# Parity-time-symmetric vector rational rogue wave solutions
in any $n$-component nonlinear Schrödinger models

Guoqiang Zhang  Department of Mathematical Sciences, Tsinghua University,
Beijing 100084, China  Liming Ling<EMAIL_ADDRESS>(corresponding author)
School of Mathematics, South China University of Technology, Guangzhou 510640,
China  Zhenya Yan<EMAIL_ADDRESS>(corresponding author) Key Lab of
Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese
Academy of Sciences, Beijing 100190, China  School of Mathematical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China  Vladimir V.
Konotop  Departamento de Física, Faculdade de Ciências, Universidade de
Lisboa, Campo Grande, Edifício C8, Lisboa 1749-016, Portugal  Centro de Física
Teórica e Computacional, Universidade de Lisboa, Campo Grande, Edifício C8,
Lisboa 1749-016, Portugal.
(Date: )

###### Abstract

The extreme events are investigated for an $n$-component nonlinear Schrödinger
($n$-NLS) system in the focusing Kerr-like nonlinear media, which appears in
many physical fields. We report and discuss the novel multi-parametric
families of vector rational rogue wave (RW) solutions featuring the parity-
time (${\cal P}{\cal T}$) symmetry, which are characterized by non-identical
boundary conditions for the components, and consistent with the degeneracy of
$n$ branches of Benjamin-Feir instability. Explicit examples of ${\cal P}{\cal
T}$-symmetric vector RWs are presented. Some parameter constraints can make
some components generate the RWs with high amplitudes due to many-body
resonant interactions. Effect of a non-integrable deformation of the model on
the excitation of vector RWs is also discussed. These results will be useful
to design the RW experiments in multi-component physical systems.

Introduction.—Recently, rogue waves (RWs), as a special type of nonlinear
waves, have been paid more and more attention to. RWs (alias freak or killer
waves) are extreme events that were originally known as deep ocean waves RW1 ;
RW2  (see also Refs. Rev1 ; Rev2 ; Rev3  and references therein). Nowadays RWs
are recognized to be a ubiquitous phenomenon that can be observed in nonlinear
physical systems of diverse nature including optical fibers prw ; rw2010 ;
Dudley2014 ; jop ; Mihalache1 ; mat ; orw , arrays of waveguides BluKonAkh ,
cavities oc , resonators Majid , superfluids superfluid , Bose-Einstein
condensates (BECs) bec-rw ; Bludov2010 , models of atmospheric physics ap ;
Iafrati , plasmas Langmur ; RW_plasma , and even finance yanfrw . When RWs are
characterized by a scalar field, a paradigmatic physical model describing them
is the NLS equation or its generalizations. If a system is characterized by
two or more (say, $n$) components, like for example, multi-mode nonlinear
waveguides nls1b ; nlsb2  or multi-component BECs nls5 , the governing
equations are often reduced to a two- or $n$-component NLS system (alias
$n$-NLS system) nnls .

It is well established that occurrence of RWs is intimately related to the
phenomenon of the modulation instability (MI) MI ; Onorato ; MI-nonlinear-MI ;
MI-nonlinear-MI-1 ; Bludov2010 ; Baronio ; Dudley2014 . A focusing scalar NLS
equation admits one branch of the spectrum featuring MI (or briefly one branch
of MI), which in its turn originates a single type of rational RWs, known as
the Peregrine soliton Peregrine1983 . Such a solution is localized both in
time and in space Majid2 ; prl2017 , and has been experimentally observed in
nonlinear optical fibers rw2010 ; Dudley2014 , plasma RW_plasma , and a water
tank RW_tank . Meantime, already 2- and 3-component NLS equations may possess
several branches of MI. Since with each branch of MI one can relate a RW
solution, one can expect that in a system with several field components
different branches of MI can give origin to distinct RWs. Such vector RWs in
2– and 3–component systems were extensively studied Onorato ; yanfrw ;
Bludov2010 ; Baronio ; Zhao2012 ; Kedziora2013 ; Baronio121318 ; yan-coupled ;
Mihalache2 . An $n$-NLS system generally admits $n$ distinct branches of MI.
Since with each branch one can relate a RW, a natural issue arises, on whether
$n$ distinct branches of MI could collapse, thus manifesting degeneracy of MI,
and thus result in a generation of a RW with greatly enhanced amplitude. A
general answer to this issue still remains open, although a few examples were
reported Zhao2012 .

In this paper, we would like to tackle the above-mentioned open and
interesting issue in a general case, and for the first time find the
analytical $n$-vector RWs, addressing the relation between the degenerated MI
and vector RWs in the $n$-NLS equation, in which one of key and difficult
points approaching the problem requires to solve an $(n+1)$-th-order
polynomial equation explicitly. Furthermore, we find that the obtained
$n$-vector RWs feature the physically interesting parity-time (${\cal P}{\cal
T}$) symmetry, resembling the scalar time-reversal RWs previously reported in
time-revers ; Chabchoub14 , and the vector RWs can generate the high
amplitudes for some parameter choices.

_The $n$-NLS models and existence of vector RWs._—We consider the wave
propagation of $n$ self-trapped beam in a slow media with Kerr-like
nonlinearity, a dimensionless $n$-NLS system for a complex-valued
$n$-component column-vector field
$\mathbf{q}(x,t)=\left(q_{1}(x,t),\ldots,q_{n}(x,t)\right)^{\rm T}$ nnls ;
scott ; yeh ; nail98 ; nail99 ; Ablowitz2004

$\mathrm{i}\frac{\partial{\bf q}}{\partial
t}+\frac{1}{2}\,\frac{\partial^{2}{\bf q}}{\partial
x^{2}}+|\mathbf{q}|^{2}\mathbf{q}=\mathbf{0},$ (1)

where $|\mathbf{q}|^{2}=|q_{1}|^{2}+\cdots+|q_{n}|^{2}$ stands for the norm of
the column-vector ${\bf q}$ and the superscript T stands for transpose. At
$n=1$ Eq. (1) is reduced to the scalar NLS equation, while at $n=2$ it is the
well-known Manakov system Manakov . Eq. (1) also has direct physical
applications, for instance, in the theory of alpha-helixes scott . Here we are
interested in vector RW solutions of Eq. (1) propagating against a constant-
amplitude carrier-wave (CW) background. The latter is considered of a general
form $\mathbf{q}_{0}(x,t)=(q_{01},\ldots,q_{0n})^{\rm T}$, where
$q_{0j}=a_{j}\mathrm{e}^{\mathrm{i}\varphi_{j}}$,
$\varphi_{j}=k_{j}x-\omega_{j}t$ are the real phases, and positive amplitudes
$a_{j}$, (real) wave-numbers $k_{j}$, and frequencies $\omega_{j}$ satisfy the
dispersion relations: $\omega_{j}=k_{j}^{2}/2-|\mathbf{a}|^{2}$ with
$\mathbf{a}=(a_{1},\ldots,a_{n})^{\mathrm{T}}$. The components of the RWs,
$q_{j}$, are all localized (against the CW background) in space and in time
and at $|x|,\,|t|\to\infty$ they have the asymptotics: $q_{j}\sim q_{0j}{\rm
e}^{-2\mathrm{i}\theta_{j}}$ with constant phases $\theta_{j}$.

To present the $n$-vector RWs Suppl  $q_{j}(x,t)=q_{0j}-4\mathrm{i}{\rm
Im}(\lambda_{0})\\!\left(\Psi\Psi^{{\dagger}}/(\Psi^{{\dagger}}\Psi)\right)_{j+1,1}$
with ‘${\dagger}$’ being the Hermitian conjugation via the Darboux transform
(see e.g., dt ; Akhmediev2009a ; Guo2012 ; he2013 ), one needs to find the
auxiliary vector function of the Lax pair Ablowitz2004 :
$\Psi=\mathbf{G}\Phi\mathbf{c}$,
$\mathbf{G}=\mathrm{diag}(1,\mathrm{e}^{\mathrm{i}\varphi_{1}},\ldots,\mathrm{e}^{\mathrm{i}\varphi_{n}})$,
the non-zero constant column-vector ${\bf c}=(c_{0},c_{1},\ldots,c_{n})^{\rm
T}$ introduces free parameters $c_{j}$ (as shown below they characterize RWs),
and $\Phi$ is an $(n+1)$-order square matrix solution of the gauge-transformed
Lax pair with constant coefficients: $\Phi_{x}=\mathrm{i}\mathbf{H}\Phi$ and
$\Phi_{t}=\mathrm{i}\left[\frac{1}{2}\mathbf{H}^{2}+\lambda_{0}\mathbf{H}-\left(|\mathbf{a}|^{2}+\frac{1}{2}\lambda_{0}^{2}\right)\mathbb{I}_{n+1}\right]\Phi$,
where $\mathbf{H}=\lambda_{0}\sigma_{3}+\begin{pmatrix}0&\mathbf{a}^{\rm T}\\\
\mathbf{a}&-\mathbf{K}\end{pmatrix}$ with
$\mathbf{K}=\mathrm{diag}\left(k_{1},\cdots,k_{n}\right)$ being a constant
matrix, and $\lambda_{0}\in\mathbb{C}$ the spectral parameter. Thus, the
construction of a desired solution ${\bf q}(x,t)$ is reduced to solving the
eigenvalue problem. Representing (for convenience) an eigenvalue of ${\bf H}$
as ${\rm i}\chi-\lambda_{0}$, i.e. regarding $\chi$ as an eigenvalue, one
obtains the characteristic equation for ${\bf H}$ in the form Suppl

$\displaystyle P(\chi):=2\lambda_{0}-{\rm
i}\chi+\sum_{j=1}^{n}\frac{a_{j}^{2}}{{\rm i}\chi+k_{j}}=0.$ (2)

We are interested in vector rational RWs of Eq. (1), which are represented by
a quotient of two polynomials of $x$ and $t$ [like in Eqs. (7) and (Parity-
time-symmetric vector rational rogue wave solutions  in any $n$-component
nonlinear Schrödinger models) shown below] Peregrine1983 . Therefore we seek
matrix solutions of the above-mentioned equations that feature algebraic
dependence on the spatial (and temporal) variables. This implies that roots of
Eq. (2) determining such solutions must have multiplicity bigger than one.
Finding such roots is a nontrivial task because effectively Eq. (2) is an
algebraic equation of order $(n+1)$, whose roots of high multiplicity may even
do not exist in a general case. Furthermore, here we consider the roots of
largest possible, i.e., $n+1$ multiplicity. Thus, in addition to Eq. (2) we
require

$P^{(s)}(\chi):=\delta_{s1}+\sum_{j=1}^{n}\frac{a_{j}^{2}}{({\rm
i}\chi+k_{j})^{s+1}}=0,$ (3)

where $s=1,\ldots,n$ and $\delta_{ij}$ is the Kronecker delta. Since
straightforward obtaining $\chi$ from Eqs. (2) and (3) seems to be impossible,
we reformulate the problem and focus on finding the non-zero amplitudes
$a_{j}$ and wave-numbers $k_{j}$ of the CW background, for which a given value
of the spectral parameter $\chi=\chi_{0}$ is the root of the multiplicity
$n+1$, i.e., $P(\chi_{0})=P^{(1)}(\chi_{0})=\cdots=P^{(n)}(\chi_{0})=0$. We
notice that $P^{(1)}(\chi)=0$ is the dispersion relation for the MI ling2019
and for $n=2,3$ it was shown Zhao2012  to coincide with the condition for the
baseband MI Baronio ; Baronio121318  of Eq. (1).

Due to the Galilean invariance of Eq. (1) the $\chi_{0}$ can be made real by
the shift of the wave-numbers $k_{j}\to k_{j}-{\rm Im}(\chi_{0})$.
Furthermore, by rescaling $a_{j}$ and $k_{j}$ one can make $\chi_{0}=1$.
Seeking the solutions of Eqs. (2) and (3) that are located on the left branch
of the hyperbola: $a_{j}^{2}-k_{j}^{2}=1$, we obtain

$\displaystyle
a_{j}=\csc\theta_{j},\,k_{j}=\cot\theta_{j},\,\theta_{j}=\frac{\pi
j}{n+1},\,\lambda_{0}=\frac{{\rm i}}{2}(n\\!+\\!1).\,\,$ (4)

Thus, we have found the phases $\theta_{j}$ of the components, which determine
the amplitudes $a_{j}$, the wave-numbers $k_{j}$, and the frequencies
$\omega_{j}$ of the CW background sustaining a vector rational RW. The
analytic expression for the respective RW is obtained by computing $\Phi$
which corresponds to the $(n+1)$-multiple eigenvalue
$\mathrm{i}\chi_{0}-\lambda_{0}={\rm i}(1-n)/2$ of ${\bf H}$, and its
substitution into the Darboux transform. The fundamental solution determined
by $\Phi(0,0)=\mathbb{I}_{n+1}$, and hence represented as $\Psi={\bf
G}\Phi{\bf c}$, generates a family of vector raiotnal RWs Suppl

$q_{j}(x,t)=q_{0j}\left[1-\frac{2\mathrm{i}\left(n+1\right)}{a_{j}}\frac{\mathbf{A}_{j+1}{\bf
c}(\mathbf{A}_{1}{\bf c})^{*}}{(\mathbf{A}{\bf c})^{\dagger}\mathbf{A}{\bf
c}}\right],$ (5)

where the asterisk denotes the complex conjugation, $\mathbf{A}_{j}$ is the
$j$-th row of the matrix function

$\mathbf{A}\\!=\\!\sum_{s=0}^{n}\sum_{m=0}^{\lfloor
n/2\rfloor}\\!\frac{(\mathrm{i}x\\!-\\!t)^{s}(\mathrm{i}t)^{m}}{2^{m}s!\,m!}\mathbf{B}^{s+2m},\,\,\mathbf{B}\\!=\\!\begin{pmatrix}{\rm
i}n&\mathbf{a}^{\rm T}\\\ \mathbf{a}&-{\rm
i}\mathbb{I}_{n}\\!-\\!\mathbf{K}\end{pmatrix}$ (6)

and $\lfloor\cdot\rfloor$ stands for the integer part.

_${\cal PT}$ -symmetric vector rogue waves._—Generally speaking, the obtained
vector RW solution (5) does not inherit the symmetry of the CW background (4)
that is symmetric with respect to simultaneous space inversion $x\to-x$ and
swapping the components performed by the matrix ${\cal
P}=(\delta_{j,n+1-j^{\prime}})_{n\times n}$. Meantime, it is possible to
impose a constraint on the parameters ensuring the ${\cal PT}$-symmetry of a
vector RW, with ${\cal T}$ being the conventional bosonic time-reversion:
$t\to-t,\,{\rm i}\to-{\rm i}$. Indeed, the formula (5) yields a ${\cal
PT}$-symmetric vector RW: ${\bf q}(x,t)={\cal PT}{\bf q}(x,t)={\cal P}{\bf
q}^{*}(x,-t)$ if $\mathbf{c}$ is required to satisfy the symmetry: ${\bf
c}=\widehat{\cal P}{\bf c}^{*}$ with $\widehat{\cal
P}=\begin{pmatrix}1&\mathbf{0}_{1\times n}\\\ \mathbf{0}_{n\times 1}&-{\cal
P}\end{pmatrix}$.

By construction $\mathbf{B}$ is a defective matrix with the $n$-th order
exceptional point for the parameter choices (4). Therefore, it is convenient
to define a set of generalized eigenvectors $\mathbf{s}_{\ell}=(\delta_{\ell
0},{a_{1}}/{\left({\rm
i}+k_{1}\right)^{\ell+1}},\ldots,{a_{n}}/{\left(\mathrm{i}+k_{n}\right)^{\ell+1}})^{\mathrm{T}}$,
$\ell=0,\ldots,n$, satisfying
$\mathbf{B}^{\ell}\mathbf{s}_{\ell}\neq\mathbf{0}$ and
$\mathbf{B}^{\ell+1}\mathbf{s}_{\ell}=\mathbf{0}$. The vector $\mathbf{c}$ can
be spanned over the basis
$\mathbf{S}=\left(\mathbf{s}_{0},\ldots,\mathbf{s}_{n}\right)$:
$\mathbf{c}=\mathbf{S}\bm{\alpha}$, where the form of the expansion
coefficient, compactly represented by the column-vector $\bm{\alpha}$, must be
chosen to ensure the validity of ${\bf c}=\widehat{\cal P}{\bf c}^{*}$. This
is achieved by
$\bm{\alpha}=\left(\varepsilon_{0}\alpha_{0},\ldots,\varepsilon_{n}\alpha_{n}\right)^{\mathrm{T}}$
with all $\alpha_{\ell}\in\mathbb{R}$, $\varepsilon_{\ell}=1$ for even
${\ell}$ and $\varepsilon_{\ell}={\rm i}$ for odd ${\ell}$. Furthermore, since
one of nonzero parameters $\alpha_{\ell}$ can be arbitrarily fixed, we set
$\alpha_{\ell}=1$ when $\alpha_{\ell+1}=\cdots=\alpha_{n}=0$. With this choice
$\mathbf{Ac}$ is a vector polynomial of degree $\ell$.

Resuming, a family of ${\cal PT}$-symmetric vector RWs (5) obtained against a
given CW background with parameters (4), can be characterized by a pair of
indexes $(n,\ell)$. The internal structure of such a RW is also characterized
by the real parameters $\alpha_{0},\ldots,\alpha_{\ell-1}$. Due to the
translational invariance, one of the parameters $\alpha_{j}$ is a trivial
spatial shift determining the location of the RW event and hence, it can be
scaled out by the choice of the system of coordinates. Other $\ell-1$
parameters determine the structure of the RW, as it is discussed below.

_Examples of vector rational rogue waves._ —The fundamental vector rational
RWs correspond to $\ell=1$ (respectively $\alpha_{\ell}=1$) in (5). Choosing
$\alpha_{0}=-1/2$ we obtain:

$\displaystyle
q_{j}(x,t)=\left(a_{j}+\frac{\mathrm{i}}{a_{j}}\frac{2(k_{j}x-t)+\mathrm{i}}{x^{2}+t^{2}+1/4}\right){\rm
e}^{\mathrm{i}(k_{j}x-\omega_{j}t)-2\mathrm{i}\theta_{j}}.$ (7)

The $(n,1)$-RW solution has no free parameter and is fully determined by the
characteristics of the CW background. A number of humps and dips, i.e. of the
most interesting events, in each component depend on the number of components
Suppl . At $n=1$ we recover the Peregrine soliton having one hump and two dips
Peregrine1983 . For $n=2$, when $\theta_{1}=\pi/3$ and $\theta_{2}=2\pi/3$
[see Eq. (4)], the RW components are related by the ${\cal PT}$-transform, and
now each component has one hump and two dips as this is illustrated for $n=2$
in Fig. 1(a). For $n\geq 3$ the components with
$j\in\left(0,(n+1)/6\right]\cup[5(n+1)/6,n]$ have two humps and one dip (for
this case to take place one must have at least $n=5$ components); the
components with
$j\in\left((n+1)/6,(n+1)/3\right)\cup\left(2(n+1)/3,5(n+1)/6\right)$ have two
humps and two dips [see Fig. 1 (c)], the components with
$j\in\left[(n+1)/3,2(n+1)/3\right]$ have one hump and two dips [see Fig. 1
(d)].

Figure 1: Profiles of the RW components (7) (panels a, c, d) and (Parity-time-
symmetric vector rational rogue wave solutions  in any $n$-component nonlinear
Schrödinger models) (panels b, e, f) for the cases: (a) $(n,\ell)=(2,1)$; (b)
$(n,\ell)=(2,2)$ with $\alpha_{0}=12.5$; (c) and (d) $(n,\ell)=(3,1)$; (e) and
(f) $(n,\ell)=(3,2)$ with $\alpha_{0}=12.5$.

Next we consider examples of $(n,2)$-RWs, whose spatial location is
parameterized by $\alpha_{1}$. We scale it out by the choice $\alpha_{1}=1/2$.
However now there exists one free parameter $\alpha_{0}$, which determines the
shape of the RWs components. From (5) we obtain

$\displaystyle
q_{j}(x,t)=\left(a_{j}+\frac{\mathrm{i}}{a_{j}}\frac{2r_{0j}-(4\alpha_{0}+1/2)r_{1j}+k_{j}+\mathrm{i}}{r+({x}^{2}-t^{2}-2\alpha_{0}-1/4)^{2}}\right)$
$\displaystyle\times{\rm
e}^{\mathrm{i}(k_{j}x-\omega_{j}t)-2\mathrm{i}\theta_{j}},\quad$ (8)

where $r=[(2{x}-1)^{2}+1]t^{2}+({x}+1/2)^{2}+1/4$,
$r_{0j}=2(k_{j}{x}-t)({x}^{2}+t^{2})+({\rm
i}-2k_{j})t^{2}+k_{j}x+[(3\mathrm{i}k_{j}-1){x}^{2}+4k_{j}{x}t-2t^{2}-2{x}-2(k_{j}-\mathrm{i})t]/(k_{j}+\mathrm{i})$
and $r_{1j}=2(k_{j}{x}+t)+(\mathrm{i}k_{j}+1)/(k_{j}+\mathrm{i})$. Fig. 1(b)
displays the fundamental (2,2)-RW with one hump and two dips, while Figs. 1(e)
and (f) exhibit the components of a (3,2)-RW. We observe that the second
component is characterized by “dominating” humps while in the first components
the dips are more pronounced.

_Polymeric ${\cal PT}$-symmetric vector rational RWs._— The diversity of
multi-component ${\cal PT}$-symmetric vector RWs (5) at large $n$ is abundant.
Indeed, for a given $n$ one has $\ell=1,\ldots,n$ different solutions
characterized by $\ell-1$ nontrivial free parameters, and hence $n(n-1)/2$
distinct types of RWs. One can expect that the strongest collective effect of
the components is achieved when they all acquire their maxima at the same
instant of time and spatial location. Without loss of generality we can look
for such a solution with the largest hump occurring at $t=0$ and $x=0$, when
$\mathbf{A}(0,0)=\mathbb{I}_{n+1}$ [see (6)]. Since one of the components of
$\mathbf{c}$ can be arbitrarily chosen, we can set $c_{0}=1$ and
$c_{j}=\mathrm{i}\tilde{c}_{j}$ for $1\leq j\leq n$, where real
$\tilde{c}_{j}$ are to be determined. Now we deduce from the general formula
(5) that the maximum of $|q_{j}(0,0)|$ is achieved if the second term in the
square brackets is real negative, and hence $\tilde{c}_{j}$ are all real
positive. In that case $q_{j}(0,0)=a_{j}+\eta_{j}$ with
$\eta_{j}=2(n+1)\tilde{c}_{j}/(1+|\tilde{\bf c}|^{2}),\,\,\tilde{\bf
c}=(\tilde{c}_{1},\ldots,\tilde{c}_{n})^{\rm T}$ is the deviation of the
amplitude of the $j$-th component from the background.

Figure 2:  (a, b) Profiles $|{\bf q}(x,t)|/|{\bf a}|$ of RW (5) for ${\bf
c}=(1,\mathrm{i}/\sqrt{n},\ldots,\mathrm{i}/\sqrt{n})^{T}$ and (a) $n=15$ or
(b) $n\in[1,500]$. (c, d) The profiles of RW (5) for ${\bf
c}=(1,0,\cdots,0,{\rm i},0,\cdots,0)^{T}$ with odd $n$ and ${\bf
c}=(1,0,\cdots,0,{\rm i}/\sqrt{2},{\rm i}/\sqrt{2},0,\cdots,0)^{T}$ with even
$n$: (c) $|q_{8}(x,t)|/a_{8}$ with the ultrahigh peak amplitude ${\rm
max}(|q_{8}|/a_{8})=17$ for $n=15$; (d) The curves of ${\rm
max}(|q_{j}(x,t)|/a_{j})=|q_{j}(0,0)|/a_{j}$ for $j=(n+1)/2$ with odd $n$ and
$j=n/2$ with even $n$ as $n\in[1,30]$.

Case A.-For the parameter ${\bf
c}=(\sqrt{n},\mathrm{i},\mathrm{i},\cdots,\mathrm{i})^{T}$, the average
amplitude $\left|\mathbf{q}(x,t)\right|\left/\right.\left|\mathbf{a}\right|$
of the polymeric RWs attain maximum at the origin, where
$\left|\mathbf{q}(x,t)\right|=\sum_{s=1}^{n}\left|q_{s}(x,t)\right|$ and
$\left|\mathbf{a}\right|=\sum_{s=1}^{n}a_{s}$. In this case, the ratio between
peak value $|\mathbf{q}(0,0)|$ and CW background $|\mathbf{a}|$ can be deduced
as

$Ra=\frac{|\mathbf{q}(0,0)|}{|\mathbf{a}|}=1+\frac{\sqrt{n}(n+1)}{\sum_{j=1}^{n}\csc\theta_{j}},$
(9)

which has the following estimates

$1+\frac{\sqrt{n}}{\ln\left(n+1\right)+1}\leq Ra\leq
1+\frac{\sqrt{n}\pi}{2\ln(n/2)},$ (10)

which implies $Ra\leq 3$ for $2\leq n\leq 11$ and $Ra\gg 3$ for $n\gg 12$, and
$Ra=\mathcal{O}\left(\sqrt{n}/\ln n\right)$ as $n\to\infty$. Fig. 2(a) shows
the distribution structure of
$\left|\mathbf{q}(x,t)\right|\left/\right.\left|\mathbf{a}\right|$ for
$15$-NLS equations. Fig. 2(b) indicates the ratio
$|\mathbf{q}(0,0)|/|\mathbf{a}|$ for $n$. Besides, the ratio between peak
value and background for the $j$-th component is
$|q_{j}(0,0)|/a_{j}=1+(n+1)/\sqrt{n}\sin\theta_{j}$.

Case B.-In this case, we consider the polymeric RW, whose middle component
attains maximum at the origin. Due to the restrictive of the time-reversal
property, two categories are studied respectively. As $n$ is odd, as long as
$\mathbf{c}=\left(1,0,\cdots,0,\mathrm{i},0,\cdots,0\right)$, the amplitude of
$(n+1)/2$-th component attains maximum $q_{m}(0,0)/a_{m}=(n+2),\,m=(n+1)/2$ at
the origin, and others are the height $a_{j}^{\prime}s$ of the backgrounds at
the origin. Fig. 2(c) shows the eighth component of polymeric RWs for $15$-NLS
equations. The red dots in Fig. 2(d) shows the values of
${q_{m}(0,0)}/{a_{m}}$ with $m=(n+1)/2$ and odd $n\in[1,30]$. As $n$ is even,
as long as
$\mathbf{c}=\left(1,0,\cdots,0,\mathrm{i}/\sqrt{2},\mathrm{i}/\sqrt{2},0,\cdots,0\right)$,
the amplitudes of $n/2$\- and $(n+1)/2$-th components attain maximum at the
origin with $m=n/2$ and
$q_{m}(0,0)/a_{m}=q_{m+1}(0,0)/a_{m+1}=1+(n+1)/\sqrt{2}\sin(\theta_{m})$, and
others are the height $a_{j}^{\prime}s$ of the corresponding backgrounds. The
blue triangles in Figs. 2(d) show the values of ${q_{m}(0,0)}/{a_{m}}$ with
$m=n/2$ and even $n\in[1,30]$.

Figure 3: Numerical simulations of non-integrable 3-NLS equation (11) with
$\delta=0.05$. Upper panes show evolution of $|q_{j}|/a_{j}\,(j=1,2)$ two non-
equal components (recall that due to the ${\cal PT}$-symmetry
$|q_{3}|=|q_{1}|$). Bottom panels show comparison of numerical solutions at
$t=0.41$ (blue solid curves) and exact RWs (red dashed curves) corresponding
to $\delta=0$, i.e., given by Eq. (7) [cf. with Figs. 1(c) and (d)].

Non-integrable $n$-NLS model.—In a general setting a multi-parametric non-
integrable system cannot be solved analytically. Even numerical study would
require particularization depending on the physical nature of the
nonlinearity. Therefore here we address a somewhat simpler question whether
${\cal PT}$-symmetric RWs of enhanced intensity can be excited in an $n$-NLS
equation, where the integrability is broken by unequal nonlinear interactions.
We explore this issue by simulating evolution of vector RWs in the
dimensionless non-integrable 3-NLS equation nail98 ; nail99 :

${\rm i}\frac{\partial q_{j_{1}}}{\partial
t}+\frac{1}{2}\frac{\partial^{2}q_{j_{1}}}{\partial
x^{2}}+\left[|q_{j_{1}}|^{2}+(1-\delta)(|q_{j_{2}}|^{2}+|q_{j_{3}}|^{2})\right]q_{j_{1}}=0,$
(11)

where $j_{1}$, $j_{2}$, and $j_{3}$ are all different and acquire values $1$,
$2$ and $3$, and $\delta$ is a small real parameter quantifying imbalance
between intra- and inter-mode interactions. However, in spite of the
$\delta$-deformation, model (11) remains ${\cal P}$-symmetric, i.e., is not
affected by swapping the components. For numerical simulations via the Fourier
spectral method we consider small deviation from the integrable limit letting
$\delta=0.05$ and propagate the “initial” condition, in the form of the exact
${\cal PT}$-symmetric vector RW (7) taken at $t=-1$ [Figs. 3 (a) and (b)].
Although the post-extreme-event evolution of the non-integrable model is
appreciably different from its integrable counterpart [cf. with Figs. 1 (c)
and (d)], the RW itself is well distinguishable in the evolution. In Figs. 3
(c) and (d) we compare the numerical solutions of (11) with the exact RW
solutions (7) at the instant of occurrence of the maximum, i.e., at $t=0.41$
in our case. We observe that the structures and especially the shapes of the
pick amplitudes are very close in both models.

Conclusions.—To conclude, we described a family of exact ${\cal PT}$-symmetric
vector rational RW solutions of an integrable system of $n$-component coupled
NLS equations. Such extreme events occur against CW backgrounds which are
characterized by different asymptotics of different components. The higher
amplitudes reached during the evolution increase with the number of the
components. The reported RWs are observable in non-integrable systems, too.
The obtained vector RW solutions may pave the way for obtaining high-amplitude
RWs in diverse multi-component physical systems including multi-mode and poly-
chromatic optical devices, as well as multi-atomic and spinor Bose-Einstein
condensates. Moreover, we also give the higher-order vector RWs and asymptotic
estimates of the $n$-NLS system in another literature long . The idea can also
be extended to other $n$-component nonlinear integrable systems (e.g.,
$n$-component Hirota equations, mKdV equations, complex mKdV equations,
higher-order NLS equations, KP equations, and etc.) long . Being restricted to
all attractive interactions, our study left open such questions as obtaining
high-intensity RWs in presence of attractive interactions, as well as
establishing direct links between ${\cal PT}$-symmetric vector RWs and the
modulation instability of the governing system.

Acknowledgments.-G.Z. acknowledges support from China Postdoctoral Science
Foundation (Grant No.2019M660600). L.L. acknowledges support from the National
Natural Science Foundation of China (Grant No.11771151), the Guangzhou Science
and Technology Program of China (Grant No.201904010362), and the Fundamental
Research Funds for the Central Universities of China (Grant No.2019MS110).
Z.Y. acknowledges support from the the National Natural Science Foundation of
China (Grant Nos.11925108 and 11731014). V.V.K. acknowledges financial support
from the Portuguese Foundation for Science and Technology (FCT) under Contract
no. UIDB/00618/2020.

## References

  * (1) M. Onorato, A. R. Osborne, M. Serio, and S. Bertone, Phys. Rev. Lett. 86, 5831 (2001); C. Kharif and E. Pelinovsky, Eur. J. Mech. B/Fluids, 22, 603 (2003).
  * (2) P. Müller, C. Garrett, and A. Osborne, Oceanography 18, 66 (2005).
  * (3) C. Kharif, E. Pelinovsky, and A. Slunyaev, Rogue Waves in the Ocean (Springer, New York, 2009).
  * (4) M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. T. Arecchi, Phys. Rep. 528, 47 (2013).
  * (5) J. M. Dudley, G. Genty, A. Mussot, A. Chabchoub, and F. Dias, Nature Rev. Phys. 1, 675 (2019).
  * (6) D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, Nature (London) 450, 1054 (2007).
  * (7) B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, Nature Phys. 6, 790 (2010).
  * (8) P. Dubard, P. Gaillard, C. Klein, and V. B. Matveev, Eur. Phys. J. Special Topics, 185, 247 (2010).
  * (9) J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, Nature Photon. 8, 755 (2014).
  * (10) N. Akhmediev, J. M. Dudley, D. R. Solli, and S. Turitsyn, J. Opt. 15, 060201 (2013).
  * (11) G. Yang, Y. Wang, Z. Qin, B. A. Malomed, D. Mihalache, and L. Li, Phys. Rev. E 90, 062909 (2014).
  * (12) P. Suret, R. El Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, and S. Bielawski, Nature Commun. 7, 13136 (2016).
  * (13) Y. V. Bludov, V. V. Konotop, and N. Akhmediev, Opt. Lett. 19, 3015 (2009).
  * (14) A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, Phys. Rev. Lett. 103, 173901 (2009).
  * (15) F. Leo, S. Coen, P. Kockaert, P. Emplit, M. Haelterman, A. Mussot, and M. Taki, Phys. Lett. A 379, 1934 (2015).
  * (16) A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin, and P. V. E. McClintock, Phys. Rev. Lett. 101, 065303 (2008).
  * (17) Yu. V. Bludov, V. V. Konotop, and N. Akhmediev, Phys. Rev. A 80, 033610 (2009); Z. Yan, V. V. Konotop, and N. Akhmediev, Phys. Rev. A 82, 036610 (2010).
  * (18) Y. V. Bludov, V. V. Konotop, and N. Akhmediev, Eur. Phys. J. Special Topics 185, 169 (2010).
  * (19) L. Stenflo and M. Marklund, J. Plasma Phys. 76, 293 (2010).
  * (20) A. Iafrati, A. Babanin, and M. Onorato, Phys. Rev. Lett. 110, 184504 (2013).
  * (21) W. M. Moslem, Phys. Plasmas 18, 032301 (2011).
  * (22) H. Bailung, S. K. Sharma, and Y. Nakamura, Phys. Rev. Lett. 107, 255005 (2011).
  * (23) Z. Yan, Commun. Theor. Phys. 54, 947 (2010); Z. Yan, Phys. Lett. A 375, 4274 (2011).
  * (24) A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford University Press, Oxford, 1995).
  * (25) N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, London, 1997).
  * (26) L. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford University Press, Oxford, 2003).
  * (27) Y. Nogami and C. S. Warke, Phys. Lett. 59A, 251 (1976).
  * (28) T. B. Benjamin and J. E. Feir, J. Fluid Mech. 27, 417 (1967).
  * (29) M. Onorato, A. R. Osborne, and M. Serio, Phys. Rev. Lett. 96, 014503 (2006).
  * (30) V. E. Zakharov and L. A. Ostrovsky, Physica (Amsterdam) 238D, 540 (2009); V. E. Zakharov and A. A. Gelash, Phys. Rev. Lett. 111, 054101 (2013).
  * (31) M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, Phys. Rev. Lett. 107, 253901 (2011).
  * (32) F. Baronio, M. Conforti, A. Degasperis, S. Lombardo, M. Onorato, and S. Wabnitz, Phys. Rev. Lett. 113, 034101 (2014).
  * (33) D. Peregrine, J. Aust. Math. Soc. B, Appl. Math. 25, 16 (1983).
  * (34) N. Akhmediev, A. Ankiewicz, and M. Taki, Phys. Lett. A 373, 675 (2009).
  * (35) A. Tikan, C. Billet, G. El, A. Tovbis, M. Bertola, T. Sylvestre, F. Gustave, S. Randoux, G. Genty, P. Suret, and J. M. Dudley, Phys. Rev. Lett. 119, 033901 (2017).
  * (36) A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, Phys. Rev. Lett. 106, 204502 (2011).
  * (37) B. Guo and L. Ling, Chin. Phys. Lett. 28, 110202 (2011); L.-C. Zhao and J. Liu, J. Opt. Soc. Am. B 29, 3119 (2012); L.-C. Zhao and J. Liu, Phys. Rev. E 87, 013201 (2013).
  * (38) D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, Phys. Rev. E 88, 013207 (2013).
  * (39) F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, Phys. Rev. Lett. 109, 044102 (2012); F. Baronio, M. Conforti, A. Degasperis, and S. Lombardo, Phys. Rev. Lett. 111, 114101 (2013).
  * (40) G. Zhang, Z. Yan, X. Wen, and Y. Chen, Phys. Rev. E 95, 042201 (2017); G. Zhang and Z. Yan, Physica (Amsterdam) 366D, 27 (2018); G. Zhang, Z. Yan, and L. Wang, Proc. R. Soc. A 475, 20180625 (2019).
  * (41) S. Chen and D. Mihalache, J. Phys. A: Math. Theor. 48, 215202 (2015).
  * (42) A. Przadka, S. Feat, P. Petitjeans, V. Pagneux, A. Maurel, and M. Fink, Phys. Rev. Lett. 109, 064501 (2012).
  * (43) A. Chabchoub and M. Fink, Phys. Rev. Lett. 112, 124101 (2014).
  * (44) A. C. Scott, Phys. Scr. 25, 651 (1982).
  * (45) C. Yeh and L. Bergman, Phys. Rev. E 57, 2398 (1998).
  * (46) N. Akhmediev, W. Królikowski, and A. W. Snyder, Phys. Rev. Lett. 81, 4632 (1998).
  * (47) A. A. Sukhorukov and N. Akhmediev, 83, 4736 (1999).
  * (48) M. J. Ablowitz, B. Prinari, and A. Trubatch, Discrete and continuous nonlinear Schrödinger systems (Cambridge University Press, Cambridge, 2004).
  * (49) S. V. Manakov, Zh. Eksp. Teor. Fiz. 67, 543 (1974) [Sov. Phys. JETP 38, 248 (1974)].
  * (50) The details of calculations are presented in the Supplemental Material.
  * (51) V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons (Springer-Verlag, Berlin, 1991).
  * (52) N. Akhmediev, A. Ankiewicz, and J. Soto-Crespo, Phys. Rev. E 80, 026601 (2009).
  * (53) B. Guo, L. Ling, and Q. Liu, Phys. Rev. E 85, 026607 (2012); L. Ling, L.-C. Zhao, and B. Guo, Nonlinearity 28, 3243 (2015).
  * (54) J. He, H. Zhang, L. Wang, K. Porsezian, and A. Fokas, Phys. Rev. E 87, 052914 (2013).
  * (55) L. Ling, L.-C. Zhao, Z. Yang, and B. Guo, Phys. Rev. E 93, 022217 (2017); L. Ling and L.-C. Zhao, Commun. Nonlinear Sci. Numer. Simulat. 72, 449 (2019).
  * (56) G. Zhang, L. Ling, and Z. Yan, preprint (2020).