A Revisit to the ABS H2 Equation

A Revisit to the ABS H2 Equation

Aye Aye CHO, Maebel MESFUN and Da-Jun ZHANG

A.A. Cho, M. Mesfun and D.-J. Zhang

Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Received June 25, 2021, in final form October 13, 2021; Published online
October 18, 2021

In this paper we revisit the Adler–Bobenko–Suris H2 equation. The H2 equation
is linearly related to the $S^{(0,0)}$ and $S^{(1,0)}$ variables in the Cauchy
matrix scheme. We elaborate the coupled quad-system of $S^{(0,0)}$ and
$S^{(1,0)}$ in terms of their 3-dimensional consistency, Lax pair, bilinear
form and continuum limits. It is shown that $S^{(1,0)}$ itself satisfies a
9-point lattice equation and in continuum limit $S^{(1,0)}$ is related to the
eigenfunction in the Lax pair of the Korteweg–de Vries equation.

H2 equation; consistent around cube; Cauchy matrix approach; continuum limit;
KdV equation

35Q51; 35Q55; 37K60

##  1 Introduction

In the past two decades the research of discrete integrable systems has
undergone a true development (e.g., see [7] and the references therein). The
property called multi-dimensional consistency (MDC) plays a central role in
understanding integrability of discrete systems [1, 3, 13, 16]. The MDC means
a lattice equation can be consistently embedded into a higher dimensional
system. For quadrilateral equations, this property is geometrically described
as the consistency around a cube (CAC). Adler, Bobenko and Suris (ABS)
classified all affine linear quadrilateral equations that are consistent-
around-cube (CAC), $D_{4}$ symmetric and possess the so-called tetrahedron
property [1]. In their full list there are amazingly 9 equations: namely, H1,
H2, H3δ, A1δ, A2, Q1δ, Q2, Q3δ and Q4. Most of the equations in the ABS list
are (or related to) equations known before the list was found. One of the new
equation is H2, which is also special because only for H2 and H3δ their
continuous counterpart are not found in terms of Miwa’s coordinates [18].

H2 equation reads [1]

$\displaystyle\big{(}y-\widehat{\widetilde{y}}\big{)}\big{(}\widetilde{y}-\widehat{y}\big{)}+\big{(}p^{2}-q^{2}\big{)}\big{(}y+\widetilde{y}+\widehat{y}+\widehat{\widetilde{y}}-p^{2}-q^{2}\big{)}=0,$
(1.1)

where $p$, $q$ are spacing parameters in $n$\- and $m$-direction,
respectively, and tilde-hat notations stand for shifts in $n$\- and
$m$-direction (see equation (2.1) for definition). In this paper, we will
revisit H2 equation with the help of H1 (i.e., the lattice potential
Korteweg–de Vries equation (lpKdV) [15])

$\displaystyle\big{(}z-\widehat{\widetilde{z}}\big{)}\big{(}\widehat{z}-\widetilde{z}\big{)}=p^{2}-q^{2}.$
(1.2)

Note that H1 has a background solution $z_{0}=pn+qm$. By removing the
background from the equation, i.e., introducing $z=z_{0}-u$ so that $u$ has a
zero background, the resulting equation [15]

$\displaystyle\big{(}p+q+u-\widehat{\widetilde{u}}\big{)}\big{(}p-q+\widehat{u}-\widetilde{u}\big{)}=p^{2}-q^{2}$
(1.3)

is ready to yield the potential KdV equation in its continuum limit. H2 has
also a background solution $y_{0}=z_{0}^{2}$ [8] but it is hard to remove it
for H2 equation. This is why so far the continuous counterpart of H2 is
unknown.

The main contents of the paper are the following. We will start from the
following relations obtained from Cauchy matrix approach (see equation (2.6)),

$\displaystyle w+\widetilde{w}=p\widetilde{u}-pu+u\widetilde{u},$ (1.4a)
$\displaystyle w+\widehat{w}=q\widehat{u}-qu+u\widehat{u},$ (1.4b)

which provide a non-auto Bäcklund transformation (BT) between H1 and H2 with

$\displaystyle y=z_{0}^{2}-2z_{0}u+2w,$ (1.5a) $\displaystyle z=z_{0}-u.$
(1.5b)

From (1.4) we will derive a quadrilateral coupled system (in Section 3.1)

$\displaystyle\big{(}p+q+u-\widehat{\widetilde{u}}\big{)}\big{(}\widetilde{w}-\widehat{w}\big{)}-(p-q)\big{(}w-\widehat{\widetilde{w}}\big{)}=0,$
(1.6a)
$\displaystyle\big{(}p-q+\widehat{u}-\widetilde{u}\big{)}\big{(}w-\widehat{\widetilde{w}}\big{)}-(p+q)\big{(}\widetilde{w}-\widehat{w}\big{)}=0,$
(1.6b)

which is CAC and therefore is integrable. As a consequence we will have a one-
component 9-point equation of $w$,

$\displaystyle(p+q)\left(\frac{\widetilde{w}-\widehat{w}}{w-\widehat{\widetilde{w}}}+\frac{\widehat{\widehat{\widetilde{w}}}-\widehat{\widetilde{\widetilde{w}}}}{\widehat{\widetilde{w}}-\widehat{\widehat{\widetilde{\widetilde{w}}}}}\right)+(p-q)\left(\frac{\widehat{w}-\widehat{\widehat{\widetilde{w}}}}{\widehat{\widehat{w}}-\widehat{\widetilde{w}}}+\frac{\widetilde{w}-\widehat{\widetilde{\widetilde{w}}}}{\widetilde{\widetilde{w}}-\widehat{\widetilde{w}}}\right)=0.$
(1.7)

Considering the relation (1.5) we call (1.6) or (1.7) an alternative of H2. In
this paper we will focus on the coupled system (1.6) to investigate its
integrability, bilinear form, continuum limits, etc. It turns out that the
continuous counterpart of (1.6) consists of the potential KdV equation and the
time-part in the Lax pair of the (potential) KdV equation. Note that with some
transformations the system (1.6) will be equivalent to a
$\mathrm{H}1\times\mathrm{H}2$ two-component system which is first derived in
[11] (see equation (3.9)).

The paper is arranged as follows. In Section 2 we recall the Cauchy matrix
approach and derive the BT (1.4). Then we derive the coupled quad-system (1.6)
and investigate its 3D consistency and Lax pairs in Section 3, and present its
bilinear form and Casoratian solutions in Section 4. After that, Section 5
consists of the investigation of the continuous counterpart of the quad-system
(1.6), including their continuum limits and Cauchy matrix approach in
continuous case, which leads to a connection between $S^{(1,0)}$ equation and
the KdV equation. Finally, concluding remarks are given in Section 6. There is
an appendix section to explore higher order equations in the continuum limits.

##  2 Cauchy matrix scheme

###  2.1 Preliminary

For a function $U$ of $(n,m)\in\mathbb{Z}^{2}$, we introduce shorthand to
express it and its shifts, such as

$\displaystyle U\doteq U(n,m),\quad\ \widetilde{U}\doteq U(n+1,m),\quad\
\widehat{U}\doteq U(n,m+1),\quad\ \widehat{\widetilde{U}}\doteq U(n+1,m+1).$
(2.1)

Spacing parameters $p$ and $q$ serve as spacing parameters of $n$-direction
and $m$-direction, respectively. These settings are depicted as in Figure
1($a$). In 3-dimensional case, we introduce the third direction $l$ together
with its spacing parameter $r$, and denote the shift by $\overline{U}\doteq
U(n,m,l+1)$. We also introduce shift operators $E_{n}$, $E_{m}$ and $E_{l}$
such that $E_{n}U=\widetilde{U}$, $E^{2}_{m}U=\widehat{\widehat{U}}$, etc.

($a$)$\widehat{U}$$\widehat{\widetilde{U}}$$q$$U$$\widetilde{U}$$p$(275,2708)(4075,2708)
(3075,3633)(3075,0) (275,908)(4075,908) (1275,3633)(1275,0)

($b$)$\widetilde{\overline{U}}$$\overline{U}$$\widehat{\overline{U}}$$\widehat{\widetilde{U}}$$U$$r$$q$$p$$\widehat{\widetilde{\overline{U}}}$$\widetilde{U}$$\widehat{U}$(1275,2708)(3075,2708)
(1275,2708)(450,1883) (450,1883)(450,83) (3075,2708)(2250,1883)
(450,1883)(2250,1883) (3075,2633)(3075,908) 60.000(1275,908)(450,83)
60.000(1275,908)(3075,908) (2250,1883)(2250,83) (450,83)(2250,83)
(3075,908)(2250,83) 60.000(1275,2633)(1275,908)

Figure 1: ($a$): The points on which the equation is defined and ($b$): the
consistency cube.

A quadrilateral equation

$Q\big{(}U,\widetilde{U},\widehat{U},\widehat{\widetilde{U}};p,q\big{)}=0$

is CAC means the equation can be consistently embedded on six faces of the
cube in Figure 1($b$). In other words, given initial values $U$,
$\widetilde{U}$, $\widehat{U}$, $\overline{U}$, the value of
$\widehat{\widetilde{\overline{U}}}$ is unique although it will be determined
by the top-, front- and right-side, three equations.

###  2.2 Cauchy matrix approach to (1.4)

To derive the coupled system (1.4), we consider the Sylvester equation

$\displaystyle\boldsymbol{K}\boldsymbol{M}+\boldsymbol{M}\boldsymbol{K}=\boldsymbol{r}{\boldsymbol{c}}^{\rm
T},$ (2.2)

where $\boldsymbol{M},\boldsymbol{K}\in\mathbb{C}_{N\times N}$,
$\boldsymbol{r}$ and $\boldsymbol{c}$ are column vectors in $\mathbb{C}_{N}$
defined as

$\boldsymbol{r}=(r_{1},r_{2},\dots,r_{N})^{\rm
T},\qquad\boldsymbol{c}=(c_{1},c_{2},\dots,c_{N})^{\rm T}.$

Let us sketch some known results related to this equation.

######  Proposition 2.1.

For given $\boldsymbol{r}$, $\boldsymbol{c}$ and $\boldsymbol{K}$, if
$\boldsymbol{K}$ and $-\boldsymbol{K}$ do not share eigenvalues, then
$\boldsymbol{M}$ will be uniquely determined by the equation (2.2) [17]. In
addition, explicit formulae of $\boldsymbol{M}$ have been constructed in [22]
with respect to canonical forms of $\boldsymbol{K}$.

######  Proposition 2.2 ([22]).

Introduce

$\displaystyle S^{(i,j)}=\boldsymbol{c}^{\rm
T}{\boldsymbol{K}}^{j}(\boldsymbol{I}+\boldsymbol{M})^{-1}\boldsymbol{K}^{i}\boldsymbol{r},\qquad
i,j\in\mathbb{Z},$ (2.3)

where the elements $\boldsymbol{r}$, $\boldsymbol{c}$, $\boldsymbol{K}$ and
$\boldsymbol{M}$ satisfy the Sylvester equation (2.2) and $\boldsymbol{I}$ is
the $N$-th order unit matrix. For given $\boldsymbol{r}$ and $\boldsymbol{c}$,
the matrix $\boldsymbol{K}$ and any matrix similar to it give rise to the same
$S^{(i,j)}$, and the symmetric property

$\displaystyle S^{(i,j)}=S^{(j,i)}$ (2.4)

holds.

The Cauchy matrix approach is a method to construct and study integrable
equations by means of the Sylvester-type equations. In this approach
integrable equations are presented as closed forms of $S^{(i,j)}$ and its
shifts (or derivatives). It is first systematically used in [14] to
investigate ABS equations and later developed in [20, 22] to more general
cases.

To derive coupled system (1.4), we impose dispersion relation on
$\boldsymbol{r}$ as

$\displaystyle(p\boldsymbol{I}-\boldsymbol{K})\widetilde{\boldsymbol{r}}=(p\boldsymbol{I}+\boldsymbol{K})\boldsymbol{r},\qquad(q\boldsymbol{I}-\boldsymbol{K})\widehat{\boldsymbol{r}}=(q\boldsymbol{I}+\boldsymbol{K})\boldsymbol{r},$

and let $\boldsymbol{c}$ be a constant column vector, where $\boldsymbol{K}$
is a constant $N\times N$ matrix. Next, making use of the results obtained in
[22], we have the following shift relations for $\boldsymbol{M}$,

$\displaystyle\widetilde{\boldsymbol{M}}(p\boldsymbol{I}+\boldsymbol{K})-(p\boldsymbol{I}+\boldsymbol{K})\boldsymbol{M}=\widetilde{\boldsymbol{r}}\boldsymbol{c}^{\rm
T},$
$\displaystyle(p\boldsymbol{I}-\boldsymbol{K})\widetilde{\boldsymbol{M}}-\boldsymbol{M}(p\boldsymbol{I}-\boldsymbol{K})=\boldsymbol{r}\boldsymbol{c}^{\rm
T},$
$\displaystyle\widehat{\boldsymbol{M}}(q\boldsymbol{I}+\boldsymbol{K})-(q\boldsymbol{I}+\boldsymbol{K})\boldsymbol{M}=\widehat{\boldsymbol{r}}\boldsymbol{c}^{\rm
T},$
$\displaystyle(q\boldsymbol{I}-\boldsymbol{K})\widehat{\boldsymbol{M}}-\boldsymbol{M}(q\boldsymbol{I}-\boldsymbol{K})=\boldsymbol{r}\boldsymbol{c}^{\rm
T},$

from which one arrives at (see equations (27) in [22])

$\displaystyle
p\widetilde{S}^{(i,j)}-\widetilde{S}^{(i,j+1)}=pS^{(i,j)}+S^{(i+1,j)}-\widetilde{S}^{(i,0)}S^{(0,j)},$
$\displaystyle
q\widehat{S}^{(i,j)}-\widehat{S}^{(i,j+1)}=qS^{(i,j)}+S^{(i+1,j)}-\widehat{S}^{(i,0)}S^{(0,j)}.$

Now, taking $(i,j)=(0,0)$ in the above recurrence relation, defining

$\displaystyle u=S^{(0,0)},\qquad w=S^{(1,0)},$ (2.5)

and making use of the symmetric property (2.4), we have

$\displaystyle w+\widetilde{w}=p\widetilde{u}-pu+u\widetilde{u},$ (2.6a)
$\displaystyle w+\widehat{w}=q\widehat{u}-qu+u\widehat{u},$ (2.6b)

i.e., the coupled system (1.4).

###  2.3 Consistent triplet

Suppose that we have a coupled system

$\displaystyle f\big{(}u,v,\widetilde{u},\widetilde{v},p\big{)}=0,\qquad
f\big{(}u,v,\widehat{u},\widehat{v},q\big{)}=0.$ (2.7)

The compatibility with respect to $v$ (i.e.,
$\widehat{\widetilde{v}}=\widetilde{\widehat{v}}$) gives rise to a
quadrilateral equation of $u$,

$\displaystyle
Q\big{(}u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}};p,q\big{)}=0,$
(2.8)

while the compatibility for $u$ yields a quadrilateral equation of $v$,

$\displaystyle
G\big{(}v,\widetilde{v},\widehat{v},\widehat{\widetilde{v}};p,q\big{)}=0.$
(2.9)

When this happens we say that equations (2.7), (2.8) and (2.9) compose a
consistent triplet. Note that such a triplet can be extended to the lattice
systems defined on larger stencils, not necessary to be restricted to the
quadrilateral case. In such a triplet, equation (2.7) acts as a BT to connect
the other two equations (2.8) and (2.9), and due to the compatibility, any
pair of solutions $(u,v)$ of (2.7) will provide solutions to (2.8) and (2.9).
Consistent triplets have played a useful role in constructing rational
solutions for a number of quadrilateral equations [21].

Using the transformation (1.5), the coupled system (2.6) is written as

$\displaystyle y+\widetilde{y}-p^{2}=2z\widetilde{z},$ (2.10a) $\displaystyle
y+\widehat{y}-q^{2}=2z\widehat{z}.$ (2.10b)

Note that (1.5) has been implied from equation (5.26) in [14]. By checking
compatibility of $z$ and $y$, one immediately find that $y$ and $z$ have to
satisfy H2 (1.1) and H1 (1.2), respectively. Thus we have the following.

######  Proposition 2.3.

H2 equation (1.1), H1 (1.2) and the system (2.10) compose a consistent
triplet.

This means that solving (2.10) yields solutions to H1 and H2. Based on this
fact in Section 4 we will bilinearize (2.10) and get a bilinear form for H2
which is simpler than the one given in [8]. Note that the fact of (2.10) being
a BT for H1 and H2 was also found in [2] using the CAC property of H2 and the
degeneration relation from H2 to H1.

##  3 Coupled system (1.6)

The coupled system (2.6) is a set of 3-point equations. Initial values can be
given on any line $n+m=l$, where $l$ can be arbitrary integers. However, the
system can only evolve towards down-left from the initial-value line. Initial
values can also be given on a vertical line or on a horizontal line, but
neither of them can yield a bidirectional evolution. Compared with (2.6), the
quad-system (1.6) is not only CAC, but also allows bidirectional evolution
starting from initial values given on some staircase. Next, we derive (1.6)
from (2.6) and investigate its integrability.

###  3.1 Derivation

First, in light of the compatibility $E_{n}E_{m}w=E_{m}E_{n}w$, from (2.6) we
immediately get H1 (lpKdV) equation (1.3). Next, eliminating $w$ from (2.6) by
$\eqref{BT-1b}$–$\eqref{BT-1a}$ and by
$\widehat{\eqref{BT-1a}}$–$\widetilde{\eqref{BT-1b}}$, respectively, we get

$\displaystyle\widehat{w}-\widetilde{w}+p\widetilde{u}-q\widehat{u}=\big{(}p-q+\widehat{u}-\widetilde{u}\big{)}u$
(3.1)

and

$\displaystyle\widehat{w}-\widetilde{w}+p\widehat{u}-q\widetilde{u}=\big{(}p-q+\widehat{u}-\widetilde{u}\big{)}\widehat{\widetilde{u}}.$
(3.2)

Meanwhile, subtracting (2.6b) from $\widehat{\eqref{BT-1a}}$ and subtracting
(2.6a) from $\widetilde{\eqref{BT-1b}}$, respectively, yield

$\displaystyle
w-\widehat{\widetilde{w}}+p\widehat{\widetilde{u}}+qu=\big{(}p+q+u-\widehat{\widetilde{u}}\big{)}\widehat{u}$
(3.3)

and

$\displaystyle
w-\widehat{\widetilde{w}}+q\widehat{\widetilde{u}}+pu=\big{(}p+q+u-\widehat{\widetilde{u}}\big{)}\widetilde{u}.$
(3.4)

Then, by multiplying $\widehat{\widetilde{u}}$ to (3.1) and $u$ to (3.2) and
eliminating their right-hand sides, we arrive at

$\displaystyle\big{(}\widehat{w}-\widetilde{w}\big{)}\big{(}u-\widehat{\widetilde{u}}\big{)}=\big{(}p\widetilde{u}-q\widehat{u}\big{)}\widehat{\widetilde{u}}+\big{(}q\widetilde{u}-p\widehat{u}\big{)}u,$

and similarly, from (3.3) and (3.4) we have

$\displaystyle\big{(}\widehat{u}-\widetilde{u}\big{)}\big{(}w-\widehat{\widetilde{w}}\big{)}=\big{(}p\widetilde{u}-q\widehat{u}\big{)}\widehat{\widetilde{u}}+\big{(}q\widetilde{u}-p\widehat{u}\big{)}u.$

It then directly follows that

$\displaystyle\big{(}w-\widehat{\widetilde{w}}\big{)}\big{(}\widetilde{u}-\widehat{u}\big{)}=\big{(}\widetilde{w}-\widehat{w}\big{)}\big{(}u-\widehat{\widetilde{u}}\big{)}.$
(3.5)

Finally, combining the above equation and the lpKdV equation (1.3),
eliminating the term $\widetilde{u}-\widehat{u}$ yields (1.6a) and eliminating
the term ${u}-\widehat{\widetilde{u}}$ gives rise to (1.6b).

Now we have obtained the coupled system (1.6), together with H1 (1.3) and a
relation (3.5). The above derivation indicates the following more precise
connection of them.

######  Proposition 3.1.

Among the following four equations,

$\displaystyle\big{(}p+q+u-\widehat{\widetilde{u}}\big{)}\big{(}\widetilde{w}-\widehat{w}\big{)}-(p-q)\big{(}w-\widehat{\widetilde{w}}\big{)}=0,$
(3.6a)
$\displaystyle\big{(}p-q+\widehat{u}-\widetilde{u}\big{)}\big{(}w-\widehat{\widetilde{w}}\big{)}-(p+q)\big{(}\widetilde{w}-\widehat{w}\big{)}=0,$
(3.6b)
$\displaystyle\big{(}w-\widehat{\widetilde{w}}\big{)}\big{(}\widetilde{u}-\widehat{u}\big{)}=\big{(}\widetilde{w}-\widehat{w}\big{)}\big{(}u-\widehat{\widetilde{u}}\big{)},$
(3.6c)
$\displaystyle\big{(}p+q+u-\widehat{\widetilde{u}}\big{)}\big{(}p-q+\widehat{u}-\widetilde{u}\big{)}=p^{2}-q^{2},$
(3.6d)

any two of them can be derived from the other two.

###### Proof.

Let us sketch the proof. First, eliminating $w$ from (3.6a) and (3.6b) yields
the lpKdV equation (3.6d). Rewriting (3.6d) as

$\displaystyle
p+q+u-\widehat{\widetilde{u}}=-(p-q)\frac{u-\widehat{\widetilde{u}}}{\widehat{u}-\widetilde{u}}$
(3.7)

and then substituting its left-hand-side into (3.6a) give rise to (3.6c).
Thus, (3.6c) and (3.6d) are derived if (3.6a) and (3.6b) hold.

Next, assume (3.6a) and (3.6c) are known. Eliminating
$\widetilde{w}-\widehat{w}$ from them yields (3.7), which is (3.6d). Note that
(3.6d) can also be written as

$\displaystyle
u-\widehat{\widetilde{u}}=\frac{-(p+q)\big{(}\widehat{u}-\widetilde{u}\big{)}}{p-q+\widehat{u}-\widetilde{u}}.$
(3.8)

Substituting its left-hand-side into (3.6c) we get (3.6b).

If we assume (3.6a) and (3.6d) are known, we may eliminate
$p+q+u-\widehat{\widetilde{u}}$ from them and get equation (3.6b). Then, as we
described before, (3.6c) can be obtained when (3.6a) and (3.6b) hold.

If we start from (3.6c) and (3.6d), we can first rewrite (3.6d) as

$\displaystyle\widehat{u}-\widetilde{u}=\frac{-(p-q)\big{(}u-\widehat{\widetilde{u}}\big{)}}{p+q+u-\widehat{\widetilde{u}}},$

and then eliminate $\widehat{u}-\widetilde{u}$ using (3.6c). This gives rise
to (3.6a), while (3.6b) is a consequence of eliminating
$u-\widehat{\widetilde{u}}$ from (3.8) and (3.6c).

Other cases can be proved similarly. ∎

In addition, eliminating $u$ from (1.6) yields the nine-point single-component
equation (1.7). Thus we have the following result parallel to Proposition 2.3.

######  Proposition 3.2.

H1 equation (1.3), the nine-point equation (1.7) and the system (1.4) compose
a consistent triplet.

Besides, we also note that starting from the coupled system (1.6) and making
use of (1.5) and (2.10), one can get another coupled system in terms of $z$
and $y$,

$\displaystyle\big{(}\widetilde{y}-\widehat{y}\big{)}\big{(}\widehat{\widetilde{z}}-z\big{)}+\big{(}p^{2}-q^{2}\big{)}\big{(}\widehat{\widetilde{z}}+z\big{)}=0,$
(3.9a)
$\displaystyle\big{(}y-\widehat{\widetilde{y}}\big{)}\big{(}\widetilde{z}-\widehat{z}\big{)}+\big{(}p^{2}-q^{2}\big{)}\big{(}\widetilde{z}+\widehat{z}\big{)}=0.$
(3.9b)

In fact, from (1.5) we have

$u=z_{0}-z,\qquad w=\frac{1}{2}\big{(}y+z_{0}^{2}-2z_{0}z\big{)}.$

Substituting these into (1.6a), and then successfully using (2.10a) to
eliminate $y$, using $\widetilde{\eqref{BT-2b}}$ to eliminate
$\widehat{\widetilde{y}}$ and using (1.2) to eliminate $\widehat{z}$, finally
we arrive at (3.9a). Equation (3.9b) can be derived in a similar way. This
coupled system was first derived in [11], known as the
$\mathrm{H}1\times\mathrm{H}2$ 2-component vertex equation. Using (1.5) and
(2.10), the coupled system (1.6) can be recovered from (3.9) as well.

###  3.2 Integrability: CAC and Lax pair

In this part, let us focus on the integrability of the coupled system (1.6).
In light of Proposition 3.1, in practice, we consider the equivalent equations
(3.6b) and (3.6d), i.e.,

$\displaystyle\big{(}p-q+\widehat{u}-\widetilde{u}\big{)}\big{(}w-\widehat{\widetilde{w}}\big{)}-(p+q)\big{(}\widetilde{w}-\widehat{w}\big{)}=0,$
(3.10a)
$\displaystyle\big{(}p+q+u-\widehat{\widetilde{u}}\big{)}\big{(}p-q+\widehat{u}-\widetilde{u}\big{)}=p^{2}-q^{2}.$
(3.10b)

Let $U=(u,w)^{\rm T}$ and put the above system on the six faces of the cube
(see Figure 1($b$)). Given initial values $U$, $\widetilde{U}$, $\widehat{U}$,
$\overline{U}$, we have

$\displaystyle\widehat{\widetilde{u}}=u-\frac{(p+q)\big{(}\widetilde{u}-\widehat{u}\big{)}}{p-q+\widehat{u}-\widetilde{u}},$
(3.11a)
$\displaystyle\widetilde{\overline{u}}=u-\frac{(p+r)\big{(}\widetilde{u}-\overline{u}\big{)}}{p-r+\overline{u}-\widetilde{u}},$
(3.11b)
$\displaystyle\widehat{\overline{u}}=u-\frac{(r+q)\big{(}\overline{u}-\widehat{u}\big{)}}{r-q+\widehat{u}-\overline{u}},$
(3.11c)

and

$\displaystyle\widehat{\widetilde{w}}=w-\frac{(p+q)\big{(}\widetilde{w}-\widehat{w}\big{)}}{p-q+\widehat{u}-\widetilde{u}},$
(3.12a)
$\displaystyle\widetilde{\overline{w}}=w-\frac{(p+r)\big{(}\widetilde{w}-\overline{w}\big{)}}{p-r+\overline{u}-\widetilde{u}},$
(3.12b)
$\displaystyle\widehat{\overline{w}}=w-\frac{(r+q)\big{(}\overline{w}-\widehat{w}\big{)}}{r-q+\widehat{u}-\overline{u}}.$
(3.12c)

Then, with these in hand, we find that the value of
$\widehat{\widetilde{\overline{U}}}$ is same no matter we calculate it from
the equation on the top, or the front, or the right face, where

$\widehat{\widetilde{\overline{u}}}=\frac{B}{A},\qquad\widehat{\widetilde{\overline{w}}}=\frac{C}{A},$

with

$\displaystyle A={}$
$\displaystyle\big{(}p^{2}-q^{2}\big{)}\overline{u}+\big{(}r^{2}-p^{2}\big{)}\widehat{u}+\big{(}q^{2}-r^{2}\big{)}\widetilde{u}+(q-r)(p-q)(p-r),$
$\displaystyle B={}$
$\displaystyle(p-q)(p+r)(q+r)\overline{u}+(p+q)(q+r)(r-p)\widehat{u}+(p+q)(p+r)(q-r)\widetilde{u}$
$\displaystyle{}+\big{(}r^{2}-q^{2}\big{)}\overline{u}\widehat{u}+\big{(}p^{2}-r^{2}\big{)}\overline{u}\widetilde{u}+\big{(}q^{2}-p^{2}\big{)}\widetilde{u}\widehat{u},$
$\displaystyle C={}$
$\displaystyle(p+r)(q+r)(p-q+\widehat{u}-\widetilde{u})\overline{w}+(p+q)(q+r)(r-p+\widetilde{u}-\overline{u})\widehat{w}$
$\displaystyle{}+(p+q)(p+r)(q-r+\overline{u}-\widehat{u})\widetilde{w}.$

Thus, the system (3.10) (or (1.6)) is CAC. Note that the lpKdV equation
(3.10b) itself is CAC as well. This also indicates that the 9-point equation
(1.7) can be embedded into 3-dimensional space and is consistent around a
large cube consisting of 4 elementary cubes.

To obtain a Lax pair, following the standard procedure (cf. [1, 4, 13]), we
introduce

$\overline{u}=\frac{g}{f},\qquad\overline{w}=\frac{h}{f},$

and rewrite (3.11b), (3.11c), (3.12b) and (3.12c), which leads us to the
matrix system

$\displaystyle\widetilde{\Phi}=\mathcal{N}\Phi,\qquad\widehat{\Phi}=\mathcal{M}\Phi,$
(3.13)

where $\Phi=(f,g,h)^{\rm T}$,

$\displaystyle\mathcal{N}=\begin{pmatrix}p-s-\widetilde{u}&1&0\\\
(p-s-\widetilde{u})u-(p+s)\widetilde{u}&p+s+u&0\\\
(p-s-\widetilde{u})w-(p+s)\widetilde{w}&w&p+s\end{pmatrix}$

and

$\displaystyle\mathcal{M}=\begin{pmatrix}q-s-\widehat{u}&1&0\\\
(q-s-\widehat{u})u-(q+s)\widehat{u}&q+s+u&0\\\
(q-s-\widehat{u})w-(q+s)\widehat{w}&w&q+s\end{pmatrix}\\!.$

Compatibility of (3.13), i.e.,
$\widehat{\widetilde{\Phi}}=\widetilde{\widehat{\Phi}}$ yields the four
equations in (3.6). In light of Proposition 3.1, (3.13) provides a Lax pair
for any two equations of (3.6), including (1.6).

##  4 Bilinear formulation

Now that both the coupled system (1.6) and H2 equation (1.1) are the
consequence of the coupled equation (1.4), we can get a bilinear form for both
(1.6) and H2 (1.1) from (1.4). To achieve that, we introduce transformation

$\displaystyle u=\frac{g}{f},\qquad w=\frac{\theta}{2f},$ (4.1)

from which the coupled system (1.4) is bilinearized as

$\displaystyle
2p\big{(}\widetilde{f}g-f\widetilde{g}\big{)}-2g\widetilde{g}+\widetilde{\theta}f+\theta\widetilde{f}=0,$
(4.2a) $\displaystyle
2q\big{(}\widehat{f}g-f\widehat{g}\big{)}-2g\widehat{g}+\widehat{\theta}f+\theta\widehat{f}=0.$
(4.2b)

In the following we present Casoratian solutions to the above bilinear
equations. We introduce [8]

$\displaystyle\psi_{i}(n,m,l)=\varrho^{+}_{i}k_{i}^{l}(p+k_{i})^{n}(q+k_{i})^{m}+\varrho^{-}_{i}(-k_{i})^{l}(p-k_{i})^{n}(q-k_{i})^{m},$
(4.3)

where $\varrho^{\pm}_{i}$ and $k_{i}$ are constants, and define a vector

$\displaystyle\psi(n,m,l)=\big{(}\psi_{1}(n,m,l),\psi_{2}(n,m,l),\dots,\psi_{N}(n,m,l)\big{)}^{\rm
T}.$

A Casoratian composed by $\psi$ w.r.t shift in $l$ is a determinant defined as
(we skip $n$, $m$ notations for convenience)

$\displaystyle|\psi(l),\psi(l+1),\psi(l+2),\dots,\psi(l+N-1)|=|0,1,2,\dots,N-1|=\big{|}\widehat{N-1}\big{|}.$

In a similar way one can introduce
$|0,1,2,\dots,N-2,N|=\big{|}\widehat{N-2},N\big{|}$, etc. Note that the
shorthand $\big{|}\widehat{N-1}\big{|}$ is a conventional notation in
expression of Casoratian or Wronskian [6] and we do not think it makes any
confusion with the hat-shift for $m$-direction in context. A useful identity
in verifying solution in Caosratian/Wronskian form is [6]

$\displaystyle|\boldsymbol{K},\boldsymbol{a},\boldsymbol{b}||\boldsymbol{K},\boldsymbol{c},\boldsymbol{d}|-|\boldsymbol{K},\boldsymbol{a},\boldsymbol{c}||\boldsymbol{K},\boldsymbol{b},\boldsymbol{d}|+|\boldsymbol{K},\boldsymbol{a},\boldsymbol{d}||\boldsymbol{K},\boldsymbol{b},\boldsymbol{c}|=0,$
(4.4)

where $\boldsymbol{K}$ is a $N\times(N-2)$ matrix, and $\boldsymbol{a}$,
$\boldsymbol{b}$, $\boldsymbol{c}$ and $\boldsymbol{d}$ are $N$th-order column
vectors.

Now let us present solutions to the bilinear equation (4.2).

######  Theorem 4.1.

The equations (4.2) provide a bilinear form for H2 equation (1.1) via the
transformation (1.5a) together with (4.1). Equations (4.2) have the following
solution

$\displaystyle f=\big{|}\widehat{N-1}\big{|},\qquad
g=\big{|}\widehat{N-2},N\big{|},$ (4.5a)
$\displaystyle\theta=\big{|}\widehat{N-3},N-1,N\big{|}+\big{|}\widehat{N-2},N+1\big{|},$
(4.5b)

composed by the vector $\psi$ with entries (4.3), in terms of shifts in $l$.

###### Proof.

Let us first prove (4.2b). Introduce

$\displaystyle h=\big{|}\widehat{N-2},N+1\big{|},\qquad
s=\big{|}\widehat{N-3},N-1,N\big{|}$ (4.6)

and rewrite (4.2b) as

$\displaystyle A+B=0,$

where

$\displaystyle A=\widehat{f}(h+qg)-\widehat{g}(g+qf)+f\widehat{s},\qquad
B=f\big{(}\widehat{h}-q\widehat{g}\big{)}-g\big{(}\widehat{g}-q\widehat{f}\big{)}+\widehat{f}s.$

Next, we prove both $A$ and $B$ are zero when $f$, $g$, $h$, $s$ are
Casoratians defined above. Note that with $\psi$ given by (4.3) as elementary
entries, shifts of $f$, $g$, $h$ satisfy (see formulae given in Appendix in
[8])

$\displaystyle
q^{N-2}\underset{\text{$\widehat{\phantom{\cdot}}$}}{f}=-\big{|}\widehat{N-2},\underset{\text{$\widehat{\phantom{\cdot}}$}}{\psi}(N-2)\big{|},$
$\displaystyle
q^{N-2}\big{(}\underset{\text{$\widehat{\phantom{\cdot}}$}}{h}+q\underset{\text{$\widehat{\phantom{\cdot}}$}}{g}\big{)}=-\big{|}\widehat{N-3},N,\underset{\text{$\widehat{\phantom{\cdot}}$}}{\psi}(N-2)\big{|},$
$\displaystyle
q^{N-2}\big{(}\underset{\text{$\widehat{\phantom{\cdot}}$}}{g}+q\underset{\text{$\widehat{\phantom{\cdot}}$}}{f}\big{)}=-\big{|}\widehat{N-3},N-1,\underset{\text{$\widehat{\phantom{\cdot}}$}}{\psi}(N-2)\big{|}.$

Then we have

$\displaystyle q^{N-2}\underset{\text{$\widehat{\phantom{\cdot}}$}}{A}={}$
$\displaystyle
q^{N-2}\big{[}f\big{(}\underset{\text{$\widehat{\phantom{\cdot}}$}}{h}+q\underset{\text{$\widehat{\phantom{\cdot}}$}}{g}\big{)}-g\big{(}\underset{\text{$\widehat{\phantom{\cdot}}$}}{g}+q\underset{\text{$\widehat{\phantom{\cdot}}$}}{f}\big{)}+\underset{\text{$\widehat{\phantom{\cdot}}$}}{f}s\big{]}$
$\displaystyle={}$
$\displaystyle-\big{|}\widehat{N-1}\big{|}\big{|}\widehat{N-3},N,\underset{\text{$\widehat{\phantom{\cdot}}$}}{\psi}(N-2)\big{|}+\big{|}\widehat{N-2},N\big{|}\big{|}\widehat{N-3},N-1,\underset{\text{$\widehat{\phantom{\cdot}}$}}{\psi}(N-2)\big{|}$
$\displaystyle-\big{|}\widehat{N-2},\underset{\text{$\widehat{\phantom{\cdot}}$}}{\psi}(N-2)\big{|}\big{|}\widehat{N-3},N-1,N\big{|},$

which vanishes in light of the identity (4.4), where
$\boldsymbol{K}=\big{(}\widehat{N-3}\big{)}$, $\boldsymbol{a}=\psi(N-2)$,
$\boldsymbol{b}=\psi(N-1)$, $\boldsymbol{c}=\psi(N)$ and
$\boldsymbol{d}=\underset{\text{$\widehat{\phantom{\cdot}}$}}{\psi}(N-2)|$.
$B=0$ is the equation (4.21e) in [8], which is already proved. Thus, equation
(4.2b) holds. Equation (4.2a) can be proved in a similar way. ∎

The proof of the this theorem indicates that the bilinear equations (4.2) can
further be decoupled.

######  Corollary 4.2.

The bilinear form (4.2) can be decoupled into

$\displaystyle\widetilde{f}(h+pg)-\widetilde{g}(g+pf)+f\widehat{s}=0,$ (4.7a)
$\displaystyle
f\big{(}\widetilde{h}-p\widetilde{g}\big{)}-g\big{(}\widetilde{g}-p\widetilde{f}\big{)}+\widetilde{f}s=0,$
(4.7b) $\displaystyle\widehat{f}(h+qg)-\widehat{g}(g+qf)+f\widehat{s}=0,$
(4.7c) $\displaystyle
f\big{(}\widehat{h}-q\widehat{g}\big{)}-g\big{(}\widehat{g}-q\widehat{f}\big{)}+\widehat{f}s=0,$
(4.7d)

which admit Casoratian solutions given by (4.5a) and (4.6) with entries (4.3).
This provides an alternative bilinear form for the coupled system (1.4),
therefore both (4.2) and (4.7) can serve as bilinear forms for H2 equation
(1.1) and the coupled system (1.6) or (3.10).

In addition, since $\varrho^{\pm}_{i}$ in (4.3) are arbitrary parameters
independent of $(n,m,l)$, we introduce a dummy variable $x$ and replace
$\varrho^{\pm}_{i}$ by ${\rm e}^{\pm k_{i}x}\varrho^{\pm}_{i}$, and then we
have

$\displaystyle\psi_{i}(x,l)=\varrho^{+}_{i}{\rm
e}^{k_{i}x}k_{i}^{l}(p+k_{i})^{n}(q+k_{i})^{m}+\varrho^{-}_{i}{\rm
e}^{-k_{i}x}(-k_{i})^{l}(p-k_{i})^{n}(q-k_{i})^{m},$ (4.8)

with which we redefine $\psi(x,l)=(\psi_{1},\psi_{2},\dots,\psi_{N})^{\rm T}$.
Obviously,

$\displaystyle\psi(x,l+1)=\partial_{x}\psi(x,l).$

Thus we have

$f=\big{|}\widehat{N-1}\big{|}=\big{|}\psi(x,l),\partial_{x}\psi(x,l),\partial_{x}^{2}\psi(x,l),\dots,\partial_{x}^{N-1}\psi(x,l)\big{|}=\big{|}\widehat{N-1}\big{|}_{[x]},$

where the subscript $[x]$ means the determinant is a Wronskian composed by
$\psi(x,l)$ and its derivatives w.r.t. $x$. Besides, for the Casoratians $g$
and $\theta$ defined in (4.5), we have

$g=f_{x}=\big{|}\widehat{N-2},N\big{|}_{[x]},\qquad\theta=f_{xx}=\big{|}\widehat{N-3},N-1,N\big{|}_{[x]}+\big{|}\widehat{N-2},N+1\big{|}_{[x]}.$

Thus we reach the following.

######  Corollary 4.3.

In terms of Wronskians composed by $\psi(x,l)$ with entries (4.8), the
bilinear system (4.2) is rewritten as

$\displaystyle\big{(}D^{2}_{x}+2pD_{x}\big{)}f\cdot\widetilde{f}=0,$ (4.9a)
$\displaystyle\big{(}D^{2}_{x}+2qD_{x}\big{)}f\cdot\widehat{f}=0,$ (4.9b)

where $D$ is the Hirota bilinear operator defined as [10]

$D_{x}^{i}D_{t}^{j}f(x,t)\cdot
g(x,t)=(\partial_{x}-\partial_{x^{\prime}})^{i}(\partial_{t}-\partial_{t^{\prime}})^{j}f(x,t)g(x^{\prime},t^{\prime})|_{x^{\prime}=x,t^{\prime}=t}.$

##  5 Continuous counterparts

###  5.1 Continuum limits

Let us start to investigate continuum limits of the coupled system (3.10) via
a two-step scheme, which is based on the deformation of the H2 plane wave
factor [8]:

$\displaystyle\rho(n,m)=\biggl{(}\frac{p-k}{p+k}\biggr{)}^{n}\biggl{(}\frac{q-k}{q+k}\biggr{)}^{m}$
$\displaystyle\longrightarrow{\rm
e}^{-2k\xi+o(\frac{1}{p})}\biggl{(}\frac{q-k}{q+k}\biggr{)}^{m}$
$\displaystyle\longrightarrow{\rm
e}^{-2k(\xi+\eta)-\frac{2k^{3}}{3q^{2}}\eta+o\left(\frac{1}{q^{3}}\right)},$

where

$\xi=\frac{n}{p},\qquad\eta=\frac{m}{q}$

which are finite when $n\to\infty$, $p\to\infty$ and $m\to\infty$,
$q\to\infty$.

In the first step, after Taylor expanding the coupled system (3.10) into power
series of $\frac{1}{p}$, the leading term in each equation w.r.t.
$\frac{1}{p}$ gives rise to

$\displaystyle\partial_{\xi}\big{(}w+\widehat{w}\big{)}+\big{(}w-\widehat{w}\big{)}\big{(}u-\widehat{u}+2q\big{)}=0,$
(5.1a)
$\displaystyle\partial_{\xi}\big{(}u+\widehat{u}\big{)}+\big{(}q+u-\widehat{u}\big{)}^{2}-q^{2}=0,$
(5.1b)

where for convenience we still employ $u$ and $w$ to denote functions of
$(\xi,m)$ in this step. Note that equation (5.1a) yields a form
$u-\widehat{u}=A(w)$, and substituting it into equation (5.1b) yields a form
$\partial_{\xi}(u+\widehat{u})=B(w)$, where $A(w)$ and $B(w)$ are some
functions of $w$, $\widehat{w}$ and their derivatives w.r.t. $\xi$. Then,
eliminating $u$ from them we get a differential-difference equation for $w$,
which is

$\displaystyle\frac{1}{\big{(}w-\widehat{w}\big{)}^{2}}\Big{[}2q\big{(}w-\widehat{w}\big{)}\partial_{\xi}\big{(}w+\widehat{w}\big{)}+\partial_{\xi}\big{(}w^{2}-\widehat{w}^{2}\big{)}+\big{(}\partial_{\xi}\big{(}w+\widehat{w}\big{)}\big{)}^{2}-\big{(}w-\widehat{w}\big{)}\partial_{\xi\xi}\big{(}w+\widehat{w}\big{)}\Big{]}$
$\displaystyle\qquad{}+\frac{1}{\big{(}\widehat{w}-\widehat{\widehat{w}}\big{)}^{2}}\Big{[}{-}2q\big{(}\widehat{w}-\widehat{\widehat{w}}\big{)}\partial_{\xi}\big{(}\widehat{w}+\widehat{\widehat{w}}\big{)}+\partial_{\xi}\big{(}\widehat{w}^{2}-\widehat{\widehat{w}}{}^{2}\big{)}-\big{(}\partial_{\xi}\big{(}\widehat{w}+\widehat{\widehat{w}}\big{)}\big{)}^{2}$
$\displaystyle\qquad\hphantom{+\frac{1}{\big{(}\widehat{w}-\widehat{\widehat{w}}\big{)}^{2}}\Big{[}}{}-\big{(}\widehat{w}-\widehat{\widehat{w}}\big{)}\partial_{\xi\xi}\big{(}\widehat{w}+\widehat{\widehat{w}}\big{)}\Big{]}=0.$
(5.2)

In the second step, first, expanding (5.1) in terms of $\frac{1}{q}$, then,
introducing

$\displaystyle x=\xi+\eta,\qquad t=\frac{\eta}{12q^{2}},$

and reorganizing the expansion w.r.t. $(x,t)$ and the derivatives w.r.t. them,
finally, from the leading terms we get

$\displaystyle w_{t}-6w_{x}u_{x}-w_{xxx}=0,$ (5.3a) $\displaystyle
u_{t}-6u_{x}^{2}-u_{xxx}=0,$ (5.3b)

which contains the potential KdV equation (5.3b) and its companion (5.3a).
Again, one may eliminate $u$ from the system and obtain an equation of $w$:

$\displaystyle
w_{t}[6w_{2x}^{3}-10w_{x}w_{2x}w_{3x}+3w_{x}^{2}w_{4x}]+2w_{t}^{2}w_{x}w_{2x}-3w_{xt}w_{x}[w_{x}w_{t}+2w_{2x}^{2}-2w_{x}w_{3x}]$
$\displaystyle\qquad{}+3w_{x}^{2}w_{2x}w_{2xt}-2w_{x}^{3}w_{3xt}+w_{2t}w_{x}^{3}+w_{2x}w_{3x}[8w_{x}w_{3x}-6w_{2x}^{2}]$
$\displaystyle\qquad{}+6w_{x}w_{4x}[w_{2x}^{2}-w_{x}w_{3x}]-3w_{x}^{2}w_{2x}w_{5x}+w_{x}^{3}w_{6x}=0,$
(5.4)

where the shorthand $w_{kx}$ means $\partial^{k}_{x}w$.

Later we will have a closer look at this system in Section 5.3. We also note
that higher order equations in the continuum limit of the coupled quad-system
(3.10) will be explored in Appendix.

###  5.2 Cauchy matrix approach to (5.3)

A continuous version of the Cauchy matrix approach to the KdV equation has
been developed in [20]. In the following we derive the coupled system (5.3)
using this approach.

####  5.2.1 Derivation

We will start from the Sylvester equation (2.2) and the function $S^{(i,j)}$
defined as in (2.3), while in this case $\boldsymbol{M}$, $\boldsymbol{r}$ and
$\boldsymbol{c}^{\rm T}$ depends on $(x,t)$. An auxiliary vector
$\boldsymbol{u}^{(i)}$ is introduced as

$\displaystyle\boldsymbol{u}^{(i)}=(\boldsymbol{I}+\boldsymbol{M})^{-1}\boldsymbol{K}^{i}\boldsymbol{r},\qquad
i\in\mathbb{Z}.$

Let us first recall some useful relations obtained in [20]. Proposition 2.2
holds as well in the continuous case.

######  Proposition 5.1.

  1. $1.$

$S^{(i,j)}$ satisfies the recursive relation for $i,j\in\mathbb{Z}$,

$\displaystyle
S^{(i,j+2k)}=S^{(i+2k,j)}-\sum_{l=0}^{2k-1}(-1)^{l}S^{(2k-1-l,j)}S^{(i,l)},\qquad
k=1,2,\dots,$

and in particular, when $k=1$ we have

$\displaystyle S^{(i,j+2)}=S^{(i+2,j)}-S^{(i,0)}S^{(1,j)}+S^{(i,1)}S^{(0,j)}.$
(5.5)

  2. $2.$

$\boldsymbol{K}$ and any matrix similar to it lead to same $S^{(i,j)}$.

  3. $3.$

The symmetric property holds:

$\displaystyle S^{(i,j)}=S^{(j,i)}.$

In addition, a relation for $\boldsymbol{u}^{(i)}$ is [20]

$\displaystyle(\boldsymbol{I}+\boldsymbol{M})\boldsymbol{K}^{2k}\boldsymbol{u}^{(i)}=\boldsymbol{K}^{2k+i}\boldsymbol{r}-\sum_{l=0}^{2k-1}(-1)^{l}\boldsymbol{K}^{2k-1-l}\boldsymbol{r}\boldsymbol{c}^{\rm
T}\boldsymbol{K}^{l}\boldsymbol{u}^{(i)}.$

Multiplied $(\boldsymbol{I}+\boldsymbol{M})^{-1}$ from the left, one first has

$\boldsymbol{K}^{2k}\boldsymbol{u}^{(i)}=(\boldsymbol{I}+\boldsymbol{M})^{-1}\boldsymbol{K}^{2k+i}\boldsymbol{r}-\sum_{l=0}^{2k-1}(-1)^{l}\bigl{(}(\boldsymbol{I}+\boldsymbol{M})^{-1}\boldsymbol{K}^{2k-1-l}\boldsymbol{r}\bigr{)}\boldsymbol{c}^{\rm
T}\boldsymbol{K}^{l}\boldsymbol{u}^{(i)},$

and then, in light of the definitions of $\boldsymbol{u}^{(i)}$ and
$S^{(i,l)}$, it follows that

$\displaystyle\boldsymbol{u}^{(2k+i)}=\boldsymbol{K}^{2k}\boldsymbol{u}^{(i)}+\sum_{l=0}^{2k-1}(-1)^{l}\boldsymbol{u}^{(2k-1-l)}S^{(i,l)}.$
(5.6)

Introducing dispersion relation (cf. [20])

$\displaystyle\boldsymbol{r}_{x}=\boldsymbol{K}\boldsymbol{r},\qquad$
$\displaystyle\boldsymbol{c}_{x}=\boldsymbol{K}^{\rm T}\boldsymbol{c},$
$\displaystyle\boldsymbol{r}_{t}=4\boldsymbol{K}^{3}\boldsymbol{r},\qquad$
$\displaystyle\boldsymbol{c}_{t}=4\big{(}\boldsymbol{K}^{\rm
T}\big{)}^{3}\boldsymbol{c}.$

one has (cf. [20])

$\displaystyle\boldsymbol{u}_{x}^{(i)}=\boldsymbol{u}^{(i+1)}-S^{(i,0)}\boldsymbol{u}^{(0)},$
(5.7a)
$\displaystyle\boldsymbol{u}_{t}^{(i)}=4\big{(}\boldsymbol{u}^{(i+3)}-S^{(i,0)}\boldsymbol{u}^{(2)}-S^{(i,2)}\boldsymbol{u}^{(0)}+S^{(i,1)}\boldsymbol{u}^{(1)}\big{)},$
(5.7b) $\displaystyle
S_{x}^{(i,j)}=S^{(i+1,j)}+S^{(i,j+1)}-S^{(i,0)}S^{(0,j)},$ $\displaystyle
S_{t}^{(i,j)}=4\big{(}S^{(i+3,j)}+S^{(i,j+3)}+S^{(i,1)}S^{(1,j)}-S^{(i,0)}S^{(2,j)}-S^{(i,2)}S^{(0,j)}\big{)},$

and

$\displaystyle
S_{xx}^{(i,j)}=S^{(i+2,j)}+S^{(i,j+2)}-2S^{(i+1,0)}S^{(0,j)}-2S^{(i,0)}S^{(0,j+1)}+2S^{(i+1,j+1)}$
$\displaystyle\hphantom{S_{xx}^{(i,j)}=}{}-S^{(i,0)}S^{(1,j)}-S^{(i,1)}S^{(0,j)}+2S^{(i,0)}S^{(0,0)}S^{(0,j)},$
$\displaystyle
S_{xxx}^{(i,j)}=S^{(i+3,j)}+S^{(i,j+3)}+3S^{(i+2,j+1)}+3S^{(i+1,j+2)}-3S^{(i+2,0)}S^{(0,j)}$
$\displaystyle\hphantom{S_{xxx}^{(i,j)}=}{}-3S^{(i,0)}S^{(0,j+2)}-6S^{(i+1,0)}S^{(0,j+1)}-3S^{(i+1,0)}S^{(1,j)}-3S^{(i,1)}S^{(0,j+1)}$
$\displaystyle\hphantom{S_{xxx}^{(i,j)}=}{}-S^{(i,2)}S^{(0,j)}-S^{(i,0)}S^{(2,j)}-3S^{(i+1,1)}S^{(0,j)}-3S^{(i,0)}S^{(1,j+1)}$
$\displaystyle\hphantom{S_{xxx}^{(i,j)}=}{}+6S^{(i+1,0)}S^{(0,0)}S^{(0,j)}+6S^{(i,0)}S^{(0,0)}S^{(0,j+1)}-2S^{(i,1)}S^{(1,j)}+3S^{(i,0)}S^{(0,0)}S^{(1,j)}$
$\displaystyle\hphantom{S_{xxx}^{(i,j)}=}{}+6S^{(i,0)}S^{(1,0)}S^{(0,j)}+3S^{(i,1)}S^{(0,0)}S^{(0,j)}-6S^{(i,0)}S^{(0,0)^{2}}S^{(0,j)}.$

From the above general formulae, for $u$ and $w$ defined as in (2.5), one has

$\displaystyle u_{x}=2S^{(1,0)}-u^{2},$ (5.8a) $\displaystyle
u_{t}=4\big{(}2S^{(3,0)}-2uS^{(2,0)}+S^{(1,0)^{2}}\big{)},$ (5.8b)
$\displaystyle
u_{xxx}=2S^{(3,0)}+6S^{(2,1)}-8uS^{(2,0)}-14S^{(1,0)^{2}}-6uS^{(1,1)}+24u^{2}S^{(1,0)}-6u^{4},$
(5.8c)

and

$\displaystyle w_{x}=S^{(2,0)}+S^{(1,1)}-uw,$ (5.9a) $\displaystyle
w_{t}=4\big{(}S^{(4,0)}+S^{(1,3)}+wS^{(1,1)}-wS^{(2,0)}-uS^{(1,2)}\big{)},$
(5.9b) $\displaystyle
w_{xx}=S^{(3,0)}+3S^{(1,2)}-2uS^{(2,0)}-uS^{(1,1)}-3w^{2}+2u^{2}w,$ (5.9c)
$\displaystyle
w_{xxx}=S^{(4,0)}+4S^{(1,3)}+3S^{(2,2)}-3uS^{(3,0)}-13wS^{(2,0)}-4uS^{(2,1)}-8wS^{(1,1)}$
$\displaystyle\hphantom{w_{xxx}=}{}+15uw^{2}+6u^{2}S^{(2,0)}+3u^{2}S^{(1,1)}-6u^{3}w.$
(5.9d)

The function $u$ obeys the potential KdV equation (cf. [20])

$\displaystyle u_{t}-6(u_{x})^{2}-u_{xxx}=0,$ (5.10)

while $w$ satisfies

$\displaystyle
w_{t}-6u_{x}w_{x}-w_{xxx}=3\big{(}{-}S^{(2,2)}+S^{(4,0)}+uS^{(3,0)}+u^{2}S^{(1,1)}-wS^{(2,0)}-uw^{2}\big{)}.$

The right hand side vanishes by using the identity (5.5) for $(i,j)=(2,0)$ and
$(1,0)$. Thus, we obtain equation (5.3a), i.e.,

$\displaystyle w_{t}-6u_{x}w_{x}-w_{xxx}=0.$ (5.11)

Note that equation (5.8a) indicates a Miura-type transformation

$\displaystyle 2w=u_{x}+u^{2},$ (5.12)

which connects the potential KdV equations (5.10) and (5.11). For this we have
the following.

######  Proposition 5.2.

In light of relation (5.12), the two equations in the system (5.3) are
connected as

$\displaystyle
w_{t}-6u_{x}w_{x}-w_{xxx}=\frac{1}{2}(\partial_{x}+2u)\big{[}u_{t}-6(u_{x})^{2}-u_{xxx}\big{]}.$
(5.13)

####  5.2.2 Lax Pair

A Lax pair for the coupled system (5.3) can be constructed as well from the
Cauchy matrix approach (cf. [7]). Let us consider the relation (5.7) with
$i=0$ and $i=1$, i.e.,

$\displaystyle\boldsymbol{u}_{x}^{(0)}=\boldsymbol{u}^{(1)}-u\boldsymbol{u}^{(0)},$
(5.14a)
$\displaystyle\boldsymbol{u}_{x}^{(1)}=\boldsymbol{u}^{(2)}-w\boldsymbol{u}^{(0)},$
(5.14b)

and

$\displaystyle\boldsymbol{u}_{t}^{(0)}=4\big{(}\boldsymbol{u}^{(3)}-S^{(0,0)}\boldsymbol{u}^{(2)}-S^{(0,2)}\boldsymbol{u}^{(0)}+S^{(0,1)}\boldsymbol{u}^{(1)}\big{)},$
(5.15a)
$\displaystyle\boldsymbol{u}_{t}^{(1)}=4\big{(}\boldsymbol{u}^{(4)}-S^{(1,0)}\boldsymbol{u}^{(2)}-S^{(1,2)}\boldsymbol{u}^{(0)}+S^{(1,1)}\boldsymbol{u}^{(1)}\big{)}.$
(5.15b)

Then we do the following. Denote the first entries of $\boldsymbol{u}^{(0)}$
and $\boldsymbol{u}^{(1)}$ by $\phi_{1}$ and $\phi_{2}$, respectively. Since
in the complex field matrix $\boldsymbol{K}$ is always similar to a lower
triangular matrix, in light of Proposition 5.1, without loss of generality, we
assume in the following $\boldsymbol{K}$ is lower triangular and denote the
entry $K_{1,1}$ in $\boldsymbol{K}$ by $\lambda$. Make use of the recursion
relation (5.6) to express $\boldsymbol{u}^{(2)}$, $\boldsymbol{u}^{(3)}$ and
$\boldsymbol{u}^{(4)}$ in terms of $\boldsymbol{u}^{(0)}$ and
$\boldsymbol{u}^{(1)}$. Then, for $\Phi=(\phi_{1},\phi_{2})^{\rm T}$, from
(5.14) and (5.15), and after the above treatment we arrive at

$\displaystyle\Phi_{x}=L_{1}\Phi,\qquad\Phi_{t}=L_{2}\Phi,$

where

$\displaystyle L_{1}=\begin{pmatrix}-u&1\\\ \lambda^{2}-2w&u\end{pmatrix}\\!,$
$\displaystyle
L_{2}=4\begin{pmatrix}-\lambda^{2}u-w_{x}&\lambda^{2}+2w-u^{2}\\\
\lambda^{4}-2\lambda^{2}w-\frac{1}{2}w_{xx}-uw_{x}&\lambda^{2}u+w_{x}\end{pmatrix}\\!,$

and we have made use of (5.9a) and (5.9c) to express $S^{(1,1)}$, $S^{(0,2)}$,
$S^{(1,2)}$ and $S^{(0,3)}$ in terms of $u$, $w$ and their derivatives.

The compatibility $\Phi_{xt}=\Phi_{tx}$ gives rise to

$\displaystyle L_{1_{t}}-L_{2_{x}}+[L_{1},L_{2}]=0,$

i.e., in explicit form, gives the following equations

$\displaystyle
4\lambda^{2}\big{(}u^{2}+u_{x}-2w\big{)}-u_{t}-8w\big{(}u^{2}-2w\big{)}-4uw_{x}+2w_{xx}=0,$
$\displaystyle u^{2}-2w+u_{x}=0,$ $\displaystyle
w_{t}-6u_{x}w_{x}-w_{xxx}+4w_{x}\big{(}u^{2}+u_{x}-2w\big{)}=0.$

This leads to three equations

$\displaystyle 2w=u^{2}+u_{x},$ (5.16a) $\displaystyle
u_{t}-6(u_{x})^{2}-u_{xxx}=0,$ (5.16b) $\displaystyle
w_{t}-6u_{x}w_{x}-w_{xxx}=0,$ (5.16c)

where (5.16a) is nothing but the Miura transformation (5.8a), i.e., (5.12),
equation (5.16c) is a consequence of (5.16b) and (5.16a) in light of
Proposition 5.2.

###  5.3 Connection to the KdV

Equation (5.3a) is the continuous counterpart of equation (1.6a). To reveal
more links between this equation and the KdV equation, let us recall some
known results about the KdV equation. The potential KdV equation (5.3b) has a
Lax pair111Note that equation (5.3b) is a potential version of the usual KdV
equation $v_{t}=6vv_{x}+v_{xxx}$ with $v=2u_{x}$.

$\displaystyle\phi_{xx}+2u_{x}\phi=\lambda\phi,$ (5.17a)
$\displaystyle\phi_{t}=\phi_{xxx}+3(\lambda+2u_{x})\phi_{x}.$ (5.17b)

Through the transformation

$\displaystyle u=(\ln f)_{x}$

the potential KdV equation (5.3b) is bilinearized as [9]

$\displaystyle\big{(}D_{x}D_{t}-D_{x}^{4}\big{)}f\cdot f=0,$ (5.18)

from which one can construct a bilinear Bäcklund transformation (BT) [10]

$\displaystyle D^{2}_{x}f\cdot f^{\prime}=\lambda ff^{\prime},$ (5.19a)
$\displaystyle\big{(}D_{t}-D_{x}^{3}-3\lambda D_{x}\big{)}f\cdot
f^{\prime}=0,$ (5.19b)

and if $f$ is a solution to the bilinear KdV equation (5.18), so is
$f^{\prime}$. Introducing

$\displaystyle\phi=\frac{f^{\prime}}{f},$

one can recover the Lax pair (5.17) from the BT (5.19) [10].

Note that by the transformation

$\displaystyle w=\frac{f^{\prime}}{f},\qquad u=(\ln f)_{x},$

equation (5.3a) is bilinearized as

$\displaystyle\big{(}D_{t}-D_{x}^{3}\big{)}f\cdot f^{\prime}=0,$

which belongs to a deformed or modified bilinear BT of the KdV equation [5]:

$\displaystyle\big{(}D_{x}^{2}+2\gamma D_{x}\big{)}f\cdot f^{\prime}=0,$
(5.20a) $\displaystyle\big{(}D_{t}-D_{x}^{3}\big{)}f\cdot f^{\prime}=0,$
(5.20b)

where $\gamma$ is a new Bäcklund parameter. Introducing

$\displaystyle\psi=\frac{f^{\prime}}{f},\qquad u=(\ln f)_{x},$

from the above deformed bilinear BT, one finds

$\displaystyle\psi_{xx}-2\gamma\psi_{x}+2u_{x}\psi=0,$ (5.21a)
$\displaystyle\psi_{t}=\psi_{xxx}+6u_{x}\psi_{x},$ (5.21b)

which is another Lax pair of the potential KdV equation (5.3b). The two Lax
pairs, (5.17) and (5.21), are connected by the gauge transformation

$\displaystyle\phi={\rm e}^{-(\gamma
x+4\gamma^{3}t)}\psi,\qquad\gamma^{2}=\lambda.$

Now we get the links between equation (5.3a) and the potential KdV equation
(5.3b): the time-part (5.21b) of the Lax pair (5.21) is nothing but the
equation (5.3a), which means $w$ is in some sense the eigenfunction of the
(potential) KdV equation; if, based on the idea in [12], we consider the
transformation $f\to f^{\prime}$ as a shift in a discrete independent
variable, e.g., $f^{\prime}=\widetilde{f}$, the transformation (5.20a) is
converted to the bilinear equation (4.9a) with $p=\gamma$.

##  6 Concluding remarks

We have revisited H2 equation (1.1) with the help of the Cauchy matrix
approach and H1 equation. The function $y$ in H2 can be expressed as (1.5),
where $u=S^{(0,0)}$ and $w=S^{(1,0)}$ are variables defined in the Cauchy
matrix approach and $u$, $w$ satisfy the coupled system (1.4). This coupled
system provides a BT between the $u$-equation, H1 (1.3), and the 9-point
$w$-equation (1.7). They compose a consistent triplet. To investigate
integrability of the 9-point equation (1.7) of $w$, we derived from (1.4) a
coupled quad-system (1.6), or equivalently, equation (3.10), which is CAC and
allows a Lax pair (3.13). This fact indicates the integrability of the 9-point
equation (1.7). The coupled system (1.6) and the
$\mathrm{H}1\times\mathrm{H}2$ system (3.9) derived in [11] are connected
through (1.5) and (2.10).

H2 is a consequence of the coupled system (1.4). Since in a consistent triplet
the BT determines the other two equations in the triplet, in light of
Propositions 2.3 and 3.2, one may consider the 9-point equation (1.7) is an
alternative of H2 after removing the background $z^{2}_{0}$ and the term
$z_{0}u$ from $y$. Again, due to the consistent triplets, the bilinear form
(4.2) can serve as a bilinear form for H2 (1.1), the coupled quad-system (1.6)
and the 9-point equation (1.7).

As for the continuous counterparts, in continuum limits the coupled quad-
system (3.10) gives rise to the semi-discrete system (5.1), which implies
equation (5.2) for only $w$; and the full continuum limit leads (5.1) to
(5.3). The two equations in (5.3) are connected via the Miura transformation
(5.12) by (5.13). Eliminating $u$ from (5.3) yields equation (5.4) for $w$
only. In addition to the Mirua-type connection with the potential KdV
equation, equation (5.3a) appears as the $t$-part in the gauged Lax pair
(5.21) of the potential KdV equation (5.3b), which indicates that $w$ can be
considered as an eigenfunction (in the Lax pair) of the (potential) KdV
equation. Note that we derived the gauged Lax pair from a deformed bilinear BT
(5.20), in which (5.20a) and the bilinear equation (4.9a) share the same form
if we consider $f^{\prime}$ as a shift $\widetilde{f}$. We also note that
higher order equations in the continuum limit of the coupled quad-system
(3.10) will be investigated in Appendix A.

##  Appendix A Higher order equations in the continuum limits

In [19] Wiersma and Capel established a scheme to achieve higher order
equations in continuum limits, and they obtained the (potential) KdV hierarchy
from the continuum limit of the lpKdV equation (1.3). In their scheme,
discrete equations are handled in the (skew) coordinates

$(N=n+m,m),$

and in the first round, take

$\displaystyle\delta=q-p\to 0,\qquad m\to\infty,\qquad\tau_{1}=\delta
m\quad(\mathrm{being~{}finite}).$ (A.1)

For the plane wave factor $\rho$ given in Section 5.1, it turns out that

$\displaystyle\rho$
$\displaystyle=\biggl{(}\frac{p-k}{p+k}\biggr{)}^{n}\biggl{(}\frac{q-k}{q+k}\biggr{)}^{m}=\biggl{(}\frac{p-k}{p+k}\biggr{)}^{N}\biggl{(}1+\frac{2\delta
k}{(p+\delta+k)(p-k)}\biggr{)}^{m}$
$\displaystyle\longrightarrow\biggl{(}\frac{p-k}{p+k}\biggr{)}^{N}\exp\biggl{(}\frac{2k\tau_{1}}{p^{2}-k^{2}}\biggr{)}.$

The resulting semi-discrete equation usually has an evolution form

$\displaystyle\partial_{\tau_{1}}V_{N}=f\big{(}V_{N},{\rm
e}^{j\partial_{N}}V_{N}\big{)},$ (A.2)

where we adopt the notation ${\rm e}^{j\partial_{N}}V_{N}=V_{N+j}$. In the
full continuum limit, infinitely many independent variables
$(t_{1}=x,t_{3},t_{5},\dots)$ are introduced through

$\rho=\biggl{(}\frac{p-k}{p+k}\biggr{)}^{N}\exp\biggl{(}\frac{2k\tau_{1}}{p^{2}-k^{2}}\biggr{)}=\exp\Biggl{(}{\sum^{\infty}_{l=0}k^{2l+1}t_{2l+1}}\Biggr{)},$

where

$\displaystyle
t_{2l+1}=\frac{2}{p^{2l+1}}\bigg{(}\frac{-N}{2l+1}+\frac{\tau_{1}}{p}\bigg{)}.$

It then follows that

$\displaystyle\partial_{N}=\sum^{\infty}_{l=1}\frac{-2}{(2l+1)p^{2l+1}}\partial_{t_{2l+1}}$
(A.3a) and
$\displaystyle\partial_{\tau_{1}}=\sum^{\infty}_{l=1}\frac{2}{p^{2l+2}}\partial_{t_{2l+1}}=\partial_{p}\partial_{N}.$
(A.3b)

Replacing $\partial_{\tau_{1}}$ and $\partial_{N}$ in the equation (A.2) with
the above operators and expanding the equation in terms of $1/p$, a hierarchy
of equations will be obtained from the coefficients of $1/p^{j}$ in the
expansion.

With this scheme, the lpKdV equation (3.10b) is rewritten in terms of $(N,m)$
as

$\displaystyle\big{(}p+q+V-\widehat{\widetilde{\widetilde{V}}}\big{)}\big{(}p-q+\widehat{\widetilde{V}}-\widetilde{V}\big{)}=p^{2}-q^{2},$

where $V=V(N,m)=u(n,m)$. In light of (A.1), it gives rise to a semi-discrete
pKdV equation

$\displaystyle\partial_{\tau_{1}}V_{N}=1-\frac{2p}{2p+{\rm
e}^{-\partial_{N}}V_{N}-{\rm e}^{\partial_{N}}V_{N}}.$ (A.4)

In the full limit, redefining $V=V(t_{1}=x,t_{3},t_{5},\dots)$ and using
(A.3), the pKdV hierarchy is obtained by expanding (A.4) in terms of $1/p$.
The first two equations from the expansion are

$\displaystyle V_{t_{3}}+3(V_{x})^{2}-V_{xxx}=0$ (A.5)

and

$\displaystyle
V_{t_{5}}-10(V_{x})^{3}+5(V_{xx})^{2}+10V_{x}V_{xxx}-V_{xxxxx}=0,$ (A.6)

which are the 3rd-order potential KdV (pKdV) equation and 5th-order pKdV
equation. For more details one may refer to [19].

For the equation (3.10a), in the coordinates $(N,m)$ it is rewritten as

$\displaystyle\big{(}p-q+\widehat{\widetilde{V}}-\widetilde{V}\big{)}\big{(}W-\widehat{\widetilde{\widetilde{W}}}\big{)}-(p+q)\big{(}\widetilde{W}-\widehat{\widetilde{W}}\big{)}=0,$

where $W=W(N,m)=w(n,m)$. The semi-discrete equation obtained in light of (A.1)
is

$\displaystyle\partial_{\tau_{1}}W_{N}=\frac{{\rm e}^{-\partial_{N}}W_{N}-{\rm
e}^{\partial_{N}}W_{N}}{2p+{\rm e}^{-\partial_{N}}V_{N}-{\rm
e}^{\partial_{N}}V_{N}},$

where we have replaced $\partial_{\tau_{1}}V_{N}$ in the equation using (A.4).
In the full limit of the above equation, the first two equations from the
leading terms are

$\displaystyle W_{t_{3}}=W_{xxx}-3V_{x}W_{x}$ (A.7)

and

$\displaystyle
W_{t_{5}}=W_{xxxxx}-5V_{x}W_{xxx}-5V_{xx}W_{xx}-5V_{xxx}W_{x}+10(V_{x})^{2}W_{x},$
(A.8)

where $W=W(t_{1}=x,t_{3},t_{5},\dots)$.

Noting that (A.5) and (5.16b) are connected by $V=-2u$, we introduce a gauge-
transformed Schrödinger spectral problem (compared with (5.21a))

$\displaystyle W_{xx}-2\gamma W_{x}-V_{x}W=0,$ (A.9)

where $W$ serves as the eigenfunction and $\gamma$ for the spectral parameter.
Then, it turns out that the pKdV equation (A.5) is a result of the
compatibility of (A.9) and (A.7), i.e., $(W_{xx})_{t_{3}}=(W_{t_{3}})_{xx}$,
and the compatibility of (A.9) and (A.8) yields the 5th-order pKdV equation
(A.6). The same correspondence holds for the 7th-order equations obtained from
the full continuum limits.

### Acknowledgments

The authors are grateful to the referees for their invaluable comments. This
project is supported by the NSF of China (Nos. 11631007 and 11875040) and
Science and technology innovation plan of Shanghai (No. 20590742900).

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# Graphs of Maximal Energy with Fixed Maximal Degree

Octavio Arizmendi
Centro de Investigación en Matemáticas
Guanajauto, Mexico. Email<EMAIL_ADDRESS>

Jorge Fernandez Hidalgo
Universidad Nacional Autónoma de Mexico
Mexico City, Mexico. Email<EMAIL_ADDRESS>
###### Abstract

We give a bound for the graph energy with given maximal degree in terms of the
second and fourth moments of a graph. In the case in which the graph is
$d$-regular we obtain the bound that is given in Van Dam, E. et al. (2014).
through elementary methods.

(accepted in MATCH)

##  1  Introduction and Statement of Results

In this paper we study the energy of a graph as defined by Gutman [3]. For a
graph $G$ on $n$ vertexes with adjacency eigenvalues
$\lambda_{1},\dots,\lambda_{n}$, the energy of the graph $G$ is given by sum
of the absolute value of its adjacency eigenvalues,

$\sum_{i=1}^{n}|\lambda_{i}|.$

Several results on bounds for the energy of a graph have been considered in
the theory. e.g. [6, 8, 12]. A tool to give general bounds is given by the so-
called spectral moments of a graph [4, 5, 7, 9, 10, 13]. The $k$-th spectral
moment of a graph is given by

$M_{k}(G)=\sum\limits_{i=1}^{n}\lambda_{i}^{k}.$

The usefulness of these quantities comes from the fact that, when $k$ is an
integer, $M_{k}$ has a combinatorial interpretation: $M_{k}(G)$ is the trace
of the $k$’th power of the adjacency matrix of $G$, and consequently, is given
by the number of closed walks of length $i$ in $G$.

Here we present a simple but effective method to bound the energy of a graph
in terms of its spectral moments which we describe in Lemma 3.1.

A direct application of this method provides upper and lower bounds for the
energy of a graph with $n$ vertexes, and maximum degree $\Delta$, in terms of
the second and fourth moments.

To state this result, we shall use the notation

$\displaystyle A=\frac{M_{4}}{\Delta^{3}},\quad B=\frac{M_{2}}{\Delta},\quad
C=\Delta n.$ (1.1)

Our main theorem is as follows.

######  Theorem 1.1.

Let $G$ be a connected graph with at least $2$ vertexes,

$E(G)\leq-\frac{B^{2}+B\sqrt{-A+B}\sqrt{-B+C}-C(A+\sqrt{-A+B}\sqrt{-B+C})}{A-2B+C}$
(1.2)

with equality if and only if $G$ is a complete graph $K_{n}$, a strongly
regular graph with $\lambda=\mu$ or the incidence graph of a symmetric
$2-(v,k,\lambda)$ design.

For regular graphs one can see that (1.2) is decreasing in $A$ (see Lemma
3.6), and thus the above theorem subsumes the main theorem of Van Dam et al.
[11].

######  Theorem 1.2 ([11]).

For a regular graph $G$ one has

$E(G)\leq n\frac{d+(d^{2}-d)\sqrt{d-1}}{d^{2}-d+1}$

with equality if and only if $G$ is the incidence graph of a symmetric
$2-(v,d,1)$ design.

We must mention that the above result was derived originally by very different
methods from the ones in this paper and we do not know if the methods from
[11], may be applied to show our main theorem.

Apart from the introduction, the paper has two more sections. Section 2 gives
the basic preliminaries needed for this paper and the proofs of the main
results are given in Section 3.

##  2 Preliminaries on graphs

###  2.1 Basic definitions

Throughout this paper, we shall work exclusively with simple finite graphs. We
shall denote the number of vertexes of the graph $G$ by $n$ and the number of
edges by $m$. We will denote the vertexes of a graph $G$ using
$v_{1},\dots,v_{n}$. We say vertex $v_{i}$ is a neighbour of $v_{j}$ if
$v_{i}$ and $v_{j}$ are adjacent (in other words if $(v_{i},v_{j})$ is an edge
of $G$). We define the degree of a vertex $v_{i}$ as its number of neighbours
and we denote this quantity by $d_{i}$. The maximum degree over all vertexes
of the graph will be denoted by $\Delta$, while the minimum degree will be
denoted by $\delta$. If $\Delta=\delta=d$ we say that $G$ is a $d$-regular
graph.

Given a graph $G$, the adjacency matrix of $G$ is the $n\times n$ symmetric
matrix $A$ such that $A_{ij}$ is $1$ if $i$ and $j$ are adjacent and $A_{i,j}$
is $0$ otherwise. Being a symmetric matrix, $A$ has $n$ real eigenvalues,
counted with multiplicity, which are called the adjacency eigenvalues of the
graph $G$, we shall sometimes simply refer to them as the eigenvalues of $G$.

###  2.2 Spectral moments

The spectral moments of the graph $G$ are the quantities $M_{k}=Tr(A^{k})$.
Since $A$ is normal then $Tr(A^{k})=\sum\limits_{i=1}^{n}\lambda_{i}^{k}$. Let
us note that the $0$-th, first and second moments are given by

$M_{0}=n,~{}M_{1}=0,\text{ and }M_{2}=0,$

let us denote the number of $4$-cycles in $G$, by $Q$ and use $Z$ for the
Zagreb index given by $Z=\sum\limits_{i=1}^{n}d_{i}^{2}$, then

$M_{4}=2Z-2m+8Q.$

Hence, with the notation of the introdution, we have,

$\displaystyle A=\frac{2Z-2m+8Q}{\Delta^{3}},\quad B=\frac{2m}{\Delta},\quad
C=\Delta n.$ (2.1)

###  2.3 Graphs with few eigenvalues

Finally, we describe the graphs which satisfy the equality in Theorems 1.1 and
1.2.

We say that a graph is strongly connected with parameters $\lambda,\mu$ if it
is regular and it satisfies: the following properties

  * •

Every two adjacent vertexes have $\lambda$ common neighbours.

  * •

Every two non adjacent vertexes have $\mu$ common neighbours.

The spectrum of a strongly connected graph with $n$ vertexes and common degree
$d$ is:

  * •

$d$ with multiplicity $1$.

  * •

$\frac{1}{2}[(\lambda-\mu)+\sqrt{(\lambda-\mu)^{2}+4(d-\mu)}]$ with
multiplicity
$\frac{1}{2}[(v-1)-\frac{2k+(v-1)(\lambda-\mu)}{\sqrt{(\lambda-\mu)^{2}+4(d-\mu)}}],$

  * •

$\frac{1}{2}[(\lambda-\mu)-\sqrt{(\lambda-\mu)^{2}+4(d-\mu)}]$ with
multiplicity
$\frac{1}{2}[(v-1)+\frac{2k+(v-1)(\lambda-\mu)}{\sqrt{(\lambda-\mu)^{2}+4(d-\mu)}}].$

We refer to a symmetric $2-(v,k,\lambda)$ design as a set $X$ with $v$
elements, along with a family $F$ consisting of $v$ subsets of $X$ (called
blocks) such that each block has $k$ elements and for any two points $x$ and
$y$ in $X$ there are exactly $\lambda$ blocks that contain both.

We define the incidence graph of a symmetric $2-(v,k,\lambda)$ design as the
graph with vertexes $F\cup X$ such that there is an edge between a block $B\in
F$ and a point $x\in X$ if and only if $x\in B$.

The spectrum of the incidence graph of a symmetric $2-(v,k,\lambda)$ design
is:

  * •

$\pm k$, each with multiplicity $1$.

  * •

$\pm\sqrt{k-\lambda}$, each with multiplicity $v-1$.

##  3 Bounds for the energy of a graph using moments

###  3.1 A general bound for the energy of a graph

The main idea comes from the following observation. Let
$P(x)=a_{m}x^{m}+\dots+a_{0}$ be a polynomial such that $P(x)\geq|x|$ for all
$x\in[-\rho,\rho]$, then

$E(G)=\sum\limits_{i=1}^{n}|\lambda_{i}|\leq\sum\limits_{i=1}^{n}P(\lambda_{i})=\sum\limits_{i=0}^{m}a_{m}M_{m}.$

Here we emphasize that in order to find useful bounds for $E(G)$ we only need
to take into consideration $x\in[-\rho,\rho]$ and not in all $\mathbb{R}$.
Since obtaining $\rho$ directly from combinatorial properties of the graph is,
in general hard we use instead $\Delta$, obtaining the following lemma, which
includes analogous lower bounds for $E(G)$.

######  Lemma 3.1.

Let G be a graph with maximum degree $\Delta$.

1) For any polynomial $P(x)=a_{m}x^{m}+\dots+a_{0}$ a polynomial such that
$P(x)\geq|x|$ on $x\in[-\Delta,\Delta]$ we have

$E(G)\leq\sum\limits_{i=0}^{m}a_{m}M_{m}.$ (3.1)

2) For any polynomial $Q(x)=a_{m}x^{m}+\dots+a_{0}$ be a polynomial
$Q(x)\leq|x|$, on $x\in[-\Delta,\Delta]$ we have

$E(G)\geq\sum\limits_{i=0}^{m}a_{m}M_{m}.$ (3.2)

Moreover, equality in (3.1) (resp. (3.2)) occurs if and only if
$P(\lambda_{i})=|\lambda_{i}|$ (resp. $Q(\lambda_{i})=|\lambda_{i}|$) for all
$i$.

The strength of the previous lemma comes from the freedom of choosing the
above polynomials, and using the information of the class of graphs studied is
very helpful as we will see in the next sections.

###  3.2 Upper bounds in terms of second and fourth moments

Let us consider the case where $P$ is a polynomial of degree $4$. Since the
absolute value is an even function considering $x$ or $x^{3}$ in the
polynomial will produce one of the sides to tilt which will worsen our
approximation. Thus we assume that $P$ is even. Then we are led consider a
polynomial of the form $P(x)=ax^{4}+bx^{2}+c.$

For this polynomial $P(x)=ax^{4}+bx^{2}+c$ we have

$Tr(P(A))=\sum\limits_{i=1}^{n}P(\lambda_{i})=aM_{4}+bM_{2}+cM_{0}=2aZ+8aQ+2(b-a)m+cn.$
(3.3)

Moreover, we see that

$\displaystyle Tr(P(A))$ $\displaystyle=$ $\displaystyle
aM_{4}+bM_{2}+cM_{0}=a(2(\sum\limits_{i=0}^{n}d_{i}^{2})-2m+8Q)+bm+cn$
$\displaystyle=$ $\displaystyle
a8Q+(b-2)m+(c+ad^{2})n=a8Q+\left(c+ad^{2}+\frac{bd}{2}-d\right)n.$

We note for further reference that when $G$ is $d$-regular we have $2m=dn$ and
$Z=nd^{2}$ which gives us

$\sum\limits_{i=1}^{n}P(\lambda_{i})=2and^{2}+8aQ+(b-a)nd+cn.$ (3.5)

Now our goal is to choose $a,b$ and $c$ which minimize the left-hand side of
(3.2) restricted to the condition that $P(x)\geq x$ for $\Delta\geq x\geq 0$.
For this we seek for polynomials that are tangent to $f(x)=x$.

######  Lemma 3.2.

Given $0<r<1$ there is a unique polynomial $P_{r}=ax^{4}+bx^{2}+c$ such that
$P_{r}(r)=r,P_{r}(1)=1$ and $P_{r}(x)\geq x$ for $x\in[0,1]$.

###### Proof.

We shall call the desired polynomial $P_{r}=ax^{4}+bx^{2}+c$. Since $P_{r}$
must be tangent to the absolute value function at $r$ the following equations
must be satisfied

$\displaystyle P_{r}(r)$ $\displaystyle=$ $\displaystyle ar^{4}+br^{2}+c=r,$
$\displaystyle P_{r}(1)$ $\displaystyle=$ $\displaystyle a+b+c=1,$
$\displaystyle P_{r}^{\prime}(r)$ $\displaystyle=$ $\displaystyle
4ar^{3}+2br=1.$

The solution to this system of equations is unique and gives us

$a=-\frac{1}{2r(r+1)^{2}},\quad b=\frac{3r^{2}+2r+1}{2r(r+1)^{2}},\quad
c=\frac{r^{2}(2r+1)}{2r(r+1)^{2}}.$ (3.6)

First we show that $P_{r}(x)>x$ for all $0<x<1$ with $x\neq r$. Consider for
this, the function $Q(x)=P(x)-x$. Its second derivative is given by
$12ax^{2}-2b$, which is decreasing and thus $Q(x)$ is convex on $(0,r_{0})$
and concave on $(r_{0},1)$, where $r_{0}=\sqrt{\frac{3r^{2}+2r+1}{6}}$ is the
unique solution to $Q^{\prime\prime}(x)=0$. Since $r\in(0,1)$, then
$r<r_{0}<1$. Since in the interval $[0,r_{0}]$ the function is convex it
reaches its minimum at the point such that $Q^{\prime}(x)=0$. This point is
$r$ and we have $Q(r)=0$. In the interval $[r_{0},1]$ the function is concave
and therefore reaches its minimum at one of $r_{0}$ and $1$. We have
$Q(r_{0})>Q(r)=0$ and we have $Q(1)=0$. So indeed $P_{r}$ is as desired.

∎

Thus from Lemma 3.1 we obtain the following bound.

######  Theorem 3.3.

Let $r\in(0,1)$. For $G$ a graph with maximum degree $\Delta$ the energy of
$G$ is bounded by

$E(G)\leq\frac{-1}{2r(r+1)^{2}}\frac{2Z-2m+8Q}{\Delta^{3}}+(\frac{3r^{2}+2r+1}{2r(r+1)^{2}})\frac{2m}{\Delta}+n\Delta\frac{r^{2}(2r+1)}{2r(r+1)^{2}}$

###### Proof.

Let $P(x)$ be as in Lemma 2.2. Then $P_{r}(x)\geq|x|$ for $x\in[-1,1]$. Now
let $P_{r,\Delta}(x)=\Delta P_{r}(x/R)$, be the dilation by $\Delta$ of the
polynomial $P_{r}$. Then $P_{r,\Delta}(x)\geq|x|$ for $x\in[-\Delta,\Delta]$.
By part 1) of Lemma 2.1 we conclude. ∎

For the following we recall the notation

$\displaystyle A=\frac{2Z-2m+8Q}{\Delta^{3}},\quad B=\frac{2m}{\Delta},\quad
C=\Delta n$

We want to minimize for $r$. For this, we claim that $A\leq B\leq C$ and that
$0\leq\frac{B-A}{C-B}\leq 1$. To show that $B\leq C$ we simply notice
$2m\leq\Delta n\Delta^{2}n$, and to prove $A\leq B$ we see it is equivalent to
$M_{4}\leq 2m\Delta^{2}$. Now, notice that for every oriented edge $u,v$ there
can be at most $\Delta^{2}$ closed walks of length $4$ starting with $u,v$.

By elementary calculus one sees that subject to the restriction $A\leq B$ and
$B\leq C$. The minimum value of the function
$\frac{-1}{2r(r+1)^{2}}A+(\frac{3r^{2}+2r+1}{2r(r+1)^{2}})B+\frac{r^{2}(2r+1)}{2r(r+1)^{2}}C$
for $r\in(0,1)$ is reached at $\frac{\sqrt{B-A}}{\sqrt{C-B}}$, giving the
value

$-\frac{B^{2}+B\sqrt{B-A}\sqrt{C-B}-C(A+\sqrt{B-A}\sqrt{C-B})}{A-2B+C}.$ (3.7)

Thus we arrive to the following theorem.

######  Theorem 3.4.

For any graph $G$ one has

$E(G)\leq-\frac{B^{2}+B\sqrt{-A+B}\sqrt{-B+C}-C(A+\sqrt{-A+B}\sqrt{-B+C})}{A-2B+C}$

with equality if and only if the spectrum of the graph is contained in
$\\{\pm\frac{\sqrt{B-A}}{\sqrt{C-B}},\pm\Delta\\}$

###### Proof.

The inequality follows from the considerations above. In order to have
equality, the polynomial $P_{\frac{\sqrt{B-A}}{\sqrt{C-B}},\Delta}(x)$ and the
function $Abs(x):=|x|$ should coincide for all eigenvalues of $G$. This is
only possible for the set $\\{\pm\frac{\sqrt{B-A}}{\sqrt{C-B}},\pm\Delta\\}.$
∎

The following propositon characterizes the graphs for which equality is
reached and thus completes the main theorem of this paper.

######  Proposition 3.5.

The only connected graphs with at least $2$ vertexes for which equality is
attained are the complete graphs $K_{n}$, strongly regular graphs with
$\lambda=\mu$ and incidence graphs of symmetric $2-(v,k,\lambda)$ designs.

###### Proof.

Equality occurs if and only if the spectra of $G$ is contained in
$\\{\pm\frac{\sqrt{B-A}}{\sqrt{C-B}},\pm\Delta\\}$.

If $\Delta$ is not an eigenvalue of $G$ then $-\Delta$ is not an eigenvalue
either, so $G$ has exactly two eigenvalues. This implies $G$ is complete,
however complete graphs have $\Delta$ as an eigenvalue. We conclude $\Delta$
is an eigenvalue, meaning $G$ is regular.

If $-\Delta$ is an eigenvalue then $G$ is bipartite and therefore has
symmetric spectra. If $G$ has exactly $2$ eigenvalues then $G$ is complete and
bipartite (So $G=K_{2}$). If $G$ has exactly $4$ eigenvalues then $G$ is a
regular bipartite graph with exactly $4$ eigenvalues and must therefore be the
incidence graph of a symmetric $2-(v,k,\lambda)$ design, as shown in [1].

If $-\Delta$ is not an eigenvalue then $G$ must have exactly $3$ eigenvalues
or exactly $2$ eigenvalues. The only connected regular graphs with exactly $3$
eigenvalues are the strongly regular graphs, as is show in [2]. Moreover
$\lambda=\mu$ is required so that one eigenvalue is the negative of the other.
Finally, if $G$ has only two eigenvalues it must be a complete graph.

It should be clear that equality is indeed achieved in all these cases, as a
polynomial $P_{r,\Delta}(x)$ can be found such that $P(x)=|x|$ for all
eigenvalues $x$ of $G$.

∎

To minimize (3.7) we notice the following.

######  Lemma 3.6.

The expression given in (3.7) is decreasing in $A$.

###### Proof.

Note that the formula
$\frac{-1}{2r(r+1)^{2}}A+(\frac{3r^{2}+2r+1}{2r(r+1)^{2}})B+\frac{r^{2}(2r+1)}{2r(r+1)^{2}}C$
is decreasing in $A$ for every value of $r\in(0,1)$. The result follows since
(3.7) is the the minimum of the expression over all $r\in(0,1)$ and thus
should also be decreasing on $A$. ∎

Thus, from (2.1), one should expect that having quadrangles should decrease
the energy.

###  3.3 $d$-regular graphs

Finally, we may recover the result in [11]. Indeed, for the case of regular
graphs we have

$\displaystyle A=\frac{2nd^{2}+nd+8Q}{d^{3}},\quad B=n,\quad C=nd.$

Lemma 3.6 shows that, for $n$ and $d$ are fixed, (3.7) is decreasing in $Q$
and then $Q=0$ gives a bound for all regular graphs. In this case the minimum
value simplifies and is reached at $\frac{1}{d}\sqrt{d-1}$.

######  Corollary 3.7.

For a regular graph $G$ one has

$E(G)\leq n\frac{d+(d^{2}-d)\sqrt{d-1}}{d^{2}-d+1}$

with equality if and only if $G$ is the incidence graph of a symmetric
$2-(v,d,1)$ design.

## Acknowledgment

O. Arizmendi was supported by a CONACYT Grant No. 222668 and by the European
Union’s Horizon 2020 research and innovation programme under the Marie
Skłodowska-Curie grant agreement 734922, during the writing of this paper.

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