# Controllable generations of several nonlinear waves in optical fibers with
third-order dispersion

Peng Gao1,2 Liang Duan3 Xian-Kun Yao1,2,4 Zhan-Ying Yang1,2,4
<EMAIL_ADDRESS>Wen-Li Yang1,2,4,5 1School of Physics, Northwest University,
710069 Xi’an, China  2Shaanxi Key Laboratory for Theoretical Physics
Frontiers, 710069 Xi’an, China  3School of Physics and Astronomy, Shanghai
Jiao Tong University, Shanghai, China  4Peng Huanwu Center for Fundamental
Theory, 710127 Xi’an, China  5Institute of Modern Physics, Northwest
University, 710069 Xi’an, China

###### Abstract

We propose a method to controllably generate six kinds of nonlinear waves on
continuous waves, including the one- and multi-peak solitons, the Akhmediev,
Kuznetsov-Ma, and Taijiri-Watanabe breathers, and stable periodic waves. In
the nonlinear fiber system with third-order dispersion, we illustrate their
generation conditions by the modified linear stability analysis, and
numerically generate them from initial perturbations on continuous waves. We
implement the quantitative control over their dynamical features, including
the wave type, velocity, periodicity, and localization. Our results may
provide an effective scheme for generating optical solitons on continuous
waves, and it can also be applied for wave generations in other various
nonlinear systems.

##  I Introduction

Wave generation has recently become a subject of intense research in nonlinear
fiber systems. Successful generations of some fundamental waves were
implemented, such as soliton Mollenauer-1980 ; Stolen-1983 ; Mollenauer-1983 ;
Emplit-1987 ; Krokel-1988 ; Weiner-1988 , breathers Dudley-2009 ; Kibler-2012
; Kibler-2015 ; Xu-2019 , and Peregrine rogue wave Kibler-2010 ; Frisquet-2013
. It benefits from the researches on wave’s exact solutions and generation
mechanisms in nonlinear Schrödinger (NLS) model Zakharov-1972 ; Kuznetsov-Ma ;
Peregrine-1983 ; Akhmediev-1986 ; Tajiri-1998 ; Akhmediev-2009 ;
Akhmediev-2009-PRE ; Ankiewicz-2011 ; Guo-2012 ; He-2013 . This famous model
describes a nonlinear fiber without high-order dispersion and nonlinear
effects, and it is also remarkably effective for other diverse physical
systems Agrawal-book ; Kevrekidis-book ; Yuen-1975 ; Lake-1977 ; Bailung-2011
.

For an ultrashort pulse, some high-order effects need to be introduced into
the NLS model to ensure its effectiveness Agrawal-book . When the coefficients
of introduced effects exhibit some proportions, the model is still integrable,
such as Hirota Hirota-1973  and Sasa-Satsuma Sasa-1991  models etc
Ankiewicz-2016 . Though it is difficult to realize these models in real
nonlinear fibers, the researches on them provide the guidance for real wave
generations. In these models, more kinds of nonlinear waves were presented in
the forms of exact solutions (especially the waves on continuous waves), like
one- and multi-peak solitons and periodic waves Ankiewicz-2010 ; Bandelow-2012
; Tao-2012 ; Chen-2013 ; Zhao-2014 ; Xu-2015 ; Liu-2015 ; Liu-2016 ; Zhao-2016
; Ankiewicz-2017 . Their presence undoubtedly enriches the types of wave
patterns, and provides new potential applications of wave generations.

The key point is how to achieve and control the generations of these waves in
real fibers. To this end, some efforts have been made to study their
generation mechanism. By the linear stability analysis, the quantitative
relation between wave generations and modulation instability in the NLS model
was given Dudley-2009 ; Zhao-2016-JOSAB . It provided successful controls on
the generations of Akhmediev breathers and Peregrine rogue waves. Moreover, as
high-order effects was introduced, more parameters were considered to describe
the features of waves, and the control over their generations was implemented
numerically in an integrable model with fourth-order effects Duan-2017 ;
Duan-2019 . Though these existing analyses are not enough to control the wave
generations in real high-order fibers, they prompt us to focus on two key
features of waves, namely, periodicity and localization. Meanwhile, the
modified linear stability analysis was presented to predict the quantitative
dynamics of a perturbed continuous wave Gao-2020 , including the periodicity
and localization of generated waves, so it provides the possibility to control
the wave generations.

In this paper, we numerically generate six kinds of fundamental waves on
continuous waves, in the NLS models with third-order dispersion. We achieve
the control over their types, periodicity, localization, and velocities by the
modified linear stability analysis. The third-order dispersion enriches the
types of generated waves, like the one- and multi-peak solitons on continuous
waves. The experimental feasibility of exciting the two solitons is discussed.

##  II Modified linear stability analysis for wave features

The optical field in nonlinear fiber can be described by the NLS equation.
Considering third-order dispersion, its dimensionless model with abnormal
group velocity dispersion is denoted as Agrawal-book

$\displaystyle
i\psi_{z}+\frac{1}{2}\psi_{tt}-\frac{i\beta_{3}}{6}\psi_{ttt}+|\psi|^{2}\psi=0,$
(1)

where $\psi(z,t)$ represents the slowly-varying complex envelope of optical
field. $z$, $t$ are respectively the evolution distance and retarded time, and
the subscripts $z$ and $t$ denote the partial derivative of variable with
respect to them in this paper. $\beta_{3}$ is corresponding to the third-order
dispersion. Though the generation conditions of some waves have been given in
some integrable models with high-order effects, difficulties still remains for
non-integrable systems, such as Eq. (1).

To realize their generations, we try an initial condition with the form of

$\begin{split}\psi_{p}(0,t)&=[1+u(0,t)]\psi_{0}(0,t)\\\
&=[1+(a_{1}e^{i\omega_{p}t}+a_{2}e^{-i\omega_{p}t})L(t)]\;a_{0}e^{i\omega_{0}t}.\end{split}$
(2)

The initial continuous wave background $\psi_{0}(0,t)$ has its amplitude
$a_{0}$ and frequency $\omega_{0}$. The initial perturbation $u(0,t)$ has a
localized envelope $L(t)$ and the modulated periodic wave with double
frequencies $\omega_{p}$ and $-\omega_{p}$, whose amplitudes are severally
$a_{1}$ and $a_{2}$ (here we assume $\omega_{p}>0$). The two-frequency form of
periodic wave provides convenience for feature analysis of perturbing wave.
Besides, $L(t)$ is a smooth localized function whose limit values at
$|t|\rightarrow\infty$ are 0. Here, we consider $L(t)={\rm{sech}}(\eta_{p}t)$,
where $\eta_{p}>0$ scales the localization of this function. There were many
studies in integrable models showing that sech-type initial perturbations can
generate fundamental waves with more standard patterns than the ones with
other types of envelopes Duan-2019 ; Conforti-2018 ; Liu-2018 ; Liu-2019 .

Now we start the analysis on dynamical features of perturbing wave. An
perturbed continuous wave is assumed as

$\displaystyle\psi_{p}(z,t)=[1+u(z,t)]\psi_{0}(z,t).$ (3)

Here, $\psi_{0}(z,t)$ is continuous wave solution of the equation (1) with the
form of $\psi_{0}(z,t)=a_{0}\exp{(i\omega_{0}t-ik_{0}z)}$, where $a_{0}$,
$\omega_{0}$, and $k_{0}$ are severally the amplitude, frequency, and
propagating constant of background wave, and
$k_{0}=-a_{0}^{2}+\omega_{0}^{2}/2+{\beta_{3}}\omega_{0}^{3}/6$. $u(z,t)$
stands for a weak perturbing wave with the condition $|u|^{2}\ll 1$.
Substituting Eq. (3) into the model (1), this condition allows us to figure
out the linear equation about perturbing wave $u(z,t)$:

$\begin{split}iu_{z}&+a_{0}^{2}u+a_{0}^{2}u^{*}+(i\omega_{0}+\frac{i\beta_{3}}{2}\omega_{0}^{2})u_{t}\\\
&+(\frac{1}{2}+\frac{\beta_{3}}{2}\omega_{0})u_{tt}+(-\frac{i\beta_{3}}{6})u_{ttt}=0.\end{split}$
(4)

We consider the perturbing wave with the form,

$\displaystyle u(z,t)=Ae^{p(z,t)}+Be^{p^{*}(z,t)}.$ (5)

where $p(z,t)$ be a complex function about coordinates with a general form.
$A$, $B$ are the amplitudes of two conjugate components, whose initial values
are equal to $a_{1}$ and $a_{2}$, respectively. Our method is applicable only
when $A$ and $B$ is small quantities in principle, due to the condition
$|u|^{2}\ll 1$. However, in our some numerical simulations, their moderate
values compared with background amplitude also admit our control over the
wave’s features. To observe the obvious wave generations, we try to set
initial perturbations comparable with the background wave, which will be
illustrated in Sec. IV. We substitute Eq. (5) into Eq. (4) and obtain a set of
two homogeneous equations for $a_{1}$ and $a_{2}^{*}$. This set has a
nontrivial solution only when the determinant of the coefficient matrix is
equal to 0. Thus, the expression of $p_{z}$ is

$\begin{split}p_{z}=-\omega_{0}p_{t}+\frac{1}{6}\beta_{3}N\pm\frac{1}{2}\sqrt{M(4a_{0}^{2}-M)},\end{split}$
(6)

where $M=-(p_{t}^{2}+p_{tt})(1+\beta_{3}\omega_{0})$ and
$N=p_{t}^{3}+p_{ttt}+3p_{t}p_{tt}-3\omega_{0}^{2}p_{t}$. Comparing Eq. (5)
with Eq. (2), one can obtain that $p(0,t)=i\omega_{p}t+\ln
L(t)=i\omega_{p}t+\ln[{\rm sech}(\eta_{p}t)]$. It indicates that, at the
initial distance $z=0$, $p_{t}=i\omega_{p}+{L_{t}}/{L}$,
$p_{tt}={(-L_{t}^{2}+LL_{tt})}/{L^{2}}$,
$p_{ttt}={(2L_{t}^{3}-3LL_{t}L_{tt}+L^{2}L_{ttt})}/{L^{3}}$, which can be
substituted into Eq. (6) for the specific expression of $p_{z}$.

The wave’s frequency and propagation constant can be represented by the rate-
of-change of phase ${\rm Im}[p]$ with $t$ and $z$ respectively,

$\displaystyle\omega={\rm Im}[p_{t}],~{}~{}K=-{\rm Im}[p_{z}].$ (7)

They severally scale the wave’s periodicity at $t$ and $z$ directions.
Considering that the shape of wave envelope is related to the real part of
$p$, one can describe the wave’s localization at $t$ and $z$ directions by the
rate-of-change of ${\rm Re}[p]$, namely,

$\displaystyle\eta={\rm Re}[p_{t}],~{}~{}G=-{\rm Re}[p_{z}].$ (8)

Meanwhile, the periodicity manifests as interference fringe formed by
interaction between background and perturbing waves; the localization
manifests as localized envelope on background wave. We can respectively give
the velocities of fringe and localized envelope,

$\begin{split}V=K/\omega,\quad\Lambda=G/\eta.\end{split}$ (9)

Up to now, the wave’s periodicity, localization, and velocities are described
by the above six functions about $t$, but they are not convenient to adjust
the features of wave. Thus, we consider the relationship between $\eta_{p}$
and $\eta$:

$\displaystyle\lim\limits_{t\to\pm\infty}\eta=\mp\eta_{p}.$ (10)

It indicates that the localization of wave envelope can be described precisely
by the limit of function $\eta$ when $t$ approaches positive or negative
infinity. Therefore, we introduce a subscript ($+$ or $-$) of function $f$ to
denote its limit value when $t\rightarrow+\infty$ or $t\rightarrow-\infty$ :

$f_{\pm}=\lim\limits_{t\rightarrow\pm\infty}f.$

It provides convenient limit values to depict the six features mentioned
above, namely, $\omega_{\pm}$, $\eta_{\pm}$, $K_{\pm}$, $G_{\pm}$, $V_{\pm}$
and $\Lambda_{\pm}$. For the initial condition (2), we have
$\omega_{+}=\omega_{-}=\omega_{p}$ and $\eta_{+}=-\eta_{-}=-\eta_{p}$. We
usually also have $K_{+}=K_{-}$ and $G_{+}=-G_{-}$ in our used model (1),
which leads to $V_{+}=V_{-}$ and $\Lambda_{+}=\Lambda_{-}$ (except for the
generation of one-peak soliton mentioned below). Meanwhile, to make $V_{\pm}$
and $\Lambda_{\pm}$ describe more accurately the propagation of whole wave,
the distributions of $V$ and $\Lambda$ on $t$ need to be constant
approximately. It requires that the envelope of initial perturbation is weakly
localized, namely, $\eta_{p}<1$.

Figure 1: (Color online) Riemann surfaces of (a) ${\rm Re}[p_{z}]$ and (b)
${\rm Im}[p_{z}]$ with respect to $t$ and $\omega_{p}$. Branches $+$ (green
surface) and $-$ (blue surface) respectively denote the function $p_{z}$ with
$+$ and $-$ before the square-root sign. The branches of ${\rm Re}[p_{z}]$ (or
${\rm Im}[p_{z}]$) is not continuous about $t$ when $\omega_{p}>\omega_{\rm
Re}$ (or $\omega_{p}>\omega_{\rm Im}$). The parameters are set as
$\beta_{3}=0.1$, $\eta_{p}=0.5$, $a_{0}=1$, and $\omega_{0}=0$.

There exists an important problem: $p_{z}$ in Eq. (6) is a multi-value complex
function so has two branches on its Riemann surfaces, $p_{z}^{\rm B+}$ and
$p_{z}^{\rm B-}$, which indicates that the limit values of above six functions
when $t\rightarrow+\infty$ and $t\rightarrow-\infty$ may locate on different
branches. For an example, when $\beta_{3}=0.1$, $\eta_{p}=0.5$, $a_{0}=1$, and
$\omega_{0}=0$, the Riemann surfaces of ${\rm Re}[p_{z}]$ and ${\rm
Im}[p_{z}]$ with respect to $t$ and $\omega_{p}$ are shown in Fig. 1. The
green or blue surfaces respectively denote the branches when the sign before
the square-root sign in Eq. (6) is $+$ or $-$, called Branch $+$ or Branch $-$
here. In Fig. 1 (a), there is a red cross line between Branches $+$ and $-$,
and this line ends when $\omega_{p}=\omega_{\rm Re}$. In the case of
$\omega_{p}>\omega_{\rm Re}$, the two branches are not continuous at $t=0$, so
one needs to change the branch for its limit values at $t\rightarrow+\infty$
and $t\rightarrow-\infty$. In the case of $\omega_{p}<\omega_{\rm Re}$, the
cross line disappears so the branch is not changed. Similarly, Fig. 1 (b)
shows the point where the cross line between the two branches ends, assumed as
$\omega_{\rm Im}$. The continuity about $t$ keeps in the range of
$\omega_{p}<\omega_{\rm Im}$ and disappears when $\omega_{p}>\omega_{\rm Im}$.
To distinguish the two classes of results from two different continuous
surfaces, we call the waves on Branch $+$ when $t\rightarrow+\infty$ Mode I,
and the ones on Branch $-$ when $t\rightarrow+\infty$ Mode II. They are
respectively denoted by the superscripts (I) and (II). The relation between
the modes and branches is shown in Tab. 1. Note that this relation is
applicable under the parameters in our example, which demonstrates an example
way to deal with different branches. The cases with other parameters may admit
various distributions of branches so require recalculations for the relations.

Table 1: An example relation between the modes and branches Cases |  Mode I |  Mode II
---|---|---
$\omega_{p}<\omega_{\rm Re}$ |  ${\rm Re}[p_{z}]^{\rm(I)}_{\pm}={\rm Re}[p_{z}]^{\rm B+}_{\pm}$ |  ${\rm Re}[p_{z}]^{\rm(II)}_{\pm}={\rm Re}[p_{z}]^{\rm B-}_{\pm}$
$\omega_{p}>\omega_{\rm Re}$ |  ${\rm Re}[p_{z}]^{\rm(I)}_{\pm}={\rm Re}[p_{z}]^{\rm B\pm}_{\pm}$ |  ${\rm Re}[p_{z}]^{\rm(II)}_{\pm}={\rm Re}[p_{z}]^{\rm B\mp}_{\pm}$
$\omega_{p}<\omega_{\rm Im}$ |  ${\rm Im}[p_{z}]^{\rm(I)}_{\pm}={\rm Im}[p_{z}]^{\rm B+}_{\pm}$ |  ${\rm Im}[p_{z}]^{\rm(I)}_{\pm}={\rm Im}[p_{z}]^{\rm B-}_{\pm}$
$\omega_{p}>\omega_{\rm Im}$ |  ${\rm Im}[p_{z}]^{\rm(I)}_{\pm}={\rm Im}[p_{z}]^{\rm B\pm}_{\pm}$ |  ${\rm Im}[p_{z}]^{\rm(I)}_{\pm}={\rm Im}[p_{z}]^{\rm B\mp}_{\pm}$

As different modes can admit different dynamical features of perturbing waves,
it is necessary to implement the choice of mode. The mode choice can be
realized by setting different initial amplitudes of perturbation $A$ and $B$.
With a small $B$, one can obtain the required $A$ from the homogeneous
equation set,

$\begin{split}A^{({\rm
I,II})}={a_{0}^{2}B^{*}}/[-i\omega_{0}p_{t}+\frac{i}{6}\beta_{3}N+\frac{1}{2}M-ip_{z}^{({\rm
I,II})}-a_{0}^{2}].\end{split}$ (11)

We know that $A^{({\rm I,II})}$ is a function with respect to $t$ and could
have different limit values when $t\rightarrow+\infty$ and
$t\rightarrow-\infty$. From our experience, we need to average the values of
$A_{+}^{({\rm I,II})}$ and $A_{-}^{({\rm I,II})}$, namely,

$\begin{split}A_{\rm ini}^{({\rm I,II})}=\frac{1}{2}\;[A_{+}^{({\rm
I,II})}+A_{-}^{({\rm I,II})}].\end{split}$ (12)

It provides the required initial values of $A$ (namely $a_{1}$) when we
generate nonlinear waves corresponding to mode ${\rm I}$ or ${\rm II}$.

By adjusting $\omega_{p}$, $\eta_{p}$, $a_{0}$, $\omega_{0}$, and $\beta_{3}$,
the type, periodicity, localization and velocities of generated wave can be
controlled. To control the types of waves, we will classify the fundamental
nonlinear waves by the above six features in the next section, and for
convenience the superscript (I,II) will be omitted for a while.

##  III Nonlinear waves and their generation conditions

Periodicity and localization of perturbing waves in $t$ or $z$ directions can
be used to classify them, and we take the case of $t\rightarrow+\infty$ as an
example here. For a wave with the periodicity ($\omega_{+}$, $K_{+}$) and the
localization ($\eta_{+}$, $G_{+}$), we define two quantities,

$\begin{split}\varphi^{(t)}=\tan^{-1}\Big{[}\frac{|\omega_{+}|}{|\eta_{+}|}\Big{]},\quad\varphi^{(z)}=\tan^{-1}\Big{[}\frac{|K_{+}|}{|G_{+}|}\Big{]}.\end{split}$
(13)

Both of the two quantities have the range of $0\leq\varphi^{(t,z)}\leq\pi/2$
and they describe respectively the mixture of periodicity and localization in
$t$ and $z$ directions. On the $t$ or $z$ direction, the typical wave profiles
corresponding to different values of $\varphi^{(t,z)}$ are shown in Fig. 2
(a). There are three types of wave profiles: when $\varphi^{(t,z)}=0$, the
wave is purely localized; when $\varphi^{(t,z)}=\pi/2$, the wave is purely
periodic; when $0<\varphi^{(t,z)}<\pi/2$, the wave is periodic-localized. With
the increase of $\varphi^{(t,z)}$, the wave’s periodicity gets strong and the
localization gets weak gradually. The three types can help us to classify the
fundamental waves.

Figure 2: (Color online) (a) Typical amplitude profiles at $t$ or $z$
directions when $\varphi^{(t)}$ or $\varphi^{(z)}$ is set as different values.
The localization dominates with $\varphi^{(t,z)}$ close to $0$ while the
periodicity dominates with $\varphi^{(t,z)}$ close to $\pi/2$. (b) Feature
conditions of the six kinds of waves on $\varphi^{(t)}$-$\varphi^{(z)}$ plane.
The black square, green dotted dashed line, black circle, blue dashed line,
red solid line, and gray regions denote the conditions of SPW, AB, OPS, KMB,
MPS, and TWB, respectively.

Their classification has been discussed in some integrable models with high-
order effects Liu-2015 ; Zhao-2016 ; Duan-2017 . Here, we slightly change it
and classify them into six kinds of waves, whose feature conditions are shown
in Fig. 2 (b). The six waves are discussed below.

  * •

Stable periodic wave (SPW) is purely periodic in both $t$ and $z$ directions,
namely, $\varphi^{(t)}=\varphi^{(z)}=\pi/2$.

  * •

Akhmediev breather (AB) is purely periodic in $t$ direction and is not purely
periodic in $z$ direction, namely, $\varphi^{(t)}=\pi/2$ and
$\varphi^{(z)}\neq\pi/2$.

  * •

One-peak soliton (OPS) is purely localized in both $t$ and $z$ directions,
namely, $\varphi^{(t)}=\varphi^{(z)}=0$.

  * •

Kuznetsov-Ma breather (KMB) is purely localized in $t$ direction and is not
purely localized in $z$ direction, namely, $\varphi^{(t)}=0$ and
$\varphi^{(z)}\neq 0$.

  * •

Multi-peak soliton (MPS) is periodic-localized in both $t$ and $z$ directions.
It requires that the velocities of fringe and localized envelope are the same,
i.e., $V_{+}=\Lambda_{+}$, to make the wave shape stable in the evolution
process. This condition of velocity matching is equivalent to
$\varphi^{(t)}=\varphi^{(z)}$.

  * •

Taijiri-Watanabe breather (TWB) is periodic-localized in $t$ direction. To
ensure the existence of breathing behavior, the velocity matching needs to be
avoided, namely, $V_{+}\neq\Lambda_{+}$. It is equivalent to
$\varphi^{(t)}\neq\varphi^{(z)}$.

Note that Peregrine rogue wave is not within the scope of our discussion
though it has double localization in $t$ and $z$ directions. It is because
rogue wave is a special wave as the limit of breathers at
$\omega_{+},\eta_{+},K_{+},G_{+}\rightarrow 0$, and this limit operation leads
to a rational form of waves, which cannot be described by our assumed
perturbing wave with exponential form. In a similar way, doubly periodic wave
generation cannot be analyzed by our method due to its complex elliptic form.
Besides, the above feature conditions of waves are based on the analysis of
periodicity and localization, which is helpful for the wave classification. It
neglects the waves with some special features, namely, the SPW with $V_{+}=0$,
the OPS with $\Lambda_{+}=0$, the MPS and TWB with $V_{+}=0$ or
$\Lambda_{+}=0$. These neglected waves will be considered in the following
discussion about the generation conditions of waves.

Table 2: Generation conditions of fundamental waves Fundamental waves |  Generation conditions
---|---
Stable periodic wave (SPW) |  $\omega_{+}\neq 0$, $\eta_{+}=0$, $G_{+}=0$
Akhmediev breather (AB) |  $\omega_{+}\neq 0$, $\eta_{+}=0$, $G_{+}\neq 0$
One-peak soliton (OPS) |  $\omega_{+}=0$, $\eta_{+}\neq 0$, $K_{+}=0$
Kuznetsov-Ma breather (KMB) |  $\omega_{+}=0$, $\eta_{+}\neq 0$, $K_{+}\neq 0$
Multi-peak soliton (MPS) |  $\omega_{+}\neq 0$, $\eta_{+}\neq 0$, $V_{+}=\Lambda_{+}$
Taijiri-Watanabe breather (TWB) |  $\omega_{+}\neq 0$, $\eta_{+}\neq 0$, $V_{+}\neq\Lambda_{+}$

Let us transform the feature conditions of the six waves into their generation
conditions. The generation conditions are decided by $\omega_{+}$, $K_{+}$,
$\eta_{+}$, $G_{+}$, $V_{+}$, and $\Lambda_{+}$ to control the type and
features of generated wave and are shown in the Table 2. SPW and AB can be
generated from a purely periodic initial perturbation (namely $\omega_{p}\neq
0$ and $\eta_{p}=0$), generations of which can be distinguished by whether
localization exists in $z$ direction or not. In similar ways, OPS and KMB can
be generated from a purely localized initial perturbation (namely
$\omega_{p}=0$ and $\eta_{p}\neq 0$), generations of which can be
distinguished by whether periodicity exists in $z$ direction or not; MPS and
TWB can be generated from a localized-periodic initial perturbation (namely
$\omega_{p}\neq 0$ and $\eta_{p}\neq 0$), generations of which can be
distinguished by whether fringes an localized envelope have identical velocity
or not. According to these generation conditions, the controllable generations
of waves in the fiber system will be discussed in the next section.

##  IV Controllable generation of nonlinear waves

Figure 3: (Color online) (a1) $|G^{(\rm II)}_{+}|$ value distribution on
$\omega_{0}$-$\omega_{p}$ plane with $\eta_{p}=0$, where white and colored
regions severally denote SPW and AB generations. (b1) $|K^{(\rm II)}_{+}|$
value distribution on $\omega_{0}$-$\eta_{p}$ plane with $\omega_{p}=0$, where
white and colored regions severally denote OPS and KMB generations. (c1)
$|V^{(\rm II)}_{+}-\Lambda^{(\rm II)}_{+}|$ value distribution on
$\omega_{p}$-$\eta_{p}$ plane with $\omega_{0}=0$, where white and colored
regions denote MPS and TWB generations. The middle and lower plots illustrate
amplitude evolution for generating (a2) SPW, (a3) AB, (b2) OPS, (b3) KMB, (c2)
MPS, and (c3) TWB. Their initial parameters are set as the coordinates of
triangles or diamonds in different regions of (a1), (b1), and (c1). Black
solid and red dashed lines represent the predicted propagating velocities of
localized envelope and fringe, respectively. All the waves are generated
successfully, except the KMB. The splitting in OPS generation is related to
the non-degeneracy of modes and time signs. Other parameters are
$\beta_{3}=0.1$ and $a_{0}=1$.

When $\beta_{3}=0$, Eq. (1) is the NLS system without the third-order
dispersion. Since this system is integrable, many exact solutions describing
fundamental nonlinear waves were presented Zakharov-1972 ; Kuznetsov-Ma ;
Peregrine-1983 ; Akhmediev-1986 ; Tajiri-1998 . Among the six kinds of
fundamental waves mentioned above, only AB, KMB, and TWB solutions exist in
NLS systems. It indicates that one can use a temporal profile as initial
condition to generate perfect them. Unlike it, what we focus on in this paper
is how to controllably generate them from an initial condition with a general
form, which is more convenient to prepare. In our analysis on the NLS system,
the expressions of $K_{\pm}$ and $G_{\pm}$ are found to be wonderfully
equivalent to the coefficients describing periodicity and localization in $z$
direction given by the exact solutions of waves (which is also found in some
other integrable models). This equivalence may be related to the functional
form of perturbing wave solution we assume, as it can be considered as an
approximative solution of initial stage in the NLS model.

Considering that many waves in the NLS model have been excited in real fibers,
we focus on the system with third-order dispersion, which admits more kinds of
wave generations. Without loss of generality, the coefficient $\beta_{3}$ is
set as $0.1$, and the background amplitude is $a_{0}=1$. We take the mode II
when $t\rightarrow+\infty$ as an example to discuss the wave generations from
three types of initial perturbations.

Firstly, a purely periodic initial perturbation (with $\omega_{p}\neq 0$ and
$\eta_{p}=0$) is used to generate SPW or AB. When $\eta_{p}=0$, based on Eq.
(8), one can obtain

$\begin{split}G^{(\rm II)}_{+}=\frac{1}{2}{\rm
Re}\Big{[}\sqrt{\omega_{p}^{2}(1+\beta_{3}\omega_{0})[4a_{0}^{2}-\omega_{p}^{2}(1+\beta_{3}\omega_{0})]}\Big{]}.\end{split}$
(14)

The value of $|G^{(\rm II)}_{+}|$ is distributed on the $\omega_{0}$-$\omega$
plane to distinguish the generations of AB and SPW, as shown in Fig. 3 (a1).
The white [$|G^{(\rm II)}_{+}|=0$] and colored [$|G^{(\rm II)}_{+}|\neq 0$]
regions denote the generations of SPW and AB, respectively. We set the
parameters as the coordinates of triangle, namely,
$(\omega_{0},\omega_{p})=(4,4)$. Its amplitude evolution is shown in Fig. 3
(a2). A SPW is generated perfectly, and its velocity of fringe matches well
with the one predicted by the limit value of Eq. (9) (denoted by the red
dashed line). Then, we set $(\omega_{0},\omega_{p})=(4,1)$, which are the
coordinates of diamond in Fig. 3 (a1). Its amplitude evolution is shown in
Fig. 3 (a3), and an AB is generated as expected. Its numerical velocity of
fringe also has a good agreement with the one we predict.

Secondly, we consider a purely localized initial perturbation (with
$\omega_{p}=0$ and $\eta_{p}\neq 0$) to generate OPS or KMB. When
$\omega_{p}=0$, based on Eq. (7), one can obtain

$\begin{split}K^{(\rm II)}_{+}=\frac{1}{2}{\rm
Im}\Big{[}\sqrt{-\eta_{p}^{2}(1+\beta_{3}\omega_{0})[4a_{0}^{2}+\eta_{p}^{2}(1+\beta_{3}\omega_{0})]}\Big{]}.\end{split}$
(15)

The distribution of $|K^{(\rm II)}_{+}|$ on $\omega_{0}$-$\eta_{p}$ plane is
shown in Fig. 3 (b1) for their generation conditions. Unlike the NLS system,
the consideration of third-order dispersion admits the OPS generation, which
is denoted by the grey region [with $|K^{(\rm II)}_{+}|=0$]. The colored
region [with $|K^{(\rm II)}_{+}|\neq 0$] corresponds to KMB generation. For a
purely localized initial perturbation, its initial condition (2) becomes

$\begin{split}\psi_{p}(0,t)=[1+(A+B){\rm
sech}(\eta_{p}t)]\;a_{0}e^{i\omega_{0}t}.\end{split}$ (16)

It indicates that the generation mode cannot be selected by the combinations
of different $A$ and $B$. We set the total amplitude $A+B$ as a modest value
$0.5$ to try generating KMB or OPS. We set $(\omega_{0},\eta_{p})$ as the
triangle’s coordinates $(-12,0.5)$, and the amplitude evolution plot is shown
in Fig. 3 (b2). The initial localized envelope splits into to waves with
different velocities. The left-hand wave has a stable shape in the propagation
process, so we consider it as a OPS; the right-hand wave gets wider and wider
with the increasing distance, and it has a typical shape of dispersive shock
wave. Compared with our analysis result, the wave splitting indicates the
generation of non-degenerate waves for different modes and time signs. The
velocities of two waves need to be predicted by $\Lambda_{+}^{(\rm II)}$ and
$\Lambda_{-}^{(\rm II)}$, which are shown in Fig. 3 (b2). They have good
agreements with the numerical evolution of the two waves. Then, we change
$(\omega_{0},\eta_{p})$ into the diamond’s coordinates $(-4,0.5)$ in Fig. 3
(b1). Its amplitude evolution plot is shown in Fig. 3 (b3). Before the
appearance of second breathing period of KMB, the spontaneous oscillation
emerges and breaks the generation of KMB, which indicates that the generation
of KMB is not successful. The spontaneous oscillation always has a large
influence on the generation of nonlinear waves, especially for the KMB
generation. Though some efforts have been paid on generating KMBs, generating
a KMB with more than one breathing period from a nonideal initial perturbation
is still an open problem.

Thirdly, a periodic-localized perturbation can be used to generate MPS or TWB
when the velocity matching is satisfied or not. Based on Eq. (9), one can
derive

$\begin{split}V^{(\rm
II)}_{+}=\omega_{0}+\frac{\beta_{3}}{6}(\omega_{p}^{2}-3\eta_{p}^{2}+3\omega_{0}^{2})+\frac{{\rm
Im}\Big{[}\sqrt{-P(4a_{0}^{2}+P)}\Big{]}}{2\omega_{p}},\\\ \Lambda^{(\rm
II)}_{+}=\omega_{0}+\frac{\beta_{3}}{6}(3\omega_{p}^{2}-\eta_{p}^{2}+3\omega_{0}^{2})-\frac{{\rm
Re}\Big{[}\sqrt{-P(4a_{0}^{2}+P)}\Big{]}}{2\eta_{p}},\end{split}$ (17)

where $P=(\eta_{p}-i\omega_{p})^{2}(1+\beta_{3}\omega_{0})$. When
$\omega_{0}=0$, the distribution of $|V^{(\rm II)}_{+}-\Lambda^{(\rm
II)}_{+}|$ on $\omega_{p}$-$\eta_{p}$ plane is shown in Fig. 3 (c1). The
colored [$|V^{(\rm II)}_{+}-\Lambda^{(\rm II)}_{+}|>0$] and white [$|V^{(\rm
II)}_{+}-\Lambda^{(\rm II)}_{+}|=0$] regions are respectively corresponding to
the generation of TWB and MPS. We set the initial parameters
$(\omega_{p},\eta_{p})$ as the triangle’s coordinates in the plot. Its
amplitude evolution plot is shown in Fig. 3 (c2). Despite the appearance of
auto-modulation process, a MPS is generated with a negative velocity. On the
localized envelope, many fringes can be observed and it can keep stable for a
long distance. Then, we change $(\omega_{p},\eta_{p})$ into the coordinates of
diamond in Fig. 3 (c1). The amplitude evolution plot is shown in Fig. 3 (c3).
Before the auto-modulation appears, a TWB is generated and propagates with a
constant velocity. Compared with the MPS generation, the localized envelope
and fringe’s velocities of generated TWB are obviously different, which
matches well with our prediction. Note that Eqs. (14,15,17) work only in the
cases we consider here, so they needs a recalculation when the parameter
conditions are changed.

Figure 4: (a) $|K_{(+t)}^{(\rm II)}|$ value distribution on
$\beta_{3}$-$\omega_{0}$ plane when $\omega_{p}=0$ and $\eta_{p}=0.5$, where
the white and colored regions severally denote OPS and KMB generations. (b)
Generation condition of MPS on $\beta_{3}$-$\omega_{p}$ plane when
$\omega_{0}=0$. The inset shows the dependence of amplitude’s asymmetric
degree $A/B$ on $\beta_{3}$ for MPS generation.

From the above results, the third-order dispersion has an influence on
enriching the types of nonlinear waves. The generation condition of OPS and
KMB when $\omega_{p}=0$ and $\eta_{p}=0.5$ is shown in Fig. 4 (a). With
$\beta_{3}$ increasing, the required $\omega_{0}$ for OPS generation is
getting higher from a negative infinite value. When $\beta_{3}$ approaches
$0$, the OPS generation is forbidden. The generation condition of MPS is also
shown in Fig. 4 (b) when $\omega_{0}=0$. With $\beta_{3}$ increasing, the
required $\omega_{p}$ for MPS generation is getting lower. In the limit of
$\beta_{3}\rightarrow 0$, the required $\omega_{p}$ has an unreachable
infinite value. This result agrees well with the case of NLS model. Meanwhile,
the asymmetric degree of initial amplitude is defined by $A/B$, and its
corresponding value for MPS generation is shown in the inset. The asymmetric
degree increases from $0$ and then decreases with $\beta_{3}$ increasing, and
it has a maximal value around $0.2$. Its low asymmetric degree indicates the
amplitude of component with frequency $\omega_{p}$ is far less than the one
with frequency $-\omega_{p}$. When $\beta_{3}$ is close to $0$, one can obtain
$A\rightarrow 0$, which implies that the required perturbation for MPS
generation has quasi-single frequency.

##  V Experimental feasibility of multi- and one-peak soliton excitations

Figure 5: (Color online) (a) Power evolution plot of MPS generation when
$a_{noise}=0.001$. (b) Its power profile at $Z=0\,{\rm km}$ (black curve) and
$Z=10\,{\rm km}$ (red curve) when $a_{noise}=0.01$. (c) The same to (b) except
$a_{noise}=0.05$. (d) Power evolution plot of OPS generation when
$a_{noise}=0.001$. (e) Its power profile at $Z=0\,{\rm km}$ (black curve) and
$Z=10\,{\rm km}$ (red curve) when $a_{noise}=0.01$. (f) The same to (e) except
$a_{noise}=0.05$. The power mentioned here is the normalized power,
$|A(Z,T)|^{2}/[P_{0}\exp(-\alpha Z)]$. $\Lambda_{\rm MPS}$ and $\Lambda_{\rm
OPS}$ are the envelope velocities of MPS and OPS, respectively.

Now, we focus on the experimental feasibility for exciting multi- and one-
peak solitons. By considering $A=\sqrt{P_{0}}\psi$,
$T=\sqrt{|\beta^{(2)}|/(\gamma P_{0})}t$, and $Z=(\gamma P_{0})^{-1}z$, the
model (1) can be transformed into a dimensional model. When the fiber loss is
considered, the model is

$\displaystyle
iA_{Z}-\frac{\beta^{(2)}}{2}A_{TT}-\frac{i\beta^{(3)}}{6}A_{TTT}+\gamma|A|^{2}A+i\frac{\alpha}{2}A=0,$
(18)

which describes the optical wave evolution in real fibers. Here, $P_{0}$,
$\gamma$, $\beta^{(2)}$, $\beta^{(3)}$, and $\alpha$ scale respectively the
input power, the nonlinearity, the group velocity dispersion, the third-order
dispersion, and the fiber loss. The relation between the two coefficients of
third-order dispersion is $\beta_{3}=\sqrt{\gamma
P_{0}/|\beta^{(2)}|^{3}}\beta^{(3)}$. We set $\beta^{(2)}=-0.5065\,{\rm
ps^{2}/km}$, $\beta^{(3)}=0.1\,{\rm ps^{3}/km}$, $\gamma=1.3\,{\rm
W^{-1}km^{-1}}$, $\alpha=4.2\times 10^{-3}\,{\rm km^{-1}}$, and the input
power $P_{0}=0.1\,{\rm W}$. The initial condition is still the continuous wave
perturbed by a sech-type envelope with two-frequency carrier waves,

$\begin{split}A_{p}(0,T)=\sqrt{P_{0}}e^{i\Omega_{0}T}[1+(A_{1}e^{i\Omega_{p}T}+A_{2}e^{-i\Omega_{p}T}){\rm{sech}}(H_{p}T)].\end{split}$
(19)

The phase of continuous wave is modulated periodically by the frequency
$\Omega_{0}$. The relative periodicity and localization of perturbation are
scaled by $\Omega_{p}$ and $H_{p}$, respectively. When we consider the
influence of random noise on wave excitations, the initial condition becomes

$\begin{split}A_{noise}(0,T)=A_{p}(0,T)(1+a_{noise}{\rm
Random}[-1,1]),\end{split}$ (20)

where $a_{noise}$ is the relative amplitude of random noise, and the function,
${\rm Random}[a,b]$, can produce a random value between $a$ and $b$ at every
numerical time point.

For the excitation of MPS, we set $\Omega_{0}=-461.22\,{\rm GHz}$,
$\Omega_{p}=520.97\,{\rm GHz}$, $H_{p}=38.44\,{\rm GHz}$, $A_{1}=0.03$, and
$A_{2}=0.5$, based on the above analysis. When $a_{noise}=0.001$, the power
evolution plot is shown in Fig. 5 (a). Its color scale denotes the normalized
power, $|A(Z,T)|^{2}/[P_{0}\exp(-\alpha Z)]$. One can see that a MPS
propagating stably is excited. When the noise amplitude is increased, its
profiles at the initial and final distances are compared in Fig. 5 (b) and
(c). The good agreements between them indicates that the MPS keeps robust
against increased noises. Then, we try the excitation of OPS by setting
$\Omega_{0}=-768.71\,{\rm GHz}$, $\Omega_{p}=0\,{\rm GHz}$, $H_{p}=38.44\,{\rm
GHz}$, and $A_{1}+A_{2}=0.5$. This set is locating on the critical line
between OPS and KMB regions in Fig. 3 (b1), which avoids the splitting of
initial perturbation. Its power evolution plot is shown in Fig. 5 (d) when
$a_{noise}=0.001$. A OPS propagates stably with a constant velocity. Its
initial and final profiles are compared under different $a_{noise}$ in Fig. 5
(e) and (f). When $a_{noise}=0.01$, small fluctuations appear, and the final
profile agrees well with the initial one. When $a_{noise}=0.05$, though the
fluctuation in the final profile is amplified, the profile of soliton is still
kept well. These results indicates that a MPS has a better robustness than a
OPS when a noise is considered.

##  VI Conclusion

In conclusion, we numerically generate six kinds of nonlinear waves, and
control their type, velocities, periodicity, and localization, in the NLS
model with third-order dispersion. The generation condition of these waves is
illustrated by means of the modified linear stability analysis method. They
are generated from a cosine-type initial perturbation with a sech-type
envelope on continuous waves. The velocities of their localized envelope and
fringes are predicted successfully. Besides, the introduction of third-order
dispersion is found to admit the generation of OPS and MPS so riches the kinds
of waves. The results can provide the guidance for wave generations in real
optical fibers and other nonlinear systems, like Bose-Einstein condensates,
water, and plasmas.

## Acknowledgement

This work was supported by National Natural Science Foundation of China
(Contact No. 11875220; 11947301).

## References

  * (1) L. F. Mollenauer, R. H. Stolen, J. P. Gordon. Phys. Rev. Lett. 45, 1095 (1980).
  * (2) R. H. Stolen, L. F. Mollenauer, and W. J. Tomlinson, Opt. Lett. 8, 186 (1983).
  * (3) L. F. Mollenauer, R. H. Stolen, J. P. Gordon, and W. J. Tomlinson, Opt. Lett. 8, 289 (1983).
  * (4) P. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, and A. Barthelemy, Opt. Commun. 62, 374 (1987).
  * (5) D. Krökel, N. J. Halas, G. Giuliani, and D. Grischkowsky, Phys. Rev. Lett. 60, 29 (1988).
  * (6) A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Krischner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1988).
  * (7) J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, Opt. Express 17, 21497 (2009).
  * (8) B. Kibler, J. Fatome, C. Finot, G. Millot, G. Genty, B. Wetzel, N. Akhmediev, F. Dias, and J. M. Dudley, Sci. Rep. 2, 463 (2012).
  * (9) B. Kibler, A. Chabchoub, A. Gelash, N. Akhmediev, and V. E. Zakharov, Phys. Rev. X 5, 041026 (2015).
  * (10) G. Xu, A. Gelash, A. Chabchoub, V. Zakharov, and B. Kibler, Phys. Rev. Lett. 122, 084101 (2019).
  * (11) B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, Nat. Phys. 6, 790 (2010).
  * (12) B. Frisquet, B. Kibler, and G. Millot, Phys. Rev. X 3, 041032 (2013).
  * (13) V. Zakharov and A. Shabat, Soviet physics JETP 34, 62 (1972).
  * (14) E. A. Kuznetsov, Sov. Phys. Dokl. 22, 507 (1977); Y. C. Ma, Stud. Appl. Math. 60, 43 (1979).
  * (15) D. H. Peregrine, J. Aust. Math. Soc. Ser. B 25, 16 (1983).
  * (16) N. Akhmediev and V. I. Korneev, Theor. Math. Phys. 69, 1089 (1986).
  * (17) M. Tajiri and Y. Watanabe, Phys. Rev. E 57, 3510 (1998).
  * (18) N. Akhmediev, A. Ankiewicz, and M. Taki, Phys. Lett. A 373, 675 (2009).
  * (19) N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, Phys. Rev. E 80, 026601 (2009).
  * (20) A. Ankiewicz, D. J. Kedziora, and N. Akhmediev, Phys. Lett. A 375, 2782 (2011).
  * (21) B. L. Guo, L. M. Ling, and Q. P. Liu, Phys. Rev. E 85, 026607 (2012).
  * (22) J. S. He, H. R. Zhang, L. H. Wang, K. Porsezian, and A. S. Fokas, Phys. Rev. E 87, 052914 (2013).
  * (23) G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, New York, 2007).
  * (24) P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-Gonz$\rm{\acute{a}}$lez, Emergent Nonlinear Phenomena in Bose-Einstein Condensates: Theory and Experiment (Springer, New York, 2008).
  * (25) H. C. Yuen and B. M. Lake, Phys. Fluids 18, 956 (1975).
  * (26) B. M. Lake, H. C. Yuen, H. Rungaldier, and W. E. Ferguson, J. Fluid Mech. 83, 49 (1977).
  * (27) H. Bailung, S. K. Sharma, and Y. Nakamura, Phys. Rev. Lett. 107, 255005 (2011).
  * (28) R. Hirota, J. Math. Phys. 14, 805 (1973).
  * (29) N. Sasa and J. Satsuma, J. Phys. Soc. Jpn. 60, 409 (1991).
  * (30) A. Ankiewicz, D. J. Kedziora, A. Chowdury, U. Bandelow, and N. Akhmediev, Phys. Rev. E 93, 012206 (2016).
  * (31) A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, Phys. Rev. E 81, 046602 (2010).
  * (32) U. Bandelow and N. Akhmediev, Phys. Rev. E 86, 026606 (2012).
  * (33) Y. S. Tao and J. S. He, Phys. Rev. E 85, 026601 (2012).
  * (34) S. H. Chen, Phys. Rev. E 88, 023202 (2013).
  * (35) L. C. Zhao, S. C. Li, and L. M. Ling, Phys. Rev. E 89, 023210 (2014).
  * (36) T. Xu, M. Li, and L. Li, Eur. Phys. Lett. 109, 30006 (2015).
  * (37) C. Liu, Z. Y. Yang, L. C. Zhao, and W. L. Yang, Phys. Rev. E 91, 022904 (2015).
  * (38) C. Liu, Z. Y. Yang, L. C. Zhao, L. Duan, G. Y. Yang, and W. L. Yang, Phys. Rev. E 94, 042221 (2016).
  * (39) L. C. Zhao, S. C. Li, and L. M. Ling, Phys. Rev. E 93, 032215 (2016).
  * (40) A. Ankiewicz and N. Akhmediev, Phys. Rev. E 96, 012219 (2017).
  * (41) L. C. Zhao and L. Ling, J. Opt. Soc. Am. B 33, 850 (2016).
  * (42) L Duan, L. C. Zhao, W. H. Xu, C. Liu, Z. Y. Yang, and W. L. Yang, Phys. Rev. E 95, 042212 (2017).
  * (43) L. Duan, Z. Y. Yang, P. Gao, and W. L. Yang, Phys. Rev. E 99, 012216 (2019).
  * (44) P. Gao, C. Liu, L. C. Zhao, Z. Y. Yang, and W. L. Yang, Phys. Rev. E 102, 022207 (2020).
  * (45) M. Conforti, S. Li, G. Biondini, and S. Trillo, Opt. Lett. 43, 5291 (2018).
  * (46) C. Liu, Z. Y. Yang, and W. L. Yang, Chaos 28, 083110 (2018).
  * (47) C. Liu, Z. Y. Yang, W. L. Yang, and N. Akhmediev, J. Opt. Soc. Am. B 36, 1294 (2019).
  * (48) J. Yang, Nonlinear waves in integrable and nonintegrable systems (Siam, 2010).