# Rogue wave patterns in the nonlinear Schrödinger equation

Bo Yang and Jianke Yang  Department of Mathematics and Statistics, University
of Vermont, Burlington, VT 05405, USA

###### Abstract

Rogue wave patterns in the nonlinear Schrödinger equation are analytically
studied. It is shown that when an internal parameter in the rogue waves (which
controls the shape of initial weak perturbations to the uniform background) is
large, these waves would exhibit clear geometric structures, which are formed
by Peregrine waves in shapes such as triangle, pentagon, heptagon and nonagon,
with a possible lower-order rogue wave at its center. These rogue patterns are
analytically determined by the root structures of the Yablonskii-Vorob’ev
polynomial hierarchy, and their orientations are controlled by the phase of
the large parameter. It is also shown that when multiple internal parameters
in the rogue waves are large but satisfy certain constraints, similar rogue
patterns would still hold. Comparison between true rogue patterns and our
analytical predictions shows excellent agreement.

##  1 Introduction

The name of rogue waves first appeared in oceanography, where it referred to
large spontaneous and unexpected water wave excitations that are a threat even
to big ships Ocean_rogue_review ; Pelinovsky_book . Later, their counterparts
in optics were also reported Solli_Nature ; Wabnitz_book . Due to their
physical importance, rogue waves have received intensive theoretical and
experimental studies in the past decade. On the theoretical front, analytical
expressions of rogue waves have been derived in a wide variety of integrable
physical models, such as the nonlinear Schrödinger (NLS) equation for wave-
packet propagation in the ocean and optical systems Benney ; Peregrine ;
AAS2009 ; DGKM2010 ; KAAN2011 ; GLML2012 ; OhtaJY2012 , the derivative NLS
equations for circularly polarized nonlinear Alfvén waves in plasmas and
short-pulse propagation in a frequency-doubling crystal Kaup_Newell ;
KN_Alfven1 ; Wise2007 ; KN_rogue_2011 ; KN_rogue_2013 ; YangDNLS2019 , the
Manakov equations for light transmission in randomly birefringent fibers
Menyuk ; BDCW2012 ; ManakovDark ; LingGuoZhaoCNLS2014 ; Chen_Shihua2015 ;
ZhaoGuoLingCNLS2016 , and the three-wave resonant interaction equations
Ablowitz_book ; BaroDegas2013 ; DegasLomba2013 ; ChenSCrespo2015 ;
WangXChenY2015 ; ZhangYanWen2018 . On the experimental front, various rogue
wave solutions in the NLS equation and defocusing Manakov equations have been
observed in water tanks and optical fibers Tank1 ; Tank2 ; Fiber1 ; Fiber2 ;
Fiber3 . In these experiments, intimate knowledge of theoretical rogue wave
solutions in the underlying nonlinear wave equations was utilized, which
highlights the importance of theoretical developments on rogue waves for
practical rogue wave verifications and predictions.

The study of rogue wave patterns is important as this information allows for
the prediction of later rogue wave events from earlier wave forms. Although
graphs of low-order rogue waves have been plotted for many integrable
equations, and simple patterns such as triangles and rings have been reported,
richer patterns arising from high-order rogue wave solutions have received
little attention. For the NLS equation, preliminary investigations on rogue
wave patterns were made in KAAN2011 ; HeFokas ; KAAN2013  through Darboux
transformation and numerical simulations. It was observed in KAAN2011  that if
a $N$-th order rogue wave exhibits a single-shell ring structure, then the
center of the ring is a $(N-2)$-th order rogue wave. This observation was
explained analytically in HeFokas . In KAAN2013 , it was observed that NLS
rogue patterns could be classified according to the order of the rogue waves
and the parameter shifts applied to the Akhmediev breathers in the rogue-wave
limit. This latter observation allowed the authors to extrapolate the shapes
of rogue waves beyond order six, where numerical plotting of rogue waves
became difficult. However, an analytical and quantitative prediction of NLS
rogue patterns at arbitrary orders is still nonexistent.

In this article, we analytically investigate rogue wave patterns in the NLS
equation at arbitrary rogue-wave orders. We show that if any internal
parameter in the rogue waves (which controls the shape of initial small
perturbations to the uniform background) is large, then these waves would
exhibit clear geometric structures, which comprise Peregrine (fundamental)
rogue waves forming shapes such as triangle, pentagon, heptagon and nonagon,
with a possible lower-order rogue wave at the center. These rogue patterns are
analytically predicted by the root structures of the Yablonskii-Vorob’ev
polynomial hierarchy, and their orientations are controlled by the phase of
the large internal parameter. We also show that if multiple internal
parameters in the rogue waves are large but satisfy certain constraints, then
the same rogue patterns would still hold. These results reveal a deep
connection between rogue patterns and the Yablonskii-Vorob’ev polynomials, and
drastically improve our analytical understanding and quantitative description
of rogue events. As a small application of our analytical results, the
numerical observation in KAAN2011  on single-shell ring structures is
explained. Comparison between true rogue patterns and our analytical
predictions is also presented, and excellent agreement is observed.

This paper is structured as follows. In Sec. 2, we present a simplified
bilinear expression of general rogue waves in the NLS equation, as well as the
Yablonskii-Vorob’ev polynomial hierarchy and its root structure, which we will
utilize in later texts. In Sec. 3, we present our main theorems on rogue wave
patterns when a single internal parameter in the rogue waves is large. In Sec.
4, we graphically illustrate rogue patterns under a large parameter and
compare these patterns to our theoretical predictions. In Sec. 5, we prove the
theorems stated in Sec. 3. In Sec. 6, we generalize our analytical results to
cases where multiple internal parameters in the rogue waves are large but meet
certain constraints. Sec. 7 concludes the paper.

##  2 Preliminaries

The nonlinear Schrödinger (NLS) equation

$\textrm{i}u_{t}+\frac{1}{2}u_{xx}+|u|^{2}u=0$ (1)

arises in numerous physical situations such as water waves and optics Benney ;
Ablowitzbook ; Kivsharbook . In this article, we consider its rogue wave
solutions, which are rational solutions which approach a constant-amplitude
continuous-wave background as $x,t\to\pm\infty$. Since this equation admits
Galilean and scaling invariances, we can set the boundary conditions of these
rogue waves as

$u(x,t)\to e^{\textrm{i}t},\quad x,t\to\infty,$ (2)

without any loss of generality.

###  2.1 Improved bilinear expressions of general rogue waves

Analytical expressions for general rogue waves in the NLS equation have been
derived in DGKM2010 ; GLML2012 ; OhtaJY2012  by various methods. However,
those expressions are not the best for our solution analysis. Here, we present
a simpler expression for these solutions, which can be derived by
incorporating a new parameterization YangDNLS2019  into bilinear rogue waves
of Ref. OhtaJY2012 . These simpler expressions of rogue waves are given by the
following theorem.

> Theorem 1. The general NLS rogue waves under boundary conditions (2) are
>
> $\displaystyle u_{N}(x,t)=\frac{\sigma_{1}}{\sigma_{0}}e^{\textrm{i}t},$ (3)
>
> where the positive integer $N$ represents the order of the rogue wave,
> $\sigma_{n}$ is a $N\times N$ Gram determinant
>
> $\sigma_{n}=\det_{\begin{subarray}{l}1\leq i,j\leq
> N\end{subarray}}\left(\begin{array}[]{c}\phi_{2i-1,2j-1}^{(n)}\end{array}\right),$
> (4)
>
> the matrix elements in $\sigma_{n}$ are defined by
>
> $\phi_{i,j}^{(n)}=\sum_{\nu=0}^{\min(i,j)}\frac{1}{4^{\nu}}\hskip
> 1.70709ptS_{i-\nu}(\textbf{\emph{x}}^{+}(n)+\nu\textbf{\emph{s}})\hskip
> 1.70709ptS_{j-\nu}(\textbf{\emph{x}}^{-}(n)+\nu\textbf{\emph{s}}),$ (5)
>
> vectors
> $\textbf{\emph{x}}^{\pm}(n)=\left(x_{1}^{\pm},x_{2}^{\pm},\cdots\right)$ are
> defined by
>
> $\displaystyle x_{1}^{\pm}=x\pm\textrm{i}t\pm n,\ \ \ x_{2k}^{\pm}=0,\quad
> x_{2k+1}^{+}=\frac{x+2^{2k}(\textrm{i}t)}{(2k+1)!}+a_{2k+1},\quad
> x_{2k+1}^{-}=\frac{x-2^{2k}(\textrm{i}t)}{(2k+1)!}+a_{2k+1}^{*},$ (6)
>
> with $k\geq 1$ and the asterisk * representing complex conjugation,
> $\textbf{\emph{s}}=(0,s_{2},0,s_{4},\cdots)$ are coefficients from the
> expansion
>
>
> $\displaystyle\sum_{j=1}^{\infty}s_{j}\lambda^{j}=\ln\left[\frac{2}{\lambda}\tanh\left(\frac{\lambda}{2}\right)\right],$
> (7)
>
> the Schur polynomials $S_{k}(\mbox{\boldmath$x$})$, with
> $\emph{{x}}=\left(x_{1},x_{2},\ldots\right)$, are defined by
>
>
> $\sum_{k=0}^{\infty}S_{k}(\mbox{\boldmath$x$})\epsilon^{k}=\exp\left(\sum_{k=1}^{\infty}x_{k}\epsilon^{k}\right),$
> (8)
>
> or more explicitly,
>
> $S_{k}(\mbox{\boldmath$x$})=\sum_{l_{1}+2l_{2}+\cdots+ml_{m}=k}\left(\
> \prod_{j=1}^{m}\frac{x_{j}^{l_{j}}}{l_{j}!}\right),$ (9)
>
> and $a_{2k+1}\hskip 1.42271pt(k=1,2,\cdots,N-1)$ are free irreducible
> complex constants.

This theorem will be proved in Appendix A. Since these rogue waves approach a
uniform background as $t\to-\infty$, the internal parameters $\\{a_{2k+1}\\}$
in these waves physically control the shape of initial small perturbations to
this uniform background, which in turn decide the subsequent time evolution
and the resulting pattern of rogue waves.

As we will show, these rogue wave solutions will exhibit clear and
recognizable patterns when some of these $N-1$ internal parameters
$(a_{3},a_{5},\cdots,a_{2N-1})$ get large. It turns out that the resulting
rogue patterns are determined by the root structures of the Yablonskii-
Vorob’ev polynomial hierarchy, and this polynomial hierarchy and their root
structures will be introduced next.

###  2.2 The Yablonskii-Vorob’ev polynomial hierarchy and their root
structures

Yablonskii-Vorob’ev polynomials arose in rational solutions of the second
Painlevé equation ($\mbox{P}_{\mbox{\scriptsize II}}$) Yablonskii1959 ;
Vorobev1965

$w^{\prime\prime}=2w^{3}+zw+\alpha,$ (10)

where $\alpha$ is an arbitrary constant. It has been shown that this
$\mbox{P}_{\mbox{\scriptsize II}}$ equation admits rational solutions if and
only if $\alpha=N$ is an integer. In this case, the rational solution is
unique and is given by

$\displaystyle w(z;N)=\frac{d}{dz}\ln\frac{Q_{N-1}(z)}{Q_{N}(z)},\quad N\geq
1,$ (11) $\displaystyle w(z;0)=0,\quad w(z;-N)=-w(z;N),$ (12)

and the polynomials $Q_{N}(z)$, now called the Yablonskii-Vorob’ev
polynomials, are constructed by the following recurrence relation

$Q_{N+1}Q_{N-1}=zQ_{N}^{2}-4\left[Q_{N}Q_{N}^{\prime\prime}-(Q_{N}^{\prime})^{2}\right],$
(13)

with $Q_{0}(z)=1$, $Q_{1}(z)=z$, and the prime denoting the derivative. Later,
a determinant expression for these polynomials was found in Kajiwara-Ohta1996
. Let $p_{k}(z)$ be the special Schur polynomial defined by

$\sum_{k=0}^{\infty}p_{k}(z)\epsilon^{k}=\exp\left(z\epsilon-\frac{4}{3}\epsilon^{3}\right).$
(14)

Then, Yablonskii-Vorob’ev polynomials $Q_{N}(z)$ are given by the $N\times N$
determinant Kajiwara-Ohta1996

$\displaystyle
Q_{N}(z)=c_{N}\left|\begin{array}[]{cccc}p_{1}(z)&p_{0}(z)&\cdots&p_{2-N}(z)\\\
p_{3}(z)&p_{2}(z)&\cdots&p_{4-N}(z)\\\ \vdots&\vdots&\vdots&\vdots\\\
p_{2N-1}(z)&p_{2N-2}(z)&\cdots&p_{N}(z)\end{array}\right|,$ (19)

where $c_{N}=\prod_{j=1}^{N}(2j-1)!!$, and $p_{k}(z)=0$ if $k<0$. These
polynomials are monic polynomials with integer coefficients Clarkson2003-II .
The first few Yablonskii-Vorob’ev polynomials are

$\displaystyle Q_{2}(z)=z^{3}+4,$ $\displaystyle Q_{3}(z)=z^{6}+20z^{3}-80,$
$\displaystyle Q_{4}(z)=z(z^{9}+60z^{6}+11200).$

To define the Yablonskii-Vorob’ev polynomial hierarchy, we let
$p_{k}^{[m]}(z)$ be the generalized Schur polynomial defined by

$\sum_{k=0}^{\infty}p_{k}^{[m]}(z)\epsilon^{k}=\exp\left(z\epsilon-\frac{2^{2m}}{2m+1}\epsilon^{2m+1}\right),$
(20)

where $m$ is a positive integer. Then, the Yablonskii-Vorob’ev hierarchy
$Q_{N}^{[m]}(z)$ are given by the $N\times N$ determinant Clarkson2003-II

$\displaystyle
Q_{N}^{[m]}(z)=c_{N}\left|\begin{array}[]{cccc}p^{[m]}_{1}(z)&p^{[m]}_{0}(z)&\cdots&p^{[m]}_{2-N}(z)\\\
p^{[m]}_{3}(z)&p^{[m]}_{2}(z)&\cdots&p^{[m]}_{4-N}(z)\\\
\vdots&\vdots&\vdots&\vdots\\\
p^{[m]}_{2N-1}(z)&p^{[m]}_{2N-2}(z)&\cdots&p^{[m]}_{N}(z)\end{array}\right|,$
(25)

where $p_{k}^{[m]}(z)=0$ if $k<0$. When $m=1$, $Q_{N}^{[1]}(z)$ are the
original Yablonskii-Vorob’ev polynomials $Q_{N}(z)$. When $m>1$,
$Q_{N}^{[m]}(z)$ give higher members of this polynomial hierarchy. All these
$Q_{N}^{[m]}(z)$ polynomials were conjectured to be monic polynomials with
integer coefficients as well Clarkson2003-II . The first few $Q_{N}^{[2]}(z)$
polynomials are

$\displaystyle Q_{2}^{[2]}(z)=z^{3},$ $\displaystyle
Q_{3}^{[2]}(z)=z(z^{5}-144),$ $\displaystyle
Q_{4}^{[2]}(z)=z^{10}-1008z^{5}-48384.$

These $Q_{N}^{[m]}(z)$ polynomials, through relations similar to (11)-(12),
provide the unique rational solution for the $\mbox{P}_{\mbox{\scriptsize
II}}$ hierarchy Clarkson2003-II ; Bertola2016 . It is noted that the
determinant (25) for $Q_{N}^{[m]}(z)$ is a Wronskian, because it is easy to
see from Eq. (20) that

$p_{k}^{[m]}(z)=[p_{k+1}^{[m]}]^{\prime}(z).$ (26)

Root structures of the Yablonskii-Vorob’ev polynomial hierarchy have been
studied before Fukutani ; Taneda ; Clarkson2003-II ; Miller2014 ; Bertola2016
. Regarding the zero root, its multiplicity in $Q_{N}(z)$, $Q_{N}^{[2]}(z)$
and $Q_{N}^{[3]}(z)$ was presented in Taneda ; Clarkson2003-II . Generalizing
those results, we have the following theorem.

> Theorem 2. The general Yablonskii-Vorob’ev polynomial $Q_{N}^{[m]}(z)$ has
> degree $N(N+1)/2$, and is of the form
>
> $Q_{N}^{[m]}(z)=z^{N_{0}(N_{0}+1)/2}q_{N}^{[m]}(\zeta),\quad\zeta=z^{2m+1},$
> (27)
>
> where $q_{N}^{[m]}(\zeta)$ is a polynomial with a nonzero constant term, and
> the integer $N_{0}$ is given by the equation
>
> $\displaystyle N\equiv N_{0}\hskip 2.84544pt\mbox{mod}\hskip
> 2.84544pt(2m+1),\quad\text{or}$ (28) $\displaystyle N\equiv-N_{0}-1\hskip
> 2.84544pt\mbox{mod}\hskip 2.84544pt(2m+1),$ (29)
>
> under the restriction of $0\leq N_{0}\leq m$. Due to this restriction on
> $N_{0}$, only one of Eqs. (28) and (29) can hold, and thus the resulting
> $N_{0}$ value is unique.

The proof of this theorem will be provided in Appendix B. This theorem gives
the multiplicity of the root zero in any $Q_{N}^{[m]}(z)$ polynomial. It also
shows that the root structure of $Q_{N}^{[m]}(z)$ is invariant under
$2\pi/(2m+1)$-angle rotation in the complex $z$ plane. In the particular case
of the original Yablonskii-Vorob’ev polynomials $Q_{N}(z)$ where $m=1$, the
above theorem shows that $0\leq N_{0}\leq 1$. This means that zero is either
not a root or a simple root for $Q_{N}(z)$, in agreement with previous results
in Fukutani ; Taneda .

On the determination of the unique $N_{0}$ value in the above theorem, let us
give an example. When $N=5$ and $m=4$, the $N_{0}$ value under the restriction
of $0\leq N_{0}\leq 4$ can only be found from Eq. (29) as $N_{0}=3$.

Regarding nonzero roots, it was shown in Fukutani  that for the original
Yablonskii-Vorob’ev polynomials $Q_{N}(z)$, all nonzero roots are simple. For
the higher Yablonskii-Vorob’ev polynomial hierarchy $Q_{N}^{[m]}(z)$, it was
conjectured in Clarkson2003-II  that all nonzero roots are also simple. In
view of Theorem 2, this conjecture implies that the polynomial
$Q_{N}^{[m]}(z)$ has

$N_{p}=\frac{1}{2}\left[N(N+1)-N_{0}(N_{0}+1)\right]$ (30)

nonzero simple roots. We have checked this conjecture for a myriad of $(N,m)$
values and found it to hold in all our examples. Thus, we will assume it true
and utilize it in our later analysis [see the sentence below Eq. (62) in Sec.
5].

Roots of many Yablonskii-Vorob’ev polynomials $Q_{N}^{[m]}(z)$ were plotted in
Clarkson2003-II , and highly regular and symmetric patterns were observed. Due
to the importance of these root structures to our work, we reproduce some of
those root plots in Fig. 1 for $N=6$ and $1\leq m\leq 5$. Boundaries of the
roots in $Q_{N}^{[m]}(z)$ in the large-$N$ limit have been determined in
Miller2014 ; Bertola2016 .

Figure 1: Plots of the roots of the Yablonskii-Vorob’ev polynomial hierarchy
$Q_{N}^{[m]}(z)$ for $N=6$ and $1\leq m\leq 5$.

##  3 Analytical predictions of rogue wave patterns for a single large
internal parameter

Rogue wave solutions in Theorem 1 contain $N-1$ free internal complex
parameters $a_{3},a_{5},\cdots,a_{2N-1}$. In this section, we consider
asymptotics of these rogue solutions when one of these internal parameters is
large, while the other parameters remain $O(1)$. Generalizations to cases
where multiple internal parameters are large but satisfy certain constraints
will be made in Sec. 6.

Suppose $a_{2m+1}$ is large, where $1\leq m\leq N-1$, and the other parameters
are $O(1)$. Then our results on the large-$a_{2m+1}$ asymptotics of rogue
waves in Theorem 1 are summarized by the following two theorems.

> Theroem 3. Far away from the origin, with
> $\sqrt{x^{2}+t^{2}}=O\left(|a_{2m+1}|^{1/(2m+1)}\right)$, the $N$-th order
> rogue wave $u_{N}(x,t)$ in Eq. (3) separates into $N_{p}$ fundamental
> (Peregrine) rogue waves, where $N_{p}$ is given in Eq. (30). These Peregrine
> waves are $\hat{u}_{1}(x-\hat{x}_{0},t-\hat{t}_{0})\hskip
> 1.42271pte^{\textrm{i}t}$, where
>
> $\hat{u}_{1}(x,t)=1-\frac{4(1+2\textrm{i}t)}{1+4x^{2}+4t^{2}},$ (31)
>
> and their positions $(\hat{x}_{0},\hat{t}_{0})$ are given by
>
> $\displaystyle\hat{x}_{0}+\textrm{i}\hskip
> 1.42271pt\hat{t}_{0}=z_{0}\left(-\frac{2m+1}{2^{2m}}a_{2m+1}\right)^{\frac{1}{2m+1}},$
> (32)
>
> with $z_{0}$ being any one of the $N_{p}$ simple nonzero roots of
> $Q_{N}^{[m]}(z)$. The error of this Peregrine wave approximation is
> $O(|a_{2m+1}|^{-1/(2m+1)})$. Expressed mathematically, when
> $\left[(x-\hat{x}_{0})^{2}+(t-\hat{t}_{0})^{2}\right]^{1/2}=O(1)$, we have
> the following solution asymptotics
>
>
> $u_{N}(x,t;a_{3},a_{5},\cdots,a_{2N-1})=\hat{u}_{1}(x-\hat{x}_{0},t-\hat{t}_{0})\hskip
> 1.42271pte^{\textrm{i}t}+O\left(|a_{2m+1}|^{-1/(2m+1)}\right),\quad|a_{2m+1}|\gg
> 1.$ (33)
>
> When $(x,t)$ is not in the neighborhood of any of these $N_{p}$ Peregrine
> waves, or $\sqrt{x^{2}+t^{2}}$ is larger than
> $O\left(|a_{2m+1}|^{1/(2m+1)}\right)$, $u_{N}(x,t)$ asymptotically
> approaches the constant background $e^{\textrm{i}t}$ as
> $|a_{2m+1}|\to\infty$.

> Theroem 4. In the neighborhood of the origin, where
> $\sqrt{x^{2}+t^{2}}=O(1)$, $u_{N}(x,t)$ is approximately a lower $N_{0}$-th
> order rogue wave $u_{N_{0}}(x,t)$, where $N_{0}$ is given in Theorem 2 with
> $0\leq N_{0}\leq N-2$, and $u_{N_{0}}(x,t)$ is given by Eq. (3) with its
> internal parameters $a_{3},a_{5},\cdots,a_{2N_{0}-1}$ being the first
> $N_{0}-1$ values in the parameter set $(a_{3},a_{5},\cdots,a_{2N-1})$ of the
> original rogue wave $u_{N}(x,t)$. The error of this lower-order rogue wave
> approximation $u_{N_{0}}(x,t)$ is $O(|a_{2m+1}|^{-1})$. Expressed
> mathematically, when $\sqrt{x^{2}+t^{2}}=O(1)$,
>
>
> $u_{N}(x,t;a_{3},a_{5},\cdots,a_{2N-1})=u_{N_{0}}(x,t;a_{3},a_{5},\cdots,a_{2N_{0}-1})+O\left(|a_{2m+1}|^{-1}\right),\quad|a_{2m+1}|\gg
> 1.$ (34)
>
> If $N_{0}=0$, then there will not be such a lower-order rogue wave in the
> neighborhood of the origin, and $u_{N}(x,t)$ asymptotically approaches the
> constant background $e^{\textrm{i}t}$ there as $|a_{2m+1}|\to\infty$.

These two theorems will be proved in Sec. 5.

Remark 1. Theorem 3 predicts that when $a_{2m+1}$ is large, the $N$-th order
rogue wave (3) far away from the origin comprises $N_{p}$ Peregrine waves. The
rogue pattern formed by these Peregrine waves has the same geometric shape as
the root structure of the polynomial $Q_{N}^{[m]}(z)$, and thus this rogue
pattern has $2\pi/(2m+1)$-angle rotational symmetry. The only difference
between the predicted rogue pattern and the root structure of $Q_{N}^{[m]}(z)$
is a dilation and rotation between them due to the multiplication factor on
the right side of Eq. (32). The angle of rotation is equal to the angle of the
complex number $-a_{2m+1}$ divided by $2m+1$, and the dilation factor is equal
to $[(2m+1)2^{-2m}|a_{2m+1}|]^{1/(2m+1)}$.

Remark 2. On the right side of Eq. (32), we can pick any one of the
$(2m+1)$-th root of $-(2m+1)2^{-2m}a_{2m+1}$, because roots $z_{0}$ of the
polynomial $Q_{N}^{[m]}(z)$ have $2\pi/(2m+1)$-angle rotational symmetry, see
the comment in the paragraph below Theorem 2.

As a small application of the above two theorems, we explain the numerical
observations in Ref. KAAN2011 . Under our bilinear rogue solution (3), a
$N$-th order rogue wave exhibits a ring structure when $a_{2N-1}$ is large
(see Fig. 2 in the next section). In this case, $m=N-1$, and $N_{0}=N-2$ from
Eq. (29). Then, our theory predicts that the center of this $N$-th order rogue
wave is a $(N-2)$-th order rogue wave, surrounded by $N_{p}=2N-1$ Peregrine
waves that are evenly spaced on a ring due to the $2\pi/(2m+1)$-, i.e.,
$2\pi/(2N-1)$-angle rotational symmetry (see Remark 1 above). This is
precisely what was observed in KAAN2011 .

##  4 Comparison between true rogue patterns and our analytical predictions

In this section, we compare true rogue patterns with our analytical
predictions. For this purpose, we first show in Fig. 2 true rogue wave
solutions (3) from the 2rd to 7th order, with large $a_{3}$, $a_{5}$, $a_{7}$,
$a_{9}$, $a_{11}$ and $a_{13}$ in the first to sixth columns respectively. The
specific value of the large parameter in each panel of this figure is listed
in Table 1, and the other parameters in each solution are chosen as zero.

Figure 2: True NLS rogue wave patterns
$|u_{N}(x,t;a_{3},a_{5},\cdots,a_{2N-1})|$ from solutions (3) when $N$ ranges
from $2$ to $7$ and one of the solution parameters is large (the other
parameters are set as zero). The large parameter is labeled on top of each
column, and its value for each panel is listed in Table 1. The center of each
panel is always the origin $x=t=0$, but the $(x,t)$ intervals differ slightly
from panel to panel. For instance, in the bottom row, the left-most panel has
$-18.5\leq x,t\leq 18.5$, and the right-most panel has $-16\leq x,t\leq 16$.

It is seen that these rogue waves comprise a number of Peregrine waves forming
triangular patterns for large $a_{3}$, pentagon patterns for large $a_{5}$,
heptagon patterns for large $a_{7}$, nonagon patterns for large $a_{9}$,
hendecagon (eleven-sided polygon) patterns for large $a_{11}$, and tridecagon
(thirteen-sided polygon) patterns for large $a_{13}$. In the literature,
patterns on the diagonal of Fig. 2 are sometimes called single-shell ring
structures KAAN2011 . In addition to these Peregrine waves away from the
origin, some of the rogue waves also contain a lower-order rogue wave at their
centers. For instance, for the 7-th order rogue waves in the bottom row of
Fig. 2, the first and fourth panels (with large $a_{3}$ and $a_{9}$
respectively) exhibit a Peregrine wave in their centers; the second panel
(with large $a_{5}$) exhibits a second-order rogue wave in the center; the
fifth panel (with large $a_{11}$) exhibits a third-order rogue wave in the
center; and the last panel (with large $a_{13}$) exhibits a fifth-order rogue
wave in the center. For our choices of parameters in rogue waves of Fig. 2,
these lower-order rogue waves in the center are all super-rogue waves, i.e.,
rogue waves with the highest peak amplitude of their orders. We note by
passing that the first five rows of rogue patterns in Fig. 2 resemble those
plotted in Ref. KAAN2013  from Akhmediev breathers in the rogue-wave limit,
although orientations between the two sets of patterns are very different.

Table 1: Value of the large parameter for rogue waves in Fig. 2 $N$ | $a_{3}$ | $a_{5}$ | $a_{7}$ | $a_{9}$ | $a_{11}$ | $a_{13}$
---|---|---|---|---|---|---
2 | $-100\textrm{i}$ |  |  |  |  |
3 | $-60\textrm{i}$ | $-1000\textrm{i}$ |  |  |  |
4 | $-30\textrm{i}$ | $-300\textrm{i}$ | $-3000\textrm{i}$ |  |  |
5 | $-20\textrm{i}$ | $-100\textrm{i}$ | $-2000\textrm{i}$ | $-12000\textrm{i}$ |  |
6 | $-20\textrm{i}$ | $-200\textrm{i}$ | $-2000\textrm{i}$ | $-20000\textrm{i}$ | $-80000\textrm{i}$ |
7 | $-20\textrm{i}$ | $-200\textrm{i}$ | $-2000\textrm{i}$ | $-30000\textrm{i}$ | $-100000\textrm{i}$ | $-300000\textrm{i}$

Now, we compare these true rogue patterns in Fig. 2 with our analytical
predictions. Our prediction $|u_{N}^{(p)}(x,t)|$ from Theorems 3 and 4 can be
assembled into a simple formula,

$\left|u_{N}^{(p)}(x,t)\right|=\left|u_{N_{0}}(x,t)\right|+\sum_{j=1}^{N_{p}}\left(\left|\hat{u}_{1}(x-\hat{x}_{0}^{(j)},t-\hat{t}_{0}^{(j)})\right|-1\right),$
(35)

where $\hat{u}_{1}(x,t)$ is the Peregrine wave given in (31), their positions
$(\hat{x}_{0}^{(j)},\hat{t}_{0}^{(j)})$ given by (32) with $z_{0}$ being every
one of the $N_{p}$ simple nonzero roots of $Q_{N}^{[m]}(z)$, and
$u_{N_{0}}(x,t)$ is the lower-order rogue wave in Eq. (34) whose internal
parameters $(a_{3},a_{5},\cdots,a_{2N_{0}-1})$ are the first $N_{0}-1$ values
in the parameter set $(a_{3},a_{5},\cdots,a_{2N-1})$ of the original rogue
wave $u_{N}(x,t)$. For true rogue waves in Fig. 2, all internal parameters
except for $a_{2m+1}$ were chosen as zero, and $N_{0}\leq m$ (see Theorem 2).
Then, all internal parameters in the predicted lower-order rogue wave
$u_{N_{0}}(x,t)$ at the origin are also zero.

Our predicted $(N_{p},N_{0})$ values for rogue waves of Fig. 2 are displayed
in Table 2, where $m=1,2,\cdots,6$ correspond to large
$a_{3},a_{5},\cdots,a_{13}$ respectively. These $(N_{p},N_{0})$ values provide
our predictions for the number of Peregrine waves away from the origin
$(x,t)=(0,0)$, as well as the order of the reduced rogue wave in the
neighborhood of the origin. Visual comparison between Table 2 and Fig. 2 shows
complete agreement.

Table 2: Predicted $(N_{p},N_{0})$ values for true rogue waves of Fig. 2 $N$ | $m=1$ | $m=2$ | $m=3$ | $m=4$ | $m=5$ | $m=6$
---|---|---|---|---|---|---
2 | (3, 0) |  |  |  |  |
3 | (6, 0) | (5, 1) |  |  |  |
4 | (9, 1) | (10, 0) | (7, 2) |  |  |
5 | (15, 0) | (15, 0) | (14, 1) | (9, 3) |  |
6 | (21, 0) | (20, 1) | (21, 0) | (18, 2) | (11, 4) |
7 | (27, 1) | (25, 2) | (28, 0) | (27, 1) | (22, 3) | (13, 5)

We further compare our predicted whole solutions (35) with the true solutions
of Fig. 2 for the same sets of $(a_{3},a_{5},\cdots)$ parameter values. These
predicted whole solutions (35) are displayed in Fig. 3, with identical $(x,t)$
intervals as in Fig. 2’s true solutions. It is seen that the predicted
patterns are strikingly similar to the true ones. In particular, since our
predicted Peregrine locations (32) in the $(x,t)$ plane are given by all the
non-zero roots of the Yablonskii-Vorob’ev polynomials $Q_{N}^{[m]}(z)$,
multiplied by a fixed complex constant, predicted patterns formed by these
Peregrine waves then are simply the root structures of these Yablonskii-
Vorob’ev polynomials under certain rotation and dilation, as is evident by
comparing predicted rogue waves in Fig. 3 to the Yablonskii-Vorob’ev root
structures in Fig. 1 for $N=6$. These predicted Peregrine patterns clearly
match the true ones in Fig. 2 very well. This visual agreement shows the deep
connection between NLS rogue patterns and root structures of the Yablonskii-
Vorob’ev hierarchy, as our theorem 3 predicts.

Regarding our predictions $u_{N_{0}}(x,t)$ for centers of the rogue waves
$u_{N}(x,t)$ in Fig. 2, we can show that our bilinear rogue wave solution (3)
in Theorem 1 with all internal parameters set as zero gives the super-rogue
wave. This means that our predictions $u_{N_{0}}(x,t)$ for the centers of true
rogue waves are all lower-order super-rogue waves, which agree with centers of
true solutions shown in Fig. 2.

Figure 3: Analytical predictions (35) for true rogue waves in Fig. 2. The $x$
and $t$ intervals here are identical to those in Fig. 2.

Theorem 3 reveals that the orientation of the rogue pattern formed by
Peregrine waves is controlled by the phase of the large parameter $a_{2m+1}$.
Specifically, the rogue-pattern orientation is the one of the root pattern of
$Q_{N}^{[m]}(z)$ rotated by an angle of $\mbox{arg}(-a_{2m+1})/(2m+1)$, where
“arg” represents the argument (angle) of a complex number. To check this
prediction, we choose the 4-th order pentagon-shaped rogue waves, where
$a_{5}$ is large and the other parameters are set as zero. For three choices
of the $a_{5}$ value with the same modulus but different arguments, namely,
$500e^{-\textrm{i}\pi/3}$, $500e^{\textrm{i}\pi/3}$ and
$500e^{4\textrm{i}\pi/3}$, true rogue patterns from solutions (3) are
displayed in the upper row of Fig. 4. As expected, orientations of these
pentagon patterns indeed change as the argument of $a_{5}$ varies. Using our
formula (32), predicted locations of Peregrine waves in the rogue pattern are
shown in the lower row of Fig. 4. Comparison of the upper and lower rows of
Fig. 4 shows that the predicted orientations are in perfect agreement with the
true ones.

Figure 4: Orientations of 4-th order pentagon-shaped rogue waves with
$a_{5}=500e^{-\textrm{i}\pi/3}$ (left column), $500e^{\textrm{i}\pi/3}$
(middle column) and $500e^{4\textrm{i}\pi/3}$ (right column) respectively; the
other parameters in the rogue solutions are zero. Upper row: true rogue
patterns from solutions (3); lower row: predicted locations of Peregrine waves
from Eq. (32).

Next, we make quantitative comparisons between true rogue waves and our
predictions for large $a_{2m+1}$, and verify the error decay rate of
$O(|a_{2m+1}|^{-1/(2m+1)})$ for the prediction of Peregrine-wave locations far
away from the origin in Theorem 3, and the error decay rate of
$O(|a_{2m+1}|^{-1})$ for the prediction of the lower-order rogue wave at the
center in Theorem 4.

For the quantitative comparison on Peregrine-wave locations away from the
origin, we choose two patterns of 3rd-order rogue waves. One is a triangle
pattern from large $a_{3}$, and we set arg$(a_{3})=-\pi/4$; and the other is a
pentagon pattern from large $a_{5}$, and we set $a_{5}$ to be real positive.
In each pattern, we choose all other parameters of the rogue wave solutions to
be zero. These triangul and pentagon patterns are shown schematically in Fig.
5(a, c) respectively. In each of these two patterns, we pick one of its
Peregrine waves, which is marked by an arrow, and quantitatively compare its
true $(x_{0},t_{0})$ location with our analytical prediction (32) as $|a_{3}|$
or $|a_{5}|$ increases. Here, the true location of the Peregrine wave is
defined as the $(x_{0},t_{0})$ location where this Peregrine wave attains its
maximum amplitude, and the error of our asymptotic prediction $(\hat{x}_{0},\
\hat{t}_{0})$ in Eq. (32) is defined as

$\mbox{error of Peregrine
location}=\sqrt{\left(\hat{x}_{0}-x_{0}\right)^{2}+\left(\hat{t}_{0}-t_{0}\right)^{2}}.$

These errors of Peregrine locations versus $|a_{3}|$ or $|a_{5}|$ are plotted
as solid lines in panels (b) and (d) of Fig. 5 for the triangular and pentagon
patterns respectively. For comparison, the decay rates of $|a_{3}|^{-1/3}$ and
$|a_{5}|^{-1/5}$ are also displayed in these panels as dashed lines. We see
that these errors of Peregrine locations indeed decay at the rate of
$|a_{2m+1}|^{-1/(2m+1)}$, thus confirming the analytical error estimates (33)
in Theorem 3.

Figure 5: Decay of errors in our prediction (32) for the Peregrine location as
$|a_{3}|$ or $|a_{5}|$ increases. (a) A triangle pattern of 3rd-order rogue
waves when $|a_{3}|$ is large and arg$(a_{3})=-\pi/4$. (b) Error versus
$|a_{3}|$ for the Peregrine location marked by an arrow in (a). (c) A pentagon
pattern of 3rd-order rogue waves when $|a_{5}|$ is large with
arg$(a_{5})=-\pi/4$. (b) Error versus $|a_{5}|$ for the Peregrine location
marked by an arrow in (c).

To quantitatively compare our prediction in Theorem 4 on the lower-order rogue
wave at the center with the true solution, we choose a fifth-order rogue wave
$u_{5}(x,t)$ with large $a_{9}$ and the other internal parameters set as zero.
This $|u_{5}(x,t)|$ solution with $a_{9}=-5000\textrm{i}$ is displayed in Fig.
6(a). The center region of this wave marked by a dashed-line box in panel (a)
is amplified and replotted in panel (b). In the present case, $N=5$ and $m=4$.
Since $5\equiv-4\hskip 2.84544pt\mbox{mod}\hskip 2.84544pt9$, we get $N_{0}=3$
from Eq. (29). Thus, according to Theorem 4, this $u_{5}(x,t)$ solution
contains a 3rd-order rogue wave $u_{3}(x,t)$ in its center region, where all
internal parameters $(a_{3},a_{5})$ in this $u_{3}(x,t)$ solution are zero.
Such a $u_{3}(x,t)$ solution is a third-order super rogue wave. This predicted
$|u_{3}(x,t)|$ solution is displayed in Fig. 6(c), with the same $(x,t)$
internals as in the true center-region solution displayed in panel (b).
Visually, this predicted center solution in (c) is identical to the true
center solution in (b). Quantitatively, we have also obtained the errors in
our predicted solution $u_{3}(x,t)$ at $x=t=0.5$ of the center region as
$a_{9}$ increases in magnitude with arg$(a_{9})=-\pi/2$. Our error is defined
as

$\mbox{error of center region
prediction}=\left|u_{5}(x,t)-u_{3}(x,t)\right|_{x=t=0.5}.$

The dependence of this error on $|a_{9}|$ is plotted in Fig. 6(d). Comparison
of this error decay with the $|a_{9}|^{-1}$ decay [shown as a dashed line in
panel (d)] indicates that this error is indeed of $O(|a_{9}|^{-1})$,
confirming the error prediction (34) in Theorem 4.

Figure 6: Decay of errors in our prediction $u_{3}(x,t)$ for the center region
of the rogue wave $u_{5}(x,t)$ with large $a_{9}$. (a) A true 5-th order rogue
wave $|u_{5}(x,t)|$ with $a_{9}=-5000\textrm{i}$ and the other parameters
being zero; the $(x,t)$ intervals here are $-12\leq x,t\leq 12$. (b) Zoomed-in
plot of the center region of the true solution marked by a dashed-line box in
panel (a). (c) Our prediction $|u_{3}(x,t)|$ for the center region with the
same $(x,t)$ intervals as in (b). (d) Error decay of our predicted solution at
the $(x,t)$ location of $(0.5,0.5)$ as $a_{9}$ increases in size with
arg$(a_{9})=-\pi/2$.

##  5 Proofs for the analytical results in Sec. 3

In this section, we prove the analytical predictions on NLS rogue patterns in
Theorems 3 and 4 of Sec. 3. Our proof is based on an asymptotic analysis of
the rogue wave solution (3), or equivalently, the determinant $\sigma_{n}$ in
Eq. (4), in the large $a_{2m+1}$ limit.

Proof of Theorem 3. When $a_{2m+1}$ is large and the other parameters $O(1)$
in the rogue wave solution (3), at $(x,t)$ where
$\sqrt{x^{2}+t^{2}}=O\left(|a_{2m+1}|^{1/(2m+1)}\right)$, by denoting

$\lambda=a_{2m+1}^{-1/(2m+1)}$ (36)

and recalling the expression of Schur polynomials in Eq. (9), we have

$\displaystyle
S_{k}(\textbf{\emph{x}}^{+}(n)+\nu\textbf{\emph{s}})=S_{k}\left(x_{1}^{+},\nu
s_{2},x_{3}^{+},\nu
s_{4},\cdots\right)=\lambda^{-k}S_{k}\left(x_{1}^{+}\lambda,\nu
s_{2}\lambda^{2},x_{3}^{+}\lambda^{3},\nu s_{4}\lambda^{4},\cdots\right)$
$\displaystyle\sim\lambda^{-k}S_{k}\left[(x+\textrm{i}t)\lambda,0,\cdots,0,1,0,\cdots\right]=S_{k}\left(x+\textrm{i}t,0,\cdots,0,a_{2m+1},0,\cdots\right).$
(37)

Thus,

$S_{k}(\textbf{\emph{x}}^{+}(n)+\nu\textbf{\emph{s}})\sim
S_{k}(\textbf{v}),\quad\quad|a_{2m+1}|\gg 1,$ (38)

where

$\textbf{v}=(x+\textrm{i}t,0,\cdots,0,a_{2m+1},0,\cdots).$ (39)

From the definition of Schur polynomials (8), $S_{k}(\textbf{v})$ is given by

$\sum_{k=0}^{\infty}S_{k}(\textbf{v})\epsilon^{k}=\exp\left[(x+\textrm{i}t)\epsilon+a_{2m+1}\epsilon^{2m+1}\right].$
(40)

Thus, it is related to the polynomial $p_{k}^{[m]}(z)$ in (20) as

$S_{k}(\textbf{v})=A^{k/(2m+1)}p_{k}^{[m]}(z),$ (41)

where

$A=-\frac{2m+1}{2^{2m}}a_{2m+1},\hskip
8.5359ptz=A^{-1/(2m+1)}(x+\textrm{i}t).$ (42)

Using these formulae, we find that

$\det_{1\leq i,j\leq
N}\left[S_{2i-j}(\textbf{\emph{x}}^{+}(n)+j\textbf{\emph{s}})\right]\sim
c_{N}^{-1}A^{\frac{N(N+1)}{2(2m+1)}}Q_{N}^{[m]}(z),\quad\quad|a_{2m+1}|\gg 1.$
(43)

Similarly,

$\det_{1\leq i,j\leq
N}\left[S_{2i-j}(\textbf{\emph{x}}^{-}(n)+j\textbf{\emph{s}})\right]\sim
c_{N}^{-1}\left(A^{*}\right)^{\frac{N(N+1)}{2(2m+1)}}Q_{N}^{[m]}(z^{*}),\quad\quad|a_{2m+1}|\gg
1.$ (44)

Hereafter, $S_{k}=0$ when $k<0$.

To proceed further, we use determinant identities and the Laplace expansion to
rewrite $\sigma_{n}$ in Eq. (4) as OhtaJY2012

$\displaystyle\sigma_{n}=\sum_{0\leq\nu_{1}<\nu_{2}<\cdots<\nu_{N}\leq
2N-1}\det_{1\leq i,j\leq
N}\left[\frac{1}{2^{\nu_{j}}}S_{2i-1-\nu_{j}}(\textbf{\emph{x}}^{+}(n)+\nu_{j}\textbf{\emph{s}})\right]\times\det_{1\leq
i,j\leq
N}\left[\frac{1}{2^{\nu_{j}}}S_{2i-1-\nu_{j}}(\textbf{\emph{x}}^{-}(n)+\nu_{j}\textbf{\emph{s}})\right].$
(45)

Since the highest order term of $a_{2m+1}$ in this $\sigma_{n}$ comes from the
index choices of $\nu_{j}=j-1$, then

$\sigma_{n}\sim|\alpha|^{2}\hskip
1.42271pt|a_{2m+1}|^{\frac{N(N+1)}{2m+1}}\left|Q_{N}^{[m]}(z)\right|^{2},\quad\quad|a_{2m+1}|\gg
1,$ (46)

where

$\alpha=2^{-N(N-1)/2}c_{N}^{-1}\left(-\frac{2m+1}{2^{2m}}\right)^{\frac{N(N+1)}{2(2m+1)}}.$
(47)

Since $\alpha$ is independent of $n$, the above equation shows that for large
$a_{2m+1}$, $\sigma_{1}/\sigma_{0}\sim 1$, i.e., the solution $u(x,t)$ is on
the unit background, except at or near $(x,t)$ locations
$\left(\hat{x}_{0},\hat{t}_{0}\right)$ where

$z_{0}=A^{-1/(2m+1)}(\hat{x}_{0}+\textrm{i}\hat{t}_{0})$ (48)

is a root of the polynomial $Q_{N}^{[m]}(z)$, and such
$\left(\hat{x}_{0},\hat{t}_{0}\right)$ locations are given by Eq. (32) in view
of Eq. (42).

Next, we show that when $(x,t)$ is in the neighborhood of each of the
$\left(\hat{x}_{0},\hat{t}_{0}\right)$ locations given by Eq. (32), i.e., when
$\left[(x-\hat{x}_{0})^{2}+(t-\hat{t}_{0})^{2}\right]^{1/2}=O(1)$, the rogue
wave $u_{N}(x,t)$ in Eq. (3) approaches a Peregrine wave
$\hat{u}_{1}(x-\hat{x}_{0},t-\hat{t}_{0})\hskip 1.42271pte^{\textrm{i}t}$ for
large $a_{2m+1}$. The asymptotic analysis above indicates that when $(x,t)$ is
in the neighborhood of $\left(\hat{x}_{0},\hat{t}_{0}\right)$, the highest
power term $|a_{2m+1}|^{\frac{N(N+1)}{2m+1}}$ in $\sigma(x,t)$ vanishes. Thus,
in order to determine the asymptotics of $u_{N}(x,t)$ in that $(x,t)$ region,
we need to derive the leading order term of $a_{2m+1}$ in Eq. (45) whose order
is lower than $|a_{2m+1}|^{\frac{N(N+1)}{2m+1}}$. For this purpose, we notice
from Eq. (5) that when $(x,t)$ is in the neighborhood of
$\left(\hat{x}_{0},\hat{t}_{0}\right)$, we have a more refined asymptotics for
$S_{k}(\textbf{\emph{x}}^{+}(n)+\nu\textbf{\emph{s}})$ as

$\displaystyle
S_{k}(\textbf{\emph{x}}^{+}(n)+\nu\textbf{\emph{s}})=\lambda^{-k}S_{k}\left(x_{1}^{+}\lambda,0,\cdots,0,1,0,\cdots\right)\left[1+O(\lambda^{2})\right]$
$\displaystyle\hskip
59.75095pt=S_{k}\left(x_{1}^{+},0,\cdots,0,a_{2m+1},0,\cdots\right)\left[1+O(\lambda^{2})\right],$
(49)

i.e.,

$S_{k}(\textbf{\emph{x}}^{+}(n)+\nu\textbf{\emph{s}})=S_{k}(\hat{\textbf{v}})\left[1+O\left(a_{2m+1}^{-2/(2m+1)}\right)\right],$
(50)

where

$\hat{\textbf{v}}=(x+\textrm{i}t+n,0,\cdots,0,a_{2m+1},0,\cdots).$ (51)

The polynomials $S_{k}(\hat{\textbf{v}})$ are related to $p_{k}^{[m]}(z)$ in
(20) as

$S_{k}(\hat{\textbf{v}})=A^{k/(2m+1)}p_{k}^{[m]}(\hat{z}),$ (52)

where $A$ is as given in Eq. (42), and
$\hat{z}=A^{-1/(2m+1)}(x+\textrm{i}t+n)$.

Now, we derive the leading order term of $a_{2m+1}$ in Eq. (45). This leading
order term comes from two index choices, one being $\nu=(0,1,\cdots,N-1)$, and
the other being $\nu=(0,1,\cdots,N-2,N)$.

With the first index choice, in view of Eqs. (50) and (52), the determinant
involving $\textbf{\emph{x}}^{+}(n)$ in Eq. (45) is

$\alpha\hskip
1.70709pta_{2m+1}^{\frac{N(N+1)}{2(2m+1)}}Q_{N}^{[m]}(\hat{z})\left[1+O\left(a_{2m+1}^{-2/(2m+1)}\right)\right],$
(53)

where $\alpha$ is given in Eq. (47). Expanding $Q_{N}^{[m]}(\hat{z})$ around
$\hat{z}=z_{0}$, where $z_{0}$ is given in Eq. (48), and recalling
$Q_{N}^{[m]}(z_{0})=0$, we have

$Q_{N}^{[m]}(\hat{z})=A^{-1/(2m+1)}\left[(x-\hat{x}_{0})+\textrm{i}(t-\hat{t}_{0})+n\right]\left[Q_{N}^{[m]}\right]^{\prime}(z_{0})\left[1+O\left(A^{-1/(2m+1)}\right)\right].$
(54)

Inserting this equation into (53) and recalling the definition of $A$ in (42),
the determinant involving $\textbf{\emph{x}}^{+}(n)$ in Eq. (45) becomes

$\left[(x-\hat{x}_{0})+\textrm{i}(t-\hat{t}_{0})+n\right]\hskip
1.70709pt\hat{\alpha}\hskip
1.70709pta_{2m+1}^{\frac{N(N+1)-2}{2(2m+1)}}\left[Q_{N}^{[m]}\right]^{\prime}(z_{0})\left[1+O\left(a_{2m+1}^{-1/(2m+1)}\right)\right],$
(55)

where $\hat{\alpha}=\alpha\hskip 1.13791pt[-(2m+1)2^{-2m}]^{-1/(2m+1)}$.
Similarly, the determinant involving $\textbf{\emph{x}}^{-}(n)$ in Eq. (45)
becomes

$\left[(x-\hat{x}_{0})-\textrm{i}(t-\hat{t}_{0})-n\right]\hskip
1.70709pt\hat{\alpha}^{*}\hskip
1.70709pt(a_{2m+1}^{*})^{\frac{N(N+1)-2}{2(2m+1)}}\left[Q_{N}^{[m]}\right]^{\prime}(z_{0}^{*})\left[1+O\left(a_{2m+1}^{-1/(2m+1)}\right)\right].$
(56)

Next, we consider the contribution in Eq. (45) from the second index choice of
$\nu=(0,1,\cdots,N-2,N)$. For this index choice, the determinant involving
$\textbf{\emph{x}}^{+}(n)$ in Eq. (45) is

$\det_{1\leq i\leq
N}\left[S_{2i-1}(\textbf{\emph{x}}^{+}),\frac{1}{2}S_{2i-2}(\textbf{\emph{x}}^{+}+\textbf{\emph{s}}),\cdots,\frac{1}{2^{N-2}}S_{2i-(N-1)}[\textbf{\emph{x}}^{+}+(N-2)\hskip
1.70709pt\textbf{\emph{s}}],\frac{1}{2^{N}}S_{2i-(N+1)}(\textbf{\emph{x}}^{+}+N\hskip
1.13791pt\textbf{\emph{s}})\right].$ (57)

Utilizing Eqs. (50)-(52), this determinant is

$2^{-N(N-1)/2-1}A^{\frac{N(N+1)-2}{2(2m+1)}}\det_{1\leq i\leq
N}\left[p^{[m]}_{2i-1}(\hat{z}),p^{[m]}_{2i-2}(\hat{z}),\cdots,p^{[m]}_{2i-(N-1)}(\hat{z}),p^{[m]}_{2i-(N+1)}(\hat{z})\right]\left[1+O\left(a_{2m+1}^{-2/(2m+1)}\right)\right].$
(58)

Recalling Eq. (26), we see that
$p^{[m]}_{2i-(N+1)}(\hat{z})=[p^{[m]}_{2i-N}]^{\prime}(\hat{z})$. Thus, the
determinant in the above expression is equal to
$c_{N}^{-1}\left[Q_{N}^{[m]}\right]^{\prime}(\hat{z})$, so that the
determinant (57) becomes

$\frac{1}{2}\hat{\alpha}\hskip
1.70709pta_{2m+1}^{\frac{N(N+1)-2}{2(2m+1)}}\left[Q_{N}^{[m]}\right]^{\prime}(\hat{z})\left[1+O\left(a_{2m+1}^{-2/(2m+1)}\right)\right].$
(59)

When $(x,t)$ is in the neighborhood of $(\hat{x}_{0},\hat{t}_{0})$, we expand
$\left[Q_{N}^{[m]}\right]^{\prime}(\hat{z})$ around $\hat{z}=z_{0}$ to reduce
this expression further to

$\frac{1}{2}\hat{\alpha}\hskip
1.70709pta_{2m+1}^{\frac{N(N+1)-2}{2(2m+1)}}\left[Q_{N}^{[m]}\right]^{\prime}(z_{0})\left[1+O\left(a_{2m+1}^{-1/(2m+1)}\right)\right].$
(60)

Similarly, the determinant involving $\textbf{\emph{x}}^{-}(n)$ in Eq. (45)
becomes

$\frac{1}{2}\hat{\alpha}^{*}\hskip
1.70709pt(a_{2m+1}^{*})^{\frac{N(N+1)-2}{2(2m+1)}}\left[Q_{N}^{[m]}\right]^{\prime}(z_{0}^{*})\left[1+O\left(a_{2m+1}^{-1/(2m+1)}\right)\right].$
(61)

Summarizing the above two contributions, we find that

$\sigma_{n}(x,t)=|\hat{\alpha}|^{2}\hskip
1.70709pt\left|\left[Q_{N}^{[m]}\right]^{\prime}(z_{0})\right|^{2}|a_{2m+1}|^{\frac{N(N+1)-2}{(2m+1)}}\left[\left(x-\hat{x}_{0}\right)^{2}+\left(t-\hat{t}_{0}\right)^{2}-(2\textrm{i})n\left(t-\hat{t}_{0}\right)-n^{2}+\frac{1}{4}\right]\left[1+O\left(a_{2m+1}^{-1/(2m+1)}\right)\right].$
(62)

Finally, we recall that nonzero roots are simple in Yablonskii-Vorob’ev
polynomials $Q_{N}(z)$ Fukutani . In addition, nonzero roots have also been
conjectured to be simple in all the Yablonskii-Vorob’ev hierarchy
$Q_{N}^{[m]}(z)$ Clarkson2003-II . Assuming this conjecture is true, then
$\left[Q_{N}^{[m]}\right]^{\prime}(z_{0})\neq 0$. This indicates that the
above leading-order asymptotics for $\sigma_{n}(x,t)$ never vanishes.
Therefore, when $a_{2m+1}$ is large and $(x,t)$ in the neighborhood of
$\left(\hat{x}_{0},\hat{t}_{0}\right)$, we get from (62) that

$u_{N}(x,t)=\frac{\sigma_{1}}{\sigma_{0}}e^{\textrm{i}t}=e^{\textrm{i}t}\left(1-\frac{4[1+2\textrm{i}(t-\hat{t}_{0})]}{1+4(x-\hat{x}_{0})^{2}+4(t-\hat{t}_{0})^{2}}\right)+O\left(a_{2m+1}^{-1/(2m+1)}\right),$
(63)

which is a Peregrine wave $\hat{u}_{1}(x-\hat{x}_{0},t-\hat{t}_{0})\hskip
1.42271pte^{\textrm{i}t}$, and the error of this Peregrine prediction is
$O\left(a_{2m+1}^{-1/(2m+1)}\right)$. Theorem 3 is then proved.

Proof of Theorem 4. To analyze the large-$a_{2m+1}$ behavior of the rogue wave
$u_{N}(x,t)$ in the neighborhood of the origin, where
$\sqrt{x^{2}+t^{2}}=O(1)$, we first rewrite the $\sigma_{n}$ determinant (4)
into a $3N\times 3N$ determinant OhtaJY2012

$\sigma_{n}=\left|\begin{array}[]{cc}\textbf{O}_{N\times N}&\Phi_{N\times
2N}\\\ -\Psi_{2N\times N}&\textbf{I}_{2N\times 2N}\end{array}\right|,$ (64)

where
$\Phi_{i,j}=2^{-(j-1)}S_{2i-j}\left[\textbf{\emph{x}}^{+}(n)+(j-1)\textbf{\emph{s}}\right]$,
and
$\Psi_{i,j}=2^{-(i-1)}S_{2j-i}\left[\textbf{\emph{x}}^{-}(n)+(i-1)\textbf{\emph{s}}\right]$.
Defining $\textbf{\emph{y}}^{\pm}$ to be the vector $\textbf{\emph{x}}^{\pm}$
without the $a_{2m+1}$ term, i.e., let

$\textbf{\emph{x}}^{+}=\textbf{\emph{y}}^{+}+(0,\cdots,0,a_{2m+1},0,\cdots),\quad\textbf{\emph{x}}^{-}=\textbf{\emph{y}}^{-}+(0,\cdots,0,a_{2m+1}^{*},0,\cdots),$
(65)

we find that the Schur polynomials of $\textbf{\emph{x}}^{\pm}$ are related to
those of $\textbf{\emph{y}}^{\pm}$ as

$S_{j}(\textbf{\emph{x}}^{+}+\nu\textbf{\emph{s}})=\sum_{i=0}^{\left[\frac{j}{2m+1}\right]}\frac{a_{2m+1}^{i}}{i!}S_{j-(2m+1)i}(\textbf{\emph{y}}^{+}+\nu\textbf{\emph{s}}),\quad
S_{j}(\textbf{\emph{x}}^{-}+\nu\textbf{\emph{s}})=\sum_{i=0}^{\left[\frac{j}{2m+1}\right]}\frac{(a_{2m+1}^{*})^{i}}{i!}S_{j-(2m+1)i}(\textbf{\emph{y}}^{-}+\nu\textbf{\emph{s}}),$
(66)

where $[a]$ represents the largest integer less than or equal to $a$. Using
this relation, we express matrix elements of $\Phi$ and $\Psi$ in Eq. (64)
through Schur polynomials
$S_{k}(\textbf{\emph{y}}^{\pm}+\nu\textbf{\emph{s}})$ and powers of $a_{2m+1}$
and $a_{2m+1}^{*}$.

We need to determine the highest power term of $a_{2m+1}$ in the determinant
(64). For that purpose, it may be tempting to retain only the highest power
term of $a_{2m+1}$ and $a_{2m+1}^{*}$ in each element of this determinant.
That does not work though because it would result in multiple rows (and
columns) which are proportional to each other, making the reduced determinant
zero. The correct way is to first judiciously remove certain leading power
terms of $a_{2m+1}$ and $a_{2m+1}^{*}$ from elements of the determinant
through row and column manipulations, so that the remaining determinant, after
retaining only the highest power term of $a_{2m+1}$ and $a_{2m+1}^{*}$ in each
element, would be nonzero. These row and column manipulations are described
below.

Suppose $N\equiv N_{0}\hskip 2.84544pt\mbox{mod}\hskip 2.84544pt(2m+1)$, i.e.,
$N=k(2m+1)+N_{0}$ for some positive integer $k$, with $0\leq N_{0}\leq m$. We
perform the following series of row operations to the matrix $\Phi$ so that
certain high-power terms of $a_{2m+1}$ in its lower rows are eliminated. In
the first round, we use the 1st to $m$-th rows of $\Phi$ to eliminate the
highest-power term $a_{2m+1}^{2\nu}$ from the $[\nu(2m+1)+1]$-th up to the
$[\nu(2m+1)+m]$-th rows for each $1\leq\nu\leq k$, so that the remaining terms
in those rows have the highest power $a_{2m+1}^{2\nu-1}$. We also use the
$(m+1)$-th to $(2m+1)$-th rows of $\Phi$ to eliminate the highest-power term
$a_{2m+1}^{2\nu+1}$ from the $[\nu(2m+1)+m+1]$-th to the $[\nu(2m+1)+2m+1]$-th
rows for each $1\leq\nu\leq k-1$, with the remaining terms in those rows
having the highest power $a_{2m+1}^{2\nu}$. In each step, the highest power
terms $a_{2m+1}^{2\nu}$ or $a_{2m+1}^{2\nu+1}$ of each row are eliminated
simultaneously, because the coefficient vector of those highest power terms in
each row below the $(2m+1)$-th is proportional to the coefficient vector of
the highest power terms in the corresponding upper row between the 1st and
$(2m+1)$-th due to the relation (66).

In the second round, we use the $(2m+1+1)$-th to $(2m+1+m)$-th rows of the
remaining matrix $\Phi$ to eliminate the highest-power term
$a_{2m+1}^{2\nu+1}$ from the $[(\nu+1)(2m+1)+1]$-th up to the
$[(\nu+1)(2m+1)+m]$-th rows for each $1\leq\nu\leq k-1$, so that the remaining
terms in those rows have the highest power $a_{2m+1}^{2\nu}$. We also use the
$(2m+1+m+1)$-th to $(2m+1+2m+1)$-th rows of $\Phi$ to eliminate the highest-
power term $a_{2m+1}^{2\nu+2}$ from the $[(\nu+1)(2m+1)+m+1]$-th up to the
$[(\nu+1)(2m+1)+2m+1]$-th rows for each $1\leq\nu\leq k-2$, with the remaining
terms in those rows having the highest power $a_{2m+1}^{2\nu+1}$. This process
is repeated $k$ rounds.

At the end of this process, the $i$-th row of the remaining matrix $\Phi$ has
the highest power $a_{2m+1}^{[(i+m)/(2m+1)]}$. Then, we keep only the highest
power terms of $a_{2m+1}$ in each row. Similar column operations are also
performed on the matrix $\Psi$. With these manipulations, we find that
$\sigma_{n}$ is asymptotically reduced to

$\sigma_{n}=\beta\hskip
1.70709pt|a_{2m+1}|^{k^{2}(2m+1)+k(2N_{0}+1)}\left|\begin{array}[]{cc}\textbf{O}_{N\times
N}&\widetilde{\Phi}_{N\times 2N}\\\ -\widetilde{\Psi}_{2N\times
N}&\textbf{I}_{2N\times
2N}\end{array}\right|\left[1+O\left(a_{2m+1}^{-1}\right)\right],$ (67)

where $\beta$ is a $(m,N)$-dependent nonzero constant, matrices
$\widetilde{\Phi}_{N\times 2N}$ and $\widetilde{\Psi}_{2N\times N}$ have the
structures

$\widetilde{\Phi}_{N\times
2N}=\left(\begin{array}[]{ccc}{\textbf{L}}_{(N-N_{0})\times(N-N_{0})}&\textbf{O}_{(N-N_{0})\times
2N_{0}}&\textbf{O}_{(N-N_{0})\times(N-N_{0})}\\\
{\textbf{M}}_{N_{0}\times(N-N_{0})}&\widehat{\Phi}_{N_{0}\times
2N_{0}}&\textbf{O}_{N_{0}\times(N-N_{0})}\end{array}\right),$ (68)
$\widetilde{\Psi}_{2N\times
N}=\left(\begin{array}[]{cc}{\textbf{U}}_{(N-N_{0})\times(N-N_{0})}&\widehat{{\textbf{M}}}_{(N-N_{0})\times
N_{0}}\\\ \textbf{O}_{2N_{0}\times(N-N_{0})}&\widehat{\Psi}_{2N_{0}\times
N_{0}}\\\ \textbf{O}_{(N-N_{0})\times(N-N_{0})}&\textbf{O}_{(N-N_{0})\times
N_{0}}\end{array}\right),$ (69)
${\textbf{L}}_{i,j}=S_{i-j}\left[\textbf{\emph{y}}^{+}+(j-1)\textbf{\emph{s}}\right],\quad{\textbf{U}}_{i,j}=S_{j-i}\left[\textbf{\emph{y}}^{-}+(i-1)\textbf{\emph{s}}\right],$
(70)
$\displaystyle\widehat{\Phi}_{i,j}=2^{-(j-1)}S_{2i-j}\left[\textbf{\emph{y}}^{+}(n)+(j-1+\nu_{0})\textbf{\emph{s}}\right],\quad\widehat{\Psi}_{i,j}=2^{-(i-1)}S_{2j-i}\left[\textbf{\emph{y}}^{-}(n)+(i-1+\nu_{0})\textbf{\emph{s}}\right],$
(71)

$\nu_{0}=k(2m+1)$, and M, $\widehat{{\textbf{M}}}$ are matrices of elements
$S_{j}(\textbf{\emph{y}}^{+}+\nu\textbf{\emph{s}})$ and
$S_{j}(\textbf{\emph{y}}^{-}+\nu\textbf{\emph{s}})$ respectively. Since L and
U are respectively lower triangular and upper triangular matrices with unit
elements on the diagonal in view that $S_{0}=1$ and $S_{j}=0$ for $j<0$,
$\sigma_{n}$ in Eq. (67) then is

$\sigma_{n}=\beta\hskip
1.70709pt|a_{2m+1}|^{k^{2}(2m+1)+k(2N_{0}+1)}\left|\begin{array}[]{cc}\textbf{O}_{N_{0}\times
N_{0}}&\widehat{\Phi}_{N_{0}\times 2N_{0}}\\\ -\widehat{\Psi}_{2N_{0}\times
N_{0}}&\textbf{I}_{2N_{0}\times
2N_{0}}\end{array}\right|\left[1+O\left(a_{2m+1}^{-1}\right)\right].$ (72)

Finally, we notice that
$S_{j}\left[\textbf{\emph{y}}^{\pm}+(\nu+\nu_{0})\textbf{\emph{s}}\right]$ is
related to $S_{j}\left(\textbf{\emph{y}}^{\pm}+\nu\textbf{\emph{s}}\right)$
through

$S_{j}\left[\textbf{\emph{y}}^{\pm}+(\nu+\nu_{0})\textbf{\emph{s}}\right]=\sum_{i=0}^{\left[j/2\right]}S_{2i}(\nu_{0}\textbf{\emph{s}})S_{j-2i}(\textbf{\emph{y}}^{\pm}+\nu\textbf{\emph{s}}).$
(73)

Using this relation, the determinant in (72) can be reduced to one where
$\nu_{0}$ is set to zero in the above $\widehat{\Phi}$ and $\widehat{\Psi}$
matrices given in Eq. (71). Such a determinant for $\sigma_{n}$ gives a
$N_{0}$-th order rogue wave, whose internal parameters
$(a_{3},a_{5},\cdots,a_{2N_{0}-1})$ are the first $N_{0}-1$ values in the
original parameter set $(a_{3},a_{5},\cdots,a_{2N-1})$. Thus, in the
neighborhood of the origin ,

$u_{N}(x,t;a_{3},a_{5},\cdots,a_{2N-1})=\frac{\sigma_{1}}{\sigma_{0}}e^{\textrm{i}t}=u_{N_{0}}(x,t;a_{3},a_{5},\cdots,a_{2N_{0}-1})\left[1+O\left(a_{2m+1}^{-1}\right)\right],\quad|a_{2m+1}|\gg
1,$ (74)

which means that the original $N$-th order rogue wave $u_{N}(x,t)$ is
approximated by a lower $N_{0}$-th order rogue wave $u_{N_{0}}(x,t)$, with the
approximation error $O\left(a_{2m+1}^{-1}\right)$.

If $N\equiv-N_{0}-1\hskip 2.84544pt\mbox{mod}\hskip 2.84544pt(2m+1)$ with
$0\leq N_{0}\leq m$, Eq. (74) can also be derived by similar analysis, and
thus the same conclusion holds.

Lastly, we recall that $1\leq m\leq N-1$. In addition, $0\leq N_{0}\leq m$ in
view of Theorem 2. Furthermore, when $m=N-1$, we find from Eq. (29) of Theorem
2 that $N_{0}=N-2$. As a consequence, $0\leq N_{0}\leq N-2$. Theorem 4 is then
proved.

##  6 Rogue wave patterns when multiple internal parameters are large

The rogue patterns in Theorems 3 and 4 were derived under the assumption that
only one of the internal parameters in the rogue wave solutions (3) was large,
and the other parameters were $O(1)$. It turns out that those results can be
generalized to more general parameter conditions. We discuss these
generalizations in this section.

Regarding the generalization of Theorem 3, we can show that if $a_{2m+1}$ is
large, and the other parameters $a_{3}$, $\dots$, $a_{2m-1}$, $a_{2m+3}$,
$\dots$, $a_{2N-1}$ are also large but satisfy the conditions

$a_{2j+1}=o\left(a_{2m+1}^{\frac{2j}{2m+1}}\right),\quad j\neq m,$ (75)

then, far away from the origin, with
$\sqrt{x^{2}+t^{2}}=O\left(|a_{2m+1}|^{1/(2m+1)}\right)$, the rogue wave
$u_{N}(x,t)$ still separates into $N_{p}$ Peregrine waves, whose positions
$(\hat{x}_{0},\hat{t}_{0})$ are given by Eq. (32). Expressed mathematically,
when $\left[(x-\hat{x}_{0})^{2}+(t-\hat{t}_{0})^{2}\right]^{1/2}=O(1)$, we
have

$u_{N}(x,t;a_{3},a_{5},\cdots,a_{2N-1})\longrightarrow\hat{u}_{1}(x-\hat{x}_{0},t-\hat{t}_{0})\hskip
1.42271pte^{\textrm{i}t}\hskip 11.38092pt\mbox{as}\hskip
4.26773pt|a_{2m+1}|\to\infty.$ (76)

The reason is that, when
$\sqrt{x^{2}+t^{2}}=O\left(|a_{2m+1}|^{1/(2m+1)}\right)$, under the same
notation $\lambda=a_{2m+1}^{-1/(2m+1)}$ as in Eq. (36), the condition (75)
means that $a_{2j+1}=o(\lambda^{-2j})$ for $j\neq m$. Thus,
$x_{2j+1}^{+}\lambda^{2j+1}=o(\lambda)$ for $j\neq m$. Then, in view of Eq.
(9), we have

$\displaystyle
S_{k}(\textbf{\emph{x}}^{+}(n)+\nu\textbf{\emph{s}})=S_{k}\left(x_{1}^{+},\nu
s_{2},x_{3}^{+},\nu
s_{4},\cdots\right)=\lambda^{-k}S_{k}\left(x_{1}^{+}\lambda,\nu
s_{2}\lambda^{2},x_{3}^{+}\lambda^{3},\nu s_{4}\lambda^{4},\cdots\right)$
$\displaystyle=\lambda^{-k}S_{k}\left(x_{1}^{+}\lambda,0,\cdots,0,1,0,\cdots\right)\left[1+o(\lambda)\right]=S_{k}\left(x_{1}^{+},0,\cdots,0,a_{2m+1},0,\cdots\right)\left[1+o(\lambda)\right].$
(77)

This relation is the counterpart of Eq. (5) in the proof of Theorem 3. Due to
this relation and a similar one on
$S_{k}(\textbf{\emph{x}}^{-}(n)+\nu\textbf{\emph{s}})$, the calculations in
the proof of Theorem 3 can still go through. The only difference is that the
error of the present Peregrine approximation may be different. Indeed, the
previous analysis, combined with the above equation (6), indicates that the
error of the current Peregrine approximation (76) is the largest order among
$O\left(a_{2j+1}/a_{2m+1}^{2j/(2m+1)}\right)$, where $1\leq j\leq N-1$ and
$j\neq m$. So, if $a_{2j+1}=O\left(a_{2m+1}^{(2j-1)/(2m+1)}\right)$ or smaller
for all $j\neq m$, then the error of the current Peregrine approximation (76)
would remain the same as that given in Eq. (33) of Theorem 3, i.e.,
$O\left(|a_{2m+1}|^{-1/(2m+1)}\right)$. Otherwise, this error would be larger
than $O\left(|a_{2m+1}|^{-1/(2m+1)}\right)$, which means that the error would
decay to zero slower than the rate $|a_{2m+1}|^{-1/(2m+1)}$ when $a_{2m+1}$
gets large.

Regarding the generalization of Theorem 4, we can show that if $a_{2m+1}$ is
large, and

$a_{3},\cdots,a_{2m-1}=O(1),\quad a_{2m+3},\cdots,a_{2N-1}=O(a_{2m+1}),$ (78)

then Theorem 4 remains valid. Specifically, the asymptotics (34), including
its error estimates, still holds. The proof for this is an extension of the
proof for Theorem 4, and will be presented in Appendix C.

To demonstrate these generalized results on rogue patterns, we consider an
example of a 7th order rogue wave $u_{7}(x,t)$ with parameter choices of

$a_{3}=1,\quad a_{5}\ \mbox{is large},\quad a_{7}=a_{5},\quad
a_{9}=2a_{5},\quad a_{11}=3a_{5},\quad a_{13}=4a_{5}.$ (79)

This set of parameters satisfy both conditions (75) and (78). Thus, according
to the above discussions, both Theorems 3 and 4 remain valid, including their
error estimates, since $a_{2j+1}=O\left(a_{2m+1}^{(2j-1)/(2m+1)}\right)$ or
smaller for all $j\neq m$ here. These theorems predict that far away from the
origin, this $u_{7}(x,t)$ would split into 25 Peregrine waves, whose $(x,t)$
locations are given by Eq. (32). Near the origin, this $u_{7}(x,t)$ would
reduce to a 2nd-order rogue wave $u_{2}(x,t)$ with $a_{3}=1$. To verify these
predictions, we choose $a_{5}=-200\textrm{i}$. The corresponding true rogue
wave solution $|u_{7}(x,t)|$ is plotted in Fig. 7(a), and its center region is
amplified and shown in panel (b). Our asymptotic predictions (35) from
Theorems 3 and 4 for the same $(x,t)$ intervals as in panels (a) and (b) are
displayed in panels (c) and (d) respectively. One can clearly see that our
predictions are almost indistinguishable from the true solutions.

Figure 7: A 7th order rogue wave $|u_{7}(x,t)|$ for generalized parameters
(79) with $a_{5}=-200\textrm{i}$, and its comparison with the analytical
prediction. (a) True solution, with $-20.5\leq x,t\leq 20.5$; (b) zoomed-in
plot of the center region of the true solution marked by a dashed-line box in
panel (a); (c) predicted solution with the same $(x,t)$ internals as in (a);
(d) zoomed-in plot of the center region of the predicted solution.

##  7 Conclusions and discussions

In this paper, we have analytically studied rogue wave patterns in the NLS
equation. We have shown that when one of the internal parameters in the
bilinear rogue wave solutions is large, these waves would exhibit clear
geometric structures, which comprise Peregrine rogue waves organized in shapes
such as triangle, pentagon, heptagon and nonagon, with a possible lower-order
rogue wave at its center. These rogue patterns are analytically determined by
the root structures of the Yablonskii-Vorob’ev polynomial hierarchy, and their
orientations are controlled by the phase of the large parameter. We have also
generalized these results and shown that, when multiple internal parameters in
the rogue waves are large but satisfy certain constraints [such as (75) and
(78)], then the same rogue patterns would persist. Comparison between true
rogue patterns and our analytical predictions has shown excellent agreement.
As a small application of our analytical results, the numerical observation in
KAAN2011  on single-shell ring structures has been explained. Our results
reveal the deep connection between NLS rogue wave patterns and the Yablonskii-
Vorob’ev polynomial hierarchy, and make prediction of sophisticated patterns
in higher-order NLS rogue waves possible.

It turns out that this connection between rogue wave patterns and the
Yablonskii-Vorob’ev polynomial hierarchy is not restricted to the NLS
equation. We have found that such connections persist in many other integrable
equations, such as the derivative NLS equation, the Boussinesq equation, the
Manakov equations and others. This general connection then gives rise to
universal rogue wave patterns in integrable systems. This university result
was briefly reported in YangYanguniv . Its details are beyond the scope of
this paper and will be pursued in future publications.

In this article, NLS rogue wave patterns are determined by the complex roots
of the Yablonskii-Vorob’ev polynomial hierarchy, and these roots are the pole
locations of rational solutions to the $\mbox{P}_{\mbox{\scriptsize II}}$
hierarchy (see Sec. 2.2 and Clarkson2003-II ; Bertola2016 ). Interestingly, in
very different contexts, somewhat similar results have also been reported. For
instance, in the semiclassical NLS equation after wave breaking, a sequence of
Peregrine waves appear, and their locations are determined by the poles of the
tritronquée solution to the first Painlevé ($\mbox{P}_{\mbox{\scriptsize I}}$)
equation Tovbis . In the semiclassical sine-Gordon equation with initial
conditions near the separatrix of a simple pendulum, superluminal (infinite
velocity) kinks that appear in the solution are linked to the _real_ roots of
the Yablonskii-Vorob’ev polynomials associated with rational solutions of the
$\mbox{P}_{\mbox{\scriptsize II}}$ equation Miller_sine . This connection of
wave phenomena to rational solutions of the Painlevé equations may arise again
in other wave systems in the future.

## Acknowledgment

This material is based upon work supported by the National Science Foundation
under award number DMS-1910282, and the Air Force Office of Scientific
Research under award number FA9550-18-1-0098.

## Appendix A

In this appendix, we briefly derive the bilinear rogue waves presented in
Theorem 1. These new rogue wave expressions can be obtained by applying a new
parameterization developed in Ref. YangDNLS2019  to the bilinear derivation of
rogue waves in Ref. OhtaJY2012 . Specifically, instead of the previous choice
(3.11) for the matrix element $m_{ij}^{(n)}$ in Ref. OhtaJY2012 , which we
denote as $\phi_{ij}^{(n)}$ in this paper, we now choose

$\phi_{ij}^{(n)}=\left.\frac{1}{i!}(p\partial_{p})^{i}\frac{1}{j!}(q\partial_{q})^{j}\phi^{(n)}\right|_{p=q=1},$
(80)

where

$\phi^{(n)}=\frac{(p+1)(q+1)}{2(p+q)}\left(-\frac{p}{q}\right)^{n}\exp\left(\xi+\eta+\sum_{k=1}^{\infty}a_{k}(\ln
p)^{k}+\sum_{k=1}^{\infty}b_{k}(\ln q)^{k}\right),$ (81)
$\xi=px_{1}+p^{2}x_{2},\quad\eta=qx_{1}-q^{2}x_{2}.$ (82)

and $a_{k},b_{k}$ are arbitrary complex constants. Obviously, the function
$\tau_{n}=\det_{1\leq i,j\leq N}\left(\phi_{2i-1,2j-1}^{(n)}\right)$ with the
above choice of $\phi_{ij}^{(n)}$ also satisfies the bilinear equations (3.14)
in OhtaJY2012 . Then, when we set $b_{k}=a_{k}^{*}$, $x_{1}=x$ and
$x_{2}=\textrm{i}t/2$, this $\tau_{n}$ function would satisfy the bilinear
equations (3.1) of the NLS equation in OhtaJY2012  [with $t$ switched to
$-t/2$ since the NLS equation (1) in this paper differs from that in
OhtaJY2012  by this $t$ rescaling]. Applying the same reduction technique of
OhtaJY2012  to the above new $\tau_{n}$ solution, we can remove the
differential operators in the expression (80) of its matrix element
$\phi_{ij}^{(n)}$ and reduce it to $\sigma_{n}=\det_{1\leq i,j\leq
N}\left(\phi_{2i-1,2j-1}^{(n)}\right)$, where

$\phi_{i,j}^{(n)}=\sum_{\nu=0}^{\min(i,j)}\frac{1}{4^{\nu}}\hskip
1.70709ptS_{i-\nu}(\hat{\textbf{\emph{x}}}^{+}(n)+\nu\textbf{\emph{s}})\hskip
1.70709ptS_{j-\nu}(\hat{\textbf{\emph{x}}}^{-}(n)+\nu\textbf{\emph{s}}),$ (83)

vectors
$\hat{\textbf{\emph{x}}}^{\pm}(n)=\left(x_{1}^{\pm},x_{2}^{\pm},\cdots\right)$
are defined by

$\displaystyle x_{1}^{+}=x+\textrm{i}t+n+a_{1},\quad
x_{1}^{-}=x-\textrm{i}t-n+a_{1}^{*},\quad
x_{k}^{+}=\frac{x+2^{k-1}(\textrm{i}t)}{k!}+a_{k},\quad
x_{k}^{-}=\frac{x-2^{k-1}(\textrm{i}t)}{k!}+a_{k}^{*},\quad k\geq 2,$ (84)

and $\textbf{\emph{s}}=(s_{1},s_{2},\cdots)$ are coefficients from the
expansion (7). Through a shift of the $x$ and $t$ axes, we normalize $a_{1}=0$
without loss of generality. Finally, we split the vectors
$\hat{\textbf{\emph{x}}}^{\pm}(n)$ into
$\textbf{\emph{x}}^{\pm}(n)+\textbf{\emph{w}}^{\pm}$, where
$\textbf{\emph{x}}^{\pm}(n)$ is as given in Eq. (6), and
$\textbf{\emph{w}}^{\pm}=(0,x_{2}^{\pm},0,x_{4}^{\pm},\cdots)$. Since
$\hat{\textbf{\emph{x}}}^{\pm}(n)+\nu\textbf{\emph{s}}=\textbf{\emph{x}}^{\pm}(n)+\nu\textbf{\emph{s}}+\textbf{\emph{w}}^{\pm}$,
it is easy to show from the definition of Schur polynomials (8) that

$S_{k}(\hat{\textbf{\emph{x}}}^{\pm}(n)+\nu\textbf{\emph{s}})=\sum_{j=0}^{\left[k/2\right]}S_{j}(\hat{\textbf{\emph{w}}}^{\pm})S_{k-2j}(\textbf{\emph{x}}^{\pm}(n)+\nu\textbf{\emph{s}}),$
(85)

where $\hat{\textbf{\emph{w}}}^{\pm}=(x_{2}^{\pm},x_{4}^{\pm},\cdots)$.
Rewriting the $\sigma_{n}$ solution $\det_{1\leq i,j\leq
N}\left(\phi_{2i-1,2j-1}^{(n)}\right)$ as a $3N\times 3N$ determinant (64) and
utilizing the above relation, we can apply row and column manipulations to
eliminate all terms involving $\hat{\textbf{\emph{w}}}^{\pm}$ in this
$3N\times 3N$ determinant. The remaining $3N\times 3N$ determinant then
becomes $\det_{1\leq i,j\leq N}\left(\phi_{2i-1,2j-1}^{(n)}\right)$, whose
matrix element $\phi_{ij}^{(n)}$ is as given in Theorem 1.

## Appendix B

In this appendix, we prove Theorem 2. First, we derive the multiplicity of
root zero in $Q_{N}^{[m]}(z)$. For this purpose, we define the Schur
polynomial $S^{[m]}_{k}(z;a)$ as

$\sum_{k=0}^{\infty}S^{[m]}_{k}(z;a)\epsilon^{k}=\exp\left[z\epsilon+a\hskip
1.13791pt\epsilon^{2m+1}\right],$ (86)

where $a$ is a constant. Through these Schur polynomials $S^{[m]}_{k}(z;a)$,
we define polynomials

$\displaystyle
P^{[m]}_{N}(z;a)=c_{N}\left|\begin{array}[]{cccc}S^{[m]}_{1}(z;a)&S^{[m]}_{0}(z;a)&\cdots&S^{[m]}_{2-N}(z;a)\\\
S^{[m]}_{3}(z;a)&S^{[m]}_{2}(z;a)&\cdots&S^{[m]}_{4-N}(z;a)\\\
\vdots&\vdots&\vdots&\vdots\\\
S^{[m]}_{2N-1}(z;a)&S^{[m]}_{2N-2}(z;a)&\cdots&S^{[m]}_{N}(z;a)\end{array}\right|,$
(91)

where $S^{[m]}_{k}(z;a)\equiv 0$ when $k<0$. It is easy to see that
$S^{[m]}_{k}(z;a)$ is related to the polynomial $p_{k}^{[m]}(z)$ in Eq. (20)
as

$S^{[m]}_{k}(z;a)=\hat{a}^{k/(2m+1)}p_{k}^{[m]}(\hat{z}),\quad\hat{z}\equiv\hat{a}^{-1/(2m+1)}z,\quad\hat{a}\equiv-\hskip
1.13791pt(2m+1)\hskip 1.13791pt2^{-2m}\hskip 0.85355pta.$ (92)

Thus, the polynomial $P^{[m]}_{N}(z;a)$ is related to the Yablonskii-Vorob’ev
polynomial hierarchy $Q_{N}^{[m]}(z)$ in Eq. (25) as

$P^{[m]}_{N}(z;a)=\hat{a}^{\frac{N(N+1)}{2(2m+1)}}Q_{N}^{[m]}(\hat{z}).$ (93)

This equation tells us that every term in the polynomial $P^{[m]}_{N}(z;a)$ is
a constant multiple of $z^{i}a^{j}$, where $i+(2m+1)j=N(N+1)/2$. Thus, to
determine the multiplicity of the zero root $z=0$ in $Q_{N}^{[m]}(z)$, we need
to determine the highest power term of $a$ in $P^{[m]}_{N}(z;a)$. To do so, we
utilize the relation

$S^{[m]}_{j}(z;a)=\sum_{i=0}^{\left[\frac{j}{2m+1}\right]}\frac{a^{i}}{i![j-i(2m+1)]!}z^{j-i(2m+1)},$
(94)

which can be derived by splitting the right side of Eq. (86) into a product of
two exponentials and then expanding both exponentials into Taylor series of
$\epsilon$. Using this relation, we express the matrix elements in the
determinant (91) through powers of $z$ and $a$. Then, we need to obtain the
highest power term of $a$ in the resulting determinant. This problem resembles
the derivation of the highest power term of $a_{2m+1}$ in the $\sigma_{n}$
determinant (64) during the proof of Theorem 4, where a polynomial relation
(66) similar to the above (94) was used. In this resemblance, the matrix
$P^{[m]}_{N}(z;a)$ here is the counterpart of the $\Phi_{N\times N}$ matrix in
Eq. (64), $a$ here is the counterpart of $a_{2m+1}$ in Eq. (66), and $z^{j}$
in the above equation (94) is the counterpart of
$S_{j}(\textbf{\emph{y}}^{+}+\nu\textbf{\emph{s}})$ in Eq. (66). Performing
the same row operations as described in Theorem 4 to remove certain leading
$a$-power terms in the lower rows of the determinant (91), we can show that
the highest-power term of $a$ in $P^{[m]}_{N}(z;a)$ is

$\displaystyle\rho_{0}\
a^{\frac{N^{2}+N-N_{0}^{2}-N_{0}}{2(2m+1)}}\left|\begin{array}[]{cccc}z&1&\cdots&0\\\
\frac{z^{3}}{3!}&\frac{z^{2}}{2!}&\cdots&0\\\ \vdots&\vdots&\vdots&\vdots\\\
\frac{z^{2N_{0}-1}}{(2N_{0}-1)!}&\frac{z^{2N_{0}-2}}{(2N_{0}-2)!}&\ldots&\frac{z^{N_{0}}}{N_{0}!}\end{array}\right|=\hat{\rho}_{0}\
a^{\frac{(N-N_{0})(N+N_{0}+1)}{2(2m+1)}}z^{N_{0}(N_{0}+1)/2},$ (99)

where $N_{0}$ is as given in Theorem 2, and $\rho_{0},\hat{\rho}_{0}$ are
$(m,N)$-dependent nonzero constants. This shows that the lowest power of $z$
in all terms of $P^{[m]}_{N}(z;a)$ is $N_{0}(N_{0}+1)/2$. Then, setting
$a=-2^{2m}/(2m+1)$ where $P^{[m]}_{N}(z;a)$ becomes $Q_{N}^{[m]}(z)$, the
multiplicity of the zero root in $Q_{N}^{[m]}(z)$ is $N_{0}(N_{0}+1)/2$.

To prove the form (27) of the polynomial $Q_{N}^{[m]}(z)$, we notice from the
definition (20) of the polynomial $p_{k}^{[m]}(z)$ that this polynomial admits
the symmetry

$p_{k}^{[m]}(z)=\omega^{-k}p_{k}^{[m]}(\omega z),$ (100)

where $\omega$ is any one of the $(2m+1)$-th root of 1, i.e.,
$\omega^{2m+1}=1$. This symmetry of $p_{k}^{[m]}(z)$ leads to the symmetry of
$Q_{N}^{[m]}(z)$ as

$Q_{N}^{[m]}(z)=\omega^{-N(N+1)/2}Q_{N}^{[m]}(\omega z).$ (101)

Since the multiplicity of the zero root in $Q_{N}^{[m]}(z)$ is
$N_{0}(N_{0}+1)/2$, let us write

$Q_{N}^{[m]}(z)=z^{N_{0}(N_{0}+1)/2}q_{N}^{[m]}(z),$ (102)

where $q_{N}^{[m]}(z)$ is a polynomial of $z$ with a nonzero constant term.
The symmetry (101) of the polynomial $Q_{N}^{[m]}(z)$ induces a symmetry for
$q_{N}^{[m]}(z)$ as

$q_{N}^{[m]}(z)=\omega^{(N_{0}^{2}+N_{0}-N^{2}-N)/2}q_{N}^{[m]}(\omega z).$
(103)

Since $N_{0}^{2}+N_{0}-N^{2}-N=(N_{0}-N)(N_{0}+N+1)$, and in view of the
$N_{0}$ value given in Theorem 2, we see that $(N_{0}^{2}+N_{0}-N^{2}-N)/2$ is
divisible by $2m+1$, which means $\omega^{(N_{0}^{2}+N_{0}-N^{2}-N)/2}=1$.
Thus, the above equation reduces to

$q_{N}^{[m]}(z)=q_{N}^{[m]}(\omega z).$ (104)

This symmetry of $q_{N}^{[m]}(z)$ dictates that $q_{N}^{[m]}(z)$ can only be a
polynomial of $z^{2m+1}$. Hence the form (27) of the polynomial
$Q_{N}^{[m]}(z)$ is proved.

Lastly, we derive the degree of the polynomial $Q_{N}^{[m]}(z)$ from its
definition (25). Notice from Eq. (20) that the highest-degree term of
$p^{[m]}_{k}(z)$ is $z^{k}/k!$. Retaining only this highest-degree term of
$p^{[m]}_{k}(z)$ in the determinant (25) for $Q_{N}^{[m]}(z)$ and evaluating
the simplified determinant by the same technique as that used in Ref.
OhtaJY2012 , we can readily show that the degree of the polynomial
$Q_{N}^{[m]}(z)$ is $N(N+1)/2$. Thus, Theorem 2 is proved.

## Appendix C

In this appendix, we prove the generalization of Theorem 4 presented in Sec. 6
when $a_{2m+1}$ is large and the other parameters satisfy the conditions (78).
In this parameter regime, let us denote

$a_{2m+3}=\beta_{1}\hskip 1.13791pta_{2m+1},\hskip
5.69046pta_{2m+5}=\beta_{2}\hskip 1.13791pta_{2m+1},\hskip
5.69046pt\cdots,\hskip 5.69046pta_{2N-1}=\beta_{N-m-1}\hskip
1.13791pta_{2m+1},$ (105)

where $\beta_{1},\beta_{2},\cdots,\beta_{N-m-1}$ are $O(1)$ constants. We
first split the vectors $\textbf{\emph{x}}^{\pm}$ as

$\textbf{\emph{x}}^{+}=\textbf{\emph{y}}^{+}+\textbf{\emph{a}},\quad\textbf{\emph{x}}^{-}=\textbf{\emph{y}}^{-}+\textbf{\emph{a}}^{*},$
(106)

where $\textbf{\emph{a}}=(0,\cdots,0,a_{2m+1},0,a_{2m+3},0,\cdots
0,a_{2N-1})$. Then, the Schur polynomials of $\textbf{\emph{x}}^{\pm}$ are
related to those of $\textbf{\emph{y}}^{\pm}$ as

$S_{j}(\textbf{\emph{x}}^{+}+\nu\textbf{\emph{s}})=\sum_{i=0}^{j}S_{i}(\textbf{\emph{a}})S_{j-i}(\textbf{\emph{y}}^{+}+\nu\textbf{\emph{s}}),\quad
S_{j}(\textbf{\emph{x}}^{-}+\nu\textbf{\emph{s}})=\sum_{i=0}^{j}S_{i}^{*}(\textbf{\emph{a}})S_{j-i}(\textbf{\emph{y}}^{-}+\nu\textbf{\emph{s}}).$
(107)

In view of the definition of a and the notations in (105), we readily see from
the definition of Schur polynomials that $S_{i}(\textbf{\emph{a}})$ is a
polynomial of $a_{2m+1}$ with coefficients dependent on
$(\beta_{1},\beta_{2},\cdots)$, and its highest degree in $a_{2m+1}$ is
$[i/(2m+1)]$, i.e., the largest integer less than or equal to $i/(2m+1)$.
Then, after a little manipulation and rearranging terms in the above
equations, we get

$S_{j}(\textbf{\emph{x}}^{+}+\nu\textbf{\emph{s}})=\sum_{i=0}^{\left[\frac{j}{2m+1}\right]}a_{2m+1}^{i}\sum_{k=0}^{\left[\frac{j-(2m+1)i}{2}\right]}c_{i,k}^{+}(m,\mbox{\boldmath$\beta$})\
S_{j-(2m+1)i-2k}(\textbf{\emph{y}}^{+}+\nu\textbf{\emph{s}})$ (108)

and

$S_{j}(\textbf{\emph{x}}^{-}+\nu\textbf{\emph{s}})=\sum_{i=0}^{\left[\frac{j}{2m+1}\right]}(a_{2m+1}^{*})^{i}\sum_{k=0}^{\left[\frac{j-(2m+1)i}{2}\right]}c_{i,k}^{-}(m,\mbox{\boldmath$\beta$})\
S_{j-(2m+1)i-2k}(\textbf{\emph{y}}^{-}+\nu\textbf{\emph{s}}),$ (109)

where the coefficients $c_{i,k}^{\pm}$ are dependent on $m$ and the vector
$\mbox{\boldmath$\beta$}=(\beta_{1},\beta_{2},\cdots)$, and
$c_{i,0}^{\pm}(m,\mbox{\boldmath$\beta$})=1/i!$.

These two Schur polynomial relations (108)-(109) are the counterparts of those
in Eq. (66) during the proof of Theorem 4. Using these relations, we can
perform similar row and column operations to the $3N\times 3N$ determinant in
Eq. (64) to eliminate certain high order powers of $a_{2m+1}$ and
$a_{2m+1}^{*}$. The main difference is that, a little more such eliminations
are required here, because to eliminate a certain power of $a_{2m+1}$ or
$a_{2m+1}^{*}$ in $S_{j}(\textbf{\emph{x}}^{\pm}+\nu\textbf{\emph{s}})$, one
needs to eliminate a linear combination of polynomials
$S_{j-(2m+1)i-2k}(\textbf{\emph{y}}^{\pm}+\nu\textbf{\emph{s}})$ now in view
of the above two Schur polynomial relations. However, these eliminations
follow a clear and regular pattern, so that they can always be achieved.
Another small difference is that here, the row and column operations will
produce some additional lower power terms of $a_{2m+1}$ and $a_{2m+1}^{*}$.
But those lower-power terms will eventually be discarded since we will retain
only the highest $a_{2m+1}$ and $a_{2m+1}^{*}$ power terms in each row and
column respectively. Therefore, these similar row and column operations will
still asymptotically reduce $\sigma_{n}$ to the same determinant (67) as
before, and hence the generalization of Theorem 4 stated in Sec. 6 can be
proved.

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