# Converses to generalized Conway–Gordon type congruences

Ryo Nikkuni  Department of Mathematics, School of Arts and Sciences, Tokyo
Woman’s Christian University, 2-6-1 Zempukuji, Suginami-ku, Tokyo 167-8585,
Japan<EMAIL_ADDRESS>This article is dedicated to Professor Kouki
Taniyama on his 60th birthday.

###### Abstract.

It is known that for every spatial complete graph on $n\geq 7$ vertices, the
summation of the the second coefficients of the Conway polynomials over the
Hamiltonian knots is congruent to $r_{n}$ modulo $(n-5)!$, where
$r_{n}=(n-5)!/2$ if $n=8k,8k+7$, and $0$ if $n\neq 8k,8k+7$. Especially the
case of $n=7$ is famous as Conway–Gordon $K_{7}$ theorem. In this paper,
conversely, we show that every integer $(n-5)!q+r_{n}$ is realized as the
summation of the second coefficients of the Conway polynomials over the
Hamiltonian knots in some spatial complete graph on $n$ vertices.

###### Key words and phrases:

Spatial graphs, Conway–Gordon theorem

###### 1991 Mathematics Subject Classification:

Primary 57M15; Secondary 57K10

The author was supported by JSPS KAKENHI Grant Number JP22K03297.

##  1\. Introduction

Throughout this paper we work in the piecewise linear category. An embedding
$f$ of a finite graph $G$ into ${\mathbb{R}}^{3}$ is called a spatial
embedding of $G$, and the image $f(G)$ is called a spatial graph of $G$. We
say that two spatial graphs are ambient isotopic if there exists an
orientation-preserving self-homeomorphism $\Phi$ on ${\mathbb{R}}^{3}$ such
that which sends one to the other. We call a subgraph $\gamma$ of $G$
homeomorphic to the circle a cycle of $G$. We denote the set of all cycles of
$G$ by $\Gamma(G)$. A cycle of $G$ is called a $p$-cycle if it contains
exactly $p$ vertices, and also called a Hamiltonian cycle if it contains all
vertices of $G$. We often denote an edge of $G$ connecting the vertices $u$
and $v$ by $\overline{uv}$, and also denote a $p$-cycle
$\overline{v_{1}v_{2}}\cup\overline{v_{2}v_{3}}\cup\cdots\cup\overline{v_{p-1}v_{p}}\cup\overline{v_{p}v_{1}}$
of $G$ by $[v_{1}v_{2}\cdots v_{p}]$. We denote the set of all $p$-cycles of
$G$ by $\Gamma_{p}(G)$, and the set of all pairs of two disjoint cycles of $G$
consisting of a $p$-cycle and a $q$-cycle by $\Gamma_{p,q}(G)$. For a cycle
$\gamma$ (resp. a pair of disjoint cycles $\lambda$) of $G$ and a spatial
graph $f(G)$, $f(\gamma)$ (resp. $f(\lambda)$) is none other than a knot
(resp. $2$-component link) in $f(G)$. In particular for a Hamiltonian cycle
$\gamma$ of $G$, we also call $f(\gamma)$ a Hamiltonian knot in $f(G)$.

Let $K_{n}$ be the complete graph on $n$ vertices, that is the graph
consisting of $n$ vertices $1,2,\ldots,n$ such that each pair of its distinct
vertices $i$ and $j$ is connected by exactly one edge $\overline{ij}$. Then,
in 1983, the following congruence was given, which is well-known as
Conway–Gordon $K_{7}$ theorem.

######  Theorem 1-1.

(Conway–Gordon [2]) For any spatial embedding $f$ of $K_{7}$, we have

(1.1) $\displaystyle\sum_{\gamma\in\Gamma_{7}(K_{7})}a_{2}(f(\gamma))\equiv
1\pmod{2},$

where $a_{2}$ denotes the second coefficient of the Conway polynomial.

The modulo two reduction of $a_{2}(K)$ for an oriented knot $K$ is also called
the Arf invariant of $K$. Theorem 1-1 says that a spatial graph $f(K_{7})$
always contains a nontrivial Hamiltonian knot with Arf invariant one. Even
when $n\geq 8$, by applying Theorem 1-1, Blain et al. and Hirano showed that a
spatial graph $f(K_{n})$ always contains a large number of such nontrivial
Hamiltonian knots [1], [6].

On the other hand, in 2009, an integral lift of (1.1) was given by the author
[10], and further extended to $K_{n}$ with $n\geq 8$ by Morishita and the
author in 2019 [8] (See Theorem 2-1). As a consequence, an extension of (1.1)
to general complete graphs was also given as follows, that is a constraint on
the set of all Hamiltonian knots in a spatial complete graph with arbitrary
number of vertices.111The case of $n=8$ had previously been obtained by
combining the results of Foisy [4] and Hirano [5].

######  Theorem 1-2.

(Morishita–Nikkuni [8]) Let $n\geq 7$ be an integer. For any spatial embedding
$f$ of $K_{n}$, we have

(1.2) $\displaystyle\sum_{\gamma\in\Gamma_{n}(K_{n})}a_{2}(f(\gamma))\equiv
r_{n}\pmod{(n-5)!},$

where

$\displaystyle
r_{n}=\left\\{\begin{array}[]{@{\,}lll}{\displaystyle\frac{(n-5)!}{2}}&(n\equiv
0,7\pmod{8})\\\ 0&(n\not\equiv 0,7\pmod{8}).\end{array}\right.$

Rephrasing Theorem 1-2, if there exists a spatial embedding of $K_{n}\ (n\geq
7)$ such that $\sum_{\gamma\in\Gamma_{n}(K_{n})}a_{2}(f(\gamma))=m$ then $m$
must be congruent to $r_{n}$ modulo $(n-5)!$. Our purpose in this paper is to
show that the converse is also true. This means that the congruence (1.2)
cannot be refined any further for every $n\geq 7$.

######  Theorem 1-3.

Let $n\geq 7$ be an integer. For an integer $m$, there exists a spatial
embedding $f$ of $K_{n}$ such that
$\sum_{\gamma\in\Gamma_{n}(K_{n})}a_{2}(f(\gamma))=m$ if and only if $m\equiv
r_{n}\pmod{(n-5)!}$.

For a spatial embedding $f$ of $G$ and a subset $\Gamma$ of $\Gamma(G)$, the
modulo $d$ reduction of $\sum_{\gamma\in\Gamma}a_{2}(f(\gamma))$ was denoted
by $\mu_{f}(G,\Gamma;d)$ in Shimabara [13] and the condition for it not to
depend on $f$ was investigated.222Later, Lemma 2 in [13] was found to be
incorrect and was modified in [4, Lemma 1]. Theorem 1-3 completely determines
the maximum value of $d$ such that $\mu_{f}(K_{n},\Gamma_{n}(K_{n});d)$ does
not depend on $f$, and also determines the possible values of
$\sum_{\gamma\in\Gamma_{n}(K_{n})}a_{2}(f(\gamma))$ as that time. We prove
Theorem 1-3 in the next section. Our proof is constructive, namely for any
integer $q$, we construct a spatial embedding $f$ of $K_{n}$ satisfying
$\sum_{\gamma\in\Gamma_{n}(K_{n})}a_{2}(f(\gamma))=(n-5)!q+r_{n}$.

######  Remark 1-4.

  1. (1)

If we define $r_{6}=0$, then we also will see that Theorem 1-3 is true for
$n=6$, namely every integer can be realized by
$\sum_{\gamma\in\Gamma_{6}(K_{6})}a_{2}(f(\gamma))$.

  2. (2)

It is also known that for every spatial embedding $f$ of $K_{6}$, the
summation $\sum_{\lambda\in\Gamma_{3,3}(K_{6})}{\rm lk}(f(\lambda))$ is odd,
where ${\rm lk}$ denotes the linking number in ${\mathbb{R}}^{3}$
(Conway–Gordon [2], Sachs [12]). We refer the reader to Karasev–Skopenkov [7]
for a converse of this congruence and its higher-dimensional analogue.

##  2\. Proof of Theorem 1-3

First of all, we prepare a kind of ‘canonical’ spatial embedding of $K_{n}$.
Let $h$ be a spatial embedding of $K_{n}$ constructed by taking all vertices
$1,2,\ldots,n$ of $K_{n}$ in order on the moment curve $(t,t^{2},t^{3})\
(t\geq 0)$ in ${\mathbb{R}}^{3}$ and connecting every pair of two distinct
vertices by a straight line segment, see Fig. 2.1 for $n=7,8$. In [8], this is
called a standard rectilinear spatial embedding of $K_{n}$, and $h(K_{n})$ is
also ambient isotopic to a canonical book presentation of $K_{n}$ introduced
in [3]. We denote $\sum_{\gamma\in\Gamma_{n}(K_{n})}a_{2}(h(\gamma))$ by
$c_{n}$. Let $V=\\{i_{1},i_{2},\ldots,i_{q}\\}$ be the set of $q$ vertices of
$K_{n}$ satisfying $i_{1}<i_{2}<\cdots<i_{q}$. Then $V$ induces a subgraph of
$K_{n}$ isomorphic to $K_{q}$. We denote this subgraph by
$K_{q}[V]=K_{q}[i_{1},i_{2},\ldots,i_{q}]$. Then for any $V$, the spatial
subgraph $h(K_{q}[V])$ of $h(K_{n})$ is ambient isotopic to $h(K_{q})$.

Figure 2.1. Spatial graph $h(K_{n})\ (n=7,8)$

To calculate the summation of $a_{2}$ over the Hamiltonian knots in a spatial
complete graph, let us recall a generalized Conway–Gordon type theorem.

######  Theorem 2-1.

(Morishita–Nikkuni [8]) Let $n\geq 6$ be an integer. For any spatial embedding
$f$ of $K_{n}$, we have

$\displaystyle\sum_{\gamma\in\Gamma_{n}(K_{n})}a_{2}(f(\gamma))-(n-5)!\sum_{\gamma\in\Gamma_{5}(K_{n})}a_{2}(f(\gamma))$
$\displaystyle=$
$\displaystyle\frac{(n-5)!}{2}\bigg{(}\sum_{\lambda\in\Gamma_{3,3}(K_{n})}{{\rm
lk}(f(\lambda))}^{2}-\binom{n-1}{5}\bigg{)}.$

Therefore, to know the summation of $a_{2}$ over the Hamiltonian knots in a
spatial complete graph, we only need to know the summation of $a_{2}$ over the
‘pentagon’ knots and the summation of ${\rm lk}^{2}$ over the ‘triangle-
triangle’ links.

######  Example 2-2.

([8]) For an integer $n\geq 6$, let us calculate
$c_{n}=\sum_{\gamma\in\Gamma_{n}(K_{n})}a_{2}(h(\gamma))$. As we said before,
for any collection of $q$ vertices $i_{1}<i_{2}<\cdots<i_{q}$ of $K_{n}$, the
spatial subgraph $h(K_{q}[i_{1},i_{2},\ldots,i_{q}])$ of $h(K_{n})$ is ambient
isotopic to $h(K_{q})$. Note that $h(K_{5})$ contains no nontrivial knots, and
$h(K_{6})$ contains exactly one nonsplittable link which is a Hopf link, see
Fig. 2.2. Then we have

$\displaystyle\sum_{\lambda\in\Gamma_{3,3}(K_{n})}{\rm lk}(h(\lambda))^{2}$
$\displaystyle=$
$\displaystyle\sum_{i_{1}<i_{2}<\cdots<i_{6}}\bigg{(}\sum_{\lambda\in\Gamma_{3,3}(K_{6}[i_{1},i_{2},\ldots,i_{6}])}{\rm
lk}(h(\lambda))^{2}\bigg{)}=\binom{n}{6},$
$\displaystyle\sum_{\gamma\in\Gamma_{5}(K_{n})}a_{2}(h(\gamma))$
$\displaystyle=$
$\displaystyle\sum_{i_{1}<i_{2}<\cdots<i_{5}}\bigg{(}\sum_{\gamma\in\Gamma_{5}(K_{5}[i_{1},i_{2},\ldots,i_{5}])}a_{2}(h(\gamma))\bigg{)}=0.$

Thus by Theorem 2-1, we have

$\displaystyle
c_{n}=\frac{(n-5)!}{2}\bigg{(}\binom{n}{6}-\binom{n-1}{5}\bigg{)}.$

Figure 2.2. $h(K_{q}[i_{1},i_{2},\ldots,i_{q}])\ (q=5,6)$

Let $n\geq 6$ be an integer and $k,l$ non-negative integers satisfying
$k+l\leq n-4$. Then for a non-negative integer $s$, we obtain a spatial graph
$f_{k,l}^{(s)}(K_{n})$ from $h(K_{n})$ by twisting two spatial edges
$\overline{1\ n-k}$ and $\overline{l+2\ \ n-(k+1)}$ $2s$ times as illustrated
in Fig. 2.3 and Fig. 2.4. Note that this twist is done close enough to the
vertices $n-k$ and $n-(k+1)$, and this twisted part represented by a gray
segment with $2s$ in the diagram as in Fig. 2.3 has no under crossings with
any other edge.

Figure 2.3. $f_{k,l}^{(s)}(K_{n})\ (s,k,l\geq 0,\ k+l\leq n-4)$

Figure 2.4. $2s$ twists ($s$ full twists) formed by two spatial edges

######  Lemma 2-3.

$\displaystyle\sum_{\gamma\in\Gamma_{n}(K_{n})}a_{2}\big{(}f^{(s)}_{k,l}(\gamma)\big{)}-c_{n}=(n-4)!\sigma(k,l;s)+(n-5)!\tau(k,l;s),$

where

$\displaystyle\sigma(k,l;s)$ $\displaystyle=$ $\displaystyle s(s(k+l+1)-k),$
$\displaystyle\tau(k,l;s)$ $\displaystyle=$ $\displaystyle
s\bigg{(}(1-s)(k^{2}+kl+l^{2})$
$\displaystyle+s\dbinom{k}{2}+s\dbinom{n-(k+l+4)}{2}+(s-2)\dbinom{l}{2}\bigg{)}.$

###### Proof.

By Theorem 2-1, we have

$\displaystyle\sum_{\gamma\in\Gamma_{n}(K_{n})}a_{2}\big{(}f^{(s)}_{k,l}(\gamma)\big{)}-c_{n}$
$\displaystyle=$
$\displaystyle(n-5)!\bigg{(}\sum_{\gamma\in\Gamma_{5}(K_{n})}a_{2}\big{(}f^{(s)}_{k,l}(\gamma)\big{)}-\sum_{\gamma\in\Gamma_{5}(K_{n})}a_{2}(h(\gamma))\bigg{)}$
$\displaystyle+\frac{(n-5)!}{2}\bigg{(}\sum_{\lambda\in\Gamma_{3,3}(K_{n})}{\rm
lk}\big{(}f_{k,l}^{(s)}(\lambda)\big{)}^{2}-\sum_{\lambda\in\Gamma_{3,3}(K_{n})}{\rm
lk}(h(\lambda))^{2}\bigg{)}.$

In addition, in the same way as Example 2-2, we have

$\displaystyle\sum_{\lambda\in\Gamma_{3,3}(K_{n})}{\rm
lk}\big{(}f_{k,l}^{(s)}(\lambda)\big{)}^{2}-\sum_{\lambda\in\Gamma_{3,3}(K_{n})}{\rm
lk}(h(\lambda))^{2}$ $\displaystyle=$
$\displaystyle\sum_{i_{1}<i_{2}<\cdots<i_{6}}\bigg{(}\sum_{\lambda\in\Gamma_{3,3}(K_{6}[i_{1},i_{2},\ldots,i_{6}])}{\rm
lk}\big{(}f_{k,l}^{(s)}(\lambda)\big{)}^{2}-\sum_{\lambda\in\Gamma_{3,3}(K_{6}[i_{1},i_{2},\ldots,i_{6}])}{\rm
lk}(h(\lambda))^{2}\bigg{)},$
$\displaystyle\sum_{\gamma\in\Gamma_{5}(K_{n})}a_{2}\big{(}f_{k,l}^{(s)}(\gamma)\big{)}-\sum_{\gamma\in\Gamma_{5}(K_{n})}a_{2}(h(\gamma))$
$\displaystyle=$
$\displaystyle\sum_{i_{1}<i_{2}<\cdots<i_{5}}\bigg{(}\sum_{\gamma\in\Gamma_{5}(K_{5}[i_{1},i_{2},\ldots,i_{5}])}a_{2}\big{(}f_{k,l}^{(s)}(\gamma)\big{)}-\sum_{\gamma\in\Gamma_{5}(K_{5}[i_{1},i_{2},\ldots,i_{5}])}a_{2}(h(\gamma))\bigg{)}.$

Here, we only need to consider $K_{q}[i_{1},i_{2},\ldots,i_{q}]\ (q=5,6)$
containing two spatial edges $\overline{1\ n-k}$ and $\overline{l+2\
n-(k+1)}$. First we consider $K_{6}[i_{1},i_{2},\ldots,i_{6}]$ containing
$\overline{1\ n-k}$ and $\overline{l+2\ n-(k+1)}$. There are six types of such
graphs:

  1. (1)

$A_{ij}=K_{6}[1,l+2,i,n-(k+1),n-k,j]$,

  2. (2)

$B_{ij}=K_{6}[1,i,l+2,n-(k+1),n-k,j]$,

  3. (3)

$C_{ij}=K_{6}[1,i,l+2,j,n-(k+1),n-k]$,

  4. (4)

$D_{ij}=K_{6}[1,l+2,i,j,n-(k+1),n-k]$,

  5. (5)

$E_{ij}=K_{6}[1,i,j,l+2,n-(k+1),n-k]$,

  6. (6)

$F_{ij}=K_{6}[1,l+2,n-(k+1),n-k,i,j]$.

Then $f_{k,l}^{(s)}(A_{ij})$, $f_{k,l}^{(s)}(B_{ij})$,
$f_{k,l}^{(s)}(C_{ij})$, $f_{k,l}^{(s)}(D_{ij})$, $f_{k,l}^{(s)}(E_{ij})$ and
$f_{k,l}^{(s)}(F_{ij})$ are spatial graphs as illustrated in Fig. 2.5 (1),
(2), (3), (4), (5) and (6), respectively.

Figure 2.5. (1) $f_{k,l}^{(s)}(A_{ij})$ (2) $f_{k,l}^{(s)}(B_{ij})$ (3)
$f_{k,l}^{(s)}(C_{ij})$ (4) $f_{k,l}^{(s)}(D_{ij})$ (5)
$f_{k,l}^{(s)}(E_{ij})$ (6) $f_{k,l}^{(s)}(F_{ij})$

(1) We put $\lambda_{A}=[1\ n-k\ i]\cup[j\ l+2\ n-(k+1)]$ and $\mu_{A}=[1\
n-k\ j]\cup[l+2\ n-(k+1)\ i]$. Then we can see that ${\rm
lk}\big{(}f_{k,l}^{(s)}(\lambda_{A})\big{)}^{2}=(s-1)^{2}$, ${\rm
lk}\big{(}f_{k,l}^{(s)}(\mu_{A})\big{)}^{2}=s^{2}$, and
$f_{k,l}^{(s)}(\lambda^{\prime})$ is a trivial link for any
$\lambda^{\prime}\in\Gamma_{3,3}(A_{ij})\setminus\\{\lambda_{A},\mu_{A}\\}$.
Note that there are $n-(k+l+4)$ ways to choose a vertex $i$, and there are $k$
ways to choose a vertex $j$. Then we have

$\displaystyle\sum_{i,j}\bigg{(}\sum_{\lambda\in\Gamma_{3,3}(A_{ij})}{\rm
lk}\big{(}f_{k,l}^{(s)}(\lambda)\big{)}^{2}-\sum_{\lambda\in\Gamma_{3,3}(A_{ij})}{\rm
lk}(h(\lambda))^{2}\bigg{)}$ $\displaystyle=$ $\displaystyle
k(n-(k+l+4))(s^{2}+(s-1)^{2}-1)$ $\displaystyle=$ $\displaystyle
2s(s-1)k(n-(k+l+4)).$

(2) We put $\lambda_{B}=[1\ n-k\ l+2]\cup[i\ j\ n-(k+1)]$, $\mu_{B}=[1\ n-k\
j]\cup[l+2\ n-(k+1)\ i]$ and $\nu_{B}=[1\ n-k\ i]\cup[l+2\ n-(k+1)\ j]$. Then
we can see that ${\rm lk}\big{(}f_{k,l}^{(s)}(\lambda_{B})\big{)}^{2}=1$,
${\rm lk}\big{(}f_{k,l}^{(s)}(\mu_{B})\big{)}^{2}={\rm
lk}\big{(}f_{k,l}^{(s)}(\nu_{B})\big{)}^{2}=s^{2}$, and
$f_{k,l}^{(s)}(\lambda^{\prime})$ is a trivial link for any
$\lambda^{\prime}\in\Gamma_{3,3}(B_{ij})\setminus\\{\lambda_{B},\mu_{B},\nu_{B}\\}$.
Note that there are $l$ ways to choose a vertex $i$, and there are $k$ ways to
choose a vertex $j$. Then we have

(2.5)
$\displaystyle\sum_{i,j}\bigg{(}\sum_{\lambda\in\Gamma_{3,3}(B_{ij})}{\rm
lk}\big{(}f_{k,l}^{(s)}(\lambda)\big{)}^{2}-\sum_{\lambda\in\Gamma_{3,3}(B_{ij})}{\rm
lk}(h(\lambda))^{2}\bigg{)}=2s^{2}kl.$

(3) Note that there are $l$ ways to choose a vertex $i$, and there are
$n-(k+l+4)$ ways to choose a vertex $j$. Then in the same way as (2), we have

$\displaystyle\sum_{i,j}\bigg{(}\sum_{\lambda\in\Gamma_{3,3}(C_{ij})}{\rm
lk}\big{(}f_{k,l}^{(s)}(\lambda)\big{)}^{2}-\sum_{\lambda\in\Gamma_{3,3}(C_{ij})}{\rm
lk}(h(\lambda))^{2}\bigg{)}$ $\displaystyle=$ $\displaystyle
2s^{2}l(n-(k+l+4)).$

(4) Note that there are $\binom{n-(k+l+4)}{2}$ ways to choose the vertices $i$
and $j$. Then in the same way as (2), we have

(2.7)
$\displaystyle\sum_{i,j}\bigg{(}\sum_{\lambda\in\Gamma_{3,3}(D_{ij})}{\rm
lk}\big{(}f_{k,l}^{(s)}(\lambda)\big{)}^{2}-\sum_{\lambda\in\Gamma_{3,3}(D_{ij})}{\rm
lk}(h(\lambda))^{2}\bigg{)}=2s^{2}\dbinom{n-(k+l+4)}{2}.$

(5) Note that there are $\binom{l}{2}$ ways to choose the vertices $i$ and
$j$. Then in the same way as (2), we have

(2.8)
$\displaystyle\sum_{i,j}\bigg{(}\sum_{\lambda\in\Gamma_{3,3}(E_{ij})}{\rm
lk}\big{(}f_{k,l}^{(s)}(\lambda)\big{)}^{2}-\sum_{\lambda\in\Gamma_{3,3}(E_{ij})}{\rm
lk}(h(\lambda))^{2}\bigg{)}=2s^{2}\dbinom{l}{2}.$

(6) Note that there are $\binom{k}{2}$ ways to choose the vertices $i$ and
$j$. Then in the same way as (2), we have

(2.9)
$\displaystyle\sum_{i,j}\bigg{(}\sum_{\lambda\in\Gamma_{3,3}(F_{ij})}{\rm
lk}\big{(}f_{k,l}^{(s)}(\lambda)\big{)}^{2}-\sum_{\lambda\in\Gamma_{3,3}(F_{ij})}{\rm
lk}(h(\lambda))^{2}\bigg{)}=2s^{2}\dbinom{k}{2}.$

Then by combining (2), (2.5), (2), (2.7), (2.8) and (2.9) with (2), we have

$\displaystyle\frac{(n-5)!}{2}\bigg{(}\sum_{\lambda\in\Gamma_{3,3}(K_{n})}{\rm
lk}\big{(}f_{k,l}^{(s)}(\lambda)\big{)}^{2}-\sum_{\lambda\in\Gamma_{3,3}(K_{n})}{\rm
lk}(h(\lambda))^{2}\bigg{)}$ $\displaystyle=$
$\displaystyle(n-5)!s\bigg{(}(s-1)k(n-(k+l+4))+s\dbinom{k}{2}+skl$
$\displaystyle+sl(n-(k+l+4))+s\dbinom{n-(k+l+4)}{2}+s\dbinom{l}{2}\bigg{)}$
$\displaystyle=$
$\displaystyle(n-5)!s\bigg{(}((s-1)k+sl)(n-4)-((s-1)k+sl)(k+l)+skl$
$\displaystyle+s\dbinom{k}{2}+s\dbinom{n-(k+l+4)}{2}+s\dbinom{l}{2}\bigg{)}$
$\displaystyle=$
$\displaystyle(n-4)!s((s-1)k+sl)+(n-5)!s\bigg{(}-l-(s-1)(k^{2}+kl+l^{2})$
$\displaystyle-l(l-1)+s\dbinom{k}{2}+s\dbinom{n-(k+l+4)}{2}+s\dbinom{l}{2}\bigg{)}$
$\displaystyle=$ $\displaystyle(n-4)!s((s-1)k+sl)-(n-5)!sl+(n-5)!\tau(k,l;s).$

Next we consider $K_{5}[i_{1},i_{2},\ldots,i_{5}]$ containing $\overline{1\
n-k}$ and $\overline{l+2\ n-(k+1)}$. There are three types of such graphs:

  1. (1)

$G_{i}=K_{5}[1,l+2,n-(k+1),n-k,i]$,

  2. (2)

$H_{i}=K_{5}[1,i,l+2,n-(k+1),n-k]$,

  3. (3)

$I_{i}=K_{5}[1,l+2,i,n-(k+1),n-k]$.

Then $f_{k,l}^{(s)}(G_{i})$, $f_{k,l}^{(s)}(H_{i})$ and $f_{k,l}^{(s)}(I_{i})$
are spatial graphs as illustrated in Fig. 2.6 (1), (2) and (3), respectively.

Figure 2.6. (1) $f_{k,l}^{(s)}(G_{i})$ (2) $f_{k,l}^{(s)}(H_{i})$ (3)
$f_{k,l}^{(s)}(I_{i})$

(1) We put $\gamma_{G}=[1\ n-k\ i\ l+2\ n-(k+1)]$ and $\delta_{G}=[1\ n-k\
l+2\ n-(k+1)\ i]$. Then we can see that $f_{k,l}^{(s)}(\gamma_{G})$ is ambient
isotopic to the $(2,2s+1)$-torus knot, $f_{k,l}^{(s)}(\delta_{G})$ is ambient
isotopic to the $(2,2s-1)$-torus knot and $f_{k,l}^{(s)}(\gamma^{\prime})$ is
a trivial knot for any
$\gamma^{\prime}\in\Gamma_{5}(G_{i})\setminus\\{\gamma_{G},\delta_{G}\\}$. We
remark here that $a_{2}(J)=s(s+1)/2$ for the $(2,2s+1)$-torus knot $J$. Also
note that there are $k$ ways to choose a vertex $i$. Then we have

$\displaystyle\sum_{i}\bigg{(}\sum_{\gamma\in\Gamma_{5}(G_{i})}a_{2}\big{(}f_{k,l}^{(s)}(\gamma)\big{)}-\sum_{\gamma\in\Gamma_{5}(G_{i})}a_{2}(h(\gamma))\bigg{)}$
$\displaystyle=$ $\displaystyle
k\Big{(}\frac{s(s+1)}{2}+\frac{s(s-1)}{2}\Big{)}=s^{2}k.$

(2) We put $\gamma_{H}=[1\ n-k\ i\ l+2\ n-(k+1)]$ and $\delta_{H}=[1\ n-k\
l+2\ n-(k+1)\ i]$. Then we can see that each of $f_{k,l}^{(s)}(\gamma_{H})$
and $f_{k,l}^{(s)}(\delta_{H})$ is ambient isotopic to the $(2,2s+1)$-torus
knot and $f_{k,l}^{(s)}(\gamma^{\prime})$ is a trivial knot for any
$\gamma^{\prime}\in\Gamma_{5}(H_{i})\setminus\\{\gamma_{H},\delta_{H}\\}$.
Note that there are $l$ ways to choose a vertex $i$. Then we have

(2.12)
$\displaystyle\sum_{i}\bigg{(}\sum_{\gamma\in\Gamma_{5}(H_{i})}a_{2}\big{(}f_{k,l}^{(s)}(\gamma)\big{)}-\sum_{\gamma\in\Gamma_{5}(H_{i})}a_{2}(h(\gamma))\bigg{)}=s(s+1)l.$

(3) Note that there are $n-(k+l+4)$ ways to choose a vertex $i$. Then in the
same way as (1), we have

(2.13)
$\displaystyle\sum_{i}\bigg{(}\sum_{\gamma\in\Gamma_{5}(I_{i})}a_{2}\big{(}f_{k,l}^{(s)}(\gamma)\big{)}-\sum_{\gamma\in\Gamma_{5}(I_{i})}a_{2}(h(\gamma))\bigg{)}=s^{2}(n-(k+l+4)).$

Then by combining (2), (2.12) and (2.13) with (2), we have

$\displaystyle(n-5)!\bigg{(}\sum_{\gamma\in\Gamma_{n}(K_{n})}a_{2}\big{(}f^{(s)}_{k,l}(\gamma)\big{)}-\sum_{\gamma\in\Gamma_{n}(K_{n})}a_{2}(h(\gamma))\bigg{)}$
$\displaystyle=$ $\displaystyle(n-5)!s\big{(}sk+(s+1)l+s(n-(k+l+4))\big{)}$
$\displaystyle=$ $\displaystyle(n-5)!s\big{(}s(n-4)+l\big{)}$ $\displaystyle=$
$\displaystyle(n-4)!s^{2}+(n-5)!sl.$

Then by (2), (2) and (2), we have the result. ∎

As special cases of Lemma 2-3, we also have the following.

######  Lemma 2-4.

  1. (1)

If $n$ is an odd integer, then

$\displaystyle\sum_{\gamma\in\Gamma_{n}(K_{n})}a_{2}\big{(}f^{(s)}_{1,0}(\gamma)\big{)}-c_{n}\equiv(n-5)!s\pmod{(n-4)!}.$

  2. (2)

If $n$ is an even integer, then

$\displaystyle\sum_{\gamma\in\Gamma_{n}(K_{n})}a_{2}\Big{(}f^{(s)}_{\frac{n-6}{2},\frac{n-6}{2}}(\gamma)\Big{)}-c_{n}\equiv(n-5)!s\pmod{(n-4)!}.$

###### Proof.

(1) By Lemma 2-3, we have

$\displaystyle\sum_{\gamma\in\Gamma_{n}(K_{n})}a_{2}\big{(}f^{(s)}_{1,0}(\gamma)\big{)}-c_{n}$
$\displaystyle\equiv$ $\displaystyle(n-5)!\tau(1,0;s)$ $\displaystyle=$
$\displaystyle(n-5)!s\bigg{(}-(s-1)+s\dbinom{n-5}{2}\bigg{)}$ $\displaystyle=$
$\displaystyle(n-5)!s\bigg{(}\frac{(n-4)(n-7)s}{2}+1\bigg{)}$ $\displaystyle=$
$\displaystyle(n-4)!s^{2}\cdot\frac{n-7}{2}+(n-5)!s$ $\displaystyle\equiv$
$\displaystyle(n-5)!s\pmod{(n-4)!}.$

(2) By Lemma 2-3, we have

$\displaystyle\sum_{\gamma\in\Gamma_{n}(K_{n})}a_{2}\bigg{(}f^{(s)}_{\frac{n-6}{2},\frac{n-6}{2}}(\gamma)\bigg{)}-c_{n}$
$\displaystyle\equiv$
$\displaystyle(n-5)!\tau\bigg{(}\frac{n-6}{2},\frac{n-6}{2};s\bigg{)}$
$\displaystyle=$
$\displaystyle(n-5)!s\bigg{(}-(s-1)\cdot\frac{3(n-6)^{2}}{4}+(2s-2)\dbinom{\frac{n-6}{2}}{2}+s\bigg{)}$
$\displaystyle=$
$\displaystyle(n-5)!s\bigg{(}-(s-1)\cdot\frac{(n-4)(n-7)}{2}+1\bigg{)}$
$\displaystyle=$ $\displaystyle-(n-4)!\cdot\frac{s(s-1)}{2}(n-7)+(n-5)!s$
$\displaystyle\equiv$ $\displaystyle(n-5)!s\pmod{(n-4)!}.$

∎

On the other hand, we can always change the value of
$\sum_{\gamma\in\Gamma_{n}(K_{n})}a_{2}(f(\gamma))$ by $\pm(n-4)$ as follows.

######  Lemma 2-5.

For any spatial embedding $g$ of $K_{n}$, there exists a spatial embedding $f$
of $K_{n}$ such that

$\displaystyle\sum_{\gamma\in\Gamma_{n}(K_{n})}a_{2}(f(\gamma))-\sum_{\gamma\in\Gamma_{n}(K_{n})}a_{2}(g(\gamma))=\pm(n-4)!.$

###### Proof.

A delta move is a local move on a spatial graph as illustrated in Fig. 2.7
[9]. It is known that if two oriented knots $J$ and $K$ are transformed into
each other by a single delta move, then $a_{2}(J)-a_{2}(K)=\pm 1$ [11]. In
particular, a ‘band sum’ of Borromean rings as illustrated in Fig. 2.8 (1)
(resp. (2)) causes $a_{2}$ to increase (resp. decrease) by exactly one. Since
there are $(n-4)!$ Hamiltonian cycles of $K_{n}$ containing a fixed path of
length $3$, we have the result. ∎

Figure 2.7. delta move

Figure 2.8. ‘Band sum’ of Borromean rings

###### Proof of Theorem 1-3.

We show that for an integer $m=(n-5)!q+r_{n}$, there exists a spatial
embedding $f$ of $K_{n}$ such that
$\sum_{\gamma\in\Gamma_{n}(K_{n})}a_{2}(f(\gamma))=m$. Since $c_{n}\equiv
r_{n}\pmod{(n-5)!}$ by Theorem 1-2, we have that $m=(n-5)!q^{\prime}+c_{n}$
for some integer $q^{\prime}$. Here, for this integer $q^{\prime}$, there
uniquely exists an integer $q^{\prime\prime}$ such that
$q^{\prime}=(n-4)q^{\prime\prime}+s\ (s\in\\{0,1,\ldots,n-5\\})$. Thus we have

$\displaystyle
m=(n-5)!((n-4)q^{\prime\prime}+s)+c_{n}=(n-4)!q^{\prime\prime}+(n-5)!s+c_{n}.$

Then by combining Lemma 2-4 with Lemma 2-5, there exists a spatial embedding
$f$ of $K_{n}$ such that

$\displaystyle\sum_{\gamma\in\Gamma_{n}(K_{n})}a_{2}(f(\gamma))=(n-4)!q^{\prime\prime}+(n-5)!s+c_{n}=m.$

This completes the proof. ∎

## References

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  * [5] Y. Hirano, Knotted Hamiltonian cycles in spatial embeddings of complete graphs, Docter Thesis, Niigata University, 2010.
  * [6] Y. Hirano, Improved lower bound for the number of knotted Hamiltonian cycles in spatial embeddings of complete graphs, J. Knot Theory Ramifications 19 (2010), 705–708.
  * [7] R. Karasev and A. Skopenkov, Some ‘converses’ to intrinsic linking theorems, arXiv:math.GT/2008.02523.
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  * [9] H. Murakami and Y. Nakanishi, On a certain move generating link-homology, Math. Ann. 284 (1989), 75–89.
  * [10] R. Nikkuni, A refinement of the Conway-Gordon theorems, Topology Appl. 156 (2009), 2782–2794.
  * [11] M. Okada, Delta-unknotting operation and the second coefficient of the Conway polynomial, J. Math. Soc. Japan 42 (1990), 713–717.
  * [12] H. Sachs, On spatial representations of finite graphs, Finite and infinite sets, Vol. I, II (Eger, 1981), 649–662, Colloq. Math. Soc. Janos Bolyai, 37, North-Holland, Amsterdam, 1984\.
  * [13] M. Shimabara, Knots in certain spatial graphs, Tokyo J. Math. 11 (1988), 405–413.


Higher order complex cubic quintic Ginzburg Landau equation : Chirped solitary
waves

Naresh Saha 111e-mail<EMAIL_ADDRESS>Barnana Roy222e-mail :
<EMAIL_ADDRESS>
Physics & Applied Mathematics Unit,
Indian Statistical Institute,
Kolkata - 700108, India.

Avinash Khare 333e-mail<EMAIL_ADDRESS>
Department of Physics,
Sabitribai Phule Pune University,
Pune - 411007, India.

Abstract

Propagation characteristics of the chirped dissipative solitary waves are
investigated within the framework of higher order complex cubic quintic
Ginzburg Landau equation. Potentially rich set of exact chirped dissipative
pulses, such as, bright, dark, grey, antidark, kink, antikink are derived in
the presence of the self steepening, self frequency shift and nonlinear
gain/loss. The linear stability results are corroborated by the direct
numerical simulations. The effect of the variation of model parameters on
physical quantities like the speed, amplitude and chirping are explored.

##  1 Introduction

Dissipative solitons [1] are a class of localized structures of the
nonconservative nonlinear system that exists due to the balance between the
dispersion (diffraction) and a conservative nonlinear effect and also between
the linear or the nonlinear loss and amplification. The concept of dissipative
solitons is applicable to the solid state lasers [2] as well as to the fiber
optics communication systems for the description of pulse propagation in
fibres in the presence of linear and nonlinear gain and spectral filtering
[3], pulse generation in fiber lasers with additive pulse mode-locking [4, 2],
or pulse propagation in hollow core photonic crystal fibers filled with
resonant gases [5]. The basic mathematical model for the description of all
these phenomena in optics is the complex cubic-quintic Ginzburg-Landau
equation (CCQGLE). [6]. Dissipative solitons have been observed experimentally
in [7]. Apart from supporting various localized solutions like stationary
solitons, sources, sinks, moving solitons and fronts [8], the CCQGLE also
posseses pulsating, creeping, erupting and chaotic solitons [9]. Exact chirped
soliton solutions e.g. fixed and arbitrary amplitude, flat top solitons as
well as some special pulse solutions with sharp peak at the edges and a dip in
the center etc. of CCQGLE is given in [10].
For studies of ultrashort pulses in optical fibers, CCQGLE should include
higher order terms like third order dispersion (TOD), fourth order dispersion
(FOD) and nonlinear gradient terms like the self steepening (SS), self
frequency shift (SFS), dispersion of the nonlnear refractive index and the
nonlinear gain [11]. It has been shown in [12] that these higher order effects
have marked impact on the dynamics of the pulsating, creeping, erupting and
chaotic solutions. Specifically, for some range of the parameter values, the
periodic as well as the chaotic behavior of some of these pulses is eliminated
and they are transformed into fixed shape solutions. The transition from
stationary to the pulsating solutions of CCQGLE under the influence of the
nonlinear gain, its saturation and higher order effects like the SS, TOD and
intrapulse Raman scattering (IRS) has been analyzed in [13]. Impact of the
SFS, SS and TOD on the soliton explosion has been studied in [14]. The role of
higher order nonlinear and dispersions on the periodic (non chaotic) and
chaotic exploding solitons is investigated in [15]. The effect of the
nonlinear gradient terms on breathing dissipative solitons is studied in [16].
It is shown that even small nonlinear gradient terms can make the soliton
either to breathe periodically or chaotically on only one side or to spread
rapidly. The influence of TOD, IRS and SS on the localized fixed shape soliton
solution of CCQGLE is studied with the help of the soliton perturbation theory
in [17]. These perturbations are found to affect the speed but not the
structural stability of the pulses which behave like solitons. Existence of
several specific families of robust stationary and oscillating bound states of
solitons in CCQGLE with TOD is shown in [18]. The effect of the nonlinear
gradient terms on the localized states in CCQGLE is investigated in [11]. It
is found that these terms affect the speed of the states as well as cause an
asymmetry. Modulational instability of the continuous wave solution to the
higher order CCQGLE has been studied in [19]. An exact analytic front solution
for CCQGLE with higher order terms is given in [20] and the necessary
stability condition is derived.
The influence of higher order dispersion on a chirped pulse oscillator has
been studied in [21]. It is shown that while positive FOD enhances the
oscillator stability, the odd negative higher order dispersions weaken the
stability but broaden the spectrum. The influence of strong stimulated Raman
scattering (SRS) on the formation of a complex of the bound dissipative
soliton and Raman pulse of comparable energy is shown in [22]. The chirped
dissipative soliton stability under the influence of quantum noise and SRS is
studied in [23]. Also the formation of a new kind of dissipative soliton
namely, Raman soliton for the CCQGLE in the presence of SRS is reported. The
latter significantly affects the energy scalability of the Raman soliton.
Chirped lambert W-kink solitons, and optical shock type solitons are obtained
in [24] for CCQGLE in the presence of IRS. The nonlinear fiber laser has been
modeled using CCQGLE including SS and IRS effects in [25]. It is shown that
above a minimum value of the IRS effect, it is possible to find two chirped
solitary pulses for the laser system. The smaller one belongs to the
dispersion managed regime whereas the larger one belongs to the so called
similariton regime. Chirped soliton like pulse solution to the higher order
extended Ginzburg Landau equation with the FOD is obtained in [26]. The
existence and the stability conditions of the model parameters have been
analyzed.
As it stands, the study of propagation dynamics of the chirped solitary wave
solutions to CCQGLE including SS and SFS along with other nonlinear gradient
terms, appears to be not that well studied in the literature. As CCQGLE is
considered as one of the paradigms for the localized state formation in mode
locked lasers [14], it is very important to know the range of values of the
coefficients for which the CCQGLE has stable solutions. This can be dealt with
by finding exact solutions and studying their stability. This has motivated us
to find exact chirped solitary wave solutions to the CCQGLE with SS, SFS and
other nonlinear gradient terms. We report a variety of dissipative pulses,
namely, the bright, dark, grey, antidark, kink, and antikink (subject to some
parametric conditions). The novelty of the present formulation lies in the
existence of grey, antidark and antikink dissipative solitary waves. To the
best of our knowledge, these have not been reported so far in the literature.
The results of linear stability analysis have been corroborated by direct
numerical simulation. It is found that the solitary waves have stable
evolution up to a distance of approximately $1000$ units along the direction
of propagation beyond which the instability in the propagation of all the
solitary waves is seen. Also the effect of variation of the independent model
parameters on the speed, amplitude and the chirping are explored.

##  2 Analytical Procedure

We consider the following generalized CCQGLE [12]

$i\psi_{z}+\frac{p_{r}}{2}\psi_{tt}+ip_{i}\psi_{tt}+|\psi|^{2}\psi+iq_{i}|\psi|^{2}\psi+(c_{r}+ic_{i})|\psi|^{4}\psi=i(m_{r}+im_{i})(|\psi|^{2}\psi)_{t}+i(n_{r}+in_{i})|\psi|^{2}_{t}\psi$
(1)

where in the optical context, $\psi(z,t)$ is the normalized envelope of the
optical field, $z$ is the normalized propagation distance and $t$ is the
retarded time in the frame moving with the pulse. $p_{r}=\pm 1$ is the group
velocity dispersion coefficient, $p_{i}$ describes the spectral filtering,
$c_{r}$ is a higher order correction term to the nonlinear refractive index
(quintic Kerr effect), $q_{i}$ accounts for nonlinear gain [and (or)
absorption processes], $c_{i}$ describes the saturable effects of the
nonlinear gain [and (or absorption)], $m_{r}$ is associated with the SS [27,
28], $m_{i}$ is associated with the frequency dependency of the nonlinear
gain/loss, $n_{i}$ is the intrapulse Raman scattering which is responsible for
the soliton SFS [27, 29] while $n_{r}$ accounts for the delayed nonlinear
gain/loss response.

We proceed to search for chirped solutions of Eqn.(1) in the form

$\psi(z,t)=\rho(\xi)e^{i[\chi(\xi)-\kappa z]}$ (2)

where $\xi=t-uz$ is the traveling coordinate, $\rho$, $\chi$ are real
functions of $\xi$ and $\kappa$ is a real constant. Here $u=\frac{1}{v}$, $v$
being the group velocity of the wave packet. The corresponding chirp is given
by $\delta\omega(t,z)=-\frac{\partial}{\partial t}[\chi(\xi)-\kappa
z]=-\frac{d}{d\xi}\chi(\xi)$. Substitution of Eqn.(2) into Eqn.(1) followed by
a separation of the real and the imaginary parts yield the following two
coupled equations in $\rho$ and $\chi$.

$u\chi^{{}^{\prime}}\rho+\kappa\rho+\frac{p_{r}}{2}\rho^{{}^{\prime\prime}}-\frac{p_{r}}{2}\rho\chi^{{}^{\prime}2}-p_{i}\rho\chi^{{}^{\prime\prime}}-2p_{i}\rho^{{}^{\prime}}\chi^{{}^{\prime}}+\rho^{3}+c_{r}\rho^{5}+m_{r}\chi^{{}^{\prime}}\rho^{3}+(3m_{i}+2n_{i})\rho^{2}\rho^{{}^{\prime}}=0$
(3)
$-u\rho^{{}^{\prime}}+p_{i}\rho^{{}^{\prime\prime}}-p_{i}\rho\chi^{{}^{\prime}2}+p_{r}\rho^{{}^{\prime}}\chi^{{}^{\prime}}+\frac{p_{r}}{2}\rho\chi^{{}^{\prime\prime}}+q_{i}\rho^{3}+c_{i}\rho^{5}-3m_{r}\rho^{2}\rho^{{}^{\prime}}+m_{i}\rho^{3}\chi^{{}^{\prime}}-2n_{r}\rho^{2}\rho^{{}^{\prime}}=0$
(4)

To solve the coupled Eqns.(3) and (4) we choose

$\chi^{{}^{\prime}}=\alpha\rho^{2}+\beta$ (5)

so that the frequency chirping is given by
$\delta\omega(t,z)=-\chi^{{}^{\prime}}(\xi)=-(\alpha\rho^{2}+\beta)$ where
$\alpha$ and $\beta$ are the nonlinear and constant chirp parameters
respectively. Using Eqn.(5) in Eqns.(3) and (4) yield

$\begin{split}(-\alpha^{2}\frac{p_{r}}{2}+c_{r}+m_{r}\alpha)\rho^{5}+(\alpha
u-\alpha\beta p_{r}+m_{r}\beta+1)\rho^{3}+(\beta
u+\kappa-\frac{p_{r}}{2}\beta^{2})\rho\\\
+\frac{p_{r}}{2}\rho^{{}^{\prime\prime}}+(-4\alpha
p_{i}+3m_{i}+2n_{i})\rho^{2}\rho^{{}^{\prime}}+(-2\beta
p_{i})\rho^{{}^{\prime}}=0\end{split}$ (6)
$\begin{split}(-\alpha^{2}p_{i}+c_{i}+m_{i}\alpha)\rho^{5}+(-2\alpha\beta
p_{i}+m_{i}\beta+q_{i})\rho^{3}-p_{i}\beta^{2}\rho+p_{i}\rho^{{}^{\prime\prime}}\\\
+(2\alpha p_{r}-3m_{r}-2n_{r})\rho^{2}\rho^{{}^{\prime}}+(-u+\beta
p_{r})\rho^{{}^{\prime}}=0\end{split}$ (7)

The following relations among the various model parameters

$\begin{split}\alpha=\frac{3m_{r}+2n_{r}}{2p_{r}},~{}\beta=\frac{u}{p_{r}},~{}p_{i}=0,~{}q_{i}=-\frac{m_{i}u}{p_{r}}\\\
c_{i}=-\frac{m_{i}(3m_{r}+2n_{r})}{2p_{r}},~{}~{}m_{i}=-\frac{2n_{i}}{3}\end{split}$
(8)

reduce Eqns.(6) and (7) into a single equation

$\rho^{{}^{\prime\prime}}+a_{1}\rho^{5}+a_{2}\rho^{3}+a_{3}\rho=0,$ (9)
$\displaystyle\text{where
}a_{1}=\frac{1}{4p_{r}^{2}}[(3m_{r}+2n_{r})(m_{r}-2n_{r})+8c_{r}p_{r}]$
$\displaystyle
a_{2}=\frac{2}{p_{r}^{2}}[p_{r}+m_{r}u],~{}~{}a_{3}=\frac{1}{p_{r}^{2}}[u^{2}+2kp_{r}].$

Eqn.(8) reveals that the nonlinear chirp parameter $\alpha$ depends on the
model coefficients of the group velocity dispersion, SS and the delayed
nonlinear gain while $\beta$ depends on the group velocity dispersion and the
velocity $u$ of the nonlinear mode. It is easy to check, with the help of the
parametric conditions for the existence of all the solutions presented below,
that the velocity depends on SS, delayed nonlinear gain and the quintic Kerr
effect. So the chirping can be controlled by changing these model
coefficients. The feasibility of the proposed solutions in real applications
depend on fixing the model parameters $q_{i}$, $c_{i}$, $n_{i}$ obeying
Eq.(8). In a ring fiber laser mode locked through nonlinear polarization
rotation [30], the coefficient $q_{i}$ can be adjusted to any value by a
choice of the combination of the angles of the phase plates. Accordingly, for
a given value of the solution parameters $u$ and $\alpha$, the values of
$m_{i}$, $c_{i}$ and $n_{i}$ can be theoretically adjusted by using the
appropriate relations in Eqn.(8). Now Eqn.(9) can be mapped to $\phi^{6}$
field equation [31] which enables us to obtain exact chirped dissipative
solitary wave solutions of Eqn.(1).

###  2.1 Chirped solitary waves

For all the solutions to (9) presented below, there are different parametric
conditions on $a_{1},a_{2},a_{3}$ for their existence [31]. The minimum number
of nonzero model parameters required for their existence are $p_{r}$, $c_{r}$
and $m_{r}$ (or $n_{r}$).
I. Bright:
Eqn. (9) admits the solution [31]

$\rho=\frac{A~{}\rm{sech}(B\xi+\xi_{0})}{\sqrt{1-D~{}\rm{tanh}^{2}(B\xi+\xi_{0})}}\,.$
(10)

provided

$D<1\,,~{}~{}B^{2}=-a_{3},~{}a_{2}A^{2}=2(1+D)B^{2},~{}~{}\frac{3a_{2}^{2}}{4a_{3}a_{1}}=\frac{(1+D)^{2}}{D}\,.$
(11)

We consider the solution when $0<D<1$ and in that case
$a_{1},a_{3}<0,a_{2}>0$. It is worth pointing out that in view of the identity
$\frac{\sinh(2\Delta)}{\sinh^{2}(\Delta)+\cosh^{2}(x)}=\tanh(x+\Delta)-\tanh(x-\Delta)$,
the solution (10) can be re-expressed as a superposition of a kink and an
anti-kink solution, i.e.

$\rho=\frac{\sqrt{2\coth(2\Delta)}B}{\sqrt{a_{2}}}[\tanh(P+\Delta)-\tanh(P-\Delta)]^{1/2}\,,$
(12)

where $D=\tanh^{2}(\Delta),P=B\xi+\xi_{0}$. The dependence of the velocity,
amplitude and chirp profile on different model parameters are shown in Fig.1.
Figs.1(a) and (b) shows respectively, that the amplitude and chirping
increases (decreases) as $m_{r}$ ($n_{r}$) increases but the velocity
decreases (increases) as $m_{r}$ ($n_{r}$) increases. For increasing $c_{r}$,
the chirping increases but the velocity and amplitude decreases (Fig.1(c)).
Fig.1(b) and Fig.1(c) demonstrate respectively the switching from positive
(negative) chirping to negative (positive) chirping as the $n_{r}$ ($c_{r}$)
increases.

(a)  (b)  (c)

(d)  (e)

Figure 1: For the solution(10) with condition (11) and $p_{r}=1$ (a), (b), (c)
are plots of $|u|$ (blue dotted line), $A$ (red dashed line), $\delta\omega$
(green solid line) at $\xi=5,~{}\xi_{0}=0$ vs $m_{r},~{}n_{r},~{}c_{r}$
respectively for the parameter values (a)
$n_{r}=9.9,~{}c_{r}=0.5,~{}D=0.9,~{}\kappa=-5.5$, (b)
$m_{r}=0.4,~{}c_{r}=0.02,~{}D=0.9,~{}\kappa=-7.5$, (c)
$m_{r}=0.6,n_{r}=0.2,~{}D=0.9,~{}\kappa=-7.5$. (d) Plot of eigen frequency and
(e) propagation of nonlinear mode for
$m_{r}=0.05\,,n_{r}=0.015,n_{i}=0.015,~{}D=0.8,~{}u=2.0,~{}\kappa=-2.0005$.

II. Dark:

$\rho=\frac{A~{}\rm{tanh}(B\xi+\xi_{0})}{\sqrt{1-D~{}\rm{tanh}^{2}(B\xi+\xi_{0})}}$
(13)

is an exact solution to eqn.(9) provided

$\begin{split}D<1\,,~{}~{}(3D-2)B^{2}=-a_{3},\\\
a_{2}A^{2}=2(3D-1)(1-D)B^{2},~{}\frac{3a_{2}^{2}}{4a_{3}a_{1}}=\frac{(3D-1)^{2}}{D(3D-2)}.\end{split}$
(14)

We choose $0<D<\frac{1}{3}$ and in that case $a_{1},a_{2}<0,~{}~{}a_{3}>0$.

The amplitude and chirping initially decrease as $m_{r}$ increases and start
increasing when a particular value of $m_{r}$ is reached while the velocity
initially decreases as $m_{r}$ increases but saturates for higher values of
$m_{r}$ as shown in (Fig.2(a)). When $n_{r}$ increases, the velocity and
amplitude decrease, while the chirping increases at first, but then decreases
as $n_{r}$ approaches a certain value (Fig.2(b)). The velocity, amplitude and
chirping, all decrease as $c_{r}$ increases (Fig.2(c)).

(a)  (b)  (c)

(d)  (e)

Figure 2: For the solution(13) with condition (14) and $p_{r}=-1,~{}D=0.2$ ,
(a), (b), (c) are plots of $|u|$ (blue dotted line), $A$ (red dashed line),
$\delta\omega$ (green solid line) at $\xi=5,~{}\xi_{0}=0$ vs
$m_{r},~{}n_{r},~{}c_{r}$ respectively for the parameter values (a)
$n_{r}=9.5,~{}c_{r}=10.5,~{}\kappa=-0.01$, (b)
$m_{r}=0.5,~{}c_{r}=1,\kappa=0.1$, (c) $m_{r}=0.5,n_{r}=0.5,\kappa=0.1$. (d)
Plot of eigen frequency and (e) propagation of nonlinear mode for
$m_{r}=2.5,~{}n_{r}=0.05,~{}n_{i}=0.1,~{}u=0.12,~{}\kappa=0.0042$.

III. Kink and antikink solitary waves
(a).Kink:

$\rho=A\sqrt{1+\rm{tanh}(B\xi+\xi_{0})}$ (15)

is an exact solution to eqn.(9) if the following constraints are satisfied:

$B^{2}=-a_{3},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}A^{2}=-\frac{2a_{3}}{a_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{2}^{2}=\frac{16}{3}a_{3}a_{1}$
(16)

(b). Antikink:

$\rho=A\sqrt{1-\rm{tanh}(B\xi+\xi_{0})}$ (17)

is an exact solution to eqn.(9) provided the constraints given by the eqn.(16)
are satisfied.
The amplitude and chirp profiles of solutions (15) and (17) for different
model parameters are shown in Fig.3 and Fig.4 respectively. The velocity
decreases as $m_{r}$ increases (Figs.3(a)) but the velocity increases for
increasing values of $n_{r}$ and $c_{r}$ (Figs.3(b), (c)). Fig.3(a) shows
that, as $m_{r}$ increases, the chirping increases while the amplitude
initially increases slightly and then saturates for higher values of $m_{r}$.
For increasing $n_{r}$ (Fig.3(b)) ($c_{r}$, Fig.3(c)), the amplitude and
chirping, both decrease as $n_{r}$ ($c_{r}$) increases but saturate for higher
values of $n_{r}$ ($c_{r}$).

(a)  (b)  (c)

(d)  (e)

Figure 3: For the solution(15) with conditions (16) with $p_{r}=-1$ (a), (b),
(c) are plots of $|u|$ (blue dotted line), $A$ (red dashed line),
$\delta\omega$ (green solid line) at $\xi=5,~{}\xi_{0}=0$ vs
$m_{r},~{}n_{r},~{}c_{r}$ respectively for $\kappa=5.5$ and the other
parameters are (a) $n_{r}=9.8,~{}c_{r}=9.0$, (b) $m_{r}=1.0,~{}c_{r}=0.5$, (c)
$m_{r}=1.0,n_{r}=1.0$. (d) Plot of eigen frequency and (d) propagation of
nonlinear mode for
$m_{r}=11.26\,,n_{r}=-9.01,n_{i}=0.1,~{}u=0.1199,~{}\kappa=0.0073$.

When $m_{r}$ increases, the chirping initially increases, but starts
decreasing as $m_{r}$ reaches a certain value (Fig.4(a)). The chirping
increases as $n_{r}$ and $c_{r}$ increase, as demonstrated in Figs. 4(b) and
4(c) respectively. As qualitative behaviours of the speed and amplitude of the
antikink solitary wave (17) with respect to $m_{r},n_{r},c_{r}$ are the same
as those of the kink solution (15) we refrain ourselves from giving the plots.

(a)  (b)  (c)

(d)  (e)

Figure 4: For the solution(17) with conditions (16) taking $p_{r}=-1$ (a),
(b), (c) are plots of $\delta\omega$ (green solid line) at
$\xi=5,~{}\xi_{0}=0$ vs $m_{r},~{}n_{r},~{}c_{r}$ respectively for
$\kappa=5.5$ and other parameters are (a) $n_{r}=9.8,~{}c_{r}=9.0$, (b)
$m_{r}=1.0,~{}c_{r}=0.5$, (c) $m_{r}=1.0,n_{r}=1.0$. (d) plot of eigen
frequency and (e) propagation of nonlinear mode for
$m_{r}=15.0\,,n_{r}=-6.0,n_{i}=1.5,~{}u=0.0733,~{}\kappa=0.0027$.

IV. Grey and Antidark solitary waves
Yet another solution to eqn.(9) is

$\rho=\frac{A}{\sqrt{1-D~{}\rm{tanh}^{2}(B\xi+\xi_{0})}}$ (18)

subject to the following constraints

$\begin{split}D<1\,,~{}~{}(3-2D)B^{2}=-a_{3}D\\\
Da_{2}A^{2}=2(3-D)(1-D)B^{2},~{}\frac{3a_{2}^{2}}{4a_{3}a_{1}}=\frac{(3-D)^{2}}{3-2D}\,.\end{split}$
(19)

Depending on the value of $D$ one gets grey or antidark solitary waves.
(a) Grey: For this, the parametric conditions are

$0<D<1,~{}~{}~{}~{}~{}~{}~{}a_{1}<0,~{}~{}~{}~{}~{}~{}~{}~{}a_{2}>0,~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{3}<0$
(20)

(b). Antidark: In this case the constraints are

$D<0,~{}~{}~{}~{}~{}~{}a_{1}>0,~{}~{}~{}~{}~{}~{}a_{2}<0,~{}~{}~{}~{}~{}a_{3}>0$
(21)

For the grey solitary wave (20), the velocity decreases for increasing $m_{r}$
(Fig.5(a)) and $n_{r}$ (Fig.5(b)) but it increases as $c_{r}$ increases
(Fig.5(c)). The amplitude decreases for increasing $m_{r}$ (Fig.5(a)), $n_{r}$
(Fig.5(b)), $c_{r}$ (Fig.5(c)). The chirping increases as $m_{r}$ increases
(Fig.5(a)) but it decreases as $c_{r}$ increases (Fig.5(c)). As $n_{r}$
increases, the chirping initially decreases then after increasing slightly, it
again decreases (Fig.5(b)).
Fig.6(a) shows that the velocity and chirping of the antidark solitary wave
(21) increases as $m_{r}$ increases while the amplitude initially decreases
but starts increasing when a certain value of $m_{r}$ is reached. The
amplitude decreases as $n_{r}$ (Fig.6(b)), $c_{r}$ (Fig.6(c)) increases. The
velocity slightly increases as $n_{r}$ and $c_{r}$ increase but it saturates
for higher values of $n_{r},c_{r}$(Figs.6(b), 6(c)). The chirping initially
increases but subsequently decreases for increasing $n_{r}$ but it saturates
for higher values of $n_{r}$ (Fig.6(b)) while for increasing $c_{r}$, chirping
initially decreases and then it saturates (Fig.6(c)).

(a)  (b)  (c)

(d)  (e)

Figure 5: For the solution(18) with conditions (19) and (20) with
$p_{r}=-1,~{}D=0.85$, (a), (b), (c) are plots of $|u|$ (blue dotted line), $A$
(red dashed line), $\delta\omega$ (green solid line) at $\xi=5,~{}\xi_{0}=0$
vs $m_{r},~{}n_{r},~{}c_{r}$ respectively for the parameters (a)
$n_{r}=9.8,~{}c_{r}=5.5,~{}\kappa=8.5$, (b)
$m_{r}=0.8,~{}c_{r}=1.5,~{}\kappa=0.5$, (c)
$m_{r}=1.5,~{}n_{r}=0.8,~{}\kappa=8.5$ respectively, (d) Plot of eigen
frequency and (e) propagation of nonlinear mode for
$m_{r}=12.74\,,n_{r}=-9.01,n_{i}=-0.5,~{}u=0.0973,~{}\kappa=0.0049$.

(a)  (b)  (c)

(d)  (e)

Figure 6: For the solution(18) with conditions (19) and (21) taking
$p_{r}=-1,D=-0.3$ (a), (b), (c) are plots of $|u|$ (blue dotted line), $A$
(red dashed line), $\delta\omega$ (green solid line) vs
$m_{r},~{}n_{r},~{}c_{r}$ respectively at $\xi=5,~{}\xi_{0}=0$. Other
parameters are $n_{r}=-4.9,~{}c_{r}=-13,~{}\kappa=0.1$ in (a),
$m_{r}=0.3,~{}c_{r}=0.5,~{}\kappa=8.5$ in (b),
$m_{r}=0.3,~{}n_{r}=0.5,~{}\kappa=8.5$ in (c) respectively. (d) Plot of eigen
frequency and (e) propagation of nonlinear mode for
$m_{r}=4.74\,,n_{r}=-2.01,n_{i}=0.05,~{}u=0.0021,~{}\kappa=-0.000497$.

##  3 Linear Stability Analysis

For checking the linear stability of the nonlinear modes $\psi(z,t)$, the
perturbation is taken as

$\psi(z,t)=[\rho(\xi)+\epsilon\\{F_{\lambda}(\xi)e^{i\lambda
z}+G_{\lambda}^{*}(\xi)e^{-i\lambda^{*}z}\\}]e^{i[\chi(\xi)-\kappa z]}$ (22)

where $\epsilon<<1$ and $F_{\lambda}(\xi)$ and $G_{\lambda}(\xi)$ are the
perturbation eigenfunctions corresponding to eigen frequency $\lambda$.
Substituting (22) into Eqn.(1) and then linearizing with respect to
$\epsilon$, we obtain the eigenvalue problem as

$\left(\begin{array}[]{cc}L_{1}&L_{2}\\\ -L_{2}^{*}&-L_{1}^{*}\\\
\end{array}\right)\left(\begin{array}[]{c}F_{\lambda}(\xi)\\\
G_{\lambda}(\xi)\\\
\end{array}\right)=\lambda\left(\begin{array}[]{c}F_{\lambda}(\xi)\\\
G_{\lambda}(\xi)\\\ \end{array}\right),$ (23)

where

$\displaystyle\begin{split}L_{1}=\frac{p_{r}}{2}\frac{d^{2}}{d\xi^{2}}+[-iu+ip_{r}(\alpha\rho^{2}+\beta)-2i(m_{r}+im_{i})\rho^{2}-i(n_{r}+in_{i})\rho^{2}]\frac{d}{d\xi}+u\alpha\rho^{2}\\\
+u\beta+\kappa+\frac{p_{r}}{2}(2i\alpha\rho\rho^{{}^{\prime}}-\alpha^{2}\rho^{4}-2\alpha\beta\rho^{2}-\beta^{2})+2(1+iq_{i})\rho^{2}+3(c_{r}+ic_{i})\rho^{4}\\\
-i(m_{r}+im_{i})[4\rho\rho^{{}^{\prime}}+2i(\alpha\rho^{2}+\beta)\rho^{2}]-3i(n_{r}+in_{i})\rho\rho^{{}^{\prime}},\end{split}$
$\displaystyle\begin{split}L_{2}=-i[(m_{r}+n_{r})+i(m_{i}+n_{i})]\rho^{2}\frac{d}{d\xi}+(1+iq_{i})\rho^{2}+2(c_{r}+ic_{i})\rho^{4}\\\
-i(m_{r}+im_{i})[2\rho\rho^{{}^{\prime}}+i(\alpha\rho^{2}+\beta)\rho^{2}]-i(n_{r}+in_{i})\rho\rho^{{}^{\prime}}.\end{split}$

The nonlinear wave is propagating in the positive $z$ direction. So, for
$\lambda=\lambda_{r}+i\lambda_{i}$, (22) shows that the nonlinear mode will
have stable propagation if $\lambda_{i}$ is nonnegative. The linear stability
analysis has been corroborated by direct numerical simulation adding $10\%$
random noise to the exact solutions.

The eigen frequency spectra for the bright (10), dark (13), kink (15),
antikink (17), grey (18) with conditions (20) and antidark (18) with
conditions (21) are depicted in Figs. 1(d), 2(d), 3(d), 4(d), 5(d), 6(d)
respectively. It is seen that for the chosen parameter values, the
eigenfrequency spectrum of all the solutions has negative imaginary part. This
is why the instability in the propagation of all the nonlinear modes is seen
after having stable propagation upto approximately 1000 units.

##  4 Summary:

The complex cubic quintic Ginzburg Landau equation in the presence of the
nonlinear gradient terms like SS, SFS and the nonlinear gain/loss is
considered. This system models localized state formation in mode locked lasers
[14]. A potentially rich set of exact solitary waves such as, bright, dark,
antidark, grey, kink, antikink (obeying certain parametric conditions) are
derived. It is found that chirping of all the nonlinear modes can be
controlled by varying self steepening, delayed nonlinear gain, and quintic
Kerr effect. The novelty of the present formulation lies in the existence of
the grey, antidark, antikink dissipative solitary waves.The stability of the
obtained solitary waves are investigated by linear stability analysis and the
results are corroborated by direct numerical simulations. The solitary waves
exhibit stable evolution up to a distance of approximately $1000$ units along
the direction of propagation for the chosen parameter values. The effect of
the variation of the model parameters (independent) on the physical quantities
such as, the amplitude, speed and the chirping are explored. The results
presented here may find applications in ultrashort pulse propagation in
optical transmission line, in the design of optical fiber amplifiers, optical
pulse compressors, dynamics of multimode lasers, and parametric oscillators
[14]. For future study, it would be interesting to obtain chirped dissipative
solitary waves in the presence of higher order quintic, septimal gradient
terms and the associated nonlinear gain/loss. This has relevance in obtaining
high repetition rate (e.g. beyond ultrashort, attosecond) pulse sources based
on the fiber technology [32].

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