-1},$

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

$\displaystyle
P_{t}=P_{\tau}+P_{y}\int^{x}\big{(}n^{\frac{1}{3}}(t,\xi)\big{)}_{t}\mathrm{d}\xi=P_{\tau}+\frac{1}{3}P_{y}\partial_{x}^{-1}n^{-\frac{2}{3}}n_{t}=P_{\tau}+\frac{1}{3}P_{y}\partial_{y}^{-1}n^{-1}n_{t}.$

On the other hand, it follows from (3.8) that

$\displaystyle
P_{t}=\frac{1}{8}n^{-\frac{3}{2}}n_{t}\big{(}1-4\partial_{x}^{2}\big{)}n^{-\frac{1}{6}}+\frac{1}{24}n^{-\frac{1}{2}}\big{(}1-4\partial_{x}^{2}\big{)}n^{-\frac{7}{6}}n_{t}$
$\displaystyle\hphantom{P_{t}}{}=-\frac{1}{2}Pn^{-1}n_{t}+\frac{1}{24}n^{-\frac{1}{2}}\big{(}1-4\partial_{x}^{2}\big{)}n^{-\frac{7}{6}}n_{t}.$

From the preceding equations and using the formula (3.12), we have

$\displaystyle
P_{\tau}=\left(-\frac{1}{3}P_{y}\partial_{y}^{-1}-\frac{1}{2}P-\frac{1}{6}\big{(}P+\partial_{y}^{2}\big{)}\right)n^{-1}n_{t}=\frac{1}{6}\bar{\mathcal{L}}\partial_{y}^{-1}n^{-1}n_{t}.$
(3.18)

Now, suppose $n(t,x)$ is the solution of the (DP)-l equation

$\displaystyle
n_{t}=G_{-l}=6\big{(}\mathcal{D}\mathcal{L}^{-1}\big{)}^{l-1}\mathcal{D}n^{-\frac{2}{3}},\qquad
l=1,2,\ldots,$ (3.19)

and $n(t,x)$ is related to $P(\tau,y)$ according to (3.7) and (3.8). Then
plugging (3.19) into (3.18), one finds that the corresponding function
$P(\tau,y)$ satisfies

$\displaystyle
P_{\tau}=\bar{\mathcal{L}}\partial_{y}^{-1}n^{-1}\big{(}\mathcal{D}\mathcal{L}^{-1}\big{)}^{l-1}\mathcal{D}n^{-\frac{2}{3}}=\bar{\mathcal{L}}\big{(}\bar{\mathcal{D}}\bar{\mathcal{L}}\big{)}^{l-1}\partial_{y}^{-1}n^{-1}\mathcal{D}n^{-\frac{2}{3}}$
$\displaystyle\hphantom{P_{\tau}}{}=\big{(}\bar{\mathcal{L}}\bar{\mathcal{D}}\big{)}^{l-1}\bar{\mathcal{L}}\bar{\mathcal{D}}\cdot
0=\big{(}\bar{\mathcal{L}}\bar{\mathcal{D}}\big{)}^{l-1}\bar{\mathcal{L}}\frac{\delta\bar{\mathcal{E}}_{0}}{\delta
P}=\bar{G}_{l}.$

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 $l\geq 1$, inserting the formula

$\displaystyle
n_{t}=G_{l}=\big{(}\mathcal{L}\mathcal{D}^{-1}\big{)}^{l-1}\mathcal{L}\frac{\delta\mathcal{E}_{0}}{\delta
n}=\big{(}\mathcal{L}\mathcal{D}^{-1}\big{)}^{l-1}\mathcal{L}\cdot\frac{9}{2}$

for the (DP)l equation into (3.18) yields

$\displaystyle
P_{\tau}=\frac{3}{4}\bar{\mathcal{L}}\partial_{y}^{-1}n^{-1}\big{(}\mathcal{L}\mathcal{D}^{-1}\big{)}^{l-1}\mathcal{L}\cdot
1.$

Therefore, referring back to the form of the operator $\mathcal{D}$ and using
the identity (3.17), we deduce that

$\displaystyle\big{(}\bar{\mathcal{L}}\bar{\mathcal{D}}\big{)}^{l}P_{\tau}=\frac{3}{4}\bar{\mathcal{L}}\big{(}\bar{\mathcal{D}}\bar{\mathcal{L}}\big{)}^{l}\partial_{y}^{-1}n^{-1}\big{(}\mathcal{L}\mathcal{D}^{-1}\big{)}^{l-1}\mathcal{L}\cdot
1$
$\displaystyle\hphantom{\big{(}\bar{\mathcal{L}}\bar{\mathcal{D}}\big{)}^{l}P_{\tau}}{}=\frac{3}{4}\bar{\mathcal{L}}\partial_{y}^{-1}n^{-1}\mathcal{D}\cdot
1=-\frac{3}{4}\big{(}\partial_{y}^{2}+4P+2P_{y}\partial_{y}^{-1}\big{)}\cdot
0=0,$

which shows that $P(\tau,y)$ is a solution for the (KK)-l equation (3.11). We
thus have proved that for each $l\geq 1$, 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. ∎

###  3.3 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 $\mathcal{L}$ and $\mathcal{D}$ defined in (3.9) in hand, the
corresponding recursive formula formally defines an infinite hierarchy of
Hamiltonian conservation laws $\mathcal{E}_{l}$ determined by

$\displaystyle\mathcal{L}\frac{\delta\mathcal{E}_{l-1}}{\delta
n}=\mathcal{D}\frac{\delta\mathcal{E}_{l}}{\delta n},\qquad l\in\mathbb{Z}.$
(3.20)

For the KK equation, its Hamiltonian conservation laws $\bar{\mathcal{E}}_{l}$
can be determined by the generalized bi-Hamiltonian (bi-Dirac) structure

$\displaystyle\bar{\mathcal{D}}\bar{\mathcal{L}}\frac{\delta\bar{\mathcal{E}}_{l-1}}{\delta
P}=\frac{\delta\bar{\mathcal{E}}_{l}}{\delta P},\qquad l\in\mathbb{Z},$

with $\bar{\mathcal{L}}$ and $\bar{\mathcal{D}}$ given by (3.10).

Before proving the main theorem for the correspondence between the two
hierarchies of Hamiltonian conservation laws $\\{\mathcal{E}_{l}\\}$ and
$\\{\bar{\mathcal{E}}_{l}\\}$, two lemmas regarding their variational
derivatives are in order.

######  Lemma 3.4.

Let $\\{\mathcal{E}_{l}\\}$ and $\\{\bar{\mathcal{E}}_{l}\\}$ be the
hierarchies of Hamiltonian conservation laws of the DP and KK equations,
respectively. Then, for each $l\in\mathbb{Z}$, their corresponding variational
derivatives are related according to the following identity

$\displaystyle\frac{\delta\mathcal{E}_{l}}{\delta
n}=6\mathcal{L}^{-1}n\partial_{y}\frac{\delta\bar{\mathcal{E}}_{-(l+2)}}{\delta
P}.$ (3.21)

###### Proof.

We first consider the case of $l\leq-2$. Since

$\displaystyle\frac{\delta\mathcal{E}_{-2}}{\delta
n}=\mathcal{L}^{-1}\mathcal{D}\frac{\delta\mathcal{E}_{-1}}{\delta
n}=6\mathcal{L}^{-1}\mathcal{D}n^{-\frac{2}{3}}=6\mathcal{L}^{-1}n\partial_{y}\bar{\mathcal{D}}\cdot
0,$

then the fact $\bar{\mathcal{D}}\cdot 0=\delta\bar{\mathcal{E}}_{0}/\delta P$
reveals that (3.21) holds for $l=-2$.

We proceed by induction on $l$. Assume that (3.21) holds when $l=k$, namely

$\displaystyle\frac{\delta\mathcal{E}_{k}}{\delta
n}=6\mathcal{L}^{-1}n\partial_{y}\frac{\delta\bar{\mathcal{E}}_{-(k+2)}}{\delta
P}.$

From the recursive formula (3.20) and by the assumption,

$\displaystyle\frac{\delta\bar{\mathcal{E}}_{-(k+1)}}{\delta
P}=\bar{\mathcal{D}}\bar{\mathcal{L}}\frac{\delta\bar{\mathcal{E}}_{-(k+2)}}{\delta
P}=\frac{1}{6}\bar{\mathcal{D}}\bar{\mathcal{L}}\partial_{y}^{-1}n^{-1}\mathcal{L}\frac{\delta\mathcal{E}_{k}}{\delta
n}=\frac{1}{6}\bar{\mathcal{D}}\bar{\mathcal{L}}\partial_{y}^{-1}n^{-1}\mathcal{L}\mathcal{D}^{-1}\mathcal{L}\frac{\delta\mathcal{E}_{k-1}}{\delta
n}.$

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

$\displaystyle\frac{\delta\bar{\mathcal{E}}_{-1}}{\delta
P}=\frac{1}{6}\partial_{y}^{-1}n^{-1}\mathcal{L}\frac{\delta\mathcal{E}_{-1}}{\delta
n}.$

Note that $\bar{\mathcal{E}}_{-1}$ is a Casimir functional for Hamiltonian
operator $\bar{\mathcal{L}}$, it suffices to show that

$\displaystyle\bar{\mathcal{L}}\partial_{y}^{-1}n^{-1}\mathcal{L}\frac{\delta\mathcal{E}_{-1}}{\delta
n}=0.$

Indeed, from the definition of the operator $\mathcal{L}$ and the formula
(3.13), we have

$\displaystyle\bar{\mathcal{L}}\partial_{y}^{-1}n^{-1}\mathcal{L}n^{-\frac{2}{3}}=n^{-\frac{1}{3}}\partial_{x}\cdot
1=0,$

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

$\displaystyle\frac{\delta\mathcal{E}_{k+1}}{\delta
n}=\mathcal{D}^{-1}\mathcal{L}\frac{\delta\mathcal{E}_{k}}{\delta
n}=6\mathcal{D}^{-1}n\partial_{y}\frac{\delta\bar{\mathcal{E}}_{-(k+2)}}{\delta
P}=6\mathcal{D}^{-1}n\partial_{y}\bar{\mathcal{D}}\bar{\mathcal{L}}\frac{\delta\bar{\mathcal{E}}_{-(k+3)}}{\delta
P}$ $\displaystyle\hphantom{\frac{\delta\mathcal{E}_{k+1}}{\delta
n}}{}=6n^{-\frac{2}{3}}\partial_{x}^{-1}n^{-\frac{1}{3}}\big{(}\partial_{x}-\partial_{x}^{3}\big{)}n^{-\frac{1}{3}}\frac{\delta\bar{\mathcal{E}}_{-(k+3)}}{\delta
P}=6\mathcal{L}^{-1}n\partial_{y}\frac{\delta\bar{\mathcal{E}}_{-(k+3)}}{\delta
P},$

which completes the induction step, and thereby proves the lemma. ∎

######  Lemma 3.5.

Let $n(t,x)$ and $P(\tau,y)$ be related by the transformations (3.7) and
(3.8). If $\mathcal{E}(n)=\bar{\mathcal{E}}(P)$, then

$\displaystyle\frac{\delta\mathcal{E}}{\delta
n}=\frac{1}{6}n^{-\frac{2}{3}}\partial_{y}^{-1}\bar{\mathcal{L}}\frac{\delta\bar{\mathcal{E}}}{\delta
P},$ (3.22)

where $\bar{\mathcal{L}}$ is the Hamiltonian operator (3.10) admitted by the
KK equation (3.3).

###### Proof.

As the first step, in view of (3.8), for convenience, we introduce the
notation

$\displaystyle\tilde{F}[n(t,x)]\equiv-n^{-\frac{1}{2}}\left(\frac{1}{4}-\partial_{x}^{2}\right)n^{-\frac{1}{6}}=P(\tau,y).$

Evaluating the Fréchet derivative of $\tilde{F}[n]$ produces

$\displaystyle\mathcal{D}_{\tilde{F}[n]}=\frac{1}{6}n^{-\frac{5}{3}}-\frac{1}{2}n^{-\frac{3}{2}}\big{(}n^{-\frac{1}{6}}\big{)}_{xx}-\frac{1}{6}n^{-\frac{1}{2}}\big{(}n^{-\frac{7}{6}}\big{)}_{xx}$
$\displaystyle\hphantom{\mathcal{D}_{\tilde{F}[n]}}{}=-\frac{1}{2}n^{-1}P-\frac{1}{6}\big{(}P+\partial_{y}^{2}\big{)}n^{-1}=-\frac{1}{6}\big{(}4P+\partial_{y}^{2}\big{)}n^{-1}.$

On the other hand, we get

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\epsilon}\Big{|}_{\epsilon=0}^{y\,{\rm
f\/ixed}}\tilde{F}[n+\epsilon\rho]=\mathcal{D}_{\tilde{F}}(\rho)-\frac{1}{3}P_{y}\partial_{y}^{-1}n^{-1}\rho=\frac{1}{6}\bar{\mathcal{L}}\partial_{y}^{-1}n^{-1}\rho.$

Secondly, by the assumption, one has

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\epsilon}\Big{|}_{\epsilon=0}\mathcal{E}(n+\epsilon\rho)=\frac{\mathrm{d}}{\mathrm{d}\epsilon}\Big{|}_{\epsilon=0}\bar{\mathcal{E}}\big{(}\tilde{F}[n+\epsilon\rho]\big{)}.$

Furthermore, due to the fact that $\bar{\mathcal{L}}$ is skew-adjoint, the
righ-hand side of the above expression yields

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\epsilon}\Big{|}_{\epsilon=0}\bar{\mathcal{E}}\big{(}\tilde{F}[n+\epsilon\rho]\big{)}=\int\frac{\delta\bar{\mathcal{E}}}{\delta
P}\cdot\frac{\mathrm{d}}{\mathrm{d}\epsilon}\Big{|}_{\epsilon=0}^{y\,{\rm
f\/ixed}}\tilde{F}[n+\epsilon\rho]\mathrm{d}y$
$\displaystyle\hphantom{\frac{\mathrm{d}}{\mathrm{d}\epsilon}\Big{|}_{\epsilon=0}\bar{\mathcal{E}}\big{(}\tilde{F}[n+\epsilon\rho]\big{)}}{}=\int\frac{\delta\bar{\mathcal{E}}}{\delta
P}\cdot\left(\frac{1}{6}\bar{\mathcal{L}}\partial_{y}^{-1}n^{-1}\rho\right)\mathrm{d}y=\frac{1}{6}\int
n^{-\frac{2}{3}}\partial_{y}^{-1}\bar{\mathcal{L}}\frac{\delta\bar{\mathcal{E}}}{\delta
P}\rho\mathrm{d}x,$

which, combined with the definition of the variational derivative, produces
(3.22). ∎

Finally, it follows from (3.14) and (3.17) with $l=1$ that

$\displaystyle\mathcal{L}^{-1}n\partial_{y}=\mathcal{D}^{-1}n\partial_{y}\bar{\mathcal{D}}\bar{\mathcal{L}}=n^{-\frac{2}{3}}\partial_{y}^{-1}\bar{\mathcal{L}},$

which, together with (3.21), implies

$\displaystyle\frac{\delta\mathcal{E}_{l}}{\delta
n}=6n^{-\frac{2}{3}}\partial_{y}^{-1}\bar{\mathcal{L}}\frac{\delta\bar{\mathcal{E}}_{-(l+2)}}{\delta
P}.$ (3.23)

Now, we suppose that $n(t,x)$ and $P(\tau,y)$ are related by the
transformations (3.7) and (3.8). Define a functional

$\displaystyle\tilde{\mathcal{G}}_{k}(P)\equiv\mathcal{E}_{l}(n),$

for some $k\in\mathbb{Z}$. Then from Lemma 3.5, we conclude, for each
$k\in\mathbb{Z}$,

$\displaystyle\frac{\delta\mathcal{E}_{l}}{\delta
n}=\frac{1}{6}n^{-\frac{2}{3}}\partial_{y}^{-1}\bar{\mathcal{L}}\frac{\delta\tilde{\mathcal{G}}_{k}}{\delta
P},$

which, in comparision with (3.23) produces

$\displaystyle\mathcal{E}_{l}(n)=\tilde{\mathcal{G}}_{k}(P)=36\bar{\mathcal{E}}_{-(l+2)}(P).$

As a consequence, the following theorem is thereby proved.

######  Theorem 3.6.

Under the Liouville transformations (3.7) and (3.8), for each
$l\in\mathbb{Z}$, the Hamiltonian conservation law $\bar{\mathcal{E}}_{l}(P)$
of the KK equation is related to that $\mathcal{E}_{l}(n)$ of the DP equation,
according to the following identity

$\displaystyle\mathcal{E}_{l}(n)=36\bar{\mathcal{E}}_{-(l+2)}(P),\qquad
l\in\mathbb{Z}.$

For example, in the case of $l=-2$, it is inferred from (3.13) and (3.14), and
using the fact $\bar{\mathcal{D}}\cdot 0=\delta\bar{\mathcal{E}}_{0}/\delta
P=(P_{yy}+8P^{2})$, that

$\displaystyle\frac{\delta\mathcal{E}_{-2}}{\delta
n}=\mathcal{L}^{-1}\mathcal{D}\frac{\delta\mathcal{E}_{-1}}{\delta
n}=6n^{-\frac{2}{3}}\partial_{y}^{-1}\bar{\mathcal{L}}\bar{\mathcal{D}}\cdot
0=-6n^{-\frac{2}{3}}\left(P_{yyyy}+20PP_{yy}+15P_{y}^{2}+\frac{80}{3}P^{3}\right),$

with $P$ satisfying (3.8). Consequently, we arrive at

$\displaystyle\mathcal{E}_{-2}(n)=\frac{18}{5}\int
n^{\frac{1}{3}}\left(P_{yyyy}+20PP_{yy}+15P_{y}^{2}+\frac{80}{3}P^{3}\right)\mathrm{d}x$
$\displaystyle\hphantom{\mathcal{E}_{-2}(n)}{}=36\int\left(\frac{8}{3}P^{3}-\frac{1}{2}P_{y}^{2}\right)\mathrm{d}y=36\bar{\mathcal{E}}_{0}(P),$

which is in accordance with Theorem 3.6.

##  4 The relationship between the Novikov equation
and the DP equation

It has been shown in [23] that under the Miura transformations

$\displaystyle\mathcal{B}_{1}(Q,V)\equiv Q-V_{y}+V^{2}=0$ (4.1)

and

$\displaystyle\mathcal{B}_{2}(P,V)\equiv P+V_{y}+\frac{1}{2}V^{2}=0,$ (4.2)

the SK equation (2.15) and the KK equation (3.3) are respectively transformed
into the _Fordy–Gibbons–Jimbo–Miwa equation_

$\displaystyle
V_{\tau}+V_{yyyyy}-5\big{(}V_{y}V_{yyy}+V_{yy}^{2}+V_{y}^{3}+4VV_{y}V_{yy}+V^{2}V_{yyy}-V^{4}V_{y}\big{)}=0.$
(4.3)

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

$\displaystyle
T_{1}\equiv\mathcal{B}_{1,V}^{-1}\mathcal{B}_{1,Q}=(2V-\partial_{y})^{-1},$
(4.4)

where $\mathcal{B}_{1,V}$ and $\mathcal{B}_{1,Q}$ are the Fréchet derivatives
of (4.1) with respect to $V$ and $Q$, respectively. Then, the recursion
operator $\bar{\mathcal{R}}$ of the SK equation and the recursion operator
$\mathcal{R}^{*}$ of equation (4.3) satisfy

$\displaystyle\mathcal{R}^{*}=T_{1}\bar{\mathcal{R}}T_{1}^{-1}.$

Similarly, each member in the KK hierarchy admits the Miura transformation
(4.2), and its corresponding recursion operator $\widehat{\mathcal{R}}$ is
linked with the recursion operator $\mathcal{R}^{*}$ according to the identity

$\displaystyle\mathcal{R}^{*}=T_{2}\widehat{\mathcal{R}}T_{2}^{-1},$

where

$\displaystyle
T_{2}\equiv\mathcal{B}_{2,V}^{-1}\mathcal{B}_{2,P}=(V+\partial_{y})^{-1}$

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

$\displaystyle\mathcal{R}^{*}V_{\tau}=0,$ (4.5)

via the Miura transformations (4.1) and (4.2), respectively. Indeed, we have
the following result.

######  Proposition 4.1.

Assume that $V$ satisfies the equation (4.5). Then $Q=V_{y}-V^{2}$ and
$P=-V_{y}-\frac{1}{2}V^{2}$ satisfy the first negative flow of the SK
hierarchy $\bar{\mathcal{R}}Q_{\tau}=0$ and the first negative flow of the KK
hierarchy $\widehat{\mathcal{R}}P_{\tau}=0$, respectively.

###### Proof.

Thanks to (4.1), one has

$\displaystyle Q_{\tau}=V_{y\tau}-2VV_{\tau}=-T_{1}^{-1}V_{\tau}.$

This, together with (4.4), implies

$\displaystyle\bar{\mathcal{R}}Q_{\tau}=-\bar{\mathcal{R}}T_{1}^{-1}V_{\tau}=-T_{1}^{-1}\mathcal{R}^{*}V_{\tau}.$

Therefore, if $\mathcal{R}^{*}V_{\tau}=0$, then
$\bar{\mathcal{R}}Q_{\tau}=-(2V-\partial_{y})\mathcal{R}^{*}V_{\tau}=0$,
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.

######  Proposition 4.2.

Both the Novikov equation (2.1) and the DP equation (3.1) are linked with the
equation (4.5) in the following sense. If $V(\tau,y)$ is a solution of
equation (4.5), then the function $m(t,x)$ determined implicitly by the
relation

$\displaystyle
V_{y}-V^{2}=-m^{-1}\big{(}1-\partial_{x}^{2}\big{)}m^{-\frac{1}{3}},\qquad
y=\int^{x}m^{\frac{2}{3}}(t,\xi)\mathrm{d}\xi,\qquad\tau=t,$

satisfies the Novikov equation (2.1), while the function $n(t,x)$ determined
by

$\displaystyle
V_{y}+\frac{1}{2}V^{2}=\frac{1}{4}n^{-\frac{1}{2}}\big{(}1-4\partial_{x}^{2}\big{)}n^{-\frac{1}{6}},\qquad
y=\int^{x}n^{\frac{1}{3}}(t,\xi)\mathrm{d}\xi,\qquad\tau=t,$

satisfies the DP equation (3.1).

### Acknowledgements

The authors thank the referees for valuable comments and suggestions. Kang’s
research was supported by NSFC under Grant 11631007 and Grant 11471260. Liu’s
research was supported in part by NSFC under Grant 11631007 and Grant
11401471, and Ph.D. Programs Foundation of Ministry of Education of
China-20136101120017. Olver’s research was supported by NSF under Grant
DMS-1108894. Qu’s research was supported by NSFC under Grant 11631007 and
Grant 11471174.

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