#  $k$-Diophantine $m$-tuples in finite fields

Trajan Hammonds, Seoyoung kim, Steven J. Miller, Arjun nigam, Kyle Onghai,
dishant saikia, lalit M. sharma

###### Abstract.

In this paper, we define a $k$-Diophantine $m$-tuple to be a set of $m$
positive integers such that the product of any $k$ distinct positive integers
is one less than a perfect square. We study these sets in finite fields
$\mathbb{F}_{p}$ for odd prime $p$ and guarantee the existence of a
$k$-Diophantine m-tuple provided $p$ is larger than some explicit lower bound.
We also give a formula for the number of 3-Diophantine triples in
$\mathbb{F}_{p}$ as well as an asymptotic formula for the number of
$k$-Diophantine $k$-tuples.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Legendre Symbol and Their Sums
    2. 2.2 Gauss’s Lemma
    3. 2.3 Weil’s Theorem and Quadratic Character Sums
  3. 3 Proofs of the main results
    1. 3.1 Existence of $k$-Diophantine $m$-tuples
    2. 3.2 Counting 3-Diophantine Triples
      1. 3.2.1 Counting Problems
    3. 3.3 Asymptotic Formula
  4. 4 Concluding remarks
  5. A Proofs from the Preliminaries
  6. B Tables and Graphs

##  1\. Introduction

The study of Diophantine $m$-tuples can be traced to the work of Diophantus of
Alexandria, and has caught the attention of numerous leading mathematicians
since then. In the 3rd century, Diophantus observed that the set of four
numbers:
$\left\\{\frac{1}{16},\frac{33}{16},\frac{17}{4},\frac{105}{16}\right\\}$
satisfy an interesting property that the product of any two elements in the
set is one less than a rational square. This is the first example of a
rational Diophantine quadruple. In the 17th century, Fermat became interested
in finding integer solutions and eventually found the Diophantine quadruple
$\\{1,3,8,120\\}$. Euler discovered that the Diophantine quadruple given by
Fermat can be extended to form a rational Diophantine quintuple, namely
$\left\\{1,3,8,120,\frac{777480}{8288641}\right\\}$. These sets of numbers
studied by Diophantus, Fermat and Euler are now known as Diophantine
$m$-tuples, which we define below.

######  Definition 1.1.

Let $S$ be a set of $m$ positive integers $\\{a_{1},a_{2},\ldots,a_{m}\\}$. If
$a_{i}a_{j}+1$ is a perfect square for all $i,j$ such that $1\leq i<j\leq m$,
then $S$ is a Diophantine $m$-tuple.

Similarly, we define a rational Diophantine $m$-tuple as follows. If $S$ is a
set of $m$ positive rationals and satisfies the same condition, it is called a
rational Diophantine $m$-tuple. For a more in-depth overview of the history of
this problem, see [Dic20, p. 513-519].

The first important result concerning the size of Diophantine $m$-tuples was
given by Baker and Davenport in 1969 [BD69]. They showed using Baker’s theory
on linear forms in logarithms of algebraic numbers that if $d$ is a positive
integer such that $\\{1,3,8,d\\}$ is a Diophantine quadruple, then $d$ has to
be 120, implying that $\\{1,3,8,120\\}$ cannot be extended to a Diophantine
quintuple. In 1979, Arkin, Hoggatt and Strauss showed that any Diophantine
triple can be extended to a Diophantine quadruple [AHS79]. In 2004, Dujella
proved that there is no Diophantine sextuple and that there are at most
finitely many Diophantine quintuples [Duj04]. In 2018, He, Togbé and Ziegler
showed that there does not exist a Diophantine quintuple [HTZ18].

In the case of rationals, no absolute upper bound for the size of rational
Diophantine $m$-tuples is known. Euler proved that there are infinitely many
rational Diophantine quintuples. In 1999, Gibbs found the first rational
Diophantine sextuple
$\left\\{\frac{11}{192},\frac{35}{192},\frac{155}{27},\frac{512}{27},\frac{1235}{48},\frac{180873}{16}\right\\}$
[Gib]. In 2017, Dujella, Kazalicki, Mikić, Szikszai proved that there are
infinitely many rational Diophantine sextuples [DKMS16]. It is not known
whether there are rational Diophantine septuples.

There are many generalizations of Diophantine $m$-tuples. One natural
generalization which has been extensively studied is if we replace the number
1 in “$a_{i}a_{j}+1$” with $n$. These sets are called _Diophantine $m$-tuples
with the property $D(n)$_. Recently, Bliznac Trebješanin and Filipin proved
that there is no $D(4)$-quintuple [BF19]. Dujella, Filipin and Fuchs proved
that there does not exist a $D(-1)$-quintuple and that there are at most
finitely many $D(-1)$-quadruples, all of them containing the element 1 [DF05,
DFF07]. Recently, Bonciocat, Cipu and Mignotte proved the nonexistence of
$D(-1)$-quadruples [BCM20].

There is an interesting connection between Diophantine $m$-tuples and elliptic
curves. If $\\{a,b,c\\}$ are assumed to form a Diophantine triple, then in
order to extend this triple to a quadruple, the task is to find an integer $x$
such that $ax+1,bx+1$ and $cx+1$ are all squares of integers. Finding a
solution $x\in\mathbb{Z}$ to the three simultaneous conditions implies that
there exists $y\in\mathbb{Z}$ such that

(1.1) $y^{2}\ =\ (ax+1)(bx+1)(cx+1);$

this equation describes an elliptic curve. Hence, extending a Diophantine
triple to a Diophantine quadruple is equivalent to finding integer solutions
of the mentioned elliptic curve. A more detailed survey on Diophantine
$m$-tuples and its connections with elliptic curves can be found at [Duj] and
[DS21].

While most of the work on Diophantine $m$-tuples has been done over integers
and rationals, Diophantine $m$-tuples may be studied over any commutative ring
with identity. Studies have been made over the ring of integers in a quadratic
field ([Fra08], [Fra09] and [FS14]) by Franušić and Soldo. In 2013, Franušić
also studied Diophantine quadruples over a cubic field [Fra13]. Recently,
Dujella and Kazalicki studied Diophantine $m$-tuples over finite fields
$\mathbb{F}_{p}$ where $p$ is an odd prime in [DK21]. They proved the
existence of a Diophantine $m$-tuple in $\mathbb{F}_{p}$ where $p$ is a prime
and $p>2^{2m-2}m^{2}$. Using character sums, they also derive expressions for
the number of Diophantine pairs, triples, and quadruples in $\mathbb{F}_{p}$
for given prime $p$, and provide an asymptotic formula for the number of
Diophantine $m$-tuples. In recent years, there has been a lot of activity on
Diophantine $m$-tuples and its generalizations. To get an extensive list of
papers on Diophantine $m$-tuples, we refer the interested reader to Dujella’s
list of references: https://web.math.pmf.unizg.hr/~duje/ref.html.

We study a generalization of Diophantine $m$-tuples called $k$-Diophantine
$m$-tuples.

######  Definition 1.2.

Let $S$ be a set of $m$ positive integers $\\{a_{1},a_{2},\ldots,a_{m}\\}$. If
$1+\prod_{j=i_{1}}^{i_{k}}a_{j}$ is a perfect square for all
$i_{1},\ldots,i_{k}\in\\{1,2,\ldots,m\\}$ such that $1\leq
i_{1}<i_{2}<\cdots<i_{k}\leq m$, then $S$ is a $k$-Diophantine $m$-tuple.

One motivation behind studying these sets is the relationship between
$k$-Diophantine $k$-tuples and a well-known, open problem in number theory
known as Brocard’s problem. Brocard’s problem asks for all integer solutions
$(n,m)$ to the equation $n!+1=m^{2}$. It can be clearly observed that if the
elements of a $k$-Diophantine $k$-tuple are consecutive natural numbers
starting from 1, then it gives a solution for Brocard’s problem. Currently,
there are only three known pairs of numbers solving Brocard’s problem:
$(4,5),(5,11),(7,71)$. Erdős conjectured that no other solutions exist. In
1993, Overholt proved that there are only finitely many solutions to Brocard’s
problem provided that the $abc$ conjecture is true [Ove93]. Till now,
computations for $n$ up to a magnitude of $10^{15}$ have been done but yielded
no further solutions for the problem.

Moreover, just as Brocard’s problem is not a trivial exercise, the same can be
said of finding $k$-Diophantine $m$-tuples. Similar to the connection between
Diophantine triples, i.e., 2-Diophantine triples, and elliptic curves, a
connection can also be made between 3-Diophantine triples and elliptic curves.
Indeed, the problem of extending a 3-Diophantine triple $\\{a,b,c\\}$ to a
3-Diophantine quadruple $\\{a,b,c,d\\}$ is equivalent to finding integer
solutions of the elliptic curve

(1.2) $y^{2}\ =\ (abx+1)(acx+1)(bcx+1).$

Hence, for even the simpler cases of $k$ and $m$, finding $k$-Diophantine
$m$-tuples is already of the same complexity and importance as finding
integral solutions of an elliptic curve. As no efficient, general algorithm to
find integral solutions of an elliptic curve has been found yet, there is no
algorithm to find the number of ways to extend a 3-Diophantine triple to a
3-Diophantine quadruple. In fact, the same can be said about the problem of
extending $k$-Diophantine $k$-tuples to $k$-Diophantine $(k+1)$-tuples.

Inspired by the work of Dujella and Kazalicki [DK21], we studied
$k$-Diophantine $m$-tuples in finite fields $\mathbb{F}_{p}$ where $p$ is an
odd prime. We show the existence of at least one $k$-Diophantine $m$-tuple for
all primes $p$ that are sufficiently large, and give a formula for the number
of 3-Diophantine triples in $\mathbb{F}_{p}$.

In Section 2, we provide results that we need to present the proofs of our new
results. Next, we show the following theorems.

######  Theorem 1.3.

Let $m\geq k$ be an integer. If
$p>4^{\binom{m}{k-1}+1}\left(\frac{\binom{m}{k-1}}{2}+m+1\right)^{2}$ is a
prime, then there exists at least one $k$-Diophantine $m$-tuple in
$\mathbb{F}_{p}$.

Then, we prove a theorem about the number of 3-Diophantine triples in
$\mathbb{F}_{p}$.

######  Theorem 1.4.

Let $N_{3}(p)$ be the number of 3-Diophantine triples in $\mathbb{F}_{p}$. If
$p\equiv 1\text{ mod 3}$, let $a$ be an integer such that $a\equiv 2\text{ mod
3}$ and $p=a^{2}+3b^{2}$ for some integer $b>0$. Then,

(1.3) $N_{3}(p)\ =\ \begin{cases}\frac{a+1}{3}+\binom{p-1}{3}/2,&\emph{for
}p\equiv 1\pmod{3}\\\ \binom{p-1}{3}/2,&\emph{for }p\equiv
2\pmod{3}.\end{cases}$

To do this, we need to show the following.

######  Theorem 1.5.

We have

(1.4) $\\#\left\\{(a,b,c)\in\mathbb{F}_{p}^{3}\mathrel{\mathop{\mathchar
58\relax}}abc+1\equiv 0\pmod{p}\right\\}\ =\
\begin{cases}(p-2)(p-3)+4,&\emph{if }p\equiv 1\pmod{3}\\\ (p-2)(p-3),&\emph{if
}p\equiv 2\pmod{3}.\end{cases}$

Finally, we prove the following asymptotic formula for the number of
$k$-Diophantine $k$-tuples in $\mathbb{F}_{p}$ holds.

######  Theorem 1.6.

Let $N_{k}(p)$ be the number of $k$-Diophantine $k$-tuples in
$\mathbb{F}_{p}$. Then

(1.5) $N_{k}(p)\ \thicksim\ \frac{p^{k}}{k!\cdot 2}+o(p^{k}).$

##  2\. Preliminaries

###  2.1. Legendre Symbol and Their Sums

First, let us define an operation from number theory known as the Legendre
symbol.

We recall that if $a,p\in\mathbb{Z}$ with $p$ prime, $\gcd(a,p)=1$, then the
Legendre Symbol, denoted as $\Big{(}\frac{a}{p}\Big{)}$ is

(2.1) $\left(\frac{a}{p}\right)\ \coloneqq\ \begin{cases}0&\mbox{if }p\mid
a\\\ 1&\mbox{if $a$ is a quadratic residue modulo $p$}\\\ -1&\mbox{if $a$ is a
quadratic nonresidue modulo $p$}.\end{cases}$

Note: In the finite field $F_{p}$ where $p$ is an odd prime, the Legendre
symbol is equivalent to the quadratic character [LN97, p. 191].

In determining the formula for the number of 3-Diophantine triples in
$\mathbb{F}_{p}$, we relied on two well-known sums of Legendre symbols.
Consider a given polynomial $f$ with integer coefficients. The two well-known
sums are special cases of the sum

(2.2) $\sum_{x=0}^{p-1}\left(\frac{f(x)}{p}\right)..$

If $f$ is linear, then we have the following result.

######  Lemma 2.1.

For arbitrary integers $a$ and $b$, and a prime $p\nmid a$, we have

(2.3) $\sum_{x=0}^{p-1}\left(\frac{ax+b}{p}\right)\ =\ 0..$

###### Proof.

See Lemma A.1 of Appendix A. ∎

If $f$ is quadratic, then we have this next result.

######  Lemma 2.2.

For arbitrary integers $a,b,c$, and a prime $p$ such that $p\nmid a$, then

(2.4) $\sum_{x=0}^{p-1}\left(\frac{ax^{2}+bx+c}{p}\right)\ =\
\begin{cases}(p-1)\left(\frac{a}{p}\right)&\emph{if }p\mid b^{2}-4ac\\\
-\left(\frac{a}{p}\right)&\emph{otherwise}.\end{cases}$

###### Proof.

See Lemma A.2 of Appendix A. ∎

###  2.2. Gauss’s Lemma

######  Theorem 2.3.

_(Gauss)_ Let $E(\mathbb{F}_{p})\mathrel{\mathop{\mathchar
58\relax}}y^{2}=x^{3}+D$ be an elliptic curve. Then for $p\equiv 1\text{ mod
3}$

(2.5) $\\#E(\mathbb{F}_{p})\ =\ \begin{cases}p+1+2a&\emph{if $D$ is a sextic
residue mod $p$ }\\\ p+1-2a&\emph{if $D$ is cubic but not a quadratic residue
mod $p$ }\\\ p+1-a\pm 3b&\emph{if $D$ is a quadratic but not a cubic residue
mod $p$}\\\ p+1+a\pm 3b&\emph{if $D$ is neither quadratic nor cubic residue
mod $p$}\\\ \end{cases}$

where $a$ is an integer such that $a\equiv 2\text{ mod 3}$ and
$p=a^{2}+3b^{2}$ for some integer $b>0$. For $p\equiv 2\text{ mod 3}$,

(2.6) $\\#E(\mathbb{F}_{p})\ =\ p+1.$

###### Proof.

See [IR90, p. 305, Thm. 4]. ∎

###  2.3. Weil’s Theorem and Quadratic Character Sums

We first state Weil’s theorem for the estimation of character sums; we require
this result for the proof of Lemma 2.5.

######  Theorem 2.4 (Weil).

Let $\chi$ be an $n^{th}$ order non-trivial multiplicative character in the
finite field $\mathbb{F}_{q}$. Let $f(x)$ be a degree $d$ polynomial in
$\mathbb{F}_{q}$ such that $f(x)\neq k{g(x)}^{n}$ for any polynomial $g(x)$
and constant $k$ in $\mathbb{F}_{q}$. Then

(2.7)
$\mathinner{\\!\left\lvert\sum_{x\in\mathbb{F}_{q}}\chi(f(x))\right\rvert}\
\leq\ (d-1)\sqrt{q}.$

###### Proof.

c.f. [IK04, Thm. 11.23] ∎

Lemma 2.5 is needed in the proof of Lemma 2.6.

######  Lemma 2.5 ([LN97, Ex. 5.63]).

Let $a_{1},\ldots,a_{k}$ be distinct elements of $\mathbb{F}_{q}$, $q$ odd,
and let $\epsilon_{1},\ldots,\epsilon_{k}$ be $k$ given integers, each of
which is 1 or -1. Let $N(\epsilon_{1},\ldots,\epsilon_{k})$ denote the number
of $c\in\mathbb{F}_{q}$ with $\eta(c+a_{j})=\epsilon_{j}$ for $1\leq j\leq k$,
where $\eta$ is the quadratic character of $\mathbb{F}_{q}$. Then

(2.8) $N(\epsilon_{1},\dots,\epsilon_{k})\ =\
\frac{1}{2^{k}}\sum_{c\in\mathbb{F}_{q}}\left[1+\epsilon_{1}\eta(c+a_{1})\right]\cdots\left[1+\epsilon_{k}\eta(c+a_{k})\right]-A,$

where $0\leq A\leq k/2$.

###### Proof.

See Lemma A.3 of Appendix A. ∎

The final result we present in this section is necessary to prove the
existence of $k$-Diophantine $m$-tuples in Subsection 3.1.

######  Lemma 2.6 ([LN97, Ex. 5.64]).

We have

(2.9) $\left|N(\epsilon_{1},\ldots,\epsilon_{k})-\frac{q}{2^{k}}\right|\ \leq\
\left(\frac{k-2}{2}+\frac{1}{2^{k}}\right)q^{1/2}+\frac{k}{2}.$

###### Proof.

See Lemma A.4 of Appendix A. ∎

We now proceed to the main results of this paper. First, we will prove that
for all sufficiently large odd primes $p$, there exists at least one
$k$-Diophantine $m$-tuple in $\mathbb{F}_{p}$.

##  3\. Proofs of the main results

###  3.1. Existence of $k$-Diophantine $m$-tuples

Here, we prove that $k$-Diophantine $m$-tuples exist for a large enough prime.
First, we establish the case when $k=3$.

######  Theorem 3.1.

Let $m\geq 3$ be an integer. If $p>2^{m^{2}-m-2}{(m^{2}+3m+4)}^{2}$ is a
prime, then there exists at least one 3-Diophantine $m$-tuple in
$\mathbb{F}_{p}$.

###### Proof.

We prove this theorem by induction on $m$. For $m=3$ and a prime $p$ such that

(3.1) $p\ >\ 2^{3^{2}-3-2}{(3^{2}+3(3)+4)}^{2}\ =\ 7744,$

we have the 3-Diophantine triple $\\{2,3,4\\}$ in $\mathbb{F}_{p}$. Indeed,
$p\geq 5$ is large enough to guarantee the existence of this 3-Diophantine
triple. Suppose that there exists at least one $3$-Diophantine $m$-tuple in
$\mathbb{F}_{p}$. Now, we want to prove that there exists a 3-Diophantine
$(m+1)$-tuple in $\mathbb{F}_{p}$ where $p$ is a prime such that
$p>2^{m^{2}+m-2}{(m^{2}+5m+8)}^{2}$. Let us take a prime $p$ such that

$\displaystyle p$ $\displaystyle\ >\
2^{(m+1)^{2}-(m+1)-2}\\{(m+1)^{2}+3(m+1)+4)\\}^{2}$ $\displaystyle\ =\
2^{m^{2}+m-2}{(m^{2}+5m+8)}^{2}.$

Clearly, $p>2^{m^{2}-m-2}{(m^{2}+3m+4)}^{2}$. Thus, by the induction
hypothesis, there exists a 3-Diophantine $m$-tuple
$\\{a_{1},a_{2},\ldots,a_{m}\\}$ in $\mathbb{F}_{p}$. Define

(3.2) $g\ \coloneqq\ \\#\left\\{x\in\mathbb{F}_{p}\mathrel{\mathop{\mathchar
58\relax}}\left(\frac{a_{i}a_{j}x+1}{p}\right)\ =\ 1\text{ where
}i,j\in\mathbb{Z},1\leq i<j\leq m\right\\}\ =\
\\#\left\\{x\in\mathbb{F}_{p}\mathrel{\mathop{\mathchar
58\relax}}\left(\frac{x+\overline{a_{i}a_{j}}}{p}\right)\ =\
\left(\frac{\overline{a_{i}}\overline{a_{j}}}{p}\right)\right\\}$

for all $i$, $j$ such that $1\leq i<j\leq m$, where $\overline{a_{i}}$ denotes
the multiplicative inverse of $a_{i}$ in $\mathbb{F}_{p}$. We will prove that
$g-(m+1)>0$, which guarantees that there exists
$x\in\mathbb{F}_{p},x\not\in\\{0,a_{1},\ldots,a_{m}\\}$ such that
$\left(\frac{a_{i}a_{j}x+1}{p}\right)=1$ with $1\leq i<j\leq m$. By choosing
pairs in $\mathbb{F}_{p}$ in ${m\choose 2}$ ways and using Lemma 2.6,

$\displaystyle\mathinner{\\!\left\lvert g-\frac{p}{2^{{m\choose
2}}}\right\rvert}$ $\displaystyle\ \leq\ \left\\{\frac{{m\choose
2}-2}{2}+\frac{1}{2^{m\choose 2}}\right\\}\sqrt{p}+\frac{{m\choose 2}}{2}$
$\displaystyle g$ $\displaystyle\ \geq\ \frac{p}{2^{{m\choose
2}}}-\left\\{\frac{{m\choose 2}-2}{2}+\frac{1}{2^{m\choose
2}}\right\\}\sqrt{p}-\frac{{m\choose 2}}{2}$ $\displaystyle\ \geq\
\frac{p}{2^{\frac{m(m-1)}{2}}}-\left(\frac{m(m-1)-4}{4}+\frac{1}{2^{\frac{m(m-1)}{2}}}\right)\sqrt{p}-\frac{m(m-1)}{4}.$

Since

$\displaystyle\left(\frac{m(m-1)}{4}-1+\frac{1}{2^{\frac{m(m-1)}{2}}}\right)\sqrt{p}+\frac{m(m-1)}{4}+m+1$
$\displaystyle<\
\left(\frac{m^{2}-m}{4}-1+\frac{1}{2^{\frac{m(m-1)}{2}}}+\frac{1}{2^{\frac{m(m-1)}{2}+1}}\right)\sqrt{p}$
$\displaystyle=\
\left(\frac{m^{2}-m}{4}-1+\frac{3}{2^{\frac{m(m-1)}{2}+1}}\right)\sqrt{p}$
$\displaystyle<\ \frac{m(m-1)\sqrt{p}}{4}\ <\ \frac{p}{2^{\frac{m(m-1)}{2}}},$

we find, $g>m+1$. So, there exists a 3-Diophantine $(m+1)$-tuple
$\\{a_{1},\ldots,a_{m},x\\}$ in $\mathbb{F}_{p}$. ∎

We now consider the same question for arbitrary $k$. See 1.3

###### Proof.

We first prove the existence of a $k$-Diophantine $k$ tuple in
$\mathbb{F}_{p}$ by using induction on $k$. Then we proceed to prove the
theorem by using induction on $m$. The base case in the induction process of
$m$ is the case when $m=k$ i.e the existence of a $k$-Diophantine $k$-tuple
which we would have already proved.

Now, we prove that there exists a $k$-Diophantine $k$-tuple for
$p>4^{k}(3k+2)^{2}$. We prove this result by induction on $k\geq 2$. For
$p>1024$ and $k=2$, we get the Diophantine pair $\\{1,3\\}$ in
$\mathbb{F}_{p}$.

Assume the statement holds for $k\geq 2$. We consider a prime
$p>4^{k+1}(3k+5)^{2}$. Since $p>4^{k}(3k+2)^{2}$, there exists a
$k$-Diophantine $k$-tuple $\\{a_{1},a_{2},\dots,a_{k}\\}$ in $\mathbb{F}_{p}$.
Let

(3.3) $g\ \coloneqq\ \\#\left\\{x\in\mathbb{F}_{p}\mathrel{\mathop{\mathchar
58\relax}}\left(\frac{a_{1}a_{2}\dots a_{k}x+1}{p}\right)=1\right\\}.$

Let $\overline{a_{i}}$ denote the multiplicative inverse of $a_{i}$ in
$\mathbb{F}_{p}$. By Lemma 2.6,

$\displaystyle g$ $\displaystyle\ =\
\\#\left\\{x\in\mathbb{F}_{p}\mathrel{\mathop{\mathchar
58\relax}}\left(\frac{x+\overline{a_{1}a_{2}\dots a_{k}}}{p}\right)\ =\
\left(\frac{\overline{a_{1}a_{2}\dots a_{k}}}{p}\right)\right\\}$
$\displaystyle\ \geq\
\frac{p}{2^{\binom{k}{k}}}-\left(\frac{\binom{k}{k}-2}{2}+\frac{1}{2^{\binom{k}{k}}}\right)\sqrt{p}-\frac{\binom{k}{k}}{2}$
$\displaystyle\ =\ \frac{p-1}{2}$ $\displaystyle\ >\ k+1\text{ for
}p>4^{k+1}(3k+5)^{2}.$

So, there exists at least one $x\in\mathbb{F}_{p}$ such that
$\left(\frac{a_{1}a_{2}\dots a_{k}x+1}{p}\right)=1$ and hence we get a
$(k+1)$-Diophantine $(k+1)$-tuple $\\{a_{1},a_{2},\dots,a_{k},x\\}$ in
$\mathbb{F}_{p}$. Thus, there exists a $k$-Diophantine $k$-tuple in
$\mathbb{F}_{p}$ where $p>4^{k}(3k+2)^{2}$ and $k\geq 2$. Therefore, the base
case for the induction proof holds.

Let us now assume there exists at least one $k$-Diophantine $m$-tuple in
$\mathbb{F}_{p}$ for
$p>4^{\binom{m}{k-1}+1}\left(\frac{\binom{m}{k-1}}{2}+m+1\right)^{2}$. Now, we
want to prove that there exists a $k$-Diophantine $(m+1)$-tuple in
$\mathbb{F}_{p}$ where $p$ is a prime such that
$p>4^{\binom{m+1}{k-1}+1}\left(\frac{\binom{m+1}{k-1}}{2}+m+2\right)^{2}$. By
the induction hypothesis, since
$p>4^{\binom{m}{k-1}+1}\left(\frac{\binom{m}{k-1}}{2}+m+1\right)^{2}$, there
exists a $k$-Diophantine $m$-tuple $\\{a_{1},a_{2},\ldots,a_{m}\\}$ in
$\mathbb{F}_{p}$. Define

(3.4) $g\ \coloneqq\ \\#\left\\{x\in\mathbb{F}_{p}\mathrel{\mathop{\mathchar
58\relax}}\left(\frac{a_{i_{1}}a_{i_{2}}\dots
a_{i_{k-1}}x+1}{p}\right)=1\right\\}$

where
$a_{i_{1}},a_{i_{2}},\ldots,a_{i_{k-1}}\in\\{a_{1},a_{2},\ldots,a_{m}\\}$. Let
$\overline{a_{i}}$ denote the multiplicative inverse of $a_{i}$ in
$\mathbb{F}_{p}$.
By Lemma 2.6,

$\displaystyle g$ $\displaystyle\ =\
\\#\left\\{x\in\mathbb{F}_{p}\mathrel{\mathop{\mathchar
58\relax}}\left(\frac{x+\overline{a_{i_{1}}a_{i_{2}}\dots
a_{i_{k-1}}}}{p}\right)=\left(\frac{\overline{a_{i_{1}}a_{i_{2}}\dots
a_{i_{k-1}}}}{p}\right)\right\\}$ $\displaystyle\ \geq\ \frac{p}{2^{m\choose
k-1}}-\left(\frac{{m\choose k-1}-2}{2}+\frac{1}{2^{m\choose
k-1}}\right)\sqrt{p}-\frac{{m\choose k-1}}{2}.$

Now, we also see that
$p>4^{\binom{m}{k-1}+1}\left(\frac{\binom{m}{k-1}}{2}+m+1\right)^{2}$ gives

(3.5) $\frac{{m\choose k-1}}{2}+m+1\ <\ \frac{\sqrt{p}}{2^{{m\choose
k-1}+1}}.$

Using (3.5), we get

$\displaystyle\left(\frac{{m\choose k-1}-2}{2}+\frac{1}{2^{m\choose
k-1}}\right)\sqrt{p}+\frac{{m\choose k-1}}{2}+m+1$ $\displaystyle<\
\left(\frac{{m\choose k-1}}{2}-1+\frac{1}{2^{m\choose
k-1}}+\frac{1}{2^{{m\choose k-1}+1}}\right)\sqrt{p}$ $\displaystyle=\
\left(\frac{{m\choose k-1}}{2}-1+\frac{3}{2^{{m\choose
k-1}+1}}\right)\sqrt{p}$ $\displaystyle<\ \frac{{m\choose k-1}}{2}\sqrt{p}\ <\
\frac{p}{2^{m\choose k-1}}.$

Hence, we have $g>m+1$. Thus, there exists an
$x\in\mathbb{F}_{p},x\not\in\\{0,a_{1},\ldots,a_{m}\\}$ such that

$\left(\frac{a_{i_{1}}a_{i_{2}}\dots a_{i_{k-1}}x+1}{p}\right)\ =\ 1$

where
$a_{i_{1}},a_{i_{2}},\ldots,a_{i_{k-1}}\in\\{a_{1},a_{2},\ldots,a_{m}\\}$. So,
there exists a $k$-Diophantine $(m+1)$-tuple $\\{a_{1},\ldots,a_{m},x\\}$ in
$\mathbb{F}_{p}$. ∎

###  3.2. Counting 3-Diophantine Triples

A natural question to ask is exactly how many such $k$-Diophantine $m$-tuples
exist for a given $(k,m)$. The following result gives an answer for a special
case.

See 1.4

Indeed, when we compare this formula with the results obtained
computationally, we see trends in Table 1 and Figure 1 (in Appendix B) that
give us some initial confidence in the formula’s accuracy.

However, before giving the proof, we need some other results.

####  3.2.1. Counting Problems

In this subsection, we provide some lemmas that are needed to prove Theorem
1.4.

######  Lemma 3.2.

We have

(3.6) $\\#\left\\{(a,b)\in\mathbb{F}_{p}^{2}\mathrel{\mathop{\mathchar
58\relax}}a\neq b,ab+1=0\pmod{p}\right\\}\ =\ \begin{cases}p-3,&\emph{if
}p\equiv 1\pmod{4}\\\ p-1,&\emph{if }p\equiv 3\pmod{4}.\end{cases}$

###### Proof.

First, we solve the problem without the condition that $a\neq b$. As
$\mathbb{F}_{p}$ is a field, for each $a\in\mathbb{F}_{p}\setminus{\\{0\\}}$
there exists a unique $a^{-1}\in\mathbb{F}_{p}$ such that $aa^{-1}=1$. Hence,
for each $a\in\mathbb{F}_{p}$, take $b=-a^{-1}$. It follows from this
definition that $ab\equiv-1\pmod{p}$. Since each $b$ is unique for fixed $a$,
there are exactly $p-1$ pairs $(a,b)\in\mathbb{F}_{p}^{2}$ such that $ab+1=0$.

Now, with the condition $a\neq b$, notice that we need only find the odd
primes $p$ for which -1 is a quadratic residue. In other words, we wish to
find when $\left(\frac{-1}{p}\right)=1$ where $\left(\frac{a}{b}\right)$ is
the Legendre symbol. By Euler’s Criterion, this is equivalent to asking when
$(-1)^{\frac{p-1}{2}}=1$. This implies that -1 is a quadratic residue modulo
$p$ if and only if $\frac{p-1}{2}$ is even. Since $\frac{p-1}{2}$ is even when
$p\equiv 1\pmod{4}$ and odd when $p\equiv 3\pmod{4}$, we have

(3.7) $\\#\left\\{(a,b)\in\mathbb{F}_{p}^{2}\mathrel{\mathop{\mathchar
58\relax}}a\neq b,ab+1=0\pmod{p}\right\\}\ =\ \begin{cases}p-3,&\mbox{if
}p\equiv 1\pmod{4}\\\ p-1,&\mbox{if }p\equiv 3\pmod{4}.\end{cases}$

∎

See 1.5

###### Proof.

Consider fixing $ab=l$. We know that there is a unique choice of
$c\in\mathbb{F}_{p}$ such that $lc\equiv-1\pmod{p}$, namely $c=-l^{-1}$. Thus,
we begin by finding
$\\#\left\\{(a,b)\in\mathbb{F}_{p}^{2}\mathrel{\mathop{\mathchar
58\relax}}ab\equiv l\pmod{p}\right\\}$ where $l$ runs through the elements of
$\mathbb{F}_{p}\setminus{\\{0\\}}$.

We begin by finding
$\\#\left\\{(a,b)\in\mathbb{F}_{p}^{2}\mathrel{\mathop{\mathchar
58\relax}}ab\equiv l\pmod{p}\right\\}$. If $l$ is a quadratic residue modulo
$p$, then there are two pairs $(a,b)$ such that $ab=l$ and $a,b$ are not
distinct, for if $a=b=x$ is one such pair then $a=b=-x$ is the other pair. So,
if $l$ is a quadratic residue, then there are $p-3$ pairs. On the other hand,
if $l$ is a quadratic non-residue modulo $p$, then there are $p-1$ pairs such
that $ab=l$ and $a,b$ are distinct. Thus, we have that

$\displaystyle\\#\left\\{(a,b)\in\mathbb{F}_{p}^{2}\mathrel{\mathop{\mathchar
58\relax}}ab\equiv l\pmod{p}\right\\}$ $\displaystyle\ =\
(p-1)\frac{p-1}{2}+(p-3)\frac{p-1}{2}$ $\displaystyle\ =\ (p-1)(p-2).$

However, the theorem statement asks a slightly different question. Now, we
must consider when $c=a$ or $c=b$.

Case 1. $p\equiv 2\pmod{3}$:

We want to show that there are $\frac{p+1}{2}$ residues $l$ for which there
are $p-3$ distinct triples $(a,b,c)$ and $\frac{p-3}{2}$ residues $l$ for
which there are $p-5$ distinct triples $(a,b,c)$. This would imply that, for
$p\equiv 2\pmod{3}$,

$\displaystyle\\#\left\\{(a,b,c)\in\mathbb{F}_{p}^{3}\mathrel{\mathop{\mathchar
58\relax}}abc+1\equiv 0\pmod{p}\right\\}$ $\displaystyle\ =\
(p-3)\frac{p+1}{2}+(p-5)\frac{p-3}{2}$ $\displaystyle\ =\ (p-2)(p-3).$

The residues for which there are $p-5$ solutions satisfy $ab\equiv a^{2}\equiv
l\pmod{p}$ but not $abc\equiv a^{3}\equiv-1\pmod{p}$. In this case, there are
two pairs such that $a=b$ and $ab\equiv l\pmod{p}$, i.e., $(a,a)$ and
$(-a,-a)$, and two pairs such that either $b=c$ and $abc\equiv
ab^{2}\equiv-1\pmod{p}$ or $a=c$ and $abc\equiv a^{2}b\equiv-1$. Hence, we
have $p-1-4=p-5$ solutions. We know there are $\frac{p-3}{2}$ such residues by
Euler’s Criterion.

The residues, $l$, for which there are $p-3$ solutions either do not satisfy
$ab\equiv a^{2}\equiv l\pmod{p}$ or contain a pair that forms a solution of
$abc\equiv a^{3}\equiv-1\pmod{p}$ when extended by $c$. First, we restrict
ourselves to the quadratic non-residues. There exist pairs $(a,c)$ or $(b,c)$,
$a\neq b$ such that either $b=c$ and $abc\equiv-1\pmod{p}$ or $a=c$ and
$abc\equiv-1$. Since $l$ is a quadratic non-residue, $a\neq b$. We know there
are $\frac{p-1}{2}$ such quadratic non-residues. Now, for the cubic residues
of $-1$, there are $q=\gcd(3,p-1)$ solutions to the congruence
$x^{3}\equiv-1\pmod{p}$ by Euler’s criterion. For $p\equiv 2\pmod{3}$, $q=1$.
Thus, there are $\frac{p-1}{2}+1=\frac{p+1}{2}$ residues with $p-3$ solutions
to $abc+1\equiv 0\pmod{p}$.

Case 2. $p\equiv 1\pmod{3}$:

In this case, instead of there being exactly 1 solution to
$x^{3}\equiv-1\pmod{p}$, there are $q=\gcd(3,p-1)=3$ solutions. So, there are
$\frac{p-1}{2}+3=\frac{p+5}{2}$ residues for which there are $p-3$ triples
satisfying $abc+1\equiv 0\pmod{p}$, and $\frac{p-1}{2}-3=\frac{p-7}{2}$
residues for which there are $p-5$ triples satisfying $abc+1\equiv 0\pmod{p}$.
Thus,

$\displaystyle\\#\left\\{(a,b,c)\in\mathbb{F}_{p}^{3}\mathrel{\mathop{\mathchar
58\relax}}abc+1\equiv 0\pmod{p}\right\\}$ $\displaystyle\ =\
(p-3)\frac{p+5}{2}+(p-5)\frac{p-7}{2}$ $\displaystyle\ =\ (p-2)(p-3)+4.\qed$

Now we present the proof of Theorem 1.4.

###### Proof.

We have

(3.8) $12N_{3}(p)\ =\
\sum\left(1+\left(\frac{abc+1}{p}\right)^{\prime}\right)$

where the sum is evaluated over non-zero and distinct $a,b,c$, and we have
defined $\left(\frac{a}{p}\right)^{\prime}=\left(\frac{a}{p}\right)$ for
$a\neq 0$ and $\left(\frac{a}{p}\right)^{\prime}=1$ for $a=0$. Hence

$\displaystyle 12N_{3}(p)$ $\displaystyle\ =\ \sum_{c\neq 0}\,\,\,\sum_{b\neq
0,c}\,\sum_{a\neq 0,b,c}1+\sum_{c\neq 0}\,\,\,\sum_{b\neq 0,c}\,\sum_{a\neq
0,b,c}\left(\frac{abc+1}{p}\right)+\\#\left\\{(a,b,c)\in\mathbb{F}_{p}^{3}\mathrel{\mathop{\mathchar
58\relax}}abc+1\equiv 0\pmod{p}\right\\}.$

The first summand will just be $(p-1)(p-2)(p-3)$, and by Lemma 1.5 we already
have the solution for the final summand. We use Lemmas 2.1 and 2.2 to evaluate
the middle sum:

(3.9) $\sum_{c\neq 0}\,\,\,\sum_{b\neq 0,c}\,\sum_{a\neq
0,b,c}\left(\frac{abc+1}{p}\right)\ =\ -\sum_{c\neq 0}\,\,\,\sum_{b\neq
0,c}\left(\frac{1}{p}\right)+\left(\frac{b^{2}c+1}{p}\right)+\left(\frac{bc^{2}+1}{p}\right).$

This gives

$\displaystyle-\sum_{c\neq 0}\,\,\,\sum_{b\neq
0,c}1+\left(\frac{b^{2}c+1}{p}\right)+\left(\frac{bc^{2}+1}{p}\right)$
$\displaystyle\ =\ -\sum_{c\neq 0}(p-4)-2\left(\frac{c^{3}+1}{p}\right)$
$\displaystyle\ =\ -(p-1)(p-4)+2\sum_{c\neq 0}\left(\frac{c^{3}+1}{p}\right).$

Using Theorem 2.3, we can evaluate the sum $\sum_{c\neq
0}\left(\frac{c^{3}+1}{p}\right)$. In particular we get

$\displaystyle\sum_{c\neq 0}\left(\frac{c^{3}+1}{p}\right)$ $\displaystyle\ =\
\begin{cases}2a-1&\mbox{for }p\equiv 1\pmod{3}\\\ -1&\mbox{for }p\equiv
2\pmod{3}\end{cases}$ $\displaystyle\implies\sum_{c\neq 0}\,\,\,\sum_{b\neq
0,c}\,\sum_{a\neq 0,b,c}\left(\frac{abc+1}{p}\right)$ $\displaystyle\ =\
\begin{cases}-(p-2)(p-3)+4a&\mbox{for }p\equiv 1\pmod{3}\\\
-(p-2)(p-3)&\mbox{for }p\equiv 2\pmod{3}.\end{cases}$

Combining this with the result from Theorem 1.5 we get the claim. ∎

As one might imagine, counting $k$-Diophantine $k$-tuples for a general $k$ is
not that simple due to the complexity of the following problem:

_What are the total number of $k$-tuples $\\{a_{1},a_{2},\ldots,a_{k}\\}$ such
that $a_{i}$ are all distinct, non-zero and_

(3.10) $\prod_{i=1}^{k}a_{i}+1\ \equiv\ 0\mbox{ mod p}?$

###  3.3. Asymptotic Formula

While a general formula of the number of $k$-Diophantine $k$-tuples is
difficult, an asymptotic formula is well within reach. We describe the formula
in the following result.

See 1.6

###### Proof.

We know that

(3.11) $k!\cdot 2\cdot N_{k}(p)\ =\
\sideset{}{{}^{\prime}}{\sum}\left(1+\left(\frac{1+a_{1}a_{2}\ldots
a_{k}}{p}\right)^{\prime}\right)$

where the primed sum is taken over distinct and non-zero
$a_{1},a_{2},\ldots,a_{k}$ and we have defined
$\left(\frac{a}{p}\right)^{\prime}=\left(\frac{a}{p}\right)$ for $a\neq 0$ and
$\left(\frac{a}{p}\right)^{\prime}=1$ for $a=0$ as before. The main term is
$\sum 1=(p-1)(p-2)\ldots(p-k)=p^{k}+o(p^{k})$. Now

(3.12) $\sum\left(\frac{1+a_{1}a_{2}\ldots a_{k}}{p}\right)^{\prime}\ =\
\sum\left(\frac{1+a_{1}a_{2}\ldots
a_{k}}{p}\right)+\\#\left\\{(a_{1},a_{2},\ldots,a_{k})\in\mathbb{F}_{p}^{k}\mathrel{\mathop{\mathchar
58\relax}}\prod_{i=1}^{k}a_{i}+1\equiv 0\pmod{p}\right\\}.$

Using Weil’s estimate for character sums (Theorem 2.4), we note that

(3.13) $\sum\left(\frac{1+a_{1}a_{2}\ldots a_{k}}{p}\right)\ \leq\
p^{k-1}\sqrt{p}.$

We also note that

(3.14)
$\\#\left\\{(a_{1},a_{2},\ldots,a_{k})\in\mathbb{F}_{p}^{k}\mathrel{\mathop{\mathchar
58\relax}}\prod_{i=1}^{k}a_{i}+1\equiv 0\pmod{p}\right\\}\ \leq\ (p-1)^{k-1}.$

Hence

(3.15) $\sum\left(\frac{1+a_{1}a_{2}\ldots a_{k}}{p}\right)^{\prime}\ =\
o(p^{k}).$

The result follows. ∎

##  4\. Concluding remarks

In this paper, we attempted to answer two fundamental questions about
$k$-Diophantine $m$-tuples:

  1. (1)

Given a sufficiently large prime $p$, is there always a $k$-Diophantine
$m$-tuple in $\mathbb{F}_{p}$?

  2. (2)

Can we count the number of $k$-Diophantine $m$-tuples in $\mathbb{F}_{p}$ for
a given prime $p$?

We give a complete answer to $(1)$ in Theorem 3.1. While we were unable to
answer $(2)$ for an arbitrary pair $(k,m)$, we were able to come up with an
asymptotic formula for the number of $k$-Diophantine $k$-tuples for any $k$
(Theorem 1.6) and an explicit formula for $(k,m)=(3,3)$ (Theorem 1.4).

Some questions asked for the usual Diophantine $m$-tuples can be asked for
$k$-Diophantine $m$-tuples as well. For instance,

  1. (1)

Can we find a formula that, for a given $n$, allows us to count the number of
$k$-Diophantine $m$-tuples with property $D(n)$, i.e. the set where $k$-wise
products of distinct elements is $n$ less than a perfect square? What about
about $n$ less than the $t$-th power?

  2. (2)

What can be said about the existence of such tuples in other commutative rings
with unity, like Gaussian integers, integers, $p$-adic integers, polynomial
rings etc?

## Acknowledgements

This project was done as a part of the Polymath Jr. Program during the summer
2021. We would like to thank Professor Andrej Dujella for graciously granting
us access to his book Number Theory [DS21], and Rowan Mckee for writing the
computer program for calculating the number of 3-Diophantine triples.

##  Appendix A Proofs from the Preliminaries

For completeness we include proofs of some standard results about sums of
Legendre symbols.

######  Lemma A.1.

For arbitrary integers $a$ and $b$, and a prime $p\nmid a$, we have

(A.1) $\sum_{x=0}^{p-1}\left(\frac{ax+b}{p}\right)\ =\ 0..$

###### Proof.

As $p\nmid a$, $ax+b$ forms a complete set of residues modulo $p$ as $x$ runs
through the integers $0$ to $p-1$. Every such set contains $\frac{p-1}{2}$
quadratic residues and $\frac{p-1}{2}$ quadratic non-residues. Hence,

(A.2) $\sum_{x=0}^{p-1}\left(\frac{ax+b}{p}\right)\ =\
\frac{p-1}{2}-\frac{p-1}{2}\ =\ 0.\qed$

######  Lemma A.2.

For arbitrary integers $a,b,c$, and a prime $p$ such that $p\nmid a$, then

(A.3) $\sum_{x=0}^{p-1}\left(\frac{ax^{2}+bx+c}{p}\right)\ =\
\begin{cases}(p-1)\left(\frac{a}{p}\right)&\emph{if }p\mid b^{2}-4ac\\\
-\left(\frac{a}{p}\right)&\emph{otherwise}.\end{cases}$

###### Proof.

Notice that this sum is equivalently written as

(A.4)
$\left(\frac{4a}{p}\right)\sum_{x=0}^{p-1}\left(\frac{4a^{2}x^{2}+4abx+4ac}{p}\right)\
=\
\left(\frac{a}{p}\right)\sum_{x=0}^{p-1}\left(\frac{{(2ax+b)}^{2}-(b^{2}-4ac)}{p}\right)\
=\ \left(\frac{a}{p}\right)S,$

where $S=\sum_{x=0}^{p-1}\left(\frac{{(2ax+b)}^{2}-(b^{2}-4ac)}{p}\right)$.

Since the numbers $ax+b$ form a complete set of residues modulo $p$ as $x$
varies from 0 to $p-1$, we have

(A.5) $S\ =\ \sum_{l=0}^{p-1}\left(\frac{l^{2}-(b^{2}-4ac)}{p}\right).$

It is well known that $S\equiv-1\pmod{p}$ and $|S|\leq p$ [Apo98, Ex. 10.10].
From this, we obtain $S=-1,p-1$. If $S=p-1$, then $p-1$ terms in $S$ must take
the value 1 and there is exactly one term, i.e., when $l=l^{\prime}$, that
equals 0. As this $l^{\prime}$ must satisfy both
$p\mid{l^{\prime}}^{2}-(b^{2}-4ac)$ and
$p\mid{(-l^{\prime})}^{2}-(b^{2}-4ac)$, it follows that $l^{\prime}=0$ and
$p\mid b^{2}-4ac$. Conversely, if $p\mid b^{2}-4ac$, then

(A.6) $S\ =\ \sum_{l=0}^{p-1}\left(\frac{l^{2}}{p}\right)\ =\ 0+1\cdot p-1\ =\
p-1.$

Hence, $S=-1$ if and only if $p\nmid b^{2}-4ac$. Thus,

(A.7) $\sum_{x=0}^{p-1}\left(\frac{ax^{2}+bx+c}{p}\right)\ =\
\begin{cases}(p-1)\left(\frac{a}{p}\right)&\mbox{if }p\mid b^{2}-4ac\\\
-\left(\frac{a}{p}\right)&\mbox{otherwise}.\end{cases}\qed$

######  Lemma A.3 ([LN97, Ex. 5.63]).

Let $a_{1},\ldots,a_{k}$ be distinct elements of $\mathbb{F}_{q}$, $q$ odd,
and let $\epsilon_{1},\ldots,\epsilon_{k}$ be $k$ given integers, each of
which is 1 or -1. Let $N(\epsilon_{1},\ldots,\epsilon_{k})$ denote the number
of $c\in\mathbb{F}_{q}$ with $\eta(c+a_{j})=\epsilon_{j}$ for $1\leq j\leq k$,
where $\eta$ is the quadratic character of $\mathbb{F}_{q}$. Then

(A.8) $N(\epsilon_{1},\dots,\epsilon_{k})\ =\
\frac{1}{2^{k}}\sum_{c\in\mathbb{F}_{q}}\left[1+\epsilon_{1}\eta(c+a_{1})\right]\cdots\left[1+\epsilon_{k}\eta(c+a_{k})\right]-A,$

where $0\leq A\leq k/2$.

###### Proof.

Notice that, for fixed $c$, if $\eta(c+a_{j})=\epsilon_{j}$, then
$\epsilon_{j}\eta(c+a_{j})=1$; otherwise, $\epsilon_{j}\eta(c+a_{j})=-1$ for
$c+a_{j}\neq 0$ or $\eta(c+a_{j})=0$. This implies that, if
$\eta(c+a_{j})=\epsilon_{j}$ for all $j\in\mathbb{N}$ such that $1\leq j\leq
k$ and $c$ fixed, then

(A.9)
$\left[1+\epsilon_{1}\eta(c+a_{1})\right]\cdots\left[1+\epsilon_{k}\eta(c+a_{k})\right]\
=\ 2^{k}.$

Otherwise,

(A.10)
$\left[1+\epsilon_{1}\eta(c+a_{1})\right]\cdots\left[1+\epsilon_{k}\eta(c+a_{k})\right]\
=\ 0$

if $\epsilon_{i}\eta(c+a_{i})=-1$ for some $i\in\mathbb{N}$ such that $1\leq
i\leq k$, or

(A.11)
$\left[1+\epsilon_{1}\eta(c+a_{1})\right]\cdots\left[1+\epsilon_{k}\eta(c+a_{k})\right]\
=\ 2^{k-1}$

if $\epsilon_{i}\eta(c+a_{i})=0$ for some $i\in\mathbb{N}$ and
$\eta(c+a_{j})=1$ for all $j\neq i$. Note that there is at most one $a_{i}$
with the property that $\eta(c+a_{i})=0$ since $a_{1},\ldots,a_{k}$ are
distinct and $c$ is constant. Thus, we have

(A.12) $N(\epsilon_{1},\dots,\epsilon_{k})\ =\
\frac{1}{2^{k}}\sum_{c\in\mathbb{F}_{q}}\left[1+\epsilon_{1}\eta(c+a_{1})\right]\cdots\left[1+\epsilon_{k}\eta(c+a_{k})\right]-A,$

where $0\leq A\leq k/2$. ∎

######  Lemma A.4 ([LN97, Ex. 5.64]).

We have

(A.13) $\left|N(\epsilon_{1},\ldots,\epsilon_{k})-\frac{q}{2^{k}}\right|\
\leq\ \left(\frac{k-2}{2}+\frac{1}{2^{k}}\right)q^{1/2}+\frac{k}{2}.$

###### Proof.

First, we expand the product in the expression for
$N(\epsilon_{1},\ldots,\epsilon_{k})$ given above.

(A.14) $[1+\epsilon_{1}\eta(c+a_{1})]\cdots[1+\epsilon_{k}\eta(c+a_{k})]\ =\
1+\sum_{i=1}^{k}\epsilon_{i}\eta(c+a_{i})+\sum_{j\neq
i}^{k}\sum_{i=1}^{k}\epsilon_{i}\epsilon_{j}\eta(c+a_{i})\eta(c+a_{j})\\\
+\cdots+\epsilon_{1}\epsilon_{2}\cdots\epsilon_{k}\eta(c+a_{1})\eta(c+a_{2})\cdots\eta(c+a_{k}).$

By the multiplicative nature of the quadratic character, A.14 is equivalent to

$1+\sum_{i=1}^{k}\epsilon_{i}\eta(c+a_{i})+\sum_{j\neq
i}^{k}\sum_{i=1}^{k}\epsilon_{i}\epsilon_{j}\eta[(c+a_{i})(c+a_{j})]+\cdots+\epsilon_{1}\epsilon_{2}\cdots\epsilon_{k}\eta[(c+a_{1})(c+a_{2})\cdots(c+a_{k})].$

We see that the product expands into a sum of quadratic characters of
functions in $c$ of various degrees. More specifically, we find that there are
$\binom{k}{i}$ functions of degree $i$ where $1\leq i\leq k$. By Theorem 2.4,
we can show that

$\displaystyle\left|\sum_{c\in\mathbb{F}_{q}}\sum_{d=1}^{k}\eta(f_{d}(x))\right|$
$\displaystyle\ \leq\ \sum_{d=1}^{k}\binom{k}{d}(d-1)\sqrt{q}$ $\displaystyle\
=\
\sqrt{q}\left(\sum_{d=1}^{k}\binom{k}{d}d-\sum_{d=1}^{k}\binom{k}{d}\right)$
$\displaystyle\ =\ \left(k2^{k-1}-2^{k-1}\right)\sqrt{q},$

where $f_{d}$ are all the functions of degree $d$ for which the quadratic
character is evaluated. It follows that

(A.15)
$\left|N(\epsilon_{1},\ldots,\epsilon_{k})-\frac{q}{2^{k}}\right|\leq\left(\frac{k-2}{2}+\frac{1}{2^{k}}\right)q^{1/2}+\frac{k}{2}.\qed$

##  Appendix B Tables and Graphs

The tables and graphs shown are shown here to offer some computational
verification of the formula presented in Theorem 1.4 and demonstrate what the
program we used is capable of. As the formula predicts, we see in Table 1 and
Figure 1 a positive quadratic trend in the number of 3-Diophantine triples
against the size of $\mathbb{F}_{p}$. The reason we decided not to plot values
of $p$ past 200 was the computational cost of doing so and that one can see
this trend for $p<300$. The program was not only able to give the number of
$k$-Diophantine $m$-tuples in $\mathbb{F}_{p}$, but also the explicit
$m$-tuples themselves, as seen in Table LABEL:table:list.

Table 1. Number of 3-Diophantine triples in $\mathbb{F}_{p}$ for various primes $p$. $p$ | $p\equiv 1,2\pmod{3}$ | $N_{3}(p)$ |  $a$ (if $p\equiv 1\pmod{3}$) | Error term $(a+1)/3$
---|---|---|---|---
5 | 2 | 2 | - | -
7 | 1 | 11 | 2 | 1
11 | 2 | 60 | - | -
13 | 1 | 110 | -1 | 0
17 | 2 | 280 | - | -
19 | 1 | 407 | -4 | -1
23 | 2 | 770 | - | -
29 | 2 | 1638 | - | -
31 | 1 | 2031 | 2 | 1
37 | 1 | 3572 | 5 | 2
41 | 2 | 4940 | - | -
43 | 1 | 5739 | -4 | -1
101 | 2 | 80850 | - | -
229 | 1 | 97472 | 11 | 4
Figure 1. Graph of number of 3-Diophantine triples in $\mathbb{F}_{p}$ for
various $p$. Figure 2. Log-log graph of number of 3-Diophantine triples in
$\mathbb{F}_{p}$ for various $p$.

Table 2. List of all 3-Diophantine Quadruples in $\mathbb{F}_{23}$. (1, 2, 4, 6) | (1, 2, 4, 20) | (1, 2, 6, 14) | (1, 2, 11, 12) | (1, 2, 13, 15)
---|---|---|---|---
(1, 2, 13, 19) | (1, 2, 15, 20) | (1, 2, 17, 19) | (1, 2, 19, 20) | (1, 3, 4, 10)
(1, 3, 4, 21) | (1, 3, 5, 8) | (1, 3, 8, 15) | (1, 3, 15, 18) | (1, 3, 16, 19)
(1, 3, 17, 19) | (1, 3, 19, 21) | (1, 4, 6, 12) | (1, 4, 7, 20) | (1, 4, 10, 21)
(1, 4, 17, 18) | (1, 4, 17, 21) | (1, 4, 19, 20) | (1, 5, 6, 9) | (1, 5, 6, 20)
(1, 5, 8, 9) | (1, 5, 16, 19) | (1, 6, 8, 20) | (1, 6, 9, 21) | (1, 6, 12, 21)
(1, 6, 19, 20) | (1, 6, 19, 21) | (1, 6, 21, 22) | (1, 7, 9, 11) | (1, 7, 9, 18)
(1, 7, 10, 12) | (1, 7, 11, 13) | (1, 7, 13, 20) | (1, 8, 9, 21) | (1, 8, 10, 13)
(1, 8, 10, 21) | (1, 8, 15, 20) | (1, 8, 20, 22) | (1, 9, 11, 13) | (1, 9, 14, 17)
(1, 9, 17, 18) | (1, 10, 12, 14) | (1, 10, 12, 16) | (1, 10, 14, 15) | (1, 11, 12, 21)
(1, 11, 13, 16) | (1, 11, 13, 19) | (1, 11, 16, 22) | (1, 11, 17, 19) | (1, 11, 17, 21)
(1, 11, 21, 22) | (1, 12, 16, 22) | (1, 12, 21, 22) | (1, 14, 15, 17) | (1, 14, 15, 18)
(1, 14, 16, 18) | (1, 15, 20, 22) | (1, 16, 18, 22) | (1, 18, 20, 22) | (2, 3, 4, 21)
(2, 3, 4, 22) | (2, 3, 5, 8) | (2, 3, 5, 14) | (2, 3, 8, 12) | (2, 3, 8, 22)
(2, 3, 9, 20) | (2, 3, 14, 20) | (2, 3, 19, 22) | (2, 4, 5, 13) | (2, 4, 5, 15)
(2, 4, 6, 22) | (2, 4, 9, 15) | (2, 4, 9, 20) | (2, 4, 10, 15) | (2, 4, 13, 21)
(2, 4, 20, 21) | (2, 5, 7, 10) | (2, 5, 7, 14) | (2, 5, 7, 16) | (2, 5, 8, 14)
(2, 5, 12, 15) | (2, 5, 13, 16) | (2, 5, 13, 21) | (2, 5, 16, 21) | (2, 6, 7, 16)
(2, 6, 11, 22) | (2, 6, 16, 21) | (2, 7, 9, 14) | (2, 7, 9, 15) | (2, 7, 9, 18)
(2, 7, 10, 15) | (2, 7, 12, 17) | (2, 7, 16, 18) | (2, 8, 10, 13) | (2, 8, 10, 19)
(2, 8, 10, 22) | (2, 8, 11, 16) | (2, 8, 13, 16) | (2, 8, 14, 19) | (2, 8, 18, 19)
(2, 9, 14, 16) | (2, 9, 15, 18) | (2, 9, 17, 22) | (2, 10, 13, 18) | (2, 10, 19, 22)
(2, 11, 12, 15) | (2, 11, 15, 18) | (2, 11, 15, 20) | (2, 11, 16, 18) | (2, 12, 17, 22)
(2, 13, 16, 17) | (2, 13, 18, 19) | (2, 14, 19, 20) | (2, 16, 17, 21) | (2, 17, 19, 22)
(3, 4, 6, 7) | (3, 4, 6, 22) | (3, 4, 10, 11) | (3, 4, 14, 16) | (3, 4, 14, 21)
(3, 5, 10, 18) | (3, 5, 10, 20) | (3, 5, 14, 17) | (3, 5, 18, 22) | (3, 6, 7, 17)
(3, 6, 9, 20) | (3, 6, 14, 17) | (3, 6, 14, 20) | (3, 6, 15, 22) | (3, 6, 16, 17)
(3, 6, 16, 20) | (3, 6, 18, 22) | (3, 7, 8, 12) | (3, 7, 9, 19) | (3, 7, 10, 11)
(3, 7, 10, 12) | (3, 7, 17, 19) | (3, 7, 19, 22) | (3, 9, 10, 12) | (3, 9, 12, 19)
(3, 9, 17, 20) | (3, 9, 19, 21) | (3, 10, 12, 20) | (3, 10, 13, 18) | (3, 10, 18, 20)
(3, 11, 12, 15) | (3, 11, 13, 14) | (3, 11, 13, 16) | (3, 11, 14, 16) | (3, 11, 15, 21)
(3, 12, 13, 16) | (3, 12, 15, 16) | (3, 12, 15, 20) | (3, 12, 16, 20) | (3, 14, 17, 20)
(3, 15, 18, 21) | (3, 16, 17, 20) | (4, 5, 6, 7) | (4, 5, 8, 18) | (4, 5, 18, 19)
(4, 6, 8, 11) | (4, 6, 8, 17) | (4, 6, 11, 22) | (4, 6, 12, 15) | (4, 6, 12, 17)
(4, 6, 15, 17) | (4, 7, 8, 14) | (4, 7, 9, 19) | (4, 7, 19, 20) | (4, 8, 11, 17)
(4, 8, 13, 14) | (4, 8, 13, 21) | (4, 8, 18, 21) | (4, 9, 10, 11) | (4, 9, 10, 19)
(4, 9, 13, 19) | (4, 9, 13, 20) | (4, 9, 16, 20) | (4, 10, 14, 15) | (4, 10, 18, 21)
(4, 11, 12, 19) | (4, 12, 13, 14) | (4, 12, 13, 20) | (4, 12, 15, 17) | (4, 13, 21, 22)
(4, 14, 15, 16) | (4, 16, 17, 20) | (4, 16, 20, 22) | (4, 17, 20, 22) | (4, 17, 21, 22)
(4, 20, 21, 22) | (5, 6, 7, 11) | (5, 6, 9, 20) | (5, 6, 10, 12) | (5, 6, 10, 18)
(5, 6, 11, 13) | (5, 6, 11, 18) | (5, 6, 12, 18) | (5, 7, 11, 21) | (5, 7, 14, 22)
(5, 7, 16, 22) | (5, 8, 9, 15) | (5, 8, 9, 18) | (5, 8, 9, 21) | (5, 8, 11, 21)
(5, 8, 14, 15) | (5, 8, 15, 19) | (5, 8, 18, 19) | (5, 9, 11, 18) | (5, 9, 12, 16)
(5, 9, 12, 18) | (5, 9, 15, 22) | (5, 9, 16, 21) | (5, 9, 16, 22) | (5, 9, 18, 22)
(5, 9, 20, 21) | (5, 10, 17, 18) | (5, 10, 19, 21) | (5, 11, 17, 18) | (5, 12, 17, 18)
(5, 13, 16, 17) | (5, 14, 17, 22) | (5, 15, 19, 21) | (5, 15, 20, 22) | (6, 7, 11, 13)
(6, 7, 13, 21) | (6, 7, 17, 21) | (6, 8, 11, 13) | (6, 8, 12, 17) | (6, 8, 12, 19)
(6, 8, 13, 14) | (6, 8, 15, 19) | (6, 9, 15, 22) | (6, 10, 12, 16) | (6, 10, 14, 17)
(6, 10, 16, 18) | (6, 11, 15, 19) | (6, 12, 15, 17) | (6, 12, 16, 21) | (6, 13, 14, 17)
(6, 14, 18, 22) | (6, 16, 17, 21) | (6, 16, 19, 20) | (6, 18, 19, 22) | (6, 19, 21, 22)
(7, 8, 10, 12) | (7, 8, 13, 16) | (7, 8, 13, 21) | (7, 8, 15, 16) | (7, 9, 14, 19)
(7, 9, 14, 21) | (7, 9, 15, 19) | (7, 10, 11, 15) | (7, 10, 11, 22) | (7, 11, 13, 20)
(7, 11, 15, 20) | (7, 12, 13, 18) | (7, 12, 13, 20) | (7, 12, 13, 22) | (7, 12, 14, 20)
(7, 12, 17, 18) | (7, 13, 15, 16) | (7, 13, 16, 22) | (7, 13, 18, 21) | (7, 14, 19, 20)
(7, 16, 18, 22) | (7, 18, 19, 22) | (8, 9, 10, 17) | (8, 9, 10, 19) | (8, 9, 15, 19)
(8, 9, 16, 19) | (8, 10, 12, 19) | (8, 10, 13, 22) | (8, 11, 13, 16) | (8, 11, 16, 20)
(8, 12, 17, 19) | (8, 12, 18, 21) | (8, 12, 20, 21) | (8, 13, 14, 22) | (8, 14, 15, 19)
(8, 14, 18, 22) | (8, 18, 20, 22) | (9, 10, 12, 19) | (9, 11, 12, 18) | (9, 11, 13, 20)
(9, 12, 13, 14) | (9, 12, 16, 22) | (9, 13, 14, 17) | (9, 13, 14, 19) | (9, 13, 15, 19)
(9, 14, 16, 19) | (9, 14, 16, 21) | (9, 14, 17, 22) | (10, 11, 16, 18) | (10, 11, 16, 20)
(10, 11, 16, 22) | (10, 11, 17, 18) | (10, 12, 14, 20) | (10, 12, 16, 20) | (10, 13, 14, 17)
(10, 13, 17, 18) | (10, 13, 18, 20) | (10, 14, 19, 20) | (10, 15, 16, 21) | (10, 16, 18, 21)
(10, 16, 20, 22) | (10, 19, 20, 22) | (11, 12, 14, 18) | (11, 12, 15, 21) | (11, 13, 14, 19)
(11, 14, 16, 18) | (11, 15, 17, 19) | (12, 13, 15, 16) | (12, 13, 15, 17) | (12, 13, 16, 22)
(12, 13, 17, 18) | (12, 15, 16, 21) | (12, 17, 19, 22) | (13, 15, 19, 21) | (13, 15, 21, 22)
(14, 15, 16, 21) | (14, 15, 18, 21) | (14, 16, 18, 21) | (14, 17, 21, 22) | (19, 20, 21, 22)

## References

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Trajan Hammonds, Department of Mathematics, Princeton University, USA
E-mail address<EMAIL_ADDRESS>
Seoyoung Kim, Department of Mathematics and Statistics, Queen’s University,
Canada
E-mail address<EMAIL_ADDRESS>
Steven J. Miller, Department of Mathematics and Statistics, Williams College,
USA
E-mail address<EMAIL_ADDRESS>
Arjun Nigam, University of Arizona, USA
E-mail address<EMAIL_ADDRESS>
Kyle Onghai, University of California, Los Angeles, USA
E-mail address<EMAIL_ADDRESS>
Dishant Saikia, Tezpur University, India
E-mail address<EMAIL_ADDRESS>
Lalit M. Sharma, University of Delhi, India
E-mail address<EMAIL_ADDRESS>


# Kolmogorov flow: Linear Stability and Energy Transfers in a minimal low-
dimensional model

Soumyadeep Chatterjee<EMAIL_ADDRESS>Department of Physics, Indian
Institute Of Technology Kanpur, Kanpur 208016, India  Mahendra K. Verma
<EMAIL_ADDRESS>Department of Physics, Indian Institute Of Technology Kanpur,
Kanpur 208016, India

###### Abstract

In this paper, we derive a four-mode model for the Kolmogorov flow by
employing Galerkin truncation and Craya-Herring basis for the decomposition of
velocity field. After this, we perform a bifurcation analysis of the model.
Though our low-dimensional model has fewer modes than the past models, it
captures the essential features of the primary bifurcation of the Kolmogorov
flow. For example, it reproduces the critical Reynolds number for the
supercritical pitchfork bifurcation and the flow structures of the past works.
We also demonstrate energy transfers from intermediate scales to large scales.
We perform direct numerical simulations of the Kolmogorov flow and show that
our model predictions match with the numerical simulations very well.

Kolmogorov flow, low-dimensional model, linear stability, bifurcation
analysis.

††preprint: AIP/123-QED

> In the late 1950s, Kolmogorov urged the fluid community to explore the
> stability criteria of a shear flow with spatially-periodic forcing; a system
> referred to as the Kolmogorov flow. Since then, many researchers have
> attempted to address the above problem using analytical, numerical, and
> experimental tools. The leading analytical results involve infinite or a
> large number of interacting Fourier modes. The numerical calculations and
> low-dimensional models too involve many Fourier modes. In this paper, we
> construct a four-mode low-dimensional model using Galerkin truncation and
> Craya-Herring basis. Our minimal model of the Kolmogorov flow captures the
> essential features of its primary bifurcation and the critical Reynolds
> number very well. The model predictions are borne out in numerical
> simulations.

##  I INTRODUCTION

Flow instability and transition to turbulence are important problems of fluid
dynamics. Kolmogorov abstracted a simple shear flow with spatially-periodic
forcing Arnol’d and Meshalkin (1960); Obukhov (1983); Arnol’d (1991), whose
instability and bifurcation has been studied intensely over the years. Also,
the Kolmogorov flow has been experimentally realized in several setups,
including a soap filmBurgess  _et al._ (1999) and an electrolytic
fluidBondarenko, Gak, and Dolzhanskii (1979); Suri  _et al._ (2014). In this
paper, we analyze the stability of the Kolmogorov flow using a low-dimensional
model consisting of four Fourier modes. We validate the model using numerical
solutions.

Meshalkin and Sinai (1961) provided the first solution to the stability of the
Kolmogorov flow. They considered an external force per unit mass, $\gamma\sin
k_{f}y\ \hat{x}$ where $\gamma$ is the force amplitude, and $k_{f}$ is the
force wavenumber. Meshalkin and Sinai (1961) considered $k_{f}=1$, and
analyzed the stability of two-dimensional flow with $L_{x}/L_{y}=1/\alpha$.
They considered small perturbation on the fundamental stream function
(corresponding to the external force) and focused on its time variations. They
incorporated the effects of all Fourier modes and studied the stability
problem with continued fractions. An outcome of their analysis is that for
$\alpha<1$, the laminar solution becomes unstable at critical Reynolds number
$R_{c}$; and $R_{c}\rightarrow\sqrt{2}$ (for normalization of Iudovich (1965))
as $\alpha\rightarrow 0$. They observed that the laminar solution is stable
for $\alpha>1$. Using asymptotic instability analysis, Sivashinsky (1985)
showed that a periodically-forced two-dimensional plane-parallel flow becomes
unstable beyond a critical Reynolds number. He showed the secondary flow to be
chaotically self-fluctuating.

Iudovich (1965) and Marchioro (1987) extended the calculation of Meshalkin and
Sinai (1961) and concluded that the laminar flow is globally stable for
$\alpha\geq 1$. For $\alpha<1$, Iudovich (1965) proved that
$R_{c}\rightarrow\sqrt{2}$ for $\alpha\rightarrow 0$, and
$R_{c}\rightarrow\infty$ when $\alpha\rightarrow 1$. They showed that $R_{c}$
increases monotonically with $\alpha$ between $R_{c}=\sqrt{2}$ for
$\alpha\rightarrow 0$ and $R_{c}=\infty$ for $\alpha\rightarrow 1$. The
$R_{c}$ curve represents neutral stability.

Okamoto and Shōji (1993) performed a bifurcation analysis of the Kolmogorov
flow with a finite set of Fourier modes and showed supercritical pitchfork to
be the primary bifurcation. They considered $544$ modes for $\alpha>0.3$, even
more modes for $\alpha<0.3$, and observed that $R_{c}=3.011193$ for
$\alpha=0.7$. Using more sophisticated calculation, Nagatou (2004) reported
$R_{c}$ to be bracketed between $3.011528364444$ and $3.011528364446$. Later,
Okamoto (1996, 1998) extended the bifurcation diagram to larger $R$ using
path-continuation method. Matsuda and Miyatake (2002) studied the bifurcation
diagram further and derived an exact formula for the second derivatives of
their components at the bifurcation points.

The Kolmogorov flow has been simulated in experiments by inducing vortices in
magnetofluids using periodically placed electrodes. Tabeling, Perrin, and
Fauve (1987) observed supercritical pitchfork bifurcation at the instability
of the vortices. Bondarenko, Gak, and Dolzhanskii (1979) performed a similar
experiment. In another experiment, Sommeria (1986) reported the existence of
an inverse cascade due to the nonlinear interactions. Herault, Pétrélis, and
Fauve (2015) observed $1/f$ noise in the nonlinear regime of the Kolmogorov
flow. Tabeling (2002) reviewed the experiments related to the Kolmogorov flow.

Gotoh and Yamada (1987) performed instability analysis of the rhombic cells
with the stream function as $\cos kx+\cos y$, where $k$ is the aspect ratio of
the cell. Kim and Okamoto (2003) performed bifurcation and inviscid limit
analysis for the aforementioned rhombic cells. Thess (1992) studied the
effects of viscosity, linear friction and confinement on the flow. Platt,
Sirovich, and Fitzmaurice (1991) analyzed the Kolmogorov flow for $k_{f}=4$
and observed a sequence of bifurcations leading to chaos. For the same
$k_{f}$, Chen and Price (2004) studied the chaotic behavior using a truncation
model with nine modes. In addition, researchers have studied variations of the
Kolmogorov flow to three-dimensional flows Sarris  _et al._ (2007).

The Kolmogorov flow is useful not only for analyzing transition to turbulence
but also for studying the inverse cascade in two-dimensional turbulenceGreen
(1974); Gupta  _et al._ (2019); Zhang  _et al._ (2019). Green (1974) reported
that for $k>k_{f}$, kinetic energy spectrum, $E(k)\sim k^{-5}$ whereas for
$k<k_{f}$, $E(k)\sim k$. Sommeria (1986) studied the inverse cascade
experimentally and reported that the exponent to be in the range of $-4.5$ to
$-4.9$ for $k>k_{f}$. For $k<k_{f}$, direct numerical simulations (DNS) reveal
that $E(k)\sim k^{-5/3}$. For random forcing in a wavenumber band near
$k=k_{f}$, Gupta  _et al._ (2019) showed that for $k>k_{f}$, the energy
spectrum is of the form $k^{-3}\exp(-k^{2})$; the exponential part gives an
appearance of steeper spectrum compared to $k^{-3}$. Zhang  _et al._ (2019)
performed a molecular simulation using Fokker-Planck method and reported that
$E(k)\sim k^{-4}$ for $k<k_{f}$ due to condensation in the large scale
structures. For $k>k_{f}$, Zhang  _et al._ (2019) reported that
$E(k)\sim\exp(-0.2k)$. Energy condensate is observed at the large-scale due to
inverse cascade. Gallet and Young (2013) derived a mathematical model of
energy condensation in the absence of large-scale dissipation. Mishra  _et
al._ (2015) studied the condensate regime using Ekman friction. There are more
works on the Kolmogorov flow, including those by Chandler and Kerswell (2013),
Lucas and Kerswell (2014) & Fylladitakis (2018), and references therein.

In this paper, we consider incompressible Kolmogorov flow in a two-dimensional
periodic box with aspect ratio $\alpha$, and construct a low-dimensional model
with four modes. We perform bifurcation analysis of the system and derive the
critical Reynolds number for the instability. Our results are consistent with
previous ones. Besides, we also carry out direct numerical simulations of the
Kolmogorov flow for the parameter used for our model. The results from these
simulations are in good agreement with those from the low-dimensional model.
These results provide us confidence that the chosen modes are a good choice
for the Kolmogorov flow.

The outline of this paper is as follows. In Sec. II, we present the governing
equations and low-dimensional model. In Sec. III, we perform linear stability
and bifurcation analysis of the low-dimensional model. We describe the energy
transfers among the participating modes in Sec. IV. In Sec. V we present
numerical validation using direct numerical simulation.We conclude in Sec. VI.

##  II Basic formulation -

For an incompressible flow, the Navier-Stokes equation and incompressibility
condition (Verma, 2018, 2019) are

$\displaystyle\frac{\partial\mathbf{u}}{\partial
t}+\mathbf{u}\cdot\mathbf{\nabla}\mathbf{u}$ $\displaystyle=$
$\displaystyle-\mathbf{\nabla}p+\mathbf{F}_{u}+\nu\nabla^{2}\mathbf{u},$ (1)
$\displaystyle\mathbf{\nabla}\cdot\mathbf{u}$ $\displaystyle=$ $\displaystyle
0,$ (2)

where $\mathbf{u}$ and $p$ are the velocity and pressure fields respectively,
$\nu$ is the kinematic viscosity, and $\mathbf{F}_{u}$ is the acceleration due
to the external force. We consider two-dimensional Kolmogorov flow in a
doubly-periodic box of size $L_{x}\times L_{y}$. The ratio
$\alpha=L_{y}/L_{x}$ is called aspect ratio. We assume density $\rho$ to be
unity. We take

$\mathbf{F}_{u}=\gamma\sin\bigg{(}\frac{2\pi yk_{f}}{L_{y}}\bigg{)}\hat{x},$
(3)

where $\gamma$ is the amplitude of the acceleration, and $k_{f}$ is the
forcing wavenumber which we consider to be $1$.

We nondimensionalize Eqs. (1, 2) using ${L_{y}}/{2\pi}$ as the length scale
and ${2\pi\nu}/{\gamma L_{y}}$ as the time scale, and obtain the following
equations:

$\displaystyle\frac{\partial\mathbf{u}}{\partial
t}+\mathbf{u}\cdot\mathbf{\nabla}\mathbf{u}$ $\displaystyle=$
$\displaystyle-\mathbf{\nabla}p+\frac{1}{R}\sin(y)\hat{x}+\frac{1}{R}\nabla^{2}\mathbf{u},$
(4) $\displaystyle\mathbf{\nabla}\cdot\mathbf{u}$ $\displaystyle=$
$\displaystyle 0,$ (5)

where

$R=\frac{\gamma}{\nu^{2}}\bigg{(}\frac{L_{y}}{2\pi}\bigg{)}^{3},$ (6)

is the Reynolds number. A trivial stationary solution of the Eqs. (4, 5) is

$\mathbf{u}\ =\ (\sin(y),0),\quad p=\mathrm{const.}$ (7)

This solution represents a laminar flow.

For stability analysis, it is customary to work in Fourier space. In this
space, the equations get transformed to

$\displaystyle\frac{d}{dt}\mathbf{u}(\mathbf{k})+\mathbf{N_{u}}(\mathbf{k})$
$\displaystyle=$
$\displaystyle-i\mathbf{k}p(\mathbf{k})+\mathbf{F}_{u}(\mathbf{k})-\frac{1}{R}k^{2}\mathbf{u}(\mathbf{k}),$
(8) $\displaystyle\mathbf{k}.\mathbf{u}(\mathbf{k})$ $\displaystyle=$
$\displaystyle 0,$ (9)

where

$\displaystyle\mathbf{N_{u}}(\mathbf{k})$ $\displaystyle=$ $\displaystyle
i\sum\limits_{\mathbf{p}}\\{\mathbf{k}\cdot\mathbf{u}(-\mathbf{q})\\}\mathbf{u}(-\mathbf{p}),$
(10) $\displaystyle p(\mathbf{k})$ $\displaystyle=$
$\displaystyle\frac{i}{k^{2}}{\bf
k}\cdot[\mathbf{N_{u}}(\mathbf{k})-\mathbf{F_{u}}(\mathbf{k})],$ (11)

are the Fourier transforms of the nonlinear and pressure terms respectively.
Here ${\bf k=-p-q}$. Note that calculation of $\mathbf{N_{u}}(\mathbf{k})$
requires all possible wavenumber triads. We denote the Fourier mode as ${\bf
k}=(\alpha l,m)$, where $l,m$ are integers.

The Fourier transform of the external force $\frac{1}{R}\sin(y)\hat{x}$ is

$\mathbf{F}_{u}(\mathbf{k})=\frac{1}{2iR}[\delta_{k_{y},1}-\delta_{k_{y},-1}]\hat{x}.$
(12)

That is, the forcing wavenumbers are $(0,1)$ and $(0,-1)$. Near the onset of
instability, the nonlinear term ${\bf u\cdot\nabla u}$ generates other Fourier
modes. In this paper we show that a low-dimensional model having nonzero
Fourier modes at wavenumbers $\\{\mathbf{k}\ =\ (-\alpha,0)$, $\mathbf{p}\ =\
(0,1)$, $\mathbf{q}=(\alpha,-1)$, and $\mathbf{s}\ =\ (\alpha,1)\\}$
reproduces earlier results on Kolmogorov flow quite well (e.g. Iudovich
(1965)). In Sec. V we perform direct numerical numerical simulations and show
that our low-dimensional model reproduces the simulation results to a
significant degree. Due to these reasons, we work with this set of Fourier
modes. We consider the following interacting triads:

$\displaystyle{\bf k}\bigoplus{\bf p}\bigoplus{\bf
q}=(-\alpha,0)\bigoplus(0,1)\bigoplus(\alpha,-1)$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle({\bf-k})\bigoplus{\bf
p}\bigoplus({\bf-s})=(\alpha,0)\bigoplus(0,1)\bigoplus(-\alpha,-1)$
$\displaystyle=$ $\displaystyle 0,$

where $\bigoplus$ represents nonlinear interaction (see Fig. 1).

Figure 1: Schematic representation of the triad interactions in Eqs. (II,II).
Energy transfers $\sigma$ and $\gamma$ are $(r-1)/(4\sqrt{2}Rr^{2})$,
$(\alpha^{2}(r-1))/(4\sqrt{2}Rr^{2})$, where $r=R/R_{c}$.

Thus possible nonlinear interactions for the modes with wavenumbers $\bf{k}$,
$\bf{p}$, $\bf{q}$ and $\bf{s}$ are

$\displaystyle\mathbf{k}$ $\displaystyle=$
$\displaystyle[(-\alpha,-1)\bigoplus(0,1)]+[(-\alpha,1)\bigoplus(0,-1)],$ (15)
$\displaystyle\mathbf{p}$ $\displaystyle=$
$\displaystyle[(-\alpha,0)\bigoplus(\alpha,1)]+[(\alpha,0)\bigoplus(-\alpha,1)],$
(16) $\displaystyle\mathbf{q}$ $\displaystyle=$
$\displaystyle[(\alpha,0)\bigoplus(0,-1)],$ (17) $\displaystyle\mathbf{s}$
$\displaystyle=$ $\displaystyle[(\alpha,0)\bigoplus(0,1)].$ (18)

In the next section we derive the governing equations for the above Fourier
modes.

##  III Linear stability and bifurcation analysis

The derivation of the evolution equations for the Fourier modes get simplified
in Craya-Herring basis Craya (1958); Herring (1974); Lesieur (2008); Sagaut
and Cambon (2018); Verma (2019). For wavenumber ${\bf k}$, the unit vectors in
this basis are Craya (1958); Herring (1974); Lesieur (2008); Sagaut and Cambon
(2018); Verma (2019)

$\displaystyle\hat{e}_{3}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\hat{k},$ (19) $\displaystyle\hat{e}_{1}(\mathbf{k})$
$\displaystyle=$
$\displaystyle\frac{\hat{k}\times\hat{n}}{|\hat{k}\times\hat{n}|},$ (20)
$\displaystyle\hat{e}_{2}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\hat{e}_{3}(\mathbf{k})\times\hat{e}_{1}(\mathbf{k}),$ (21)

where $\hat{k}$ is the unit vector along $\mathbf{k}$, and $\hat{n}=\hat{z}$.
For all the wavenumbers under consideration, $\hat{e}_{1}$ lie on the plane,
while $\hat{e}_{2}$ are perpendicular to the plane. Hence, for the present 2D
flow, $u_{2}=0$ for all the modes. In addition, $u_{3}=0$ due to
incompressibility condition (Eq. (9)). Therefore,

$\mathbf{u}({\mathbf{k}})=u_{1}(\mathbf{k})\hat{e}_{1}(\mathbf{k}).$ (22)

Explicitly, the unit vectors $\hat{e}_{1}$’s for the four wavenumbers
($\mathbf{k}$,$\mathbf{p}$, $\mathbf{q}$, $\mathbf{s}$) are

$\displaystyle\hat{e}_{1}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\hat{y},$ (23) $\displaystyle\hat{e}_{1}(\mathbf{p})$
$\displaystyle=$ $\displaystyle\hat{x},$ (24)
$\displaystyle\hat{e}_{1}(\mathbf{q})$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{\alpha^{2}+1}}\hat{x}-\frac{\alpha}{\sqrt{\alpha^{2}+1}}\hat{y},$
(25) $\displaystyle\hat{e}_{1}(\mathbf{s})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\alpha^{2}+1}}\hat{x}-\frac{\alpha}{\sqrt{\alpha^{2}+1}}\hat{y}.$
(26)

Using Eqs. (23-26), we derive the evolution equations for $u_{1}$’s as

$\displaystyle\frac{d}{dt}u_{1}(\mathbf{k})$ $\displaystyle=$
$\displaystyle-\bigg{(}\frac{\alpha^{2}}{\sqrt{1+\alpha^{2}}}i\bigg{)}(u^{*}_{1}(\mathbf{p})u^{*}_{1}(\mathbf{q})+u^{*}_{1}(\mathbf{s})u_{1}(\mathbf{p}))$
(27) $\displaystyle-\bigg{(}\frac{\alpha^{2}}{R}\bigg{)}u_{1}(\mathbf{k}),$
$\displaystyle\frac{d}{dt}u_{1}(\mathbf{p})$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{1}{\sqrt{1+\alpha^{2}}}i\bigg{)}(u^{*}_{1}(\mathbf{k})u^{*}_{1}(\mathbf{q})-u_{1}(\mathbf{k})u_{1}(\mathbf{s}))$
(28)
$\displaystyle+\frac{1}{2iR}-\bigg{(}\frac{1}{R}\bigg{)}u_{1}(\mathbf{p}),$
$\displaystyle\frac{d}{dt}u_{1}(\mathbf{q})$ $\displaystyle=$
$\displaystyle-\bigg{(}\frac{1-\alpha^{2}}{\sqrt{1+\alpha^{2}}}i\bigg{)}u^{*}_{1}(\mathbf{k})u^{*}_{1}(\mathbf{p})-\bigg{(}\frac{1+\alpha^{2}}{R}\bigg{)}u_{1}(\mathbf{q}),$
$\displaystyle\frac{d}{dt}u_{1}(\mathbf{s})$ $\displaystyle=$
$\displaystyle-\bigg{(}\frac{1-\alpha^{2}}{\sqrt{1+\alpha^{2}}}i\bigg{)}u^{*}_{1}(\mathbf{k})u_{1}(\mathbf{p})-\bigg{(}\frac{1+\alpha^{2}}{R}\bigg{)}u_{1}(\mathbf{s}).$

The steady-state solutions of the above equations are

$\displaystyle S_{0}:\begin{cases}u_{1}(\mathbf{k})=0,\\\
u_{1}(\mathbf{p})=\frac{1}{2i},\\\ u_{1}(\mathbf{q})=0,\\\
u_{1}(\mathbf{s})=0,\end{cases}$ (31) $\displaystyle
S_{1}:\begin{cases}u_{1}(\mathbf{k})=-\frac{1}{\sqrt{2}r}\sqrt{r-1},\\\
u_{1}(\mathbf{p})=-\frac{1}{2r}i,\\\
u_{1}(\mathbf{q})=-\frac{\sqrt{1+\alpha^{2}}}{2R}\sqrt{r-1},\\\
u_{1}(\mathbf{s})=\frac{\sqrt{1+\alpha^{2}}}{2R}\sqrt{r-1},\end{cases}$ (32)

and

$\displaystyle
S_{2}:\begin{cases}u_{1}(\mathbf{k})=\frac{1}{\sqrt{2}r}\sqrt{r-1},\\\
u_{1}(\mathbf{p})=-\frac{1}{2r}i,\\\
u_{1}(\mathbf{q})=\frac{\sqrt{1+\alpha^{2}}}{2R}\sqrt{r-1},\\\
u_{1}(\mathbf{s})=-\frac{\sqrt{1+\alpha^{2}}}{2R}\sqrt{r-1},\end{cases}$ (33)

where $r=R/R_{c}$ with

$R_{c}\ =\ \frac{\sqrt{2}(1+\alpha^{2})}{\sqrt{1-\alpha^{2}}}.$ (34)

The solution $S_{0}$ is valid for all $r$, while $S_{1}$ and $S_{2}$ are
defined only for $r>1$. Also,
$u_{1}(\mathbf{k}),u_{1}(\mathbf{q}),u_{1}(\mathbf{s})$ modes of $S_{1}$ have
opposite signs compared to $S_{2}$. See Fig. 2 for an illustration of steady
$u_{1}(\mathbf{k})$ for $\alpha=0.7$; the figure exhibits a transition from
$S_{0}$ to $S_{1}$ or $S_{2}$ at $r=1$. For $r>1$, the system follows either
$S_{1}$ branch or $S_{2}$ branch depending on the initial condition. In
subsequent subsections, we show that $S_{0}$ is the only stable solution for
$r<1$, while $S_{1}$ and $S_{2}$ are the stable solutions for $r>1$. For
$r>1$, the solution $S_{0}$ is unstable. Another important point to note that
$R_{c},S_{1},S_{2}$ are not defined for $\alpha>1$, hence $S_{0}$ is the only
solution for $\alpha>1$.

Figure 2: (color online) For $\alpha\ =\ 0.7$ , $u_{1}(\mathbf{k})$ as a
function of $r$.

In the following subsections, we analyze the stability of $S_{0}$, $S_{1}$,
and $S_{2}$.

###  III.1 Stability of laminar solution $S_{0}$

First, we analyse the stability of the laminar solution, $S_{0}$. For the
same, we linearize Eqs. (27-III) around $S_{0}$ and obtain the following
equations:

$\displaystyle\frac{d}{dt}\tilde{u}_{1}(\mathbf{k})$ $\displaystyle=$
$\displaystyle-\bigg{(}\frac{\alpha^{2}}{2\sqrt{1+\alpha^{2}}}\bigg{)}(\tilde{u}^{*}_{1}(\mathbf{s})-\tilde{u}^{*}_{1}(\mathbf{q}))-\bigg{(}\frac{\alpha^{2}}{R}\bigg{)}\tilde{u}_{1}(\mathbf{k}),$
$\displaystyle\frac{d}{dt}\tilde{u}_{1}(\mathbf{p})$ $\displaystyle=$
$\displaystyle-\frac{1}{R}\tilde{u}_{1}(\mathbf{p}),$ (36)
$\displaystyle\frac{d}{dt}\tilde{u}_{1}(\mathbf{q})$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{(1-\alpha^{2})}{2\sqrt{1+\alpha^{2}}}\bigg{)}\tilde{u}^{*}_{1}(\mathbf{k})-\bigg{(}\frac{1+\alpha^{2}}{R}\bigg{)}\tilde{u}_{1}(\mathbf{q}),$
(37) $\displaystyle\frac{d}{dt}\tilde{u}_{1}(\mathbf{s})$ $\displaystyle=$
$\displaystyle-\bigg{(}\frac{(1-\alpha^{2})}{2\sqrt{1+\alpha^{2}}}\bigg{)}\tilde{u}^{*}_{1}(\mathbf{k})-\bigg{(}\frac{1+\alpha^{2}}{R}\bigg{)}\tilde{u}_{1}(\mathbf{s}),$
(38)

where $\tilde{u}_{1}(\mathbf{k})$, $\tilde{u}_{1}(\mathbf{p})$,
$\tilde{u}_{1}(\mathbf{q})$ and $\tilde{u}_{1}(\mathbf{s})$ represent
fluctuations in $S_{0}$, and they are complex quantities. Hence we split them
into real and imaginary parts:

$\displaystyle\tilde{u}_{1}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\Re[\tilde{u}_{1}(\mathbf{k})]+i\Im[\tilde{u}_{1}(\mathbf{k})],$
(39) $\displaystyle\tilde{u}_{1}(\mathbf{p})$ $\displaystyle=$
$\displaystyle\Re[\tilde{u}_{1}(\mathbf{p})]+i\Im[\tilde{u}_{1}(\mathbf{p})],$
(40) $\displaystyle\tilde{u}_{1}(\mathbf{q})$ $\displaystyle=$
$\displaystyle\Re[\tilde{u}_{1}(\mathbf{q})]+i\Im[\tilde{u}_{1}(\mathbf{q})],$
(41) $\displaystyle\tilde{u}_{1}(\mathbf{s})$ $\displaystyle=$
$\displaystyle\Re[\tilde{u}_{1}(\mathbf{s})]+i\Im[\tilde{u}_{1}(\mathbf{s})].$
(42)

Using Eqs. (III.1-38) we derive the following matrix equation:

$\frac{d}{dt}\mathbb{U}=\mathbb{A}\mathbb{U},$ (43)

where

$\displaystyle\mathbb{A}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}-\frac{A}{R}&0&0&0&B&0&-B&0\\\
0&-\frac{A}{R}&0&0&0&-B&0&B\\\ 0&0&-\frac{1}{R}&0&0&0&0&0\\\
0&0&0&-\frac{1}{R}&0&0&0&0\\\ D&0&0&0&-\frac{C}{\mathrm{R}}&0&0&0\\\
0&-D&0&0&0&-\frac{C}{\mathrm{R}}&0&0\\\
-D&0&0&0&0&0&-\frac{C}{\mathrm{R}}&0\\\
0&D&0&0&0&0&0&-\frac{C}{\mathrm{R}}\end{pmatrix},$ $\displaystyle\mathbb{U}$
$\displaystyle=$
$\displaystyle\begin{pmatrix}\Re[\tilde{u}_{1}(\mathbf{k})]\\\
\Im[\tilde{u}_{1}(\mathbf{k})]\\\ \Re[\tilde{u}_{1}(\mathbf{p})]\\\
\Im[\tilde{u}_{1}(\mathbf{p})]\\\ \Re[\tilde{u}_{1}(\mathbf{q})]\\\
\Im[\tilde{u}_{1}(\mathbf{q})]\\\ \Re[\tilde{u}_{1}(\mathbf{s})]\\\
\Im[\tilde{u}_{1}(\mathbf{s})]\end{pmatrix}.$ (44)

Here, $\mathbb{A}$ is a $8\times 8$ matrix with $A=\alpha^{2}$,
$B=\alpha^{2}/(2\sqrt{1+\alpha^{2}})$, $C=1+\alpha^{2}$ and
$D=(1-\alpha^{2})/(2\sqrt{1+\alpha^{2}})$. The solution $\mathbb{U}(t)$ is a
linear combination of $e^{\lambda t}$, where $\lambda$’s are the eigenvalues
of $\mathbb{A}$:

$\displaystyle\lambda_{1}$ $\displaystyle=$ $\displaystyle-\frac{1}{R},$ (45)
$\displaystyle\lambda_{2}$ $\displaystyle=$ $\displaystyle-\frac{C}{R},$ (46)
$\displaystyle\lambda_{3}$ $\displaystyle=$
$\displaystyle-\frac{A+C+\sqrt{(A-C)^{2}+8BDR^{2}}}{2R},$ (47)
$\displaystyle\lambda_{4}$ $\displaystyle=$
$\displaystyle-\frac{A+C-\sqrt{(A-C)^{2}+8BDR^{2}}}{2R}.$ (48)

It is easy to show that $\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$ are
negative for all $\alpha$ and ${R}$. However, $\lambda_{4}$ changes sign from
negative to positive at the following condition, called neutral stability
condition Chandrasekhar (2013):

$\sqrt{(A-C)^{2}+8BD\mathrm{R}^{2}}=A+C,~{}~{}~{}\mathrm{or}~{}~{}R=R_{c},$
(49)

where $R_{c}$ is given by Eq. (34).

In Fig. 3 we exhibit the $(R,\alpha)$ phase diagram, the $R_{c}$ curve, as
well as the regions of stability and instability. As shown in the figure,
$R_{c}$ increases monotonically with $\alpha$, and $R_{c}\rightarrow\infty$ as
$\alpha\rightarrow 1$. Also $R_{c}\rightarrow\sqrt{2}$ as $\alpha\rightarrow
0$. The figure shows that the system is stable below the $R=R_{c}$ curve and
yields the laminar solution ($S_{0}$), and unstable otherwise. Note that for
$\alpha>1$, the laminar solution is stable for all $R$.

Figure 3: (color online) Phase diagram $(R,\alpha)$ constructed using
$R=R_{c}$ curve, also called neutral stability curve (the red curve). The
figure also includes the steady-state results of the DNS. The green circles
represent laminar solution ($S_{0}$), but the blue triangles represent the
steady vortex flow ($S_{1}$ or $S_{2}$).

In the next section we show that for $\alpha<1$ and $R>R_{c}$, the solutions
$S_{1}$ and $S_{2}$ are the stable solutions.

###  III.2 Stability analysis of $S_{1}$ and $S_{2}$

As described in Sec. III, the solutions $S_{1},S_{2}$ exist only for $r>1$. In
this subsection we show that these are stable solutions for $r>1$. For the
stability analysis, we generate the stability matrix for $S_{1}$ and $S_{2}$
by linearizing Eqs. (27-III) around $S_{1},S_{2}$. This exercise yields a set
of equations for $\tilde{u}_{1}(\bf{k})$, $\tilde{u}_{1}(\bf{p})$,
$\tilde{u}_{1}(\bf{q})$, and $\tilde{u}_{1}(\bf{s})$ similar to Eqs.
(III.1-38). Note that $\tilde{u}_{1}(\bf{k})$, $\tilde{u}_{1}(\bf{p})$,
$\tilde{u}_{1}(\bf{q})$, and $\tilde{u}_{1}(\bf{s})$ are fluctuations around
$S_{1},S_{2}$. The resulting matrix equation is

$\frac{d}{dt}\mathbb{U}=\mathbb{B}\mathbb{U},$ (50)

where

$\mathbb{B}=\begin{pmatrix}-\frac{A}{R}&0&0&\pm
C^{\prime}&D^{\prime}&0&-D^{\prime}&0\\\
0&-\frac{A}{R}&0&0&0&-D^{\prime}&0&D^{\prime}\\\ 0&0&-\frac{1}{R}&0&0&\pm
E^{\prime}&0&\pm E^{\prime}\\\ \mp H^{\prime}&0&0&-\frac{1}{R}&\pm
E^{\prime}&0&\mp E^{\prime}&0\\\ F^{\prime}&0&0&\pm
G^{\prime}&-\frac{C}{\mathrm{R}}&0&0&0\\\ 0&-F^{\prime}&\pm
G^{\prime}&0&0&-\frac{C}{\mathrm{R}}&0&0\\\ -F^{\prime}&0&0&\mp
G^{\prime}&0&0&-\frac{C}{\mathrm{R}}&0\\\ 0&F^{\prime}&\pm
G^{\prime}&0&0&0&0&-\frac{C}{\mathrm{R}}\end{pmatrix},$ (51)

and $C^{\prime}$,$D^{\prime}$,$E^{\prime}$,$F^{\prime}$,$G^{\prime}$ and
$H^{\prime}$ are $(AA^{\prime})/R$, $(BR_{c})/R$, $-(BA^{\prime}R_{c})/(AR)$,
$(DR_{c})/2R$, $(DA^{\prime}R_{c})/2R$ and $A^{\prime}/R$ respectively. Here,
$A^{\prime}=\sqrt{(R/R_{c})-1}$, and $A$, $B$, $C$, $D$ are same as those
defined in Sec. III.1.

Similar to the stability analysis for $S_{0}$, we compute the eigenvalues of
the matrix $\mathbb{B}$, which are

$\displaystyle\lambda_{1}$ $\displaystyle=$ $\displaystyle-\frac{C}{R},$ (52)
$\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle-\frac{1+C+\sqrt{A+8E^{\prime}G^{\prime}R^{2}}}{2R},$ (53)
$\displaystyle\lambda_{3}$ $\displaystyle=$
$\displaystyle-\frac{1+C-\sqrt{A+8E^{\prime}G^{\prime}R^{2}}}{2R},$ (54)
$\displaystyle\lambda_{4}$ $\displaystyle=$
$\displaystyle-\frac{A+C+\sqrt{1+8D^{\prime}F^{\prime}R^{2}}}{2R},$ (55)
$\displaystyle\lambda_{5}$ $\displaystyle=$
$\displaystyle-\frac{A+C-\sqrt{1+8D^{\prime}F^{\prime}R^{2}}}{2R},$ (56)
$\displaystyle\lambda_{6}$ $\displaystyle=$
$\displaystyle-\frac{2C}{3R}+\frac{2^{\frac{4}{3}}I^{\prime}R}{6O}+\frac{2^{\frac{2}{3}}O}{6R^{3}},$
(57) $\displaystyle\lambda_{7}$ $\displaystyle=$
$\displaystyle-\frac{2C}{3R}-\frac{2^{\frac{4}{3}}I^{\prime}R(1+\sqrt{3}i)}{12O}$
(58) $\displaystyle-\frac{2^{\frac{2}{3}}O(1-\sqrt{3}i)}{12R^{3}},$
$\displaystyle\lambda_{8}$ $\displaystyle=$
$\displaystyle-\frac{2C}{3R}-\frac{2^{\frac{4}{3}}I^{\prime}R(1-\sqrt{3}i)}{12O}$
(59) $\displaystyle-\frac{2^{\frac{2}{3}}O(1+\sqrt{3}i)}{12R^{3}}.$

In the above expressions, $I^{\prime}$ and $O$ are complicated functions of
$\alpha$ and $R$, hence they are not presented here. The eigenvalues
$\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\lambda_{5}$ are always real
and negative. However, $\lambda_{6}$, $\lambda_{7}$, $\lambda_{8}$ could
become complex, still their real parts are always negative. Thus, we
demonstrate that the solutions $S_{1},S_{2}$ are stable for $r>1$ for which
these solutions are defined. Using these observations, we also conclude that
the transition at $r=1$ (or $R=R_{c}$) follows a supercritical pitchfork
bifurcation, as illustrated in Fig. 2 for $\alpha=0.7$.

In Fig. 4 we illustrate the flow profile of the vortex pattern generated by
the modes of $S_{1}$ for $\alpha=0.7$ and $R=4.61$. Note that the the large
scale vortical flow results from the presence of all the four modes of the
model. Interestingly, the vortical flow pattern is very similar to those
presented in earlier works, e.g. Okamoto and ShojiOkamoto and Shōji (1993).
Note that $R_{c}=2.95$ for $\alpha=0.7$.

The bifurcation mentioned above indicates that for $\alpha<1$, Fourier modes
with large wavelengths get excited. In the next section, we will describe
energy transfers among the interacting Fourier modes.

Figure 4: (color online) (a) The flow patterns for the model solution $S_{1}$
with $\alpha=0.7$ and $R=4.61$. Here, $L_{x}=2\pi/\alpha$ and $L_{y}=2\pi$.
(b) The same steady flow pattern, is observed in DNS for the same parameters.

##  IV Energy transfers in the Kolmogorov flow

In this section we will quantify the energy transfers between the interacting
Fourier modes $\mathbf{u(k)}$, $\mathbf{u(p)}$, $\mathbf{u(q)}$ and
$\mathbf{u(s)}$. For the same we will employ the mode-to-mode energy transfer
formalism proposed by Dar, Verma, and Eswaran (2001) and Verma (2004). For
triad (${\bf k,p,q}$) satisfying ${\bf k+p+q}=0$, the energy transfer from
$\mathbf{u}(\mathbf{p})$ to $\mathbf{u}(\mathbf{k})$ with the mediation of
$\mathbf{u}(\mathbf{q})$ is

$\mathit{S}^{uu}(\mathbf{k}|\mathbf{p}|\mathbf{q})=-\Im[\\{\mathbf{k}\cdot\mathbf{u}(\mathbf{q})\\}\\{\mathbf{u}(\mathbf{p})\cdot\mathbf{u}(\mathbf{k})\\}].$
(60)

For the transfers between other Fourier modes, we employ the corresponding
giver, receiver and mediator Fourier modes.

For the laminar solution $S_{0}$, energy transfers among the Fourier modes
vanish due to a lack of nonzero interacting triad. For $S_{1},S_{2}$, there is
no energy exchange between the Fourier modes $\mathbf{u(k)}$ and
$\mathbf{u(p)}$, that is,

$\mathit{S}^{uu}(\mathbf{k}|\mathbf{p}|\mathbf{q})=0.$ (61)

However, there are energy transfers among other Fourier modes. They are,

$\displaystyle\mathit{S}^{uu}(\mathbf{k}|\mathbf{q}|\mathbf{p})=\mathit{S}^{uu}(\mathbf{-k}|\mathbf{-s}|\mathbf{p})$
$\displaystyle=$
$\displaystyle\gamma=\frac{\alpha^{2}(r-1)}{4\sqrt{2}Rr^{2}},$ (62)
$\displaystyle\mathit{S}^{uu}(\mathbf{q}|\mathbf{p}|\mathbf{k})=\mathit{S}^{uu}(\mathbf{-s}|\mathbf{p}|\mathbf{-k})$
$\displaystyle=$ $\displaystyle\sigma=\frac{(r-1)}{4\sqrt{2}Rr^{2}}.$ (63)

It is evident that $\gamma$ and $\sigma$ are positive because $r>1$. Also
$\sigma>\gamma$ because $\alpha<1$. These energy transfers are illustrated in
Fig. 1.

The energy transfer computations indicate that the velocity mode ${\bf
u}(0,1)$ gives energy to ${\bf u}(\alpha,-1)$, which in turn gives energy to
${\bf u}(\alpha,0)$. Since $\alpha<1$, the wavenumber $(\alpha,0)$ yields the
largest wavelength. Thus, the energy flows from intermediate scale
(corresponding to wavenumber (0,1)) to large scale (coresponding to wavenumber
$(\alpha,0)$). Hence, we conclude that the Kolmogorov flow exhibits inverse
energy cascade, contrary to the forward energy transfer observed in three-
dimensional hydrodynamic turbulence.

In the next section we will describe results from direct numerical simulation
and compare them with the results of the low-dimensional model.

##  V Comparison with Direct Numerical Simulations

In this section we will compare the results of direct numerical simulation
(DNS) of the Kolmogrov flow with the model results. We numerically solve Eqs.
(4, 5) using pseudospectral method in the domain
$[0,2\pi/\alpha]\times[0,2\pi]$ with periodic boundary conditions on all
sides. We discretize the domain into $64^{2}$ uniform grid points. We start
the simulation with initial condition,
$\\{u_{1}(\mathbf{k}),u_{1}(\mathbf{p}),u_{1}(\mathbf{q}),u_{1}(\mathbf{s})\\}=\\{-0.01,0.01,-0.01(\sqrt{1+\alpha^{2}}/\alpha),0.01(\sqrt{1+\alpha^{2}}/\alpha)\\}$,
with negative wavenumbers modes given by the corresponding complex conjugates.
The rest of the modes are zeros. We employ RK2 (second order Runge-Kutta)
scheme for time advancement with fixed time step $dt=0.01$. We employ $2/3$
rule for dealiasing.

Table 1:  For $\alpha=0.7$ and $R=4.61$, the relative amplitudes of the modes of low-dimensional model (LDM), as well as the relative amplitudes of the dominant modes of DNS. The total energy of the DNS is 0.188, and that for LDM is 0.18. The amplitudes are for the steady state at $t=440$. The table does not include the $-\textbf{k}$ modes that contain the remaining 50% of the total energy.  $\textbf{k}=(k_{x},k_{y})$ | $E({\textbf{k}})/E(\%)$ | $E({\textbf{k}})/E(\%)$
---|---|---
| (DNS) | (LDM)
($\alpha,0$) | $16.035$ | $15.975$
($0,1$) | $29.459$ | $28.563$
($\alpha,1$) | $2.197$ | $2.731$
($\alpha,-1$) | $2.197$ | $2.731$
($2\alpha,1$) | $0.053$ | -
($-2\alpha,1$) | $0.053$ | -
($2\alpha,2$) | $0.002$ | -
($3\alpha,1$) | $0.001$ | -
($\alpha,2$) | $0.001$ | -

We perform the $30$ runs for different values of $\\{\alpha,R\\}$, which are
displayed in Fig. 3 as green circles and blue triangles. We observe that all
the simulations reach steady solutions, which are either laminar solution
($S_{0}$, green dots) or vortex solution ($S_{1}$ or $S_{2}$, blue triangles).
Note that the two sets of simulations are nearly separated by the $R=R_{c}$
curve, which is the red curve in Fig. 3. These observations indicate that our
low-dimensional model captures the DNS results very well for the parameters of
Fig. 3.

The dominant Fourier modes of the DNS are the same as those of the low-
dimensional model. The other modes have much small magnitudes. The flow
profiles of the DNS and the model are very similar, consistent with the above
observations. For example, for the parameter values, $\alpha=0.7$ and
$R=4.61$, the steady-state flow profiles of the low-dimensional model and DNS
exhibited in Figure 4 are very similar. For the same parameter values, the
steady-state values of the dominant Fourier modes for the DNS and the low-
dimensional model are quite close to each other (see Table 1). For the DNS,
the nine modes (along with their complex conjugates) listed in Table 1 contain
nearly all of the total energy of the system.

Figure 5: (color online) For DNS with $\alpha=0.7$ and $R=200$: (a) the energy
spectrum $E(k)$ and (b) energy flux $\Pi(k)$. Both power law ($k^{-7}$) and
exponenital function ($\exp(-2.5k)$) [see inset of (a)] fit with the numerical
data reasonably well. $\Pi(k)$ is negative for the lowest wavenumber,
indicating inverse cascade. Also, $\Pi(k)$ is negligible for $k>3$ due to the
dominance of small wavenumber modes.

In our DNS, we do not observe solutions other than $S_{0}$, $S_{1}$, and
$S_{2}$. That is, we do not observe any secondary bifurcation in our
simulations. The DNS for $\alpha=0.7$ and $R=200$ too exhibits a steady
vortical flow structure as in Fig. 4, thus indicating absence of a secondary
bifurcation. Note, however, that Okamoto and Shōji (1993) had predicted
secondary bifurcations for $\alpha=0.98$, as well as on the unstable branch
for $\alpha=0.35$. These are specialized cases that require special initial
conditions and careful time-advancing of the DNS; hence, this investigation is
deferred for future.

For the computation of the energy spectrum and flux, we performed a DNS for
$\alpha=0.7$ and $R=200$ on a relatively higher resolution of $512^{2}$. We
obtain a steady flow at $t=1300$; at this time, the energy spectrum $E(k)$ is
very steep. Steep power law of $k^{-7}$ provides a reasonable fit to the
energy spectrum, which is consistent with the predictions of Okamoto (1996).
We also remark that the exponential function $\exp(-2.5k)$ too provides a
reasonable fit to the spectrum; this result is consistent with the arguments
that the low-dimensional systems and the dissipation-range of turbulent flows
exhibit exponential spectrumPaul  _et al._ (2009); Bershadskii (2008). Note
that Zhang  _et al._ (2019) obtained similar scaling in their simulation of
the Kolmogorov flow. See Fig. 5(a) for an illustration.

We also compute the energy flux $\Pi(k)$ for the same run. The energy flux is
negative for the smallest wavenumber sphere of radius 0.5, indicating an
inverse cascade of energy (see Fig. 5(b)). The simulation result is close to
the model result, that is, the energy transfer from $u_{1}(-\alpha,-1)$ to
$u_{1}(\alpha,0)$ shown in Fig. 1 and discussed in Sec. IV. In addition,
$\Pi(k)$ falls sharply. Thus, both the energy spectrum and flux support
earlier observations that only small wavenumber modes are active in the
Kolmogorov flow. For example, see Table 1.

We conclude in the next section.

##  VI Discussions and Conclusions

In this paper, we present a low-dimensional model that captures the essential
features of the Kolmogorov flow. The Fourier components are in the Craya-
Herring basis. We identify the fixed points of the system, and show that the
system bifurcates from the laminar solution to a new solution with vortex
structure. These solutions are consistent with earlier works based on
analytical, numerical, and experimental tools. In addition, we perform direct
numerical simulation (DNS) of the Kolmogorov flow that exhibits similar
results as the low-dimensional model.

Our low-dimensional model captures the critical Reynolds number of the
Kolmogorov flow. The model predicts that the new vortex solution remains
stable beyond $R>R_{c}$. The critical Reynolds number $R_{c}$ increases
monotonically with $\alpha$, with $R_{c}\rightarrow\sqrt{2}$ as
$\alpha\rightarrow 0$, and $R_{c}\rightarrow\infty$ as $\alpha\rightarrow 1$.
But between these two limits, the model prediction of $R_{c}$ is marginally
lower than those computed using models containing a larger number of Fourier
modes Okamoto and Shōji (1993); Nagatou (2004). Using energy transfers, we
show that in the Kolmogorov flow, the energy flows from intermediate scales to
large scales; this is contrary to the forward energy transfers in Kolmogorov’s
theory of turbulence. Thus, our model captures essential aspects of the
primary bifurcation of the Kolmogorov flow, and its results are consistent
with earlier models.

Our DNS results are very similar to those of the low-dimensional model. For
example, the flow patterns and the dominants modes of DNS are close to those
of the low-dimensional model. Both DNS and the model do not exhibit any
secondary bifurcation, indicating the robustness of the low-dimensional model.
It is interesting to note that the six-mode dynamo model of Verma  _et al._
(2008) showed very similar bifurcation, as described in this paper. It is
possible that the Kolmogorov flow with forcing at larger wavenumbers
($k_{f}>1$) may exhibit secondary bifurcation.

There are certain discrepancies between the predictions of our model and those
of earlier models. As shown by Okamoto and Shōji (1993), we expect secondary
bifurcations for $\alpha$ very close to unity, as well as on the unstable
branch for other $\alpha$’s. A verification of Okamoto and Shōji (1993)’s
predictions on secondary bifurcations using DNS requires major fine-tuning of
the initial conditions and the DNS, and it is planned for the future. Also,
our model does not capture several oscillatory solutions predicted by
Sivashinsky (1985). These issues need to be explored in the future.

In summary, our four-model model captures many of its interesting features of
the Kolmogorov flow. It also opens avenues for further explorations of the
Kolmogorov flow with $k_{f}>1$ and large Reynolds numbers.

##  VII Acknowledgements

We thank Roshan Samuel, Shashwat Bhattacharya, Mohammad Anas, Narendra Pratap,
Akanksha Gupta, Shadab Alam, and Manohar Sharma for useful discussions. This
work was supported by the research grant 6104-1 from Indo-French Centre for
the Promotion of Advanced Research (IFCPAR/CEFIPRA). Soumyadeep Chatterjee is
supported by INSPIRE fellowship (IF180094) of Department of Science &
Technology, India.

##  VIII Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  IX References

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