# Exact Jacobi elliptic solutions of the $abcd$-system

JAKE DANIELS1 and NGHIEM V. NGUYEN1

###### Abstract

In this manuscript, consideration is given to the existence of periodic
traveling-wave solutions to the $abcd$-system. This system was derived by
Bona, Saut, and Chen to describe small amplitude, long wavelength gravity
waves on the surface of water. These exact solutions are formulated in terms
of the Jacobi elliptic function cnoidal. The existence of explicit traveling-
wave solutions is very useful in theoretical investigations such as stability
of solutions, as well as other numerical analysis of the system.

††2020 AMS Subject Classification: 35A16, 35A24, 35B10.††Key words: Periodic
solutions, Jacobi elliptic functions, Cnoidal solutions, abcd
system.11footnotetext: Department of Mathematics and Statistics, Utah State
University, Logan, UT 84322-3900, USA

##  1 Introduction

Let $\Omega_{t}$ be a domain in $\mathbb{R}^{3}\times\mathbb{R}$ whose
boundary consists of two parts: the fixed surface located at $z=-h(x,y)$, and
the free surface $z=\eta(x,y,t)$. It is well-known that the three-dimensional
capillary-gravity waves on an inviscid, irrotational, and incompressible fluid
layer of uniform depth $h$ that is impermeable on the fixed surface are
governed by the Euler equations

$\displaystyle\Delta\phi=0,\quad\mbox{in}\ \Omega_{t};$ (1.1a)
$\displaystyle\eta_{t}+\phi_{x}\eta_{x}+\phi_{y}\eta_{y}-\phi_{z}=0,\quad\text{at}\
z=\eta(x,y,t);$ (1.1b)
$\displaystyle\phi_{t}+\frac{1}{2}|\nabla\phi|^{2}+gz=0,\quad\text{at}\
z=\eta(x,y,t);$ (1.1c)
$\displaystyle\phi_{x}h_{x}+\phi_{y}h_{y}+\phi_{z}=0,\quad\text{at}\
z=-h(x,y).$ (1.1d)

Here, $g$ denotes the acceleration of gravity; $\phi(x,y,z,t)$ is the velocity
potential function. The full Euler equations (1.1a)-(1.1d) are often far more
complex than necessary for many applications, so several approximation models
have been derived in its place under certain restricted physical regimes. One
such model is the $abcd$-system, introduced by Bona et al. in [4, 5], to
describe small amplitude, long wavelength gravity waves on the surface of
water

$\begin{cases}\eta_{t}+w_{x}+(w\eta)_{x}+aw_{xxx}-b\eta_{xxt}=0\,,\\\
w_{t}+\eta_{x}+ww_{x}+c\eta_{xxx}-dw_{xxt}=0\,,\end{cases}$ (1.2)

where $a,b,c,$ and $d$ are real constants and $\theta\in[0,1]$ that satisfy

$\begin{split}a+b&=\frac{1}{2}(\theta^{2}-\frac{1}{3}),\\\
c+d&=\frac{1}{2}(1-\theta^{2})\geq 0,\\\ a+b+c+d&=\frac{1}{3}.\end{split}$
(1.3)

Precisely, the regime for which the above system is derived to approximate the
Euler equations is when the maximal deviation $\alpha$ of the free surface is
small and a typical wavelength $\lambda$ is large as compared to the
undisturbed water depth $h$, such that the Stokes number
$S=\frac{\alpha\lambda^{2}}{h^{3}}$ is of order one. The functions $\eta(x,t)$
and $w(x,t)$ are real valued and $x,t\in\mathbb{R}$. By choosing specific
values for the parameters $a,b,c,$ and $d$, the system (1.2) includes a wide
range of other systems that have been derived over the last few decades such
as the classical Boussinesq system [8, 9, 10], the Kaup system [21], the
coupled Benjamin-Bona-Mahony system (BBM-system) [3], the coupled Korteweg-de
Vries system (KdV-system) [4, 5], the Bona-Smith system [6], and the
integrable version of Boussinesq system [22]. In particular, these
specializations are:

  * •

Classical Boussinesq system

$\left\\{\begin{matrix}\begin{split}&\eta_{t}+w_{x}+(w\eta)_{x}=0,\\\
&w_{t}+\eta_{x}+ww_{x}-\frac{1}{3}w_{xxt}=0;\end{split}\end{matrix}\right.$

  * •

Kaup system

$\left\\{\begin{matrix}\begin{split}&\eta_{t}+w_{x}+(w\eta)_{x}+\frac{1}{3}w_{xxx}=0,\\\
&w_{t}+\eta_{x}+ww_{x}=0;\end{split}\end{matrix}\right.$

  * •

Bona-Smith system

$\left\\{\begin{matrix}\begin{split}&\eta_{t}+w_{x}+(w\eta)_{x}-b\eta_{xxt}=0,\\\
&w_{t}+\eta_{x}+ww_{x}+c\eta_{xxx}-bw_{xxt}=0;\end{split}\end{matrix}\right.$

  * •

Coupled BBM-system

$\left\\{\begin{matrix}\begin{split}&\eta_{t}+w_{x}+(w\eta)_{x}-\frac{1}{6}\eta_{xxt}=0,\\\
&w_{t}+\eta_{x}+ww_{x}-\frac{1}{6}w_{xxt}=0;\end{split}\end{matrix}\right.$

  * •

Coupled KdV-system

$\left\\{\begin{matrix}\begin{split}&\eta_{t}+w_{x}+(w\eta)_{x}+\frac{1}{6}w_{xxx}=0,\\\
&w_{t}+\eta_{x}+ww_{x}+\frac{1}{6}\eta_{xxx}=0;\end{split}\end{matrix}\right.$

  * •

Coupled KdV-BBM system

$\left\\{\begin{matrix}\begin{split}&\eta_{t}+w_{x}+(w\eta)_{x}+\frac{1}{6}w_{xxx}=0,\\\
&w_{t}+\eta_{x}+ww_{x}-\frac{1}{6}w_{xxt}=0;\end{split}\end{matrix}\right.$

  * •

Coupled BBM-KdV system

$\left\\{\begin{matrix}\begin{split}&\eta_{t}+w_{x}+(w\eta)_{x}-\frac{1}{6}\eta_{xxt}=0,\\\
&w_{t}+\eta_{x}+ww_{x}+\frac{1}{6}\eta_{xxx}=0.\end{split}\end{matrix}\right.$

For a more detailed discussion on this, we refer our readers to the papers [4,
5].

In this manuscript, attention is given to the existence of periodic traveling-
wave solutions to the $abcd$-system. A traveling-wave solution to the system
(1.2) is a vector solution $\big{(}\eta(x,t),w(x,t)\big{)}$ of the form

$\eta(x,t)=\eta(x-\sigma t)\quad\text{and}\quad w(x,t)=w(x-\sigma t),$ (1.4)

where $\sigma$ denotes the speed of the waves. Notice that when $\sigma>0$,
one obtains a right-propagating solution, while for $\sigma<0$, one has a
left-propagating solution. This feature of bi-direction propagation of
$(\eta,w)$ is a hallmark of the $abcd$-system in its derivation and is the
main difference between system (1.2) and the one-way approximation models such
as

$u_{t}+u_{x}+\frac{1}{p}(u^{p})_{x}\pm Lu_{x}=0,\ \ \ p>0,$

(which is commonly referred to as the generalized KdV equation when
$L=\partial_{xx}$, the generalized BBM equation when $L=-\partial_{xt}$, and
simply the KdV and BBM equations, respectively, when $p=1$), and the cubic,
nonlinear Schrödinger equation (NLS)

$iu_{t}+u_{xx}+|u|^{2}u=0.$

It would also be interesting to analyze the existence of periodic solutions
when the two components propagate at different speeds, and/or when the two
components travel in opposite directions. Such scenarios are not considered in
the scope of this paper.

The most studied types of traveling-wave solutions are periodic traveling-wave
solutions and solitary-wave solutions. Periodic traveling-wave solutions are
self-explanatory, while solitary waves are smooth traveling-wave solutions
that are symmetric around a single maximum and rapidly decay to zero away from
the maximum. Though less common, the term solitary wave is sometimes used to
describe traveling-wave solutions that are symmetric around a single maximum
but approach nonzero constants as $\xi=x-\sigma t\rightarrow\pm\infty$. In
this manuscript, we consider non-trivial solutions to be solutions where both
components are non-constant, and semi-trivial solutions to be solutions where
one component is a constant. The constant solution, $(f,g)$ for any
$f,\,g\in\mathbb{R}$, is always a solution and will be referred to as the
trivial solution.

The topic of existence of explicit periodic traveling-wave solutions for many
single equations are readily available such as for the KdV equation [18],
generalized KdV equation [13], BBM equation [1], NLS equation [20], and
Whitham equation [19]. In contrast, there are far less of these results for
systems due to the complexity of the coupled equations involved. Leveling up
the scalar results to systems is no minor task, requiring new insights and new
approaches. For the classical Boussinesq system, Krishnan showed that the
periodic traveling-wave solutions exist and are given by Jacobi elliptic
functions [22]. Chen and Li [12] (see also [25]) have recently established the
periodic traveling-wave solutions for the beta derivative of the Kaup system
where the first temporal derivative $\partial_{t}$ is replaced by the
fractional derivative $\partial^{\beta}_{t}$, with $\beta\in(0,1]$, given in
terms of normal trigonometic functions as well as the Jacobi cnoidal
functions. Four systems have been put forth recently by Nguyen et al. to
describe the interaction of long and short waves in dispersive media [17, 24,
23], and periodic traveling-wave solutions given by the Jacobi elliptic
function cnoidal have been explicitly calculated for all four systems [11].

Another important subject regarding traveling-wave solutions is the stability
of these solutions. The topic of stability of periodic traveling-wave
solutions such as cnoidal waves has attracted far less consideration than that
of solitary wave solutions. Within this limited attention, explicitly known
formulae for periodic traveling-wave solutions often play an instrumental
role. Angula, Bona and Scialom [2] proved that the periodic traveling-waves
for the KdV equation are nonlinearly orbitally stable with respect to
perturbations of the same period. Central to the argument in their proof is
the fact that the cnoidal waves lie in the set of minimizers, a conclusion
that would be hard to establish without having explicit formula for the
cnoidal waves at hand. Deconinck et al. established that the KdV cnoidal waves
are spectrally stable with respect to perturbations of arbitrary period [7],
and orbitally stable with respect to sub-harmonic perturbations [16]. Again,
the explicit formulae for cnoidal waves play pivital roles in their numerical
computations.

In this paper, periodic traveling-wave solutions to the system (1.2) of the
form

$\eta(\xi)=\sum_{r=0}^{n}j_{r}\operatorname{cn}^{r}(\lambda\xi,m)\quad\text{and}\quad
w(\xi)=\sum_{r=0}^{n}k_{r}\operatorname{cn}^{r}(\lambda\xi,m)\,,$ (1.5)

are analyzed, where $j_{r},\,k_{r}\in\mathbb{R}$, $\lambda>0$, $m\in(0,1]$,
$\xi=x-\sigma t$, and $\operatorname{cn}$ denotes the Jacobi elliptic function
cnoidal. For the readers’ convenience, a brief introduction to Jacobi elliptic
functions can be found in Section 2. If $m=0$, then (1.5) reduces to a cosine
series which is not the aim of this paper and is excluded. The analysis of
(1.5) requires a case split for when $c\neq 0$ and $c=0$. For the first case,
$c\neq 0$, it is shown that non-trivial periodic traveling-wave solutions only
exist when $a^{2}+b^{2}\neq 0$, and in particular, $j_{r}=k_{r}=0$ for $r\geq
3$. When $a^{2}+b^{2}=0$, it is established that only semi-trivial periodic
traveling-wave solutions exist, and that $j_{r}=0$ for all $r\geq 1$ and
$k_{r}=0$ for $r\geq 3$. For the second case, $c=0$, it is shown that non-
trivial periodic traveling-wave solutions only exist when $b^{2}+d^{2}\neq 0$,
and that $j_{r}=0$ for $r=1,\,3$ and $r\geq 5$ and $k_{r}=0$ for $r=1$ and
$r\geq 3$.

The paper is organized as follows. In Section 2, some facts and identities of
Jacobi elliptic functions are reviewed. In Section 3, the existence of
periodic traveling-wave solutions to the system (1.2) is established. The
periodic traveling-wave solutions are then explicitly computed in Section 4.
Finally, a discussion about the obtained results is given in Section 5,
including remarks about how these cnoidal solutions limit to the solitary-wave
solutions previously established in [14, 15].

##  2 Preliminaries

For the readers’ convinience, some notions of the Jacobi elliptic functions
are briefly recalled here. Let

$v=\int_{0}^{\phi}\frac{1}{\sqrt{1-m^{2}\sin^{2}t}}dt,\ \ \ \ \ \mbox{for
$0\leq m\leq 1$}.$

Then $v=F(\phi,m)$, or equivalently, $\phi=F^{-1}(v,m)=\operatorname{am}(v,m)$
which is the Jacobi amplitude and $m$ is referred to as the Jacobi elliptic
modulus. The two basic Jacobi elliptic functions $\operatorname{cn}(v,m)$ and
$\operatorname{sn}(v,m)$ are defined as

$\operatorname{sn}(v,m)=\sin(\phi)=\sin\big{(}F^{-1}(v,m)\big{)}\ \ \ \ \
\mbox{and}\ \ \ \ \
\operatorname{cn}(v,m)=\cos(\phi)=\cos\big{(}F^{-1}(v,m)\big{)}.$

These functions are generalizations of the trigonometric and hyperbolic
functions which satisfy

$\begin{matrix}\operatorname{cn}(v,0)=\cos(v),\ \ \ \ \ \ \
\operatorname{sn}(v,0)=\sin(v),\\\
\operatorname{cn}(v,1)=\operatorname{sech}(v),\ \ \ \ \
\operatorname{sn}(v,1)=\tanh(v).\end{matrix}$

We recall the following identities:

$\left\\{\begin{matrix}\begin{split}&\operatorname{sn}^{2}(\lambda\xi,m)=1-\operatorname{cn}^{2}(\lambda\xi,m),\\\
&\operatorname{dn}^{2}(\lambda\xi,m)=1-m^{2}+m^{2}\operatorname{cn}^{2}(\lambda\xi,m),\\\
&\frac{d}{d\xi}\operatorname{cn}(\lambda\xi,m)=-\lambda\operatorname{sn}(\lambda\xi,m)\operatorname{dn}(\lambda\xi,m),\\\
&\frac{d}{d\xi}\operatorname{sn}(\lambda\xi,m)=\lambda\operatorname{cn}(\lambda\xi,m)\operatorname{dn}(\lambda\xi,m),\\\
&\frac{d}{d\xi}\operatorname{dn}(\lambda\xi,m)=-m^{2}\lambda\operatorname{cn}(\lambda\xi,m)\operatorname{sn}(\lambda\xi,m).\end{split}\end{matrix}\right.$

The following relations will be useful

$\left\\{\begin{matrix}\begin{split}&\frac{d}{d\xi}\operatorname{cn}^{r}=-r\lambda\operatorname{cn}^{r-1}\operatorname{sn}\operatorname{dn},\\\
&\frac{d^{2}}{d\xi^{2}}\operatorname{cn}^{r}=-r\lambda^{2}\big{[}(r+1)m^{2}\operatorname{cn}^{r+2}+r(1-2m^{2})\operatorname{cn}^{r}+(r-1)(m^{2}-1)\operatorname{cn}^{r-2}\big{]},\\\
&\frac{d^{3}}{d\xi^{3}}\operatorname{cn}^{r}=r\lambda^{3}\operatorname{sn}\operatorname{dn}\big{[}(r+1)(r+2)m^{2}\operatorname{cn}^{r+1}+r^{2}(1-2m^{2})\operatorname{cn}^{r-1}+(r-1)(r-2)(m^{2}-1)\operatorname{cn}^{r-3}\big{]},\\\
\end{split}\end{matrix}\right.$ (2.1)

where the argument $(\lambda\xi,m)$ has been dropped for clarity reason.

##  3 Existence of Periodic Traveling-Wave Solutions

Substituting the traveling-wave ansatz (1.4) into the abcd-system (1.2) yields
the following system of ordinary differential equations (ODE)

$\begin{cases}-\sigma\eta^{\prime}+w^{\prime}+(\eta
w)^{\prime}+aw^{\prime\prime\prime}+b\sigma\eta^{\prime\prime\prime}=0\,,\\\
-\sigma
w^{\prime}+\eta^{\prime}+ww^{\prime}+c\eta^{\prime\prime\prime}+d\sigma
w^{\prime\prime\prime}=0\,,\end{cases}$ (3.1)

where the primes denote the derivatives with respect to the moving frame
$\xi=x-\sigma t$. Replacing $\eta$ and $w$ in (3.1) with the periodic
traveling-wave ansatz (1.5) and using cnoidal derivative identities (2.1), the
following generic form is obtained

$\left\\{\begin{matrix}\begin{split}&-\lambda\operatorname{sn}(\lambda\xi,m)\operatorname{dn}(\lambda\xi,m)\sum_{q=0}^{2n-1}h_{1,q}\operatorname{cn}^{q}(\lambda\xi,m)=0,\\\
&-\lambda\operatorname{sn}(\lambda\xi,m)\operatorname{dn}(\lambda\xi,m)\sum_{q=0}^{2n-1}h_{2,q}\operatorname{cn}^{q}(\lambda\xi,m)=0,\end{split}\end{matrix}\right.$
(3.2)

where the subscript $q$ in $h_{1,q}$ and $h_{2,q}$ corresponds with the power
of $\operatorname{cn}$ in their respective equations.

The analysis is now split into two distinct cases. We will show that the the
series (1.5) will terminate after a certain number of terms. When exactly this
termination will occur is found to be dependent on the values of the
parameters $a,\,b,\,c,$ and $d$.

###  3.1 The case when $c\neq 0$

Notice that as (3.2) must hold true for all $(\lambda\xi,m)$, it must be the
case that $h_{1,q}=0$ and $h_{2,q}=0$ for all $q$. Using this, the following
theorem is established. (Recall that $j_{r}$ and $k_{r}$ are the coefficients
of $\operatorname{cn}^{r}$ as given in (1.5) for $r\in\mathbb{Z}_{\geq 0}$.)

######  Theorem 3.1.

Suppose $c\neq 0$, then the periodic traveling-wave ansatz (1.5) will take on
one of the following forms depending on the values of $a$ and $b$:

  1. (i)

If $a^{2}+b^{2}\neq 0$, then $j_{r}=k_{r}=0$ for all integers $r\geq 3$ and
(1.5) reduces to

$\eta(\xi)=j_{0}+j_{1}\operatorname{cn}(\lambda\xi,m)+j_{2}\operatorname{cn}^{2}(\lambda\xi,m)\quad\text{and}\quad
w(\xi)=k_{0}+k_{1}\operatorname{cn}(\lambda\xi,m)+k_{2}\operatorname{cn}^{2}(\lambda\xi,m)\,.$
(3.3)

  2. (ii)

If $a^{2}+b^{2}=0$, then $j_{r}=0$ for all integers $r\geq 1$ and $k_{r}=0$
for all integers $r\geq 3$. Thus, (1.5) reduces to

$\eta(\xi)=j_{0}\quad\text{and}\quad
w(\xi)=k_{0}+k_{1}\operatorname{cn}(\lambda\xi,m)+k_{2}\operatorname{cn}^{2}(\lambda\xi,m)\,.$
(3.4)

Consequently, only semi-trivial periodic traveling-wave solutions exist in
this case.

###### Proof.

(i) Let $\rho(\eta^{\prime})$ denote the largest power of $\operatorname{cn}$
in $\eta^{\prime}$. Then,

$\rho(\eta^{\prime})=\rho(w^{\prime})=n-1$,
$\rho(\eta^{\prime\prime\prime})=\rho(w^{\prime\prime\prime})=n+1$, and
$\rho((\eta w)^{\prime})=\rho(ww^{\prime})=2n-1$.

From this, if $n>2$ then $2n-1>n+1$, which implies that the coefficient of
$\operatorname{cn}^{2n-1}$ in the second equation of (3.2) will solely come
from the $ww^{\prime}$-term. Thus, $h_{2,2n-1}=nk_{n}^{2}$. Setting this
coefficient equal to zero gives $k_{n}=0$. Recalculating the above orders
yields

$\rho(\eta^{\prime})=n-1$, $\rho(w^{\prime})=n-2$,
$\rho(\eta^{\prime\prime\prime})=n+1$, $\rho(w^{\prime\prime\prime})=n$,
$\rho((\eta w)^{\prime})=2n-2$, and $\rho(ww^{\prime})=2n-3$.

After $i$ iterations, the orders become

$\rho(\eta^{\prime})=n-1$, $\rho(w^{\prime})=n-2-i$,
$\rho(\eta^{\prime\prime\prime})=n+1$, $\rho(w^{\prime\prime\prime})=n-i$,
$\rho((\eta w)^{\prime})=2n-2-i$, and $\rho(ww^{\prime})=2n-3-2i$.

This argument can be repeated until
$\rho(ww^{\prime})\leq\rho(\eta^{\prime\prime\prime})$. Now, consider the two
possibilities for $n$.

  * •

If n is even: let $z\in{\mathbb{Z}}_{\geq 1}$ such that $n=2z$. Then
$\rho(ww^{\prime})=2n-3-2i=4z-3-2i$ and
$\rho(\eta^{\prime\prime\prime})=n+1=2z+1$. Since $\rho(ww^{\prime})$ and
$\rho(\eta^{\prime\prime\prime})$ are both odd, there exists an integer $i$
such that $\rho(ww^{\prime})=\rho(\eta^{\prime\prime\prime})$. Solving for $i$
gives

$\displaystyle 4z-3-2i$ $\displaystyle=2z+1.$

Thus, $\rho(ww^{\prime})=\rho(\eta^{\prime\prime\prime})$ when $i=z-2$.
Recalculating the orders reveals

$\rho(\eta^{\prime})=2z-1$, $\rho(w^{\prime})=z$,
$\rho(\eta^{\prime\prime\prime})=2z+1$, $\rho(w^{\prime\prime\prime})=z+2$,
$\rho((\eta w)^{\prime})=3z$, and $\rho(ww^{\prime})=2z+1$.

  * •

If n is odd: let $z\in{\mathbb{Z}}_{\geq 1}$ such that $n=2z+1$. Then
$\rho(ww^{\prime})=2n-3-2i=4z-1-2i$ and
$\rho(\eta^{\prime\prime\prime})=n+1=2z+2$. Since $\rho(ww^{\prime})$ is odd
and $\rho(\eta^{\prime\prime\prime})$ is even, there will not be an integer
$i$ such that $\rho(ww^{\prime})=\rho(\eta^{\prime\prime\prime})$. So this
argument will break when $\rho(ww^{\prime})<\rho(\eta^{\prime\prime\prime})$,
which will happen when

$\displaystyle 4z-1-2i$ $\displaystyle<2z+2,$

which gives $i>z-\frac{3}{2}$. Thus, $i=z-1$ will be the first iteration such
that the above inequality is achieved. From here, the orders become

$\rho(\eta^{\prime})=2z$, $\rho(w^{\prime})=z$,
$\rho(\eta^{\prime\prime\prime})=2z+2$, $\rho(w^{\prime\prime\prime})=z+2$,
$\rho((\eta w)^{\prime})=3z+1$, and $\rho(ww^{\prime})=2z+1$.

However, $\rho(\eta^{\prime\prime\prime})>\rho(ww^{\prime})$, so from the
second equation, the coefficient of $\operatorname{cn}^{2z+2}$ will solely
come from the $\eta^{\prime\prime\prime}$-term, which implies
$h_{2,2z+2}=-c\lambda^{2}m^{2}(2z+1)(2z+2)(2z+3)j_{2z+1}$. Since $c,\lambda,m$
are not zero, it must be the case that $j_{2z+1}=0$. Recalculating the orders
reveals

$\rho(\eta^{\prime})=2z-1$, $\rho(w^{\prime})=z$,
$\rho(\eta^{\prime\prime\prime})=2z+1$, $\rho(w^{\prime\prime\prime})=z+2$,
$\rho((\eta w)^{\prime})=3z$, and $\rho(ww^{\prime})=2z+1$.

So, regardless of whether $n$ is even or odd, the same reduction of orders is
achieved. The orders of the second equation are now balanced, but the orders
in the first equation are not. Specifically, $\rho((\eta w)^{\prime})$ is
greater than any other term in the first equation. This means that the
coefficient of $\operatorname{cn}^{3z}$ in the first equation of (3.2) will
solely come from the $(\eta w)^{\prime}$-term, which implies
$h_{1,3z}=(3z+1)j_{2z}k_{z+1}$. Demanding $h_{1,3z}=0$ yields that either
$j_{2z}=0$ or $k_{z+1}=0$.

  * •

If $k_{z+1}=0$: recalculating the orders

$\rho(\eta^{\prime})=2z-1$, $\rho(w^{\prime})=z-1$,
$\rho(\eta^{\prime\prime\prime})=2z+1$, $\rho(w^{\prime\prime\prime})=z+1$,
$\rho((\eta w)^{\prime})=3z-1$, and $\rho(ww^{\prime})=2z-1$.

Like in the odd case above,
$\rho(\eta^{\prime\prime\prime})>\rho(ww^{\prime})$, so the coefficients of
$\operatorname{cn}^{2z+1}$ and $\operatorname{cn}^{2z}$ are solely given by
the $\eta^{\prime\prime\prime}$-term. This implies that
$h_{2,2z+1}=-c\lambda^{2}m^{2}(2z)(2z+1)(2z+2)j_{2z}$. Since $c,\lambda,m$ are
not zero, it must be the case that $j_{2z}=0$. Similarly, since
$h_{2,2z}=-c\lambda^{2}m^{2}(2z-1)(2z)(2z+1)j_{2z-1}$, the same logic yields
$j_{2z-1}=0$. Thus,

$\rho(\eta^{\prime})=2z-3$, $\rho(w^{\prime})=z-1$,
$\rho(\eta^{\prime\prime\prime})=2z-1$, $\rho(w^{\prime\prime\prime})=z+1$,
$\rho((\eta w)^{\prime})=3z-3$, and $\rho(ww^{\prime})=2z-1$.

  * •

If $j_{2z}=0$: recalculating the orders

$\rho(\eta^{\prime})=2z-2$, $\rho(w^{\prime})=z$,
$\rho(\eta^{\prime\prime\prime})=2z$, $\rho(w^{\prime\prime\prime})=z+2$,
$\rho((\eta w)^{\prime})=3z-1$, and $\rho(ww^{\prime})=2z+1$.

Now, $\rho(ww^{\prime})>\rho(\eta^{\prime\prime\prime})$, so it is similarly
deduced that $h_{2,2z+1}=(z+1)k_{z+1}^{2}$ which implies $k_{z+1}=0$.
Therefore,

$\rho(\eta^{\prime})=2z-2$, $\rho(w^{\prime})=z-1$,
$\rho(\eta^{\prime\prime\prime})=2z$, $\rho(w^{\prime\prime\prime})=z+1$,
$\rho((\eta w)^{\prime})=3z-2$, and $\rho(ww^{\prime})=2z-1$.

Again $\rho(\eta^{\prime\prime\prime})>\rho(ww^{\prime}),$ so $j_{2z-1}=0$.
Consequently,

$\rho(\eta^{\prime})=2z-3$, $\rho(w^{\prime})=z-1$,
$\rho(\eta^{\prime\prime\prime})=2z-1$, $\rho(w^{\prime\prime\prime})=z+1$,
$\rho((\eta w)^{\prime})=3z-3$, and $\rho(ww^{\prime})=2z-1$.

So, regardless of whether $j_{2z}=0$ or $k_{z+1}=0$, the same order reduction
is achieved. After $i$ iterations of this argument, the orders become

$\rho(\eta^{\prime})=2z-3-2i$, $\rho(w^{\prime})=z-1-i$,
$\rho(\eta^{\prime\prime\prime})=2z-1-2i$,
$\rho(w^{\prime\prime\prime})=z+1-i$, $\rho((\eta w)^{\prime})=3z-3-3i$, and
$\rho(ww^{\prime})=2z-1-2i$.

By balancing the orders, there are three instances in which this argument
breaks. From the first equation, the argument breaks when $\rho((\eta
w)^{\prime})\leq\rho(w^{\prime\prime\prime})$ or $\rho((\eta
w)^{\prime})\leq\rho(\eta^{\prime\prime\prime})$. From the second equation,
the argument will break when
$\rho(ww^{\prime})=\rho(\eta^{\prime\prime\prime})\leq\rho(w^{\prime\prime\prime})$.

$\rho((\eta w)^{\prime})\leq\rho(w^{\prime\prime\prime}):\quad 3z-3-3i\leq
z+1-i\quad\iff\quad z-2\leq i$;

$\rho((\eta w)^{\prime})\leq\rho(\eta^{\prime\prime\prime}):\quad 3z-3-3i\leq
2z-1-2i\quad\iff\quad z-2\leq i$;

$\rho(ww^{\prime})=\rho(\eta^{\prime\prime\prime})\leq\rho(w^{\prime\prime\prime}):\quad
2z-1-2i\leq z+1-i\quad\iff\quad z-2\leq i$.

So this argument will break after $(z-2)$ iterations for all three cases.
Plugging $i=z-2$ and recalculating the orders a final time yields

$\rho(\eta^{\prime})=1,\quad\rho(w^{\prime})=1,\quad\rho(\eta^{\prime\prime\prime})=3,\quad\rho(w^{\prime\prime\prime})=3,\quad\rho((\eta
w)^{\prime})=3,\quad\text{and}\quad\rho(ww^{\prime})=3.$ (3.5)

Since $c\neq 0$ and $a^{2}+b^{2}\neq 0$, the highest order terms in both
equations are balanced. Furthermore, since
$\rho(\eta^{\prime})=\rho(w^{\prime})=1$, the identities in (2.1) imply that
$\rho(\eta)=\rho(w)=2$. Therefore, if $c\neq 0$ and $a^{2}+b^{2}\neq 0$, then
$j_{r}=k_{r}=0$, for all $r\geq 3$.
(ii) If $a^{2}+b^{2}=0$, then $a=b=0$. With this, the first equation in (3.1)
is reduced to

$-\sigma\eta^{\prime}+w^{\prime}+(\eta w)^{\prime}=0.$

The orders in (3.5) are now unbalanced since $\rho((\eta
w)^{\prime})>\rho(\eta^{\prime})=\rho(w^{\prime})$. So, $h_{1,3}=4j_{2}k_{2}$
which implies either $j_{2}=0$ or $k_{2}=0$.

  * •

If $j_{2}=0$: then it follows that $h_{1,2}=3j_{1}k_{2}$, so either $j_{1}=0$
or $k_{2}=0$.

    * –

If $j_{1}=0$, then the desired result is obtained.

    * –

If $k_{2}=0$, then the orders in the second equation of (3.1) are unbalanced,
and $h_{2,1}=k_{1}^{2}$ which yields $k_{1}=0$. Now
$\rho(ww^{\prime})<\rho(\eta^{\prime\prime\prime})$ so from the same argument
used in the proof of (i), it follows that $j_{1}=0$.

  * •

If $k_{2}=0$: then $h_{1,2}=3j_{2}k_{1}$, so either $j_{2}=0$ or $k_{1}=0$.

    * –

If $j_{2}=0$, then we arrive at the same case as above, and it follows that
$j_{1}=k_{1}=0$.

    * –

If $k_{1}=0$, then $w=k_{0}$ which implies $ww^{\prime}=0$. Now, the second
equation in (3.1) becomes

$\eta^{\prime}+c\eta^{\prime\prime\prime}=0,$

but $\rho(\eta^{\prime\prime\prime})>\rho(\eta^{\prime})$ and it quickly
follows that $j_{2}=j_{1}=0$.

Therefore, when $c\neq 0$ and $a^{2}+b^{2}=0$, $j_{r}=0$ for all integers
$r\geq 1$. Moreover, the same result proved in (i) holds for $w$, that is,
$k_{r}=0$ for all integers $r\geq 3$. Thus, $\eta$ is constant so there are
only semi-trivial periodic traveling-wave solutions in this case.
∎

From the above analysis, we see that non-trivial periodic traveling-wave
solutions to (3.1) may only occur when $c\neq 0$ and $a^{2}+b^{2}\neq 0$, so
we will focus on this case. Under these conditions, $j_{r}=k_{r}=0$ for all
integers $r\geq 3$, so (3.2) reduces to

$\left\\{\begin{matrix}\begin{split}&-\lambda\operatorname{sn}(\lambda\xi,m)\operatorname{dn}(\lambda\xi,m)\sum_{q=0}^{3}h_{1,q}\operatorname{cn}^{q}(\lambda\xi,m)=0,\\\
&-\lambda\operatorname{sn}(\lambda\xi,m)\operatorname{dn}(\lambda\xi,m)\sum_{q=0}^{3}h_{2,q}\operatorname{cn}^{q}(\lambda\xi,m)=0.\end{split}\end{matrix}\right.$

Next, by demanding $h_{1,q}=0$ and $h_{2,q}=0$, we obtain the following system
of 8 equations with 9 unknowns, $\lambda,\,m,\,\sigma,\,j_{i},$ and $k_{i}$
for $i=0,1,2$:

$\begin{cases}h_{1,3}=-24\,b{\lambda}^{2}{m}^{2}\sigma\,j_{{2}}-24\,a{\lambda}^{2}{m}^{2}k_{{2}}+4\,j_{{2}}k_{{2}},\\\\[5.0pt]
h_{1,2}=-6\,b{\lambda}^{2}{m}^{2}\sigma\,j_{{1}}-6\,a{\lambda}^{2}{m}^{2}k_{{1}}+3\,j_{{1}}k_{{2}}+3\,j_{{2}}k_{{1}},\\\\[5.0pt]
h_{1,1}=16\,b{\lambda}^{2}{m}^{2}\sigma\,j_{{2}}+16\,a{\lambda}^{2}{m}^{2}k_{{2}}-8\,b{\lambda}^{2}\sigma\,j_{{2}}-8\,a{\lambda}^{2}k_{{2}}-2\,\sigma\,j_{{2}}+2\,j_{{0}}k_{{2}}\\\
\qquad+2\,j_{{1}}k_{{1}}+2\,j_{{2}}k_{{0}}+2\,k_{{2}},\\\\[5.0pt]
h_{1,0}=2\,b{\lambda}^{2}{m}^{2}\sigma\,j_{{1}}+2\,a{\lambda}^{2}{m}^{2}k_{{1}}-b{\lambda}^{2}\sigma\,j_{{1}}-a{\lambda}^{2}k_{{1}}-\sigma\,j_{{1}}+j_{{0}}k_{{1}}+j_{{1}}k_{{0}}+k_{{1}},\\\\[5.0pt]
h_{2,3}=-24\,d{\lambda}^{2}{m}^{2}\sigma\,k_{{2}}-24\,c{\lambda}^{2}{m}^{2}j_{{2}}+2\,k_{{2}}^{2},\\\\[5.0pt]
h_{2,2}=-6\,d{\lambda}^{2}{m}^{2}\sigma\,k_{{1}}-6\,c{\lambda}^{2}{m}^{2}j_{{1}}+3\,k_{{1}}k_{{2}},\\\\[5.0pt]
h_{2,1}=16\,d{\lambda}^{2}{m}^{2}\sigma\,k_{{2}}+16\,c{\lambda}^{2}{m}^{2}j_{{2}}-8\,d{\lambda}^{2}\sigma\,k_{{2}}-8\,c{\lambda}^{2}j_{{2}}-2\,\sigma\,k_{{2}}+2\,k_{{0}}k_{{2}}\\\
\qquad+k_{{1}}^{2}+2\,j_{{2}},\\\\[5.0pt]
h_{2,0}=2\,d{\lambda}^{2}{m}^{2}\sigma\,k_{{1}}+2\,c{\lambda}^{2}{m}^{2}j_{{1}}-d{\lambda}^{2}\sigma\,k_{{1}}-c{\lambda}^{2}j_{{1}}-\sigma\,k_{{1}}+k_{{0}}k_{{1}}+j_{{1}}\,.\end{cases}$
(3.6)

The exact periodic traveling-wave solutions to (3.1) could then be established
by solving the system of nonlinear equations (3.6) with the help of the
computer software Maple. As there is one degree of freedom, any of the 9
unknowns can be chosen as a free parameter. In most physical settings, it is
desirable to have the elliptic modulus $m$ as the free parameter. With this in
mind, the solutions found in the next section will be given in terms of $m$,
along with the systems’ parameters $a,\,b,\,c,$ and $d$. In Section 4.1, a
solution is found with $j_{1}=k_{1}=0$. For this special case, (3.6) is
reduced to a system of 4 equations with 7 unknowns which creates two new
degrees of freedom. In this case $\lambda$ and $\sigma$, the wavelength and
the speed of the wave respectively, will be chosen along with $m$ as the free
parameters in our solutions. Furthermore, semi-trivial solutions for the case
when $c\neq 0$ and $a^{2}+b^{2}=0$ will be presented in Section 4.3.

###  3.2 The case when $c=0$

When $c=0$, system (3.1) reduces to

$\begin{cases}-\sigma\eta^{\prime}+w^{\prime}+(\eta
w)^{\prime}+aw^{\prime\prime\prime}+b\sigma\eta^{\prime\prime\prime}=0\,,\\\
-\sigma w^{\prime}+\eta^{\prime}+ww^{\prime}+d\sigma
w^{\prime\prime\prime}=0\,.\end{cases}$ (3.7)

In addition, (3.2) will take on the form

$\left\\{\begin{matrix}\begin{split}&-\lambda\operatorname{sn}(\lambda\xi,m)\operatorname{dn}(\lambda\xi,m)\sum_{q=0}^{2n-1}\tilde{h}_{1,q}\operatorname{cn}^{q}(\lambda\xi,m)=0,\\\
&-\lambda\operatorname{sn}(\lambda\xi,m)\operatorname{dn}(\lambda\xi,m)\sum_{q=0}^{2n-1}\tilde{h}_{2,q}\operatorname{cn}^{q}(\lambda\xi,m)=0,\end{split}\end{matrix}\right.$
(3.8)

where $\tilde{h}_{1,q}=h_{1,q}\rvert_{c=0}$ and
$\tilde{h}_{2,q}=h_{2,q}\rvert_{c=0}$.

Notice that as (3.8) must hold true for all $(\lambda\xi,m)$, it must be the
case that $\tilde{h}_{1,q}=0$ and $\tilde{h}_{2,q}=0$ for each $q$. Using
this, the following theorem is established. (Recall that $j_{r}$ and $k_{r}$
are the coefficients of $\operatorname{cn}^{r}$ as given in (1.5) for
$r\in\mathbb{Z}_{\geq 0}$.)

######  Theorem 3.2.

Suppose $c=0$, then the periodic traveling-wave ansatz (1.5) will take on one
of the following forms depending on the values of $b$ and $d$:

  1. (i)

If $b^{2}+d^{2}\neq 0$, then $j_{r}=0$ for all integers $r\geq 5$, and
$k_{r}=0$ for all integers $r\geq 3$. Thus, (1.5) reduces to

$\begin{split}\eta(\xi)&=j_{0}+j_{1}\operatorname{cn}(\lambda\xi,m)+j_{2}\operatorname{cn}^{2}(\lambda\xi,m)+j_{3}\operatorname{cn}^{3}(\lambda\xi,m)+j_{4}\operatorname{cn}^{4}(\lambda\xi,m)\\\
&\qquad\text{and}\quad
w(\xi)=k_{0}+k_{1}\operatorname{cn}(\lambda\xi,m)+k_{2}\operatorname{cn}^{2}(\lambda\xi,m)\,.\end{split}$

  2. (ii)

If $b^{2}+d^{2}=0$, then $j_{r}=k_{r}=0$ for all integers $r\geq 1$.
Consequently, only the trivial solution exists in this case.

###### Proof.

(i) Let $\rho(\eta^{\prime})$ denote the largest power of $\operatorname{cn}$
in $\eta^{\prime}$. Then,

$\rho(\eta^{\prime})=\rho(w^{\prime})=n-1$,
$\rho(\eta^{\prime\prime\prime})=\rho(w^{\prime\prime\prime})=n+1$, and
$\rho((\eta w)^{\prime})=\rho(ww^{\prime})=2n-1$.

If $n>2$ then $2n-1>n+1$, which implies the coefficient of
$\operatorname{cn}^{2n-1}$ in the second equation of (3.8) will solely come
from the $ww^{\prime}$-term. Thus, $\tilde{h}_{2,2n-1}=nk_{n}^{2}$. Setting
this coefficient equal to zero gives $k_{n}=0$. Recalculating the above orders
yields

$\rho(\eta^{\prime})=n-1$, $\rho(w^{\prime})=n-2$,
$\rho(\eta^{\prime\prime\prime})=n+1$, $\rho(w^{\prime\prime\prime})=n$,
$\rho((\eta w)^{\prime})=2n-2$, and $\rho(ww^{\prime})=2n-3$.

After $i$ iterations of this argument, the orders become

$\rho(\eta^{\prime})=n-1$, $\rho(w^{\prime})=n-2-i$,
$\rho(\eta^{\prime\prime\prime})=n+1$, $\rho(w^{\prime\prime\prime})=n-i$,
$\rho((\eta w)^{\prime})=2n-2-i$, and $\rho(ww^{\prime})=2n-3-2i$.

In the proof of Theorem 3.1, it was shown that this argument can be repeated
until $\rho(ww^{\prime})\leq\rho(\eta^{\prime\prime\prime})$, but for the
$c=0$ case, the $\eta^{\prime\prime\prime}$-term is no longer present in the
second equation of (3.1). Thus, the argument will break when $\rho\big{(}(\eta
w)^{\prime}\big{)}\leq\rho(\eta^{\prime})$. Now, consider the two
possibilities for $n$.

  * •

If n is even: let $z\in{\mathbb{Z}}_{\geq 1}$ such that $n=2z$. Then
$\rho(ww^{\prime})=2n-3-2i=4z-3-2i$ and $\rho(\eta^{\prime})=n-1=2z-1$. Since
$\rho(ww^{\prime})$ and $\rho(\eta^{\prime})$ are both odd, there exists an
integer $i$ such that $\rho(ww^{\prime})=\rho(\eta^{\prime})$. Solving for $i$
gives

$\displaystyle 4z-3-2i$ $\displaystyle=2z-1.$

Thus, $\rho(ww^{\prime})=\rho(\eta^{\prime})$ when $i=z-1$. Recalculating the
orders yields

$\rho(\eta^{\prime})=2z-1$, $\rho(w^{\prime})=z-1$,
$\rho(\eta^{\prime\prime\prime})=2z+1$, $\rho(w^{\prime\prime\prime})=z+1$,
$\rho((\eta w)^{\prime})=3z-1$, and $\rho(ww^{\prime})=2z-1$.

  * •

If n is odd: let $z\in{\mathbb{Z}}_{\geq 1}$ such that $n=2z+1$. Then
$\rho(ww^{\prime})=2n-3-2i=4z-1-2i$ and $\rho(\eta^{\prime})=n-1=2z$. Since
$\rho(ww^{\prime})$ is odd and $\rho(\eta^{\prime})$ is even, there will not
be an integer $i$ such that $\rho(ww^{\prime})=\rho(\eta^{\prime})$. So this
argument will break when $\rho(ww^{\prime})<\rho(\eta^{\prime})$, which will
happen when

$\displaystyle 4z-1-2i$ $\displaystyle<2z,$

which yields $i>z-\frac{1}{2}$. Thus, $i=z$ will be the first iteration such
that the above inequality is achieved. From here, the orders become

$\rho(\eta^{\prime})=2z$, $\rho(w^{\prime})=z-1$,
$\rho(\eta^{\prime\prime\prime})=2z+2$, $\rho(w^{\prime\prime\prime})=z+1$,
$\rho((\eta w)^{\prime})=3z$, and $\rho(ww^{\prime})=2z-1$.

However, $\rho(\eta^{\prime})>\rho(ww^{\prime})$, so the coefficient of
$\operatorname{cn}^{2z}$ in the second equation will solely come from the
$\eta^{\prime}$-term, which implies $\tilde{h}_{2,2z}=(2z+1)j_{2z+1}$. It then
follows that $j_{2z+1}=0$. Recalculating the orders yields

$\rho(\eta^{\prime})=2z-1$, $\rho(w^{\prime})=z-1$,
$\rho(\eta^{\prime\prime\prime})=2z+1$, $\rho(w^{\prime\prime\prime})=z+1$,
$\rho((\eta w)^{\prime})=3z-1$, and $\rho(ww^{\prime})=2z-1$.

So, regardless of whether $n$ is even or odd, the same reduction of orders is
achieved. Now the orders of the second equation are balanced, but the orders
in the first equation are not. Specifically, $\rho((\eta w)^{\prime})$ is
greater than any other term in the first equation. This means that the
coefficient of $\operatorname{cn}^{3z-1}$ in the first equation of (3.2) will
solely come from the $(\eta w)^{\prime}$-term, which implies
$\tilde{h}_{1,3z-1}=3zj_{2z}k_{z}$. Demanding $\tilde{h}_{1,3z-1}=0$ reveals
that either $j_{2z}=0$ or $k_{z}=0$.

  * •

If $k_{z}=0$: recalculating the orders

$\rho(\eta^{\prime})=2z-1$, $\rho(w^{\prime})=z-2$,
$\rho(\eta^{\prime\prime\prime})=2z+1$, $\rho(w^{\prime\prime\prime})=z$,
$\rho((\eta w)^{\prime})=3z-2$, and $\rho(ww^{\prime})=2z-3$.

Like in the odd case above, $\rho(\eta^{\prime})>\rho(ww^{\prime})$, so the
coefficients of $\operatorname{cn}^{2z-1}$ and $\operatorname{cn}^{2z-2}$ are
solely given by the $\eta^{\prime}$-term. This implies that
$\tilde{h}_{2,2z-1}=2zj_{2z}$, so it follows that $j_{2z}=0$. Similarly since
$\tilde{h}_{2,2z-2}=(2z-1)j_{2z-1}$, the same logic yields $j_{2z-1}=0$. Thus,

$\rho(\eta^{\prime})=2z-3$, $\rho(w^{\prime})=z-2$,
$\rho(\eta^{\prime\prime\prime})=2z-1$, $\rho(w^{\prime\prime\prime})=z$,
$\rho((\eta w)^{\prime})=3z-4$, and $\rho(ww^{\prime})=2z-3$.

  * •

If $j_{2z}=0$: recalculating the orders

$\rho(\eta^{\prime})=2z-2$, $\rho(w^{\prime})=z-1$,
$\rho(\eta^{\prime\prime\prime})=2z$, $\rho(w^{\prime\prime\prime})=z+1$,
$\rho((\eta w)^{\prime})=3z-2$, and $\rho(ww^{\prime})=2z-1$.

Now, $\rho(ww^{\prime})>\rho(\eta^{\prime})$, so it is similarly deduced that
$\tilde{h}_{2,2z-1}=zk_{z}^{2}$ which implies $k_{z}=0$. Therefore,

$\rho(\eta^{\prime})=2z-2$, $\rho(w^{\prime})=z-2$,
$\rho(\eta^{\prime\prime\prime})=2z$, $\rho(w^{\prime\prime\prime})=z$,
$\rho((\eta w)^{\prime})=3z-3$, and $\rho(ww^{\prime})=2z-3$.

Again $\rho(\eta^{\prime})>\rho(ww^{\prime})$ so $j_{2z-1}=0$. Consequently,

$\rho(\eta^{\prime})=2z-3$, $\rho(w^{\prime})=z-2$,
$\rho(\eta^{\prime\prime\prime})=2z-1$, $\rho(w^{\prime\prime\prime})=z$,
$\rho((\eta w)^{\prime})=3z-4$, and $\rho(ww^{\prime})=2z-3$.

So, regardless of whether $j_{2z}=0$ or $k_{z}=0$, the same order reduction is
achieved. After $i$ iterations of this argument, the orders become

$\rho(\eta^{\prime})=2z-3-2i$, $\rho(w^{\prime})=z-2-i$,
$\rho(\eta^{\prime\prime\prime})=2z-1-2i$, $\rho(w^{\prime\prime\prime})=z-i$,
$\rho((\eta w)^{\prime})=3z-4-3i$, and $\rho(ww^{\prime})=2z-3-2i$.

By balancing the orders, there are two instances in which this argument
breaks. From the first equation, the argument breaks when $\rho((\eta
w)^{\prime})\leq\rho(\eta^{\prime\prime\prime})$, and from the second
equation, the argument will break when
$\rho(ww^{\prime})=\rho(\eta^{\prime})\leq\rho(w^{\prime\prime\prime})$.

$\rho((\eta w)^{\prime})\leq\rho(\eta^{\prime\prime\prime}):\quad 3z-4-3i\leq
2z-1-2i\quad\iff\quad z-3\leq i$;

$\rho(ww^{\prime})=\rho(\eta^{\prime})\leq\rho(w^{\prime\prime\prime}):\quad
2z-3-2i\leq z-i\quad\iff\quad z-3\leq i$.

So this argument will break after $(z-3)$ iterations for both cases. Plugging
$i=z-3$ and recalculating the orders a final time yields

$\rho(\eta^{\prime})=3,\quad\rho(w^{\prime})=1,\quad\rho(\eta^{\prime\prime\prime})=5,\quad\rho(w^{\prime\prime\prime})=3,\quad\rho((\eta
w)^{\prime})=5,\quad\text{and}\quad\rho(ww^{\prime})=3.$

Since $c\neq 0$ and $b^{2}+d^{2}\neq 0$, the highest order terms in both
equations are balanced. Furthermore, since $\rho(\eta^{\prime})=3$ and
$\rho(w^{\prime})=1$, the identities in (2.1) imply that $\rho(\eta)=4$ and
$\rho(w)=2$. Therefore, if $c\neq 0$ and $b^{2}+d^{2}\neq 0$, then $j_{r}=0$
for all $r\geq 5$ and $k_{r}=0$ for all $r\geq 3$.
(ii) If $b^{2}+d^{2}=0$, then $b=d=0$. Recall that the parameters $a,\,b,\,c,$
and $d$ must satisfy the three conditions given in (1.3). Since $b=c=d=0$, it
follows that $a=\frac{1}{3}$. With this, system (3.7) reduces to

$\begin{cases}-\sigma\eta^{\prime}+w^{\prime}+(\eta
w)^{\prime}+\dfrac{1}{3}w^{\prime\prime\prime}=0\,,\\\ -\sigma
w^{\prime}+\eta^{\prime}+ww^{\prime}=0\,.\end{cases}$

Now, $\tilde{h}_{1,5}$ and $\tilde{h}_{2,3}$ are given by

$\tilde{h}_{1,5}=6j_{4}k_{2}\quad\text{and}\quad\tilde{h}_{2,3}=4j_{4}+2k_{2}^{2}.$

Requiring these both to be zero implies that $j_{4}=k_{2}=0$. Substituting
this into the rest of the coefficients, it follows that that
$\tilde{h}_{2,2}=3j_{3}$ which implies $j_{3}=0$. Now, $\tilde{h}_{1,2}$ and
$\tilde{h}_{2,1}$ are given by

$\tilde{h}_{1,2}=-k_{1}(2\lambda^{2}m^{2}-3j_{2})\quad\text{and}\quad\tilde{h}_{2,1}={k_{1}}^{2}+2j_{2}.$

Requiring $\tilde{h}_{2,1}=0$ yields $j_{2}=-\frac{1}{2}{k_{1}}^{2}$, and
substituting this into $\tilde{h}_{1,2}$ gives

$\tilde{h}_{1,2}=-\frac{1}{2}k_{1}(4\lambda^{2}m^{2}+3{k_{1}}^{2}).$

Since $\lambda,m>0$ and $k_{1}\in\mathbb{R}$, it follows that
$4\lambda^{2}m^{2}+3{k_{1}}^{2}>0$. Thus, $k_{1}=0$, and similarly, $j_{2}=0$.
Finally $\tilde{h}_{2,0}=j_{1}$, forcing $j_{1}=0$.

Therefore, when $c\neq 0$ and $b^{2}+d^{2}=0$, it is deduced that
$j_{r}=k_{r}=0$ for all integers $r\geq 1$ which implies both $\eta$ and $w$
are constant and consequently, only the trivial solution exists.
∎

The above theorem shows that periodic traveling-wave solutions to (3.7) may
only occur when $c=0$ and $b^{2}+d^{2}\neq 0$, so we will focus on this case.
Substituting $j_{r}=0$ for all $r\geq 5$ and $k_{r}=0$ for all $r\geq 3$ into
(3.2) yields

$\left\\{\begin{matrix}\begin{split}&-\lambda\operatorname{sn}(\lambda\xi,m)\operatorname{dn}(\lambda\xi,m)\sum_{q=0}^{5}\tilde{h}_{1,q}\operatorname{cn}^{q}(\lambda\xi,m)=0,\\\
&-\lambda\operatorname{sn}(\lambda\xi,m)\operatorname{dn}(\lambda\xi,m)\sum_{q=0}^{3}\tilde{h}_{2,q}\operatorname{cn}^{q}(\lambda\xi,m)=0.\end{split}\end{matrix}\right.$

Next, by demanding $\tilde{h}_{1,q}=0$ and $\tilde{h}_{2,q}=0$, we obtain the
following system of 10 equations and 11 unknowns,
$\lambda,\,m,\,\sigma,\,j_{i}$ for $i=0,1,\dots,4$, and $k_{i}$ for $i=0,1,2$:

$\begin{cases}\tilde{h}_{1,5}=-6\,j_{{4}}\left(20\,b{\lambda}^{2}{m}^{2}\sigma-
k_{{2}}\right)\,,\\\\[5.0pt]
\tilde{h}_{1,4}=-60\,b{\lambda}^{2}{m}^{2}\sigma\,j_{{3}}+5\,j_{{3}}k_{{2}}+5\,j_{{4}}k_{{1}}\,,\\\\[5.0pt]
\tilde{h}_{1,3}=-24\,b{\lambda}^{2}{m}^{2}\sigma\,j_{{2}}+128\,b{\lambda}^{2}{m}^{2}\sigma\,j_{{4}}-24\,a{\lambda}^{2}{m}^{2}k_{{2}}-64\,b{\lambda}^{2}\sigma\,j_{{4}}-4\,\sigma\,j_{{4}}+4\,j_{{2}}k_{{2}}\\\
\qquad+4\,j_{{3}}k_{{1}}+4\,j_{{4}}k_{{0}}\,,\\\\[5.0pt]
\tilde{h}_{1,2}=-6\,b{\lambda}^{2}{m}^{2}\sigma\,j_{{1}}+54\,b{\lambda}^{2}{m}^{2}\sigma\,j_{{3}}-6\,a{\lambda}^{2}{m}^{2}k_{{1}}-27\,b{\lambda}^{2}\sigma\,j_{{3}}-3\,\sigma\,j_{{3}}+3\,j_{{1}}k_{{2}}\\\
\qquad+3\,j_{{2}}k_{{1}}+3\,j_{{3}}k_{{0}}\,,\\\\[5.0pt]
\tilde{h}_{1,1}=16\,b{\lambda}^{2}{m}^{2}\sigma\,j_{{2}}-24\,b{\lambda}^{2}{m}^{2}\sigma\,j_{{4}}+16\,a{\lambda}^{2}{m}^{2}k_{{2}}-8\,b{\lambda}^{2}\sigma\,j_{{2}}+24\,b{\lambda}^{2}\sigma\,j_{{4}}\\\
\qquad-8\,a{\lambda}^{2}k_{{2}}-2\,\sigma\,j_{{2}}+2\,j_{{0}}k_{{2}}+2\,j_{{1}}k_{{1}}+2\,j_{{2}}k_{{0}}+2\,k_{{2}}\,,\\\\[5.0pt]
\tilde{h}_{1,0}=2\,b{\lambda}^{2}{m}^{2}\sigma\,j_{{1}}-6\,b{\lambda}^{2}{m}^{2}\sigma\,j_{{3}}+2\,a{\lambda}^{2}{m}^{2}k_{{1}}-b{\lambda}^{2}\sigma\,j_{{1}}+6\,b{\lambda}^{2}\sigma\,j_{{3}}-a{\lambda}^{2}k_{{1}}-\sigma\,j_{{1}}\\\
\qquad+j_{{0}}k_{{1}}+j_{{1}}k_{{0}}+\sigma k_{{1}}^{2}\,k_{{2}}\,,\\\\[5.0pt]
\tilde{h}_{2,3}=-24\,d{\lambda}^{2}{m}^{2}\sigma\,k_{{2}}+2\,k_{{2}}^{2}+4\,j_{{4}}\,,\\\\[5.0pt]
\tilde{h}_{2,2}=-6\,d{\lambda}^{2}{m}^{2}\sigma\,k_{{1}}+3\,k_{{1}}k_{{2}}+3\,j_{{3}}\,,\\\\[5.0pt]
\tilde{h}_{2,1}=16\,d{\lambda}^{2}{m}^{2}\sigma\,k_{{2}}-8\,d{\lambda}^{2}\sigma\,k_{{2}}-2\,\sigma\,k_{{2}}+2\,k_{{0}}k_{{2}}+k_{{1}}^{2}+2\,j_{{2}}\,,\\\\[5.0pt]
\tilde{h}_{2,0}=2\,d{\lambda}^{2}{m}^{2}\sigma\,k_{{1}}-d{\lambda}^{2}\sigma\,k_{{1}}-\sigma\,k_{{1}}+k_{{0}}k_{{1}}+j_{{1}}\,.\end{cases}$
(3.9)

Numerical computation shows that if $j_{1}\neq 0$ or $j_{3}\neq 0$, then there
does not exist a solution to (3.9). Similarly, numerical computation shows if
$k_{1}\neq 0$, then the only solution to (3.9) is one such that $\sigma=0$, a
contradiction. So, we will proceed under the assumption that
$j_{1}=j_{3}=k_{1}=0$. In particular, the only non-trivial periodic traveling
wave solutions to (3.7) are of the form

$\eta(\xi)=j_{0}+j_{2}\operatorname{cn}^{2}(\lambda\xi,m)+j_{4}\operatorname{cn}^{4}(\lambda\xi,m)\\\
\quad\text{and}\quad w(\xi)=k_{0}+k_{2}\operatorname{cn}^{2}(\lambda\xi,m)\,.$
(3.10)

Moreover, this reduces (3.9) to the following system of 5 equations with 8
unknowns:

$\begin{cases}\bar{h}_{1,5}=-6\,j_{{4}}\left(20\,b{\lambda}^{2}{m}^{2}\sigma-
k_{{2}}\right)\,,\\\\[5.0pt]
\bar{h}_{1,3}=-24\,b{\lambda}^{2}{m}^{2}\sigma\,j_{{2}}+128\,b{\lambda}^{2}{m}^{2}\sigma\,j_{{4}}-24\,a{\lambda}^{2}{m}^{2}k_{{2}}-64\,b{\lambda}^{2}\sigma\,j_{{4}}-4\,\sigma\,j_{{4}}+4\,j_{{2}}k_{{2}}\\\
\qquad+4\,j_{{4}}k_{{0}}\,,\\\\[5.0pt]
\bar{h}_{1,1}=16\,b{\lambda}^{2}{m}^{2}\sigma\,j_{{2}}-24\,b{\lambda}^{2}{m}^{2}\sigma\,j_{{4}}+16\,a{\lambda}^{2}{m}^{2}k_{{2}}-8\,b{\lambda}^{2}\sigma\,j_{{2}}+24\,b{\lambda}^{2}\sigma\,j_{{4}}\\\
\qquad-8\,a{\lambda}^{2}k_{{2}}-2\,\sigma\,j_{{2}}+2\,j_{{0}}k_{{2}}+2\,j_{{2}}k_{{0}}+2\,k_{{2}}\,,\\\\[5.0pt]
\bar{h}_{2,3}=-24\,d{\lambda}^{2}{m}^{2}\sigma\,k_{{2}}+2\,{k_{{2}}}^{2}+4\,j_{{4}}\,,\\\\[5.0pt]
\bar{h}_{2,1}=16\,d{\lambda}^{2}{m}^{2}\sigma\,k_{{2}}-8\,d{\lambda}^{2}\sigma\,k_{{2}}-2\,\sigma\,k_{{2}}+2\,k_{{0}}k_{{2}}+2\,j_{{2}}\,,\\\\[5.0pt]
\bar{h}_{1,4}=\bar{h}_{1,2}=\bar{h}_{1,0}=\bar{h}_{2,2}=\bar{h}_{2,0}=0\,,\end{cases}$
(3.11)

where $\bar{h}_{1,q}=\tilde{h}_{1,q}\rvert_{j_{1}=j_{3}=k_{1}=0}$ and
$\bar{h}_{2,q}=\tilde{h}_{2,q}\rvert_{j_{1}=j_{3}=k_{1}=0}$.

The exact periodic traveling-wave solutions to (3.7) could then be established
by solving the system of nonlinear equations (3.11) with the help of the
computer software Maple. As there are three degrees of freedom, any
combination of the 8 unknowns could be chosen as free parameters. For
consistency, we will choose $\lambda$, $m$, and $\sigma$ as free parameters,
like in the previous section.

##  4 Exact Jacobi Elliptic Solutions

Finally, explicit solutions to the $abcd$-system (1.2) are given for the cases
$c\neq 0$ and $c=0$. Discussion of these solutions, and how they limit to
solitary wave solutions, will be left for the next section. In addition to the
explicit solutions being stated, graphs of these solutions will also be
provided.

###  4.1 The case when $c\neq 0$

Solving (3.6), our solution will take the form as stated by (3.3), where the
parameters are found to be given by one of the following sets.

  1. 1.

When $ac(b-6d)(3b-2d),\,c(2m^{2}-1)(b+2d)(3b+2d)<0$ and
$(2m^{2}-1)(3b+2d)(b-d)\geq 0$ with equality only when $b=d$ :

$\begin{cases}j_{{0}}=-{\dfrac{a\,(3\,b+2\,d)\,(21\,b-46\,d)+2\,c\,(3\,b-2\,d)\,(b-6\,d)}{2c\left(3\,b-2\,d\right)\left(b-6\,d\right)}}\,,\\\\[9.0pt]
j_{{1}}=-\tau_{1}\tau_{2}{\dfrac{12\,a\,m\left(3\,b+2\,d\right)\sqrt{\left(2\,{m}^{2}-1\right)\left(3\,b+2\,d\right)\left(b-d\right)}}{c\left(b-6\,d\right)\left(3\,b-2\,d\right)\left(2\,{m}^{2}-1\right)}}\,,\\\\[9.0pt]
j_{{2}}={\dfrac{9\,a\,{m}^{2}\left(3\,b+2\,d\right)}{c\left(3\,b-2\,d\right)\left(2\,{m}^{2}-1\right)}}\,,\\\\[9.0pt]
k_{{0}}=\tau_{{1}}{\dfrac{\left(21\,b+8\,c+14\,d\right)\sqrt{-2\,ac\left(b-6\,d\right)\left(3\,b-2\,d\right)}}{2\,c\left(b-6\,d\right)\left(3\,b-2\,d\right)}}\,,\\\\[9.0pt]
k_{{1}}=\tau_{2}{\dfrac{6\,m\sqrt{-2\,ac\left(b-6\,d\right)\left(3\,b-2\,d\right)\left(2\,{m}^{2}-1\right)\left(3\,b+2\,d\right)\left(b-d\right)}}{c\left(2\,{m}^{2}-1\right)\left(b-6\,d\right)\left(3\,b-2\,d\right)}}\,,\\\\[9.0pt]
k_{{2}}=-\tau_{1}{\dfrac{9\,m^{2}\left(3\,b+2\,d\right)\sqrt{-2\,ac\left(b-6\,d\right)\left(3\,b-2\,d\right)}}{c\left(2\,{m}^{2}-1\right)\left(b-6\,d\right)\left(3\,b-2\,d\right)}}\,,\\\\[9.0pt]
\lambda=\dfrac{1}{2}\sqrt{{\dfrac{-6\,\left(3\,b+2\,d\right)}{c\left(2\,{m}^{2}-1\right)\left(b+2\,d\right)}}}\,,\\\\[9.0pt]
\sigma=\tau_{1}{\dfrac{4\sqrt{-2\,ac\left(b-6\,d\right)\left(3\,b-2\,d\right)}}{\left(b-6\,d\right)\left(3\,b-2\,d\right)}}\,,\\\\[9.0pt]
m\in(\,0,1\,]\,.\end{cases}$ (4.1)

Finding this solution requires taking the square root of two terms, which
leads to two independent plus or minuses. To avoid confusion, we define

$\tau_{1}=\\{\,1,\,-1\,\\}\quad\text{and}\quad\tau_{2}=\\{\,1,\,-1\,\\},$

to denote these independent plus or minuses. Two graphs of this solution can
be found below in Figures 1(a) and 1(b), in which $\eta$ is graphed in blue
and $w$ in green. The explicit parameters chosen for each graph are given in
the captions.

(a) $\\{m=\frac{3}{4},\,a=-\frac{5}{6},\,b=1,\,c=-\frac{5}{6},\,d=1,$
$\tau_{1}=1,\,\tau_{2}=1\\}$

(b) $\\{m=\frac{1}{4},\,a=-7,\,b=2,\,c=\frac{4}{3},\,d=4,$
$\tau_{1}=-1,\,\tau_{2}=1\\}$

Figure 1: Graphs of solution (4.1)

  2. 2.

When $8ac+\sigma^{2}(b-2d)^{2}>0\,$:

$\begin{cases}j_{1}=k_{1}=0\,,\\\\[9.0pt]
j_{{0}}={\dfrac{1}{4\,c^{2}\left(4\,a\,c+\sigma^{2}\left({b}^{2}+4\,{d}^{2}\right)\pm\sigma\left(b+2\,d\right)\sqrt{8\,a\,c+\sigma^{2}\left(b-2\,d\right)^{2}}\right)}}\Big{(}-8\,{b}^{4}c\,{\lambda}^{2}{m}^{2}{\sigma}^{4}\\\\[9.0pt]
\qquad+16\,{b}^{3}c\,d\,{\lambda}^{2}{m}^{2}{\sigma}^{4}-64\,a\,{b}^{2}{c}^{2}{\lambda}^{2}{m}^{2}{\sigma}^{2}-32\,a\,b\,{c}^{2}d\,{\lambda}^{2}{m}^{2}{\sigma}^{2}-64\,a\,{c}^{2}{d}^{2}{\lambda}^{2}{m}^{2}{\sigma}^{2}\\\\[5.0pt]
\qquad+4\,{b}^{4}c\,{\lambda}^{2}{\sigma}^{4}-8\,{b}^{3}c\,d\,{\lambda}^{2}{\sigma}^{4}-64\,{a}^{2}{c}^{3}{\lambda}^{2}{m}^{2}+32\,a\,{b}^{2}{c}^{2}{\lambda}^{2}{\sigma}^{2}+16\,a\,b\,{c}^{2}d\,{\lambda}^{2}{\sigma}^{2}\\\\[5.0pt]
\qquad+32\,a\,{c}^{2}{d}^{2}{\lambda}^{2}{\sigma}^{2}+{b}^{4}{\sigma}^{4}-4\,{b}^{3}d\,{\sigma}^{4}+4\,{b}^{2}{d}^{2}{\sigma}^{4}+32\,{a}^{2}{c}^{3}{\lambda}^{2}+8\,a\,{b}^{2}c\,{\sigma}^{2}-8\,a\,b\,c\,d\,{\sigma}^{2}\\\\[5.0pt]
\qquad-4\,{b}^{2}{c}^{2}{\sigma}^{2}-16\,{c}^{2}{d}^{2}{\sigma}^{2}+8\,{a}^{2}{c}^{2}-16\,a\,{c}^{3}\mp\,\sigma\sqrt{8\,a\,c+\sigma^{2}\left(b-2\,d\right)^{2}}\Big{(}8\,{b}^{3}c\,{\lambda}^{2}{m}^{2}{\sigma}^{2}\\\\[5.0pt]
\qquad+32\,a\,b\,{c}^{2}{\lambda}^{2}{m}^{2}+32\,a\,{c}^{2}d{\lambda}^{2}{m}^{2}-4\,{b}^{3}c\,{\lambda}^{2}{\sigma}^{2}-16\,a\,b\,{c}^{2}{\lambda}^{2}-16\,a\,{c}^{2}d\,{\lambda}^{2}-{b}^{3}{\sigma}^{2}\\\\[5.0pt]
\qquad+2\,{b}^{2}d\,{\sigma}^{2}-4\,a\,b\,c+4\,b\,{c}^{2}+8\,{c}^{2}d\Big{)}\Big{)}\,,\\\\[9.0pt]
j_{2}=\dfrac{3\,\lambda^{2}m^{2}}{2\,c}\Big{(}4\,a\,c+b\,{\sigma}^{2}\left(b-2\,d\right)\pm
b\,\sigma\,\sqrt{8\,a\,c+{\sigma}^{2}\left(b-2\,d\right)^{2}}\Big{)}\,,\\\\[9.0pt]
k_{{0}}={\dfrac{1}{2\,c\left(\sigma\left(b+2\,d\right)\pm\sqrt{8\,a\,c+\sigma^{2}\left(b-2\,d\right)^{2}}\right)}}\Big{(}-8\,{b}^{2}c\,{\lambda}^{2}{m}^{2}{\sigma}^{2}-32\,c\,{d}^{2}{\lambda}^{2}{m}^{2}{\sigma}^{2}\\\\[9.0pt]
\qquad-32\,a\,{c}^{2}{\lambda}^{2}{m}^{2}+4\,{b}^{2}c\,{\lambda}^{2}{\sigma}^{2}+16\,c\,{d}^{2}{\lambda}^{2}{\sigma}^{2}+16\,a\,{\lambda}^{2}{c}^{2}-{b}^{2}{\sigma}^{2}+2\,b\,c\,{\sigma}^{2}+2\,b\,d\,{\sigma}^{2}\\\\[5.0pt]
\qquad+4\,c\,d\,{\sigma}^{2}-4\,a\,c\mp\sigma\sqrt{8\,a\,c+{\sigma}^{2}\left(b-2\,d\right)^{2}}\Big{(}8\,b\,c\,{\lambda}^{2}{m}^{2}+16\,c\,d\,{\lambda}^{2}{m}^{2}-4\,b\,c\,{\lambda}^{2}\\\\[5.0pt]
\qquad-8\,c\,d\,{\lambda}^{2}+b-2\,c\Big{)}\Big{)}\,,\\\\[9.0pt]
k_{2}=3\,\lambda^{2}m^{2}\left(\sigma\left(b+2\,d\right)\pm\sqrt{8\,a\,c+\sigma^{2}\left(b-2\,d\right)^{2}}\right)\,,\\\\[9.0pt]
\lambda>0,\,\sigma\neq 0,\,m\in(\,0,1\,]\,.\end{cases}$ (4.2)

Above, the $\pm$ and $\mp$ should be chosen such that the top or bottom signs
are chosen together. Two graphs of this solution can be found below in Figures
2(a) and 2(b), in which $\eta$ is graphed in blue and $w$ in green. The
explicit parameters chosen for each graph are given in the captions.

(a) $\\{\lambda=1,\,m=\frac{1}{\sqrt{2}},\,\sigma=1,\,a=1,\,b=-\frac{8}{3},$
$c=1,\,d=1\\}$ and the top signs of $\pm$ and $\mp$

(b)
$\\{\lambda=\frac{1}{2},\,m=\frac{1}{4},\,\sigma=-\frac{1}{3},\,a=0,\,b=-1,\,c=-\frac{2}{3},\,d=2\\}$
and the bottom signs of $\pm$ and $\mp$

Figure 2: Graphs of solution (4.2)

###  4.2 The case when $c=0$

Solving (3.11), our solution will take the form as stated by (3.10), where the
parameters are found to be given by one of the following sets.

  1. 1.

When $4b-d\neq 0\,$:

$\begin{cases}j_{0}=\dfrac{1}{9\,{\sigma}^{2}\left(4\,b-d\right)^{2}}\Big{(}-32\,b\,\lambda^{4}\sigma^{4}\left(4\,b-d\right)^{2}\left(5\,b-3\,d\right)\left(11\,{m}^{4}-11\,{m}^{2}-4\right)\\\\[9.0pt]
\qquad+3\,\sigma^{2}\left(4\,b-d\right)\big{[}3\,d-4\,b\left(3+5\,a\,\lambda^{2}\left(2\,{m}^{2}-1\right)\right)\big{]}+9\,a^{2}\Big{)}\,,\\\\[9.0pt]
j_{2}={\dfrac{20\,b\,{\lambda}^{2}{m}^{2}\left(3\,a+4\,\lambda^{2}\sigma^{2}\left(4\,b-d\right)\left(5\,b-3\,d\right)\left(2\,{m}^{2}-1\right)\right)}{3(4\,b-d)}}\,,\\\\[9.0pt]
j_{4}=-40\,b\,{\lambda}^{4}{m}^{4}{\sigma}^{2}\left(5\,b-3\,d\right)\,,\\\\[9.0pt]
k_{{0}}={\dfrac{-3\,a+\sigma^{2}\left(4\,b-d\right)\left({3}-20\,b\,{\lambda}^{2}\left(2{m}^{2}-1\right)\right)}{3\,\sigma\left(4\,b-d\right)}}\,,\\\\[9.0pt]
k_{{2}}=20\,b\,{\lambda}^{2}{m}^{2}\sigma\,,\\\\[9.0pt] \lambda>0,\,\sigma\neq
0,\,m\in(\,0,1\,]\,.\end{cases}$ (4.3)

Two graphs of this solution can be found below in Figures 3(a) and 3(b), in
which $\eta$ is graphed in blue and $w$ in green. The explicit parameters
chosen for each graph are given in the captions.

(a)
$\\{\lambda=\frac{1}{2},\,m=\frac{1}{2},\,\sigma=-2,\,a=1,b=-1,\,d=\frac{1}{3}\\}$

(b)
$\\{\lambda=1,\,m=\frac{9}{10},\,\sigma=1,\,a=0,b=\frac{1}{6},\,d=\frac{1}{6}\\}$

Figure 3: Graphs of solution (4.3)

  2. 2.

When $b-2d\neq 0\,$:

$\begin{cases}j_{{0}}={\dfrac{a^{2}-\sigma^{2}\left(b-2\,d\right)\left(b-2\,d\left(1+2\,a\,\lambda^{2}\left(2\,{m}^{2}-1\right)\right)\right)}{\sigma^{2}\left(b-2\,d\right)^{2}}}\,,\\\\[9.0pt]
j_{2}=-12\,{\dfrac{a\,d\,{\lambda}^{2}{m}^{2}}{b-2\,d}}\,,\\\\[9.0pt]
k_{{0}}={\dfrac{a+\sigma^{2}\left(b-2\,d\right)\left(1-4\,d\,{\lambda}^{2}\left(2\,{m}^{2}-1\right)\right)}{\sigma\left(b-2\,d\right)}}\,,\\\\[9.0pt]
k_{2}=12\,d\,{\lambda}^{2}{m}^{2}\sigma\,,\\\\[9.0pt] \lambda>0,\,\sigma\neq
0,\,m\in(\,0,1\,]\,.\end{cases}$ (4.4)

Two graphs of this solution can be found below in Figures 4(a) and 4(b), in
which $\eta$ is graphed in blue and $w$ in green. The explicit parameters
chosen for each graph are given in the captions.

(a) $\\{\lambda=1,\,m=\frac{1}{\sqrt{2}},\,\sigma=-1,a=-\frac{11}{3},\,b=2,$
$d=2\\}$

(b)
$\\{\lambda=2,\,m=\frac{3}{4},\,\sigma=\frac{1}{8},\,a=0,\,b=-\frac{5}{3},$
$d=2\\}$

Figure 4: Graphs of solution (4.4)

###  4.3 Semi-trivial Solutions

Substituting $a=b=0$ into (3.6) and solving, a solution of the form (3.4) is
found where

$\begin{cases}j_{0}=-1\,,\\\\[5.0pt]
k_{{0}}=-8\,d\,{\lambda}^{2}{m}^{2}\sigma+4\,d\,{\lambda}^{2}\sigma+\sigma\,,\\\\[5.0pt]
k_{1}=0\,,\\\\[5.0pt] k_{2}=12\,d\,{\lambda}^{2}{m}^{2}\sigma\,,\\\\[5.0pt]
\lambda>0,\,\sigma\neq 0,\,m\in(\,0,1\,]\,.\end{cases}$ (4.5)

One may notice that Figures 2(b) and 4(b) are graphs in which $\eta=-1$. In
fact, these are exactly graphs of solution (4.5), so additional graphs need
not be provided here. The relationship between solution (4.5) and solutions
(4.2) and (4.4) is made clear in the next section.

##  5 Conclusion

As stated at the end of Section 4.3, there is a distinct relationship between
solutions (4.2), (4.4), and (4.5). In fact, if $\sigma(b-2d)>0$, then solution
(4.4) can be retrieved by taking the limit as $c$ approaches zero of solution
(4.2) while choosing the bottom signs for $\pm$ and $\mp$. Similarly, solution
(4.5) can be recovered from solutions (4.2) or (4.4) in any of the following
ways

  * •

if $d\sigma>0$, then evaluating solution (4.2) at $a=b=0$ and choosing the top
signs for $\pm$ and $\mp$;

  * •

if $\sigma(b-2d)>0$, then evaluating solution (4.2) at $a=0$ and choosing the
bottom signs for $\pm$ and $\mp$;

  * •

if $b-2d\neq 0$, then evaluating solution (4.4) at $a=0$.

The semi-trivial solution (4.5) is also of importance. When we evaluate system
(1.2) at $\eta=-1$ and $a=0$, it collapses to a single equation. This equation
is

$w_{t}+ww_{x}-dw_{xxt}=0\,,$

which is exactly the BBM equation after a transformation. This equation has
been shown to have cnoidal wave solutions [1], so it is especially important
that we recovered a cnoidal solution for this case.

Another fundamental property of the Jacobi cnoidal function is that as $m$
approaches one, the cnoidal function limits to the hyperbolic secant function.
This is important because solitary-wave solutions are often given in terms of
hyperbolic secant functions. Thus, in principle, taking the limit of our
solutions as $m$ approaches one may retrieve solitary-wave solutions. Indeed,
for (4.2) and (4.4) we get solitary-wave solutions of the form

$\eta=\bar{j}_{0}+\bar{j}_{2}\operatorname{sech}^{2}(\bar{\lambda}(x-\bar{\sigma}t))\quad\text{and}\quad
w=\bar{k}_{0}+\bar{k}_{2}\operatorname{sech}^{2}(\bar{\lambda}(x-\bar{\sigma}t),$
(5.1)

where the bar indicates the evaluation at $m=1$. This straightforward
computation is left to the reader. Some solitary-wave solutions for these
solutions are graphed in Figures 5(a) and 5(b), in which $\eta$ is graphed in
blue and $w$ in green. The exact values chosen for the parameters are given in
the captions of each figure. In the paper [14, 15], solitary-wave solutions
were shown to be given exactly by the same form in (5.1) for cases
$\\{a=0,\,c\neq 0\\}$, $\\{b=d=0,\,a=c\\}$ and $\\{b=c=0,\,d\neq 0\\}$ where
the coefficients solve certain equations.

(a) Solution (4.2) when $\\{\lambda=1,\,m=1,\,\sigma=1,\\\
a=1,\,b=-\frac{8}{3},\,c=1,\,d=1\\}$ and the top signs
of $\pm$ and $\mp$ are chosen

(b) Solution (4.4) when $\\{\lambda=1,\,m=1,\,\sigma=-1,\\\
a=-\frac{11}{3},\,b=2,\,d=2\\}$

Figure 5: Graphs of solitary wave solutions of form (5.1)

From solution (4.3), we get a solitary-wave solution of the form

$\eta=\bar{j}_{0}+\bar{j}_{2}\operatorname{sech}^{2}(\lambda(x-\sigma
t))+\bar{j}_{4}\operatorname{sech}^{4}(\lambda(x-\sigma
t))\quad\text{and}\quad
w=\bar{k}_{0}+\bar{k}_{2}\operatorname{sech}^{2}(\lambda(x-\sigma t)),$ (5.2)

where again, the bar indicates the evaluation at $m=1$. This is graphed in
Figure 6(a), in which $\eta$ is graphed in blue and $w$ in green. The form
found in this paper (5.2) again matches the form of solitary wave solutions
found in [14, 15] for cases $\\{a=c=0\\}$ and $\\{c=d=0,\,b\neq 0\\}$. This
also re-emphasizes exactly what was found in this paper, that solutions with a
$\operatorname{cn}^{4}$ term in $\eta$ can only occur when $c=0$.

Similarly, looking at solution (4.5) as $m$ approaches one, we get a solitary-
wave solution of the form

$\eta=-1\quad\text{and}\quad
w=\bar{k}_{0}+\bar{k}_{2}\operatorname{sech}^{2}(\lambda(x-\sigma t)),$

which was also found in [14, 15] for the case $\\{a=0\\}$.

Of course, there is no one-to-one correspondence between cnoidal solutions and
solitary-wave solutions. Thus, one should not expect to establish all
solitary-wave solutions by taking the limit as the elliptic modulus $m$
approaches one. One such instance of this is found in (4.1), where a solution
of the form

$\eta=\bar{j}_{0}+\bar{j}_{1}\operatorname{sech}(\bar{\lambda}(x-\bar{\sigma}t))+\bar{j}_{2}\operatorname{sech}^{2}(\bar{\lambda}(x-\bar{\sigma}t))\quad\text{and}\quad
w=\bar{k}_{0}+\bar{k}_{1}\operatorname{sech}(\bar{\lambda}(x-\bar{\sigma}t))+\bar{k}_{2}\operatorname{sech}^{2}(\bar{\lambda}(x-\bar{\sigma}t))\,,$
(5.3)

is achieved as $m$ approaches one. This is graphed in Figure 6(b), in which
$\eta$ is graphed in blue and $w$ in green. This form of solution was not
found in [14, 15], showing the lack of correspondence. In turn, a form of
solitary-wave solutions was found in [14, 15] that was not recovered in this
paper. This solution is found when $\\{b=c=d=0,\,a>0\\}$, and takes the form

$\eta=j_{0}+j_{2}\operatorname{sech}^{2}(\lambda(x-\sigma
t))\quad\text{and}\quad w=k_{0}+k_{1}\tanh(\lambda(x-\sigma t)).$

This solution was not established in this paper because hyperbolic tangent is
not a limiting case of the Jacobi cnoidal function.

(a) Solution (4.3) when $\\{\lambda=1,\,m=1,\,\sigma=1,\\\
a=0,\,b=\frac{1}{6},\,d=\frac{1}{6}\\}$

(b) Solution (4.1) when $\\{m=1,\,a=-\frac{5}{6},\,b=1,\\\
c=-\frac{1}{6},\,d=\frac{1}{3},\,\tau_{1}=1,\,\tau_{2}=-1\\}$

Figure 6: Graphs of solitary wave solutions of form (5.2) and (5.3)

Declarations

Conflict of interest: The authors have no relevant financial or non-financial
interests to disclose.
Data availability: Data sharing is not applicable to this article as no
datasets were generated or analysed during this study.

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# Incompatibilities Between Current Practices in Statistical Data Analysis and
Differential Privacy

Joshua Snoke  RAND Corporation, Pittsburgh, PA, USA<EMAIL_ADDRESS>,  Claire
McKay Bowen  Urban Institute, Washington D.C., USA<EMAIL_ADDRESS>,  Aaron
R. Williams  Urban Institute, Washington D.C., USA<EMAIL_ADDRESS>and
Andrés F. Barrientos  Department of Statistics, Florida State University,
Tallahassee, FL, USA<EMAIL_ADDRESS>
###### Abstract.

The authors discuss their experience applying differential privacy with a
complex data set with the goal of enabling standard approaches to statistical
data analysis. They highlight lessons learned and roadblocks encountered,
distilling them into incompatibilities between current practices in
statistical data analysis and differential privacy that go beyond issues which
can be solved with a noisy measurements file. The authors discuss how
overcoming these incompatibilities require compromise and a change in either
our approach to statistical data analysis or differential privacy that should
be addressed head-on.

###### Key words and phrases:

differential privacy; statistical analysis

##  1\. A Brief History of Differential Privacy in the Wild

Researchers and data practitioners make many different claims concerning
differential privacy (DP) and its impact on statistical analysis. Some
maintain that DP provides the future for how government agencies and private
companies will release public statistics and data sets. Others argue that
pursuing DP is a mistake and will destroy how we disseminate information as we
know it. These debates often center on notions of the trade-off between
accuracy and utility, selecting privacy budgets, or the appropriate definition
of privacy loss. While these questions matter, they often fail to acknowledge
other underlying issues. While DP is a framework containing a wide variety of
implementations, there exist fundamental incompatibilities between standard
statistical analysis approaches and the possibilities within a DP
framework111One might rightfully respond that some incompatibilities are not
unique to DP and also exist for alternative statistical disclosure control
methods, and we would agree. For the sake of simplicity, we do not dig into
those questions in this piece. See Slavković and Seeman (2023) for a thorough
article on the similarities and differences between DP and other statistical
disclosure control approaches..

From the position of either a statistical analyst or a privacy practitioner,
dealing with these incompatibilities often comes across in the field as claims
that the other side is “doing it wrong.” In reality, we have two paradigms
functioning under different assumed rules and making them compatible will
require some changes to one or both frameworks. We write this perspective from
the point of view of both privacy researchers and statistical analysts who are
broadly approaching their analysis from a frequentist perspective with the
goal of statistical inference. We also use the term statistical analyst as a
catch-all term encompassing statisticians, economists, demographers, or any
data scientist working in social statistics.

Briefly considering the history of practical DP implementations, early
applications of the framework generally obscured the incompatibilities due to
the specific use-cases. For instance, numerous tech companies created
interactive or query-based DP frameworks that allowed analysts to submit a
question and receive a noisy statistic in return. Some examples include
audience engagement statistics on LinkedIn (Rogers et al., 2021), SQL queries
in Uber’s driver and rider database (Johnson et al., 2018), people’s movement
on Google Maps within certain geographic regions (Aktay et al., 2020), and
HealthKit usage on Apple products.222Differential Privacy Team, Apple. n.d.
“Learning with Privacy at Scale.” https://docsassets.developer.apple.com/ml-
research/papers/learning-with-privacy-at-scale.pdf. These applications enable
analysts to perform learning tasks or other tasks that do not rely on
uncertainty estimates or explicit hypothesis testing, since statistical
learning was the goal for the use-case.

When the U.S. Census Bureau announced their adoption of DP for the 2020 Census
data products, it significantly increased the debate between DP advocates and
DP skeptics of its applicability to demographic data. At a high level, the
2020 Disclosure Avoidance System added differentially private noise to
thousands of statistics at different census geographic level and corrected for
any inconsistencies (e.g., ensuring that population counts for census tracts
sum to the county count) using a complex post-processing method formulated in
the TopDown Algorithm. This change in statistical data privacy protection
created huge backlash from the data user community, including
lawsuits333“Alabama drops lawsuit challenging Census privacy method,”
https://apnews.com/article/alabama-lawsuitscensus-2020-redistricting-us-
census-bureau-3c6f5eacc6c5638756700ba8308c45d2 and a letter to the Director of
the U.S. Census Bureau444“Researchers ask Census to stop controversial privacy
method ,” https://apnews.com/article/census-2020-us-bureau-government-and-
politics-20e683c71eeb62ee4b7792d7d8530419 to stop the use of DP on the 2020
Census data products. While some simply urged not to add noise, constructive
critiques pointed out that statistical analysts needed the ability to adjust
for the noise in order to conduct their usual statistical analyses. In
essence, analysts needed new tools that are not part of traditional
statistical data analysis. The introduction of bias or increased uncertainty
in estimates due to privacy protections was not new to those who had been
working in statistical disclosure control. What was novel is that DP promised
to account or track the added noise rather than ignoring it. But, for some
time, it was not clear how this promise would be achieved.

In response, several leading researchers requested the Census Bureau release
the noisy measurements data set (Dwork et al., 2021). The researchers’
reasoning was that access to the noisy measurement file (NMF) would allow
analysts to account for the noise introduced from the TopDown Algorithm. In
addition, even when statistical analysts have access to the noisy measurement
file, most do not have the background understanding of DP or the computational
ability to make the required corrections. Given this challenge, some privacy
researchers organized a workshop on the “Analysis of Census Noisy Measurement
Files and Differential Privacy.” The purpose of the workshop was to convene
experts from various fields and practices within social science, demography,
public policy, statistics, and computer science to discuss the need for
implementable tools that allowed analyses on privacy preserving noise induced
data and statistics.555“Workshop on the Analysis of Census Noisy Measurement
Files and Differential Privacy,” Accessed February 14, 2023.
http://dimacs.rutgers.edu/events/details?eID=2038

The debate concerning the decennial Census and the noisy measurement file
helped illuminate the distinction between the way noise is added and the
ability to account for it in statistical analyses. Some argued that reverting
to previous statistical data privacy methods, such as swapping, would be
preferable to DP. But fundamentally these methods have the same issues as the
lack of a NMF and using them would not provide statistical analysts with any
better means of analyzing the data. Without a means of accounting for the
noise introduced to protect privacy, any statistical analyses will be biased
or falsely overpowered.

##  2\. Differentially Private Query Systems for Statistical Inference

At this point in the story, it may be tempting to argue that any issues
performing statistical analysis under DP can be handled by the development of
NMFs and tools to account for the additional noise. The widespread publicity
of the Census Bureau’s adoption of DP locked in, for many, a particular means
of achieving DP, with the corresponding solution being access to the NMF. But
these solutions only makes sense under the scenario where a set of
predetermined measurements are made and released non-interactively, such as
publishing a public data set. In fact, the example of the decennial Census is
still rather similar to applications of DP in the tech space. The underlying
data were very large and the queries were counts.

Implementing DP on other products, such as smaller surveys or interactive
query systems with the purpose of statistical inference, introduces an
entirely new set of incompatibilities. The adoption of DP for settings where
researchers make sequences of queries that encompass more complex statistical
analyses will require the field to overcome more significant barriers than
those that faced the Census Bureau.

In our work, Barrientos et al. (2021), we explore creating a differentially
private query system, known as a validation server, to allow tax researchers
to estimate simple statistics and linear regressions on confidential IRS data.
In contrast to the decennial census, this system must allow for interactive
queries and include statistic-specific uncertainty estimates for hypothesis
testing, such as confidence intervals.

When conducting that study, the first difficulty we encountered was that only
a small fraction of the published papers proposing DP algorithms for querying
common statistics, such as means and regression coefficients, provide
uncertainty estimates. Another challenge was that accurate queries required
users to input substantial information about the distribution or the range of
the underlying data which is not commonly known. Additionally, an astute user
needs to determine how to allocate their privacy budget. But, in many cases,
they may not know the complete set of queries they want to perform before
starting. Finally, designing an interactive query system that allows different
types of queries with optimal performance under DP is not guaranteed to
compose in a trivial manner, though it is theoretically possible Rogers et al.
(2016); Whitehouse et al. (2023).

While still in progress, working with an interdisciplinary team has pushed us
towards considering compromises that may dissatisfy either the data analyst or
the privacy practitioner. Under one of our proposed systems, users may develop
and test analyses on non-formally private synthetic data that represent a more
limited subset of individuals prior to submitting their analysis to the
validation server. Conversely, our server includes a much more limited set of
allowed queries than a typical tax economist would expect. In particular,
methods for working with data from complex surveys that need to incorporate
survey weights, the backbone of the federal statistical system, do not exist.

Another proposed compromise is using two sets of privacy budgets, such that
users can conduct some initial analyses prior to exhausting their budget on
estimates that can be released. On the other hand, users still carry the
burden of bringing prior information for their analyses, such as indicating
the range of possible values or spending some privacy budget to estimate it.
At this point, we do not know the full implications of these decisions or
whether they will be featured in the eventual implementation of the validation
server. We only highlight the various compromises we have wrestled through to
enable statistical analysis and DP to function together.

##  3\. Incompatibilities That Require Compromise

Based on the lessons we have learned, we offer the following general
incompatibilities between DP and normal statistical practice that must be
addressed in practice implementations. When the goal is traditional
statistical analysis, specifically inference tasks, overcoming the
incompatibilities requires compromises, either from the statistical analysts,
formal privacy practitioners, or both.

(a) Estimates for traditional statistical inference. Frequentist methods for
statistical inference rely on estimates, such as confidence intervals or
$p$-values to perform hypothesis testing. DP methods have only been shown to
provide guarantees for statistical accuracy in scenarios where the size of the
data set is large enough (e.g., Chaudhuri et al., 2011; Sheffet, 2017; Peña
and Barrientos, 2021; Sart, 2023), whereas other papers simply choose not to
evaluate statistical inference tasks. While large sample properties, also
referred to as asymptotic properties, are universally desirable, they
frequently fail to provide substantial insight into what can be expected for
specific finite sample sizes. This is because the condition of being “large
enough” is difficult to define. Additionally, post-processing can frequently
induce bias, and DP implementations have not been shown in practice to provide
estimates with amounts of noise that statistical analysts would consider
reasonable without sacrificing substantial amounts of privacy666This is
measured through the privacy parameter, most commonly $\epsilon$, associated
with the definition of differential privacy that a method satisfies..
Furthermore, effective DP methods do not exist for more complex models
involving survey weights, panel data, or methods for causal inference.

A few compromise options exist to handle this incompatibility. First, a
statistical data user can only query point estimates and decide not to perform
frequentist hypothesis testing. Alternatively, users may opt for Bayesian
approaches. While requiring privacy budgets similar to frequentist methods,
Bayesian approaches can account for the assumptions and probabilistic nature
behind DP, and they can be used for full inference about the parameters and
predictions. Specifically, Bayesian techniques allow for the simultaneous
consideration of various factors, such as the use of non-sufficient summary
statistics, assumptions about data boundedness (clamping), and noise injection
for privacy. These elements are difficult for traditional frequentist
statistics to handle simultaneously. Though Bayesian methods offer a promising
alternative, they reflect a compromise because they remain unused and
unfamiliar to many in the statistical research community. The field needs
concrete work answering this question. If we need to consider alternative
approaches to frequentist inference, how will this impact statistical
practice? If we can only conduct frequentist inference under different privacy
definitions, is accurate statistical inference possible?

(b) Control or nuisance variables. In estimating regression models, it is very
common to include control variables for which the estimates are not really of
interest to the person querying the model. Current differentially private
methods do not account for this, nor is it clear what it would look like if
they did. In regression, for example, queries that include uncertainty
estimates, namely those that perturb the sufficient statistics, add noise that
grows polynomially with the number of predictors. Other methods, such as Wang
(2018), scale better with predictors but require a Bayesian inference approach
and have not been shown to be practically implementable (Barrientos et al.,
2021). Whatever the approach, queries which return these parameters lead to
heightened noise added to estimates of interest as the number of control
variables grows, and it can be argued that this noise is extraneous even if it
only scales linearly.

Can DP be reformulated to ignore the privacy-loss due to these control
variables? Is including control variables in a regression query but not
receiving coefficients possible? How would this impact the privacy guarantee?
If so, is there still value in spending privacy budget on nuisance variables?
Conversely, can statistical practice be changed such that appropriate analyses
can be made without including control variables?

(c) Assumptions on the range of the data or other assumptions. A common
barrier for statistical analysts using DP methods is the need to place prior
bounds on the distribution of data or statistics in order to calculate a
finite sensitivity. Knowing this information or finding ways of estimating it
apart from the data is not always part of statistical practice. In many cases,
there are no good priors to help set these bounds, and the DP literature is
largely silent on this problem.

In some cases, analysts may be able to estimate the bounds under DP. For
example, Wilson et al. (2019) propose a method for automatically estimating
bounds on continuous data by taking advantage of the physical limitations of
machine precision and minimizing the amount of data clipping. This helps
select bounds without prior information, but a fundamental bias-variance
trade-off exists when privately estimating bounds Amin et al. (2019). More
problematically, the analysts cannot know where they fall on this bias-
variance trade-off without knowing the real bounds of the data. This inhibits
inferential methods to adjust for the fact that the final estimates include
uncertainty both from the noise mechanism of the final query and the noise
mechanism of the prior bound-setting query. Either statistical analysts will
need to adapt their methodology to account for this, or DP methods will need
to adapt to enable analysts to estimate the impact of privately setting
bounds.

(d) Performing exploratory data analysis. Statistical analysts commonly
explore the data using visualizations, marginal and multivariate summary
statistics, and model diagnostics. Most statistical researchers, who we assume
are not trusted to access the private data, do not know ahead of time exactly
what analyses will be run. In one sense, DP can help disincentivize bad
exploratory data analysis (EDA) practices, such as $p$-hacking. For example,
if an analyst must split their privacy budget across EDA queries, they may
limit the amount of exploration they make, potentially making it less likely
for them to find spurious results. Conversely, DP may make it more difficult
to account for multiple testing, since the final inferences should account for
the uncertainty propagated through all the analyses performed to select the
final model. Given the issues discussed earlier concerning frequentist
inference under DP and the lack of work on multiple testing, it is not clear
whether this is feasible.

Though some in the broader scientific community are moving towards pre-
specifying every model in research777For example, https://plos.org/open-
science/preregistration/., this is far from the current reality in all
disciplines. And not all EDA results in $p$-hacking. Data users need the
ability to probe assumptions or look for data abnormalities, and they may run
into serious problems without this ability. For example, without understanding
the data a user may request for a regression model where a predictor has no
variance. Under DP, this query may return random noise or a null result with
some probability, and statistical analysts are not prepared for this type of
response. Either statistical analysts will need to adapt their research
without the ability to do EDA, or DP methods need to find ways to allow EDA in
a private setting.

(e) Limited queries and the privacy budget. Finally, there are a broader set
of issues that come from performing statistical analyses with a limited
privacy budget. This is a concept both unfamiliar to statistical analysts and
carrying significant implications. For example, how should data maintainers
allocate budgets to numerous analysts? How do analysts determine how much of
their budget to allocate to multiple model specifications, robustness checks,
and the final models? What should analysts do if a journal reviewer asks for
alternative model specifications or other requests, such as reproducing the
results, and there is no more privacy budget to spend?

More challenges occur when multiple data users or analysts submit the same
analysis. As an illustrative example, imagine there are two data users (A and
B), who submit the same analysis on the same part of the confidential data,
but user A submitted before data user B. The validation server could handle
this situation in two ways.

One approach is to consider the analyses from user A and B as separate
analyses. In other words, these analyses would likely produce two different
results regardless if user A and B used the same or different privacy loss
budgets. Under this approach, both data users would not be notified about each
others’ analyses, ensuring greater confidentiality and encouraging analysts to
use the validation server results more confidently. Knowing their specific
research ideas will not be revealed mitigates concerns about being scooped or
having their ideas preempted. But producing different answers for the same
query creates communication and education problems in explaining to both data
users that their answers are valid. It is also means, from a societal point of
view, that we are sacrificing more privacy (or accuracy) to run the same
analysis twice.

The other approach is to apply the same result from data user A for data user
B. Unlike the first approach, there would be no confusion of having two
different results for the same analysis. The data users could also share the
cost of the privacy loss budget, spending less of their individual budgets.
However, both data users would be informed that the analysis was conducted
twice. We could even make this more complex and extend it to the situation
where data user B wanted a more accurate result and spend more privacy loss
budget than data user A.

These are some of many other issues we need to address. In any case, either
statistical analysts will need to develop novel means of optimally allocating
their budget for their research or DP will need to rethink budgets in
interactive query systems.

##  4\. Future Steps

We hope this perspective will serve as a means of calling out the broader set
of issues beyond problems that can be solved using a noisy measurements file.
As of now, it is not clear whether the compromise comes from the way we
perform statistical analyses, the way we implement DP, or both. We hope that
future papers on DP will wrestle through these practical questions in a larger
way than has been typically done to this point.

## Acknowledgments

This research was funded by the Alfred P. Sloan Foundation [G-2020-14024] and
National Science Foundation National Center for Science and Engineering
Statistics [49100420C0002 and 49100422C0008].

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