# Rogue Waves in (2+1)-Dimensional Three-Wave Resonant Interactions

Bo Yang1, Jianke Yang2 1 School of Mathematics and Statistics, Ningbo
University, Ningbo 315211, China
2 Department of Mathematics and Statistics, University of Vermont, Burlington,
VT 05401, U.S.A

###### Abstract

Rogue waves in (2+1)-dimensional three-wave resonant interactions are studied.
General rogue waves arising from a constant background, from a lump-soliton
background and from a dark-soliton background have been derived, and their
dynamics illustrated. For rogue waves arising from a constant background,
fundamental rogue waves are line-shaped, and multi-rogue waves exhibit
multiple intersecting lines. Higher-order rogue waves could also be line-
shaped, but they exhibit multiple parallel lines. For rogue waves arising from
a lump-soliton background, they could exhibit distinctive patterns due to
their interaction with the lump soliton. For rogue waves arising from a dark-
soliton background, their intensity pattern could feature half-line shapes or
lump shapes, which are very novel.

##  I Introduction

Three-wave interaction arises in a wide variety of physical systems, such as
water waves, nonlinear optics, plasma physics, and others Bloembergen1965 ;
Benney_Newell ; Kaupreview1979 ; Ablowitz_book ; 3wave_wateropt1 ;
3wave_wateropt2 ; 3wave_wateropt3 ; 3wave_wateropt4 ; 3wave_wateropt5 . When
the wavenumbers and frequencies of the three waves form a resonant triad, this
interaction is the strongest. In this case, the governing equations for this
interaction are integrable Zakharov1973 ; Ablowitz_Haberman1975 ; Zakharov1975
; Zakharov1976 ; Kaup1976 ; Kaup1980 ; Kaup1981 ; Kaup1981b . As a
consequence, multi-solitons in one spatial dimension and multi-lumps in two
spatial dimensions of this system have been derived Zakharov1975 ;
Zakharov1976 ; Kaup1976 ; Craik1978 ; Ablowitz_book ; Zakharov_book ;
Kaup1981b ; Gilson1998 ; Yang_Stud .

Rogue waves are an interesting class of solutions in nonlinear wave systems
that “come from nowhere and disappear with no trace” Akhmediev_2009 . These
waves have been linked to freak waves in the ocean and extreme events in
optics Ocean_rogue_review ; Pelinovsky_book ; Solli_Nature ; Wabnitz_book . In
addition, rogue waves have been observed in water tanks, optical fibers and
plasma Tank1 ; Tank2 ; Tank3 ; Tank4 ; Plasma ; Fiber1 ; Fiber2 ; Fiber3 . Due
to their mathematical novelty and physical connections, rogue waves have
attracted a lot of studies, especially in the integrable systems community,
because analytical expressions of rogue waves can be explicitly obtained when
the nonlinear wave system is integrable. So far, rogue waves have been derived
in a large number of integrable systems, such as the nonlinear Schrödinger
(NLS) equation Peregrine ; AAS2009 ; DGKM2010 ; ACA2010 ; KAAN2011 ; GLML2012
; OhtaJY2012 ; DPMVB2013 , the derivative NLS equations KN_rogue_2011 ;
KN_rogue_2013 ; CCL_rogue_Chow_Grimshaw2014 ; CCL_rogue_2017 ; YangYangDNLS ,
the Manakov system BDCW2012 ; ManakovDark ; LingGuoZhaoCNLS2014 ;
Chen_Shihua2015 ; ZhaoGuoLingCNLS2016 ; YangYang2021b , the Davey-Stewartson
equations OhtaJKY2012 ; OhtaJKY2013 , and many others AANJM2010 ; OhtaJKY2014
; ASAN2010 ; Chow ; MuQin2016 ; ClarksonDowie2017 ; YangYangBoussi ;
JCChen2018LS ; XiaoeYong2018 . In addition, rogue waves that arise from a non-
uniform background have also been reported in several wave systems Qin2015 ;
Peli2019 ; He2021 . These explicit solutions of rogue waves shed much light on
the rogue dynamics in physical systems that are governed by the underlying
integrable equations.

Rogue waves in (1+1)-dimensional three-wave resonant interactions have been
derived and analyzed as well BaroDegas2013 ; DegasLomba2013 ; ChenSCrespo2015
; WangXChenY2015 ; ZhangYanWen2018 ; YangYang3wave . Compared to rogue waves
in many other (1+1)-dimensional equations, rogue waves in this three-wave
system exhibited a much wider variety of solutions and wave patterns. This
makes us wonder, what kind of rogue wave behaviors will arise in the
(2+1)-dimensional three-wave system, i.e., the three-wave system with two
spatial dimensions instead of one.

In this article, we study general rogue wave solutions in the
(2+1)-dimensional three-wave system by the bilinear method. Rogue waves
arising from both a constant background and a non-constant background are
derived, and our solutions are expressed through Schur polynomials. We find
that for rogue waves arising from a constant background, fundamental rogue
waves are line-shaped, and multi-rogue waves display multiple intersecting
lines. Higher-order rogue waves could also be line-shaped, but would exhibit
multiple parallel lines instead of intersecting lines. For rogue waves arising
from a non-constant background, we report two different types of solutions.
One type is rogue waves that emerge from a lump-soliton background, and these
waves are rational solutions. The other type is rogue waves that emerge from a
dark-soliton background, and these waves are semi-rational solutions which
contain both rational and exponential components. We find that for the latter
type of rogue waves, their intensity can exhibit half-line shapes or lump
shapes, which are fascinating.

This paper is structured as follows. In Sec. 2, we introduce the
(2+1)-dimensional three-wave system and their constant-background solutions.
In Sec. 3, we present our rogue wave solutions that arise from a constant
background. In Sec. 4, we illustrate dynamics of these rogue waves emerging
from a constant background. In Sec. 5, we present our rogue wave solutions
that arise from a non-constant background, including from a lump-soliton
background and a dark-soliton background, and illustrate their dynamics. In
Sec. 6, we provide derivations of rogue waves presented in Secs. 3 and 5. Sec.
7 summarizes the paper.

##  II Preliminaries

The (2+1)-dimensional three-wave resonant interaction system is

$\displaystyle\left(\partial_{t}+\mathbf{V}_{1}\cdot\nabla\right)q_{1}(x,y,t)=\epsilon_{1}q_{2}^{*}(x,y,t)q_{3}^{*}(x,y,t),$
$\displaystyle\left(\partial_{t}+\mathbf{V}_{2}\cdot\nabla\right)q_{2}(x,y,t)=\epsilon_{2}q_{1}^{*}(x,y,t)q_{3}^{*}(x,y,t),$
(1)
$\displaystyle\left(\partial_{t}+\mathbf{V}_{3}\cdot\nabla\right)q_{3}(x,y,t)=\epsilon_{3}q_{1}^{*}(x,y,t)q_{2}^{*}(x,y,t).$

Here, $\nabla=\left(\partial_{x},\ \partial_{y}\right)$ is the gradient
operator in the $\mathbb{R}^{2}$ space,
$\mathbf{V}_{k}=\left(V_{k,1},V_{k,2}\right)$ are the three waves’ group
velocities along the $(x,y)$ directions, and the asterisk ‘*’ represents
complex conjugation. Adopting a coordinate system that moves at the velocity
of the third wave, we make $\mathbf{V}_{3}=(0,0)$ without loss of generality.
In addition, we assume that $\mathbf{V}_{1}$ and $\mathbf{V}_{2}$ are not
parallel to each other, i.e., $V_{11}V_{22}-V_{12}V_{21}\neq 0$. Parameters
$\epsilon_{j}$ are nonlinear coefficients that can be scaled to $\pm 1$. In
addition, one can fix $\epsilon_{1}=1$.

The above three-wave system admits plane-wave solutions

$\displaystyle
q_{1,0}(x,y,t)=\rho_{1}e^{\textrm{i}(k_{1}x+\lambda_{1}y+\omega_{1}t)},$
$\displaystyle
q_{2,0}(x,y,t)=\rho_{2}e^{\textrm{i}(k_{2}x+\lambda_{2}y+\omega_{2}t)},$ (2)
$\displaystyle
q_{3,0}(x,y,t)=\textrm{i}\rho_{3}e^{\textrm{i}(k_{3}x+\lambda_{3}y+\omega_{3}t)},$

where wave vectors $(k_{j},\lambda_{j})$ and frequencies $\omega_{j}$ satisfy
the resonance relations

$k_{1}+k_{2}+k_{3}=0,\quad\lambda_{1}+\lambda_{2}+\lambda_{3}=0,\quad\omega_{1}+\omega_{2}+\omega_{3}=0,$
(3)

and wave-amplitude parameters $(\rho_{1},\rho_{2},\rho_{3})$ satisfy the
following conditions

$\displaystyle\rho_{1}\left(\omega_{1}+V_{1,1}k_{1}+V_{1,2}\lambda_{1}\right)=-\epsilon_{1}\rho_{2}^{*}\rho_{3}^{*},$
$\displaystyle\rho_{2}\left(\omega_{2}+V_{2,1}k_{2}+V_{2,2}\lambda_{2}\right)=-\epsilon_{2}\rho_{1}^{*}\rho_{3}^{*},$
(4)
$\displaystyle\rho_{3}\left(\omega_{1}+\omega_{2}\right)=\epsilon_{3}\rho_{1}^{*}\rho_{2}^{*}.$

These plane waves have constant amplitudes in the $(x,y)$ space; so we will
call them constant-background solutions.

In this article, we assume that the three wave amplitudes
$|\rho_{1}|,|\rho_{2}|$ and $|\rho_{3}|$ are all nonzero. Then, using phase
invariance of the three-wave system, we can normalize $\rho_{1}$, $\rho_{2}$
and $\rho_{3}$ to be all real. Thus, in the later text, we assume
$(\rho_{1},\rho_{2},\rho_{3})$ real. In addition, we define three real
constants

$\gamma_{1}\equiv\epsilon_{1}\frac{\rho_{2}\rho_{3}}{\rho_{1}},\
\gamma_{2}\equiv\epsilon_{2}\frac{\rho_{1}\rho_{3}}{\rho_{2}},\
\gamma_{3}\equiv\epsilon_{3}\frac{\rho_{1}\rho_{2}}{\rho_{3}}.$ (5)

Rogue waves in the three-wave system (1) are solutions that arise from the
above constant background, or from non-constant solutions such as lumps or
dark solitons that sit on the above constant background. Such solutions will
be derived in later sections. Our solutions will be presented through
elementary Schur polynomials $S_{j}(\mbox{\boldmath$x$})$, which are defined
by

$\sum_{j=0}^{\infty}S_{j}(\mbox{\boldmath$x$})\epsilon^{j}=\exp\left(\sum_{j=1}^{\infty}x_{j}\epsilon^{j}\right),$

where $\mbox{\boldmath$x$}=(x_{1},x_{2},\cdots)$.

##  III Rogue waves that arise from a constant background

Rogue waves that arise from the constant background (2) are rational
solutions. Our general rational solutions to the three-wave system (1) are
given in two different but equivalent forms by the following two theorems.

> Theorem 1. The (2+1)-dimensional three-wave system (1) admits rational
> solutions
>
> $\displaystyle
> q_{1,N}(x,y,t)=\rho_{1}\frac{g_{1,N}}{f_{N}}e^{i(k_{1}x+\lambda_{1}y+\omega_{1}t)},$
> $\displaystyle
> q_{2,N}(x,y,t)=\rho_{2}\frac{g_{2,N}}{f_{N}}e^{i(k_{2}x+\lambda_{2}y+\omega_{2}t)},$
> (6) $\displaystyle
> q_{3,N}(x,y,t)=\textrm{i}\rho_{3}\frac{g_{3,N}}{f_{N}}e^{\textrm{i}(k_{3}x+\lambda_{3}y+\omega_{3}t)},$
>
> where $N$ is an arbitrary positive integer,
>
> $f_{N}=\tau_{0,0},\quad g_{1,N}=\tau_{1,0},\quad g_{2,N}=\tau_{0,-1},\quad
> g_{3,N}=\tau_{-1,1},$ (7) $\tau_{n,k}=\det_{\begin{subarray}{l}1\leq i,j\leq
> N\end{subarray}}\left(\begin{array}[]{c}m_{i,j}^{(n,k)}\end{array}\right),$
> (8)
>
> the matrix elements in $\tau_{n,k}$ are defined by
>
> $\displaystyle
> m_{i,j}^{(n,k)}=\frac{\left(p\partial_{p}\right)^{n_{i}}}{(n_{i})!}\frac{\left(q\partial_{q}\right)^{n_{j}}}{(n_{j})!}\left.\left[\frac{1}{p+q}\left(-\frac{p}{q}\right)^{k}\left(-\frac{p-\textrm{i}}{q+\textrm{i}}\right)^{n}e^{\Theta_{i,j}(x,y,t)}\right]\
> \right|_{p=p_{i},\ q=p_{j}^{*}},$ (9)
>
> $n_{i}$ and $n_{j}$ are arbitrary non-negative integers,
>
>
> $\displaystyle\Theta_{i,j}(x,y,t)=\left(\frac{1}{p}+\frac{1}{q}\right)z_{1}+\left(\frac{1}{p-\textrm{i}}+\frac{1}{q+\textrm{i}}\right)z_{2}+(p+q)z_{3}+\sum_{r=1}^{\infty}a_{r,i}\ln^{r}\left[\frac{p}{p_{i}}\right]+\sum_{r=1}^{\infty}a^{*}_{r,j}\ln^{r}\left[\frac{q}{p_{j}^{*}}\right],$
> (10)
>
> variables $(z_{1},z_{2},z_{3})$ are related to $(x,y,t)$ as
>
> $\displaystyle
> z_{1}=\gamma_{1}\frac{V_{22}x-V_{21}y}{V_{11}V_{22}-V_{12}V_{21}},$ (11)
> $\displaystyle
> z_{2}=\gamma_{2}\frac{V_{11}y-V_{12}x}{V_{11}V_{22}-V_{12}V_{11}},$ (12)
> $\displaystyle
> z_{3}=\gamma_{3}\left[\frac{\left(V_{12}-V_{22}\right)x+\left(V_{21}-V_{11}\right)y}{V_{11}V_{22}-V_{12}V_{21}}+t\right],$
> (13)
>
> $p_{i}$ are free non-imaginary complex constants, and $a_{r,i}\hskip
> 1.42271pt(r=1,2,\dots)$ are free complex constants.

One may notice from Eq. (9) that these solutions involve exponential factors
$e^{\Theta_{i,j}}$. But those exponential factors will cancel out when
$g_{i,N}$ is divided by $f_{N}$ in Eq. (6), and the resulting ratios are thus
only rational functions of $(x,y,t)$.

The solutions in the above theorem are given through differential operators
[see (9)]. More explicit expressions of these solutions are presented in the
following theorem.

> Theorem 2. The rational solutions in Theorem 1 are given by Eqs. (6)-(8),
> where the matrix elements $m_{i,j}^{(n,k)}$ of $\tau_{n,k}$ can be rewritten
> as
>
>
> $m_{i,j}^{(n,k)}=\sum_{\nu=0}^{\min(n_{i},n_{j})}\left(\frac{1}{p_{i}+p_{j}^{*}}\right)\left[\frac{p_{i}p_{j}^{*}}{(p_{i}+p_{j}^{*})^{2}}\right]^{\nu}\hskip
> 1.70709ptS_{n_{i}-\nu}[\textbf{\emph{x}}^{+}_{i,j}(n,k)+\nu\textbf{\emph{s}}_{i,j}]\hskip
> 1.70709ptS_{n_{j}-\nu}[\textbf{\emph{x}}^{-}_{j,i}(n,k)+\nu\textbf{\emph{s}}^{*}_{j,i}],$
> (14)
>
> vectors
> $\textbf{\emph{x}}^{\pm}_{i,j}(n,k)=\left(x_{1,i,j}^{\pm},x_{2,i,j}^{\pm},\cdots\right)$
> are
>
> $\displaystyle
> x_{r,i,j}^{+}(n,k)=\alpha_{r,i}z_{1}+\beta_{r,i}z_{2}+\frac{1}{r!}p_{i}z_{3}+ng_{r,i}+kh_{r}-c_{r,i,j}+a_{r,i},$
> (15) $\displaystyle
> x_{r,i,j}^{-}(n,k)=\alpha_{r,i}^{*}z_{1}+\beta_{r,i}^{*}z_{2}+\frac{1}{r!}p_{i}^{*}z_{3}-ng_{r,i}^{*}-kh_{r}^{*}-c_{r,i,j}^{*}+a_{r,i}^{*},$
> (16)
>
> $\textbf{\emph{s}}_{i,j}=(s_{1,i,j},s_{2,i,j},\cdots)$, coefficients
> $\alpha_{r,i}$, $\beta_{r,i}$, $g_{r,i}$, $h_{r}$, $c_{r,i,j}$ and
> $s_{r,i,j}$ are obtained from the expansions
>
>
> $\displaystyle\frac{1}{p_{i}e^{\kappa}}-\frac{1}{p_{i}}=\sum_{r=1}^{\infty}\alpha_{r,i}\kappa^{r},\quad\frac{1}{p_{i}e^{\kappa}-\rm{i}}-\frac{1}{p_{i}-\rm{i}}=\sum_{r=1}^{\infty}\beta_{r,i}\kappa^{r},$
> (17)
> $\displaystyle\ln\left(\frac{p_{i}e^{\kappa}-\rm{i}}{p_{i}-\rm{i}}\right)=\sum_{r=1}^{\infty}g_{r,i}\kappa^{r},\quad\kappa=\sum_{r=1}^{\infty}h_{r}\kappa^{r},$
> (18)
> $\displaystyle\ln\left[\frac{p_{i}e^{\kappa}+p_{j}^{*}}{p_{i}+p_{j}^{*}}\right]=\sum_{r=1}^{\infty}c_{r,i,j}\kappa^{r},\
> \ \
> \ln\left[\frac{p_{i}+p_{j}^{*}}{\kappa}\left(\frac{e^{\kappa}-1}{p_{i}e^{\kappa}+p_{j}^{*}}\right)\right]=\sum_{r=1}^{\infty}s_{r,i,j}\kappa^{r},$
> (19)
>
> $p_{i}$ are free non-imaginary complex constants, and $a_{r,i}\hskip
> 1.42271pt(r=1,2,\dots)$ are free complex constants.

The above rational solutions contain rogue waves that arise from the constant
background (2), as well as algebraic localized (lump) solitons moving on this
constant background and a mixture between these two types of solutions (see
later text). To get rogue waves that arise from the constant background, we
need to impose conditions on $p_{i}$. To derive these conditions, we consider
the fundamental rational solution, where we take $N=1$ and $n_{1}=1$ in
Theorem 2. In addition, we normalize $a_{1,1}-c_{1,1,1}=0$ by a coordinate
shift. Performing simple calculations, we can reduce the $\tau_{n,k}$ function
to

$\tau_{n,k}=m_{1,1}^{(n,k)}=\xi\bar{\xi}+\Delta,$ (20)

where

$\displaystyle\xi=ax+by+ct+\theta(n,k),\quad\bar{\xi}=a^{*}x+b^{*}y+c^{*}t-\theta^{*}(n,k),\quad\Delta=\frac{|p_{1}|^{2}}{(p_{1}+p_{1}^{*})^{2}},$
(21)

and $a$, $b$, $c$, $\theta$ are complex coefficients given by

$\displaystyle
a=\frac{1}{V_{11}V_{22}-V_{12}V_{21}}\left[-\frac{1}{p_{1}}\gamma_{1}V_{22}+\frac{p_{1}}{(p_{1}-\textrm{i})^{2}}\gamma_{2}V_{12}+p_{1}\gamma_{3}(V_{12}-V_{22})\right],$
(22) $\displaystyle
b=\frac{1}{V_{11}V_{22}-V_{12}V_{21}}\left[\frac{1}{p_{1}}\gamma_{1}V_{21}-\frac{p_{1}}{(p_{1}-\textrm{i})^{2}}\gamma_{2}V_{11}+p_{1}\gamma_{3}(V_{21}-V_{11})\right],$
(23) $\displaystyle c=\gamma_{3}p_{1},\ \ \
\theta(n,k)=k+\frac{np_{1}}{p_{1}-\textrm{i}}.$ (24)

To derive parameter conditions for rogue waves, we separate the real and
imaginary parts of the above complex variables as

$p_{1}=p_{1,1}+\textrm{i}p_{1,2},\ a=a_{1}+\textrm{i}a_{2},\
b=b_{1}+\textrm{i}b_{2},\ c=c_{1}+\textrm{i}c_{2},\
\theta=\theta_{1}+\textrm{i}\theta_{2}.$ (25)

Then, the $\tau_{n,k}$ solution (20) can be rewritten as

$\tau_{n,k}=\xi_{1}^{2}+\xi_{2}^{2}-2\textrm{i}\theta_{1}\xi_{2}+2\textrm{i}\theta_{2}\xi_{1}-\theta_{1}^{2}-\theta_{2}^{2}+\Delta,$
(26)

where

$\xi_{1}=a_{1}x+b_{1}y+c_{1}t,\quad\xi_{2}=a_{2}x+b_{2}y+c_{2}t.$ (27)

We can readily show that under the velocity assumption of
$V_{11}V_{22}-V_{12}V_{21}\neq 0$, the three ratios of
$a_{1}/a_{2},b_{1}/b_{2}$ and $c_{1}/c_{2}$ cannot be all the same. Then, the
above fundamental rational solution exhibits two distinctively different
dynamical behaviours depending on the ratio relations between $a_{1}/a_{2}$
and $b_{1}/b_{2}$.

  1. 1.

If $\frac{a_{1}}{a_{2}}\neq\frac{b_{1}}{b_{2}}$, then along the
$\left[x(t),y(t)\right]$ trajectory where

$a_{1}x+b_{1}y=-c_{1}t,\ \ \ a_{2}x+b_{2}y=-c_{2}t,$ (28)

the above $\tau_{n,k}$ is a constant. The corresponding solution
$(q_{1},q_{2},q_{3})$ is an algebraic localized lump soliton moving on the
constant background (2).

  2. 2.

If $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\neq\frac{c_{1}}{c_{2}}$, then the
$\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}$ equation yields the following
parameter condition

$\displaystyle\frac{(p_{1,2}-1)(p_{1,1}^{2}+p_{1,2}^{2})^{2}}{\gamma_{1}}-\frac{p_{1,2}[p_{1,1}^{2}+(p_{1,2}-1)^{2}]^{2}}{\gamma_{2}}-\frac{p_{1,1}^{2}+p_{1,2}^{2}-p_{1,2}}{\gamma_{3}}=0.$
(29)

In this case, the corresponding solution $(q_{1},q_{2},q_{3})$ uniformly
approaches the constant background (2) in the entire $(x,y)$ plane when
$t\to\pm\infty$. In the intermediate times, it rises to a higher amplitude.
Since $a_{1}/a_{2}=b_{1}/b_{2}$, this solution depends on $(x,y)$ through the
combination of $a_{1}x+b_{1}y$. Thus, this is a line rogue wave.

For the other (i.e., non-fundamental) rational solutions in Theorems 1-2, in
order for them to be rogue waves that arise from the constant background (2),
we need to require every parameter $p_{i}$ $(1\leq i\leq N)$ to satisfy the
above condition (29), where $p_{1,1}$ is replaced by $p_{i,1}$ and $p_{1,2}$
replaced by $p_{i,2}$, with $(p_{i,1},p_{i,2})$ being the real and imaginary
parts of the complex parameter $p_{i}$.

The condition (29) can be further simplified. When $(p_{1,1},p_{1,2})$ are
replaced by the more general $(p_{i,1},p_{i,2})$, the simplified condition can
be expressed as a quartic equation for $p_{i,1}$ as

$\chi_{0}p_{i,1}^{4}+\chi_{1}p_{i,1}^{2}+\chi_{2}=0,$ (30)

where the coefficients are

$\displaystyle\chi_{0}=(p_{i,2}-1)\gamma_{1}^{-1}-p_{i,2}\gamma_{2}^{-1},$
$\displaystyle\chi_{1}=2\left(p_{i,2}-1\right)p_{i,2}^{2}\gamma_{1}^{-1}-2p_{i,2}\left(p_{i,2}-1\right)^{2}\gamma_{2}^{-1}-\gamma_{3}^{-1},$
$\displaystyle\chi_{2}=p_{i,2}\left(p_{i,2}-1\right)\left[p_{i,2}^{3}\gamma_{1}^{-1}-\left(p_{i,2}-1\right)^{3}\gamma_{2}^{-1}-\gamma_{3}^{-1}\right].$

##  IV Dynamics of rogue waves that arise from a constant background

Now we study dynamics of rogue waves that arise from the constant background
(2) in the three-wave system (1). These solutions are given by Theorems 1-2,
with all parameters $p_{i}$ satisfying the condition (30).

###  IV.1 Fundamental rogue waves

The fundamental rogue wave is given by Eq. (20), with $p_{1}$ satisfying the
condition (30). The amplitude functions of this fundamental rogue wave can be
written more explicitly as

$|q_{i,1}(x,t)|=\left|\rho_{i}\frac{g_{i,1}}{f_{1}}\right|,\quad i=1,2,3,$
(31)

where

$\displaystyle
f_{1}(x,y,t)=m_{1,1}^{(0,0)}=(a_{1}x+b_{1}y+c_{1}t)^{2}+\frac{1}{\zeta_{0}^{2}}(a_{1}x+b_{1}y+\zeta_{0}c_{2}t)^{2}+\frac{|p_{1}|^{2}}{(p_{1}+p_{1}^{*})^{2}},$
$\displaystyle
g_{1,1}(x,y,t)=m_{1,1}^{(1,0)}=f_{1}(x,y,t)-\frac{2i\hat{\theta}_{1}}{\zeta_{0}}\left(a_{1}x+b_{1}y+\zeta_{0}c_{2}t\right)+2i\hat{\theta}_{2}(a_{1}x+b_{1}y+c_{1}t)-\hat{\theta}_{1}^{2}-\hat{\theta}_{2}^{2},$
$\displaystyle
g_{2,1}(x,y,t)=m_{1,1}^{(0,-1)}=f_{1}(x,y,t)+\frac{2i}{\zeta_{0}}\left(a_{1}x+b_{1}y+\zeta_{0}c_{2}t\right)-1,$
$\displaystyle
g_{3,1}(x,y,t)=m_{1,1}^{(-1,1)}=f_{1}(x,y,t)+\frac{2i(\hat{\theta}_{1}-1)}{\zeta_{0}}\left(a_{1}x+b_{1}y+\zeta_{0}c_{2}t\right)-2i\hat{\theta}_{2}(a_{1}x+b_{1}y+c_{1}t)-(\hat{\theta}_{1}-1)^{2}-\hat{\theta}_{2}^{2},$

parameters $(a_{1},b_{1},c_{1},c_{2})$ are given by Eqs. (22)-(25), and

$\zeta_{0}=\frac{a_{1}}{a_{2}},\quad\hat{\theta}_{1}=\Re\left(\frac{p}{p-\rm{i}}\right),\quad\hat{\theta}_{2}=\Im\left(\frac{p}{p-\rm{i}}\right).$
(32)

Here, $\Re$ and $\Im$ represent the real and imaginary parts of a complex
number.

Figure 1: A fundamental line rogue wave with parameter choices (33) and
$p=0.5+0.5\rm{i}$. In all panels, $-10\leq x,y\leq 10$.

To demonstrate this fundamental rogue wave, we choose nonlinear coefficients,
background amplitudes and velocity values as

$\epsilon_{1}=-\epsilon_{2}=\epsilon_{3}=1,\ \rho_{1}=\rho_{2}=\rho_{3}=1,\
V_{1,1}=6,\ V_{1,2}=5,\ V_{2,1}=4,\ V_{2,2}=3.$ (33)

In addition, we choose $p_{1}=0.5+0.5\rm{i}$, which satisfies the condition
(30). For these choices of parameters, the corresponding rogue wave is
displayed in Fig. 1. It is seen that this is a line rogue wave with a single
dominant peak line.

###  IV.2 Multi-rogue waves

To get multi-rogue waves, we set $N>1$ and $n_{1}=n_{2}=\cdots=n_{N}=1$ in the
rational solutions of Theorem 2, and require all $\\{p_{i}\\}$ values to
satisfy condition (30). To demonstrate, we choose $N=2$ and the same nonlinear
coefficients, background amplitudes and velocity values as in (33). In
addition, we choose

$p_{1}=-0.539770966+0.3\textrm{i},\quad p_{2}=1.904662796+0.65\textrm{i},\quad
a_{1,1}=0,\quad a_{1,2}=0.$ (34)

Notice that these $(p_{1},p_{2})$ values satisfy conditions (30), because we
obtained their real parts by solving the quartic equation (30), with their
imaginary parts set as $0.3$ and $0.65$ respectively. The corresponding two-
rogue wave solution is displayed in Fig. 2. We see that this rogue wave
features two intersecting lines, indicating that this solution describes the
interaction between two fundamental line rogue waves. Interestingly, at the
intersection point between the two line rogue waves, the $|q_{2}|$ component
has higher amplitude, but the $(|q_{1}|,|q_{3}|)$ components have lower
amplitudes.

Figure 2: A two-rogue wave solution with parameter choices (33)-(34). In these
panels, $-100\leq x\leq 0$ and $-50\leq y\leq 0$ in the first row, $-50\leq
x\leq 50$ and $-25\leq y\leq 25$ in the second to fourth rows, and $0\leq
x\leq 100$, $0\leq y\leq 50$ in the last row. These $(x,y)$ intervals in
different rows are different because the intersection point of the two line
rogue waves is moving.

###  IV.3 High-order rogue waves

Higher-order rogue waves are obtained by taking $N=1$ and $n_{1}>1$ in the
rational solutions of Theorem 2, with $p_{1}$ satisfying the condition (30).
To demonstrate, we choose $n_{1}=2$, $p_{1}=\sqrt{3}/2+\textrm{i}/2$, and

$\epsilon_{1}=-\epsilon_{2}=\epsilon_{3}=1,\ \rho_{1}=1,\ \rho_{2}=\sqrt{2},\
\rho_{3}=1,\ V_{1,1}=6,\ V_{1,2}=5,\ V_{2,1}=4,\ V_{2,2}=3,\ a_{1,1}=0,\
a_{2,1}=0.$ (35)

Note that for this $p_{1}$ value, $\sqrt{3}/2$ is a double root of the quartic
equation (30). The corresponding solution is displayed in Fig. 3.
Interestingly, this second-order rogue solution is still a line rogue wave.
However, compared to the single-peak fundamental line rogue wave of Fig. 1,
this second-order line rogue wave develops two peaks, which are most
pronounced in the $t=-1$ panels. This type of higher-order rogue waves has not
been reported before to our best knowledge.

Figure 3: A second-order rogue wave under parameter choices (35) and
$p_{1}=\sqrt{3}/2+\textrm{i}/2$. In all panels, $-16\leq x\leq 16$ and $-8\leq
y\leq 8$.

##  V Rogue waves that arise from a non-constant background

The three-wave system (1) also admits rogue waves that do not arise from the
constant background (2). We consider this type of rogue waves in this section.

###  V.1 Rogue waves that arise from lump-soliton backgrounds

In the previous rational solutions of Theorems 1-2, if only some of the
$\\{p_{i}\\}$ parameters satisfy the condition (30) but the others do not,
then we will get rogue waves that arise from lump-soliton backgrounds. To
demonstrate this type of rogue waves, we choose

$N=2,\quad n_{1}=n_{2}=1,\quad p_{1}=2,\quad p_{2}=1+{\rm{i}},\quad
a_{1,1}=0,\quad a_{1,2}=0,$ (36)

and the other parameters are as given in Eq. (33). Notice that $p_{1}$ does
not satisfy the condition (30), but $p_{2}$ does. The corresponding solution
is displayed in Fig. 4. Notice that at large negative time, the solution is a
lump soliton moving on the constant background (2). However, as time
increases, a line rogue wave arises and interacts with this lump soliton. This
interaction creates interesting intensity patterns which strongly deform the
original lump soliton. At large positive time, however, this rogue wave
disappears, and the solution goes back to a lump soliton again.

Figure 4: A rogue wave that arises from a lump-soliton background. Solution
parameters are given in Eqs. (33) and (36). In these panels, $-56\leq
x\leq-40$ and $-45\leq y\leq-33$ in the first row, $-18\leq x\leq-2$ and
$-2\leq y\leq-14$ in the second row, $-8\leq x\leq 8$ and $-6\leq y\leq 6$ in
the third row, $2\leq x\leq 18$ and $2\leq y\leq 14$ in the fourth row, and
$41\leq x\leq 57$, $36\leq y\leq 48$ in the last row. These $(x,y)$ intervals
in different rows are different because the lump soliton is moving.

###  V.2 Rogue waves that arise from line-dark-soliton backgrounds

In addition to the above rogue waves that arise from lump-soliton backgrounds,
there also exist rogue waves that arise from line-dark-soliton backgrounds.
This latter type of rogue waves are not rational solutions though. They are
semi-rational solutions which contain both rational and exponential
components. Analytical expressions of these semi-rational solutions are given
by the following theorem.

> Theorem 3. The (2+1)-dimensional three-wave system (1) admits semi-rational
> solutions (6)-(8), where the matrix elements $m_{i,j}^{(n,k)}$ of
> $\tau_{n,k}$ are given by
>
> $\displaystyle
> m_{i,j}^{(n,k)}=\pi_{i,j}+\frac{\left(p\partial_{p}\right)^{n_{i}}}{(n_{i})!}\frac{\left(q\partial_{q}\right)^{n_{j}}}{(n_{j})!}\left.\left[\frac{1}{p+q}\left(-\frac{p}{q}\right)^{k}\left(-\frac{p-\textrm{i}}{q+\textrm{i}}\right)^{n}e^{\Theta_{i,j}(x,y,t)}\right]\
> \right|_{p=p_{i},\ q=p_{j}^{*}},$ (37)
>
> or equivalently, by
>
> $\displaystyle
> m_{i,j}^{(n,k)}=\left[\sum_{\nu=0}^{\min(n_{i},n_{j})}\left(\frac{1}{p_{i}+p_{j}^{*}}\right)\left[\frac{p_{i}p_{j}^{*}}{(p_{i}+p_{j}^{*})^{2}}\right]^{\nu}\hskip
> 1.70709ptS_{n_{i}-\nu}[\textbf{\emph{x}}^{+}_{i,j}(n,k)+\nu\textbf{\emph{s}}_{i,j}]\hskip
> 1.70709ptS_{n_{j}-\nu}[\textbf{\emph{x}}^{-}_{j,i}(n,k)+\nu\textbf{\emph{s}}^{*}_{j,i}\right]$
> $\displaystyle+\pi_{i,j}\left(-\frac{p_{i}}{p_{j}^{*}}\right)^{-k}\left(-\frac{p_{i}-\textrm{i}}{p_{j}^{*}+\textrm{i}}\right)^{-n}\exp\left[-\left(\frac{1}{p_{i}}+\frac{1}{p_{j}^{*}}\right)z_{1}-\left(\frac{1}{p_{i}-\textrm{i}}+\frac{1}{p_{j}^{*}+\textrm{i}}\right)z_{2}-(p_{i}+p_{j}^{*})z_{3}\right],$
> (38)
>
> $\pi_{i,j}$ are complex constants satisfying the constraint
>
> $\pi_{i,j}=\pi^{*}_{j,i},$ (39)
>
> and all other notations are the same as in Theorems 1 and 2.

If all $\pi_{i,j}$ are zero, then the above solution reduces to the rational
solutions given in Theorems 1-2. If $n_{i}=0$ for all $1\leq i\leq N$, where
the summation term in the above equation (V.2) reduces to a constant
$1/(p_{i}+p_{j}^{*})$, we would get line dark solitons or multi-line dark
solitons. But if $\pi_{i,j}$ are not all zero, and $n_{i}$ are not all zero,
then we will get semi-rational solutions.

Rogue waves will be obtained if some of the $p_{i}$ parameters in these semi-
rational solutions satisfy the condition (30), and these rogue waves will
arise from single-line or multi-line dark solitons. We illustrate this type of
rogue waves below.

Fundamental semi-rational rogue waves are obtained when we take $N=1$,
$n_{1}=1$, $p_{1}$ satisfying the condition (30), and $\pi_{1,1}$ is a real
nonzero constant [hence satisfying the constraint (39)]. To illustrate this
type of rogue waves, we choose

$p_{1}=0.536234874+0.4{\rm{i}},\quad\pi_{1,1}=1,\quad a_{1,1}=0,$ (40)

and all other parameters are as given in Eq. (33). The corresponding solution
is displayed in Fig. 5. As is seen, at large negative time, this solution is a
fundamental line dark soliton. As time increases, a rogue wave appears, which
interacts with this dark soliton. The most interesting feature of this rogue
wave is that it is a half-line rogue wave which appears on only the upper side
of the line dark soliton. This contrasts the fundamental rogue wave arising
from a constant background in Fig. 1, which is a whole line. At large positive
time, this half-line rogue wave disappears, and the solution goes back to the
fundamental line dark soliton.

Figure 5: A fundamental semi-rational rogue wave with parameter values in Eqs.
(33) and (40). In these panels, $-8\leq x,y\leq 8$ in the first four rows, and
$-3\leq x,y\leq 13$ in the last row.

To demonstrate non-fundamental semi-rational rogue waves, we take

$N=2,\quad n_{1}=n_{2}=1,\quad\pi_{2,2}=1,\quad\pi_{1,2}=\pi_{2,1}=0,\quad
a_{1,1}=0,\quad a_{1,2}=0.$ (41)

The nonlinear coefficients, background amplitudes and velocities are taken as
in Eq. (33). We will show two solutions.

In the first non-fundamental semi-rational rogue wave, we choose

$p_{1}=-0.542592881+0.35{\rm i},\quad
p_{2}=1.904662796+0.65\rm{i},\quad\pi_{1,1}=-1.$ (42)

Both $p_{1}$ and $p_{2}$ here satisfy the condition (30). Thus, we anticipate
that this solution describes the interaction between a two-rogue-wave solution
from a constant background and a two-dark-soliton solution. This solution is
plotted in Fig. 6. It is seen that at large negative and positive times, this
solution is indeed a two-dark soliton, and at intermediate times, a rogue wave
appears and interacts with this two-dark soliton. These behaviors are
consistent with our anticipations. However, this rogue wave that appears at
intermediate times here is not a two-rogue wave from a constant background as
the one shown in Fig. 2 (which features two intersecting whole lines).
Instead, the present rogue wave consists of two half-lines which appear on the
opposite sides of the two-dark soliton. In addition, locations of these two
half-lines are also moving along the dark-soliton lines. The half-line feature
of the present rogue wave echoes that in Fig. 5, but motions of the two half-
line rogue waves (especially relative to each other) are new and fascinating
features.

Figure 6: A non-fundamental semi-rational rogue wave with parameter values
given in Eqs. (33) and (41)-(42). In these panels, $-97\leq x\leq-73$ and
$-80\leq y\leq-60$ in the first row, $-17\leq x\leq 7$ and $-13\leq y\leq 7$
in the second row, $-12\leq x\leq 12$ and $-10\leq y\leq 10$ in the third row,
$-7\leq x\leq 17$ and $-7\leq y\leq 13$ in the fourth row, and $73\leq x\leq
97$, $60\leq y\leq 80$ in the last row.

In the second non-fundamental semi-rational rogue wave, we choose

$p_{1}=1.5,\quad p_{2}=1+{\rm i},\quad\pi_{1,1}=1.$ (43)

The $p_{1}$ value here does not satisfy the condition (30), but $p_{2}$ does.
Intuitively, this $p_{1}$ value in the rational part of the semi-rational
solution signals a lump soliton, and the $p_{2}$ value signals a rogue wave —
likely a half-line rogue wave in view of Figs. 6 and 7. Thus, we might have
anticipated that this semi-rational solution would describe the interaction
between a lump soliton, a half-line rogue wave and a two-dark soliton. The
true solution is displayed in Fig. 7. One of the most surprising features of
this solution is that, at large negative time, the lump soliton is absent.
However, as time increases, this lump miraculously appears. More
interestingly, as time increases further, this lump persists but detaches
itself and moves away from the two-dark soliton. In the whole evolution
process, there is no half-line rogue wave. Instead, the lump that emerges from
the two-dark soliton acts as a rogue wave. But this lump rogue wave does not
disappear at large positive time, unlike rogue waves in previous six figures.

Figure 7: Another non-fundamental semi-rational rogue wave with parameter
values given in Eqs. (33), (41) and (43). In these panels, $-92\leq x\leq-72$
and $-75\leq y\leq-57$ in the first row, $-20\leq x\leq 0$ and $-17\leq y\leq
1$ in the second row, $-10\leq x\leq 10$ and $-9\leq y\leq 9$ in the third
row, $0\leq x\leq 20$ and $-1\leq y\leq 17$ in the fourth row, and $72\leq
x\leq 92$, $57\leq y\leq 75$ in the last row.

##  VI Proofs of Theorems 1-3

In this section, we derive the rational and semi-rational solutions presented
in Theorems 1-3.

###  VI.1 Proof of Theorem 1

Due to the boundary conditions (2), we first introduce the bilinear
transformation

$\displaystyle
q_{1}(x,y,t)=\rho_{1}\frac{g_{1}}{f}e^{i(k_{1}x+\lambda_{1}y+\omega_{1}t)},$
$\displaystyle
q_{2}(x,y,t)=\rho_{2}\frac{g_{2}}{f}e^{i(k_{2}x+\lambda_{2}y+\omega_{2}t)},$
(44) $\displaystyle
q_{3}(x,y,t)=\textrm{i}\rho_{3}\frac{g_{3}}{f}e^{\textrm{i}(k_{3}x+\lambda_{3}y+\omega_{3}t)},$

where $f(x,y,t)$ is a real function, and $g_{k}(x,y,t)\hskip
1.42271pt(k=1,2,3)$ are complex functions. Under this transformation, the
three-wave system (1) is converted into the following system of bilinear
equations

$\displaystyle\left(D_{t}+V_{1,1}D_{x}+V_{1,2}D_{y}-{\rm{i}}\gamma_{1}\right)g_{1}\cdot
f$ $\displaystyle=$ $\displaystyle-{\rm{i}}\gamma_{1}g_{2}^{*}g_{3}^{*},$
$\displaystyle\left(D_{t}+V_{2,1}D_{x}+V_{2,2}D_{y}-{\rm{i}}\gamma_{2}\right)g_{2}\cdot
f$ $\displaystyle=$ $\displaystyle-{\rm{i}}\gamma_{2}g_{1}^{*}g_{3}^{*},$ (45)
$\displaystyle\left(D_{t}-{\rm{i}}\gamma_{3}\right)g_{3}\cdot f$
$\displaystyle=$ $\displaystyle-{\rm{i}}\gamma_{3}g_{1}^{*}g_{2}^{*},$

where $D$ is Hirota’s bilinear differential operator defined by

$\displaystyle P\left(D_{x},D_{y},D_{t}\right)F(x,y,t)\cdot G(x,y,t)\equiv
P\left(\partial_{x}-\partial_{x^{\prime}},\partial_{y}-\partial_{y^{\prime}},\partial_{t}-\partial_{t^{\prime}}\right)F(x,y,t)G(x^{\prime},y^{\prime},t^{\prime})|_{x^{\prime}=x,\hskip
1.42271pty^{\prime}=y,\hskip 1.42271ptt^{\prime}=t},$

$P$ is a polynomial of $(D_{x},D_{y},D_{t})$, and constants
$(\gamma_{1},\gamma_{2},\gamma_{3})$ are as defined in Eq. (5).

Next, we introduce the coordinate transformation

$\left(\begin{array}[]{ccc}x\\\ y\\\ t\\\
\end{array}\right)=\left(\begin{array}[]{ccc}\frac{V_{1,1}}{\gamma_{1}}&\frac{V_{2,1}}{\gamma_{2}}&0\\\
\frac{V_{1,2}}{\gamma_{1}}&\frac{V_{2,2}}{\gamma_{2}}&0\\\
\frac{1}{\gamma_{1}}&\frac{1}{\gamma_{2}}&\frac{1}{\gamma_{3}}\\\
\end{array}\right)\left(\begin{array}[]{ccc}z_{1}\\\ z_{2}\\\ z_{3}\\\
\end{array}\right),$ (46)

which is equivalent to Eqs. (11)-(13) in Theorem 1. Under this coordinate
transformation, we have

$\displaystyle\partial_{t}+\mathbf{V}_{1}\cdot\nabla=\gamma_{1}\partial_{z_{1}},$
$\displaystyle\partial_{t}+\mathbf{V}_{2}\cdot\nabla=\gamma_{2}\partial_{z_{2}},$
(47) $\displaystyle\partial_{t}=\gamma_{3}\partial_{z_{3}}.$

Thus, the bilinear system (45) reduces to

$\displaystyle\left({\rm{i}}D_{z_{1}}+1\right)g_{1}\cdot
f=g_{2}^{*}g_{3}^{*},$
$\displaystyle\left({\rm{i}}D_{z_{2}}+1\right)g_{2}\cdot
f=g_{1}^{*}g_{3}^{*},$ (48)
$\displaystyle\left({\rm{i}}D_{z_{3}}+1\right)g_{3}\cdot
f=g_{1}^{*}g_{2}^{*}.$

Below, we will first construct algebraic solutions to the more general
bilinear system

$\displaystyle\left({\rm{i}}D_{z_{1}}+1\right)g_{1}\cdot f=h_{2}h_{3},$
$\displaystyle\left({\rm{i}}D_{z_{2}}+1\right)g_{2}\cdot f=h_{1}h_{3},$ (49)
$\displaystyle\left({\rm{i}}D_{z_{3}}+1\right)g_{3}\cdot f=h_{1}h_{2},$

where $h_{1},h_{2}$ and $h_{3}$ are also complex functions. Afterwards, we
will impose the reality conditon for $f$ and complex conjugation conditions

$h_{k}^{*}=g_{k},\quad k=1,2,3.$ (50)

Under these conditions, the bilinear system (49) will become the bilinear
system (48), and the corresponding algebraic solutions will give rational
solutions of the three-wave system (1) through the above variable and
coordinate transformations (44) and (46).

Gram solutions to the general bilinear system (49) have been studied in Ref.
YangYang3wave . After simple generalizations of those results, we find that
the $\tau$ function

$\tau_{n,k}=\det_{1\leq i,j\leq N}\left(m_{i,j}^{(n,k)}\right),$ (51)

where

$\displaystyle
m_{i,j}^{(n,k)}=\left.\frac{\left(p\partial_{p}\right)^{n_{i}}}{(n_{i})!}\frac{\left(q\partial_{q}\right)^{n_{j}}}{(n_{j})!}\widetilde{m}_{i,j}^{(n,k)}\right|_{p=p_{i},\
q=q_{j}},$ (52)
$\displaystyle\widetilde{m}_{i,j}^{(n,k)}=\frac{1}{p+q}\left(-\frac{p}{q}\right)^{k}\left(-\frac{p-\rm{i}}{q+\rm{i}}\right)^{n}\mathrm{e}^{\xi_{i}+\eta_{j}},$
(53)
$\displaystyle\xi_{i}=\frac{1}{p}z_{1}+\frac{1}{p-\rm{i}}z_{2}+(p-{\rm{i}})z_{3}+\xi_{0,i},$
(54)
$\displaystyle\eta_{j}=\frac{1}{q}z_{1}+\frac{1}{q+\rm{i}}z_{2}+(q+{\rm{i}})z_{3}+\eta_{0,j},$
(55)

$p_{i},q_{j}$ are arbitrary complex constants, and $\xi_{0,i}(p)$,
$\eta_{0,j}(q)$ are arbitrary functions of $p$ and $q$, respectively, would
satisfy the following bilinear system

$\displaystyle\left[\textrm{i}D_{z_{1}}+1\right]\tau_{n+1,k}\cdot\tau_{n,k}=\tau_{n,k+1}\tau_{n+1,k-1},$
$\displaystyle\left[\textrm{i}D_{z_{2}}+1\right]\tau_{n,k-1}\cdot\tau_{n,k}=\tau_{n-1,k}\tau_{n+1,k-1},$
(56)
$\displaystyle\left[\textrm{i}D_{z_{3}}+1\right]\tau_{n-1,k+1}\cdot\tau_{n,k}=\tau_{n-1,k}\tau_{n,k+1}.$

Thus, if we define

$f=\tau_{0,0},\quad g_{1}=\tau_{1,0},\quad g_{2}=\tau_{0,-1},\quad
g_{3}=\tau_{-1,1},$ (57)

and

$h_{1}=\tau_{-1,0},\quad h_{2}=\tau_{0,1},\quad h_{3}=\tau_{1,-1},$ (58)

the above bilinear system (56) would reduce to (49), and these $\tau$
functions would be solutions to that bilinear system.

To introduce explicit free parameters into the above solutions, we let

$\xi_{0,i}=\sum_{r=1}^{\infty}a_{r,i}\ln^{r}\left[\frac{p}{p_{i}}\right],\quad\eta_{0,j}=\sum_{r=1}^{\infty}b_{r,j}\ln^{r}\left[\frac{q}{q_{j}}\right],$
(59)

where $a_{r,i}$ and $b_{r,j}$ are free complex constants.

In order to reduce the bilinear system (49) to the original one in (48), we
still need to impose the $f$-reality condition as well as the complex
conjugacy conditions of $h_{i}=g_{i}^{*}$. All these conditions would be
satisfied if

$\tau_{n,k}=\left[\tau_{-n,-k}\right]^{*}.$ (60)

To realize this condition, we set

$q_{j}=p_{j}^{*},\quad b_{r,j}=a^{*}_{r,j}.$ (61)

In this case, we can readily show that

$m_{j,i}^{(-n,-k)}=\left[m_{i,j}^{(n,k)}\right]^{*}.$ (62)

Thus, the condition (60) holds, and the $\tau$ functions (57) are solutions to
the original bilinear system (48). Inserting the above
$(\xi_{0,i},\eta_{0,j})$ expressions (59) and parameter conditions (61) into
the matrix element expression (52), we then obtain the rational solutions in
Theorem 1 for the three-wave system (1).

###  VI.2 Proof of Theorem 2

Next, we remove differential operators in solutions of Theorem 1, and derive a
more explicit solution form that is given in Theorem 2.

The technique we use is similar to that developed in OhtaJY2012 ;
YangYang3wave . We first introduce the generator $\mathcal{G}$ of differential
operators $(p\partial_{p})^{i}(q\partial_{q})^{j}$ as

$\mathcal{G}=\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\frac{\kappa^{i}}{i!}\frac{\lambda^{j}}{j!}\left(p\partial_{p}\right)^{i}\left(q\partial_{q}\right)^{j}.$

For this generator, we have the identity OhtaJY2012

$\mathcal{G}F(p,q)=F(e^{\kappa}p,e^{\lambda}q),$

where $F(p,q)$ is an arbitrary function. Then,

$\displaystyle\frac{1}{(p+q)\widetilde{m}_{i,j}^{(n,k)}}\left.\mathcal{G}\widetilde{m}_{i,j}^{(n,k)}\right|_{p=p_{i},\
q=q_{j}}=\frac{1}{p_{i}e^{\kappa}+q_{j}e^{\lambda}}\left(\frac{e^{\kappa}}{e^{\lambda}}\right)^{k}\left(\frac{p_{i}e^{\kappa}-\rm{i}}{p_{i}-\rm{i}}\right)^{n}\left(\frac{q_{j}e^{\lambda}+\rm{i}}{q_{j}+\rm{i}}\right)^{-n}\exp\left(\sum_{r=1}^{\infty}(a_{r,i}\kappa^{r}+a^{*}_{r,j}\lambda^{r})\right)\times$
$\displaystyle\hskip
28.45274pt\exp\left\\{\left(\frac{1}{p_{i}e^{\kappa}}-\frac{1}{p_{i}}+\frac{1}{q_{j}e^{\lambda}}-\frac{1}{q_{j}}\right)z_{1}+\left(\frac{1}{p_{i}e^{\kappa}-\rm{i}}-\frac{1}{p_{i}-\rm{i}}+\frac{1}{q_{j}e^{\lambda}+\rm{i}}-\frac{1}{q_{j}+\rm{i}}\right)z_{2}+\left(p_{i}e^{\kappa}-p_{i}+q_{j}e^{\lambda}-q_{j}\right)z_{3}\right\\}.\hskip
28.45274pt$

The $1/(p_{i}e^{\kappa}+q_{j}e^{\lambda})$ term above can be rewritten as
YangYang3wave

$\displaystyle\frac{1}{p_{i}e^{\kappa}+q_{j}e^{\lambda}}=\frac{p_{i}+q_{j}}{(p_{i}e^{\kappa}+q_{j})(q_{j}e^{\lambda}+p_{i})}\sum_{\nu=0}^{\infty}\left[\frac{(p_{i}e^{\kappa}-p_{i})(q_{j}e^{\lambda}-q_{j})}{(p_{i}e^{\kappa}+q_{j})(q_{j}e^{\lambda}+p_{i})}\right]^{\nu}$
$\displaystyle=\frac{p_{i}+q_{j}}{(p_{i}e^{\kappa}+q_{j})(q_{j}e^{\lambda}+p_{i})}\sum_{\nu=0}^{\infty}\left(\frac{p_{i}q_{j}}{(p_{i}+q_{j})^{2}}\kappa\lambda\right)^{\nu}\left(\frac{p_{i}+q_{j}}{\kappa}\frac{e^{\kappa}-1}{p_{i}e^{\kappa}+q_{j}}\right)^{\nu}\left(\frac{p_{i}+q_{j}}{\lambda}\frac{e^{\lambda}-1}{q_{j}e^{\lambda}+p_{i}}\right)^{\nu}$
$\displaystyle=\sum_{\nu=0}^{\infty}\left(\frac{1}{p_{i}+q_{j}}\right)\left(\frac{p_{i}q_{j}}{(p_{i}+q_{j})^{2}}\kappa\lambda\right)^{\nu}\left(\frac{p_{i}+q_{j}}{p_{i}e^{\kappa}+q_{j}}\right)\left(\frac{p_{i}+q_{j}}{q_{j}e^{\lambda}+p_{i}}\right)\left(\frac{p_{i}+q_{j}}{\kappa}\frac{e^{\kappa}-1}{p_{i}e^{\kappa}+q_{j}}\right)^{\nu}\left(\frac{p_{i}+q_{j}}{\lambda}\frac{e^{\lambda}-1}{q_{j}e^{\lambda}+p_{i}}\right)^{\nu}.$

Using the above two equations as well as expansions (17)-(19), together with
the conjugation relation $q_{j}=p_{j}^{*}$, we get

$\frac{1}{(p+q)\widetilde{m}_{i,j}^{(n,k)}}\left.\mathcal{G}\widetilde{m}_{i,j}^{(n,k)}\right|_{p=p_{i},\
q=q_{j}}=\sum_{\nu=0}^{\infty}\left(\frac{1}{p_{i}+q_{j}}\right)\left(\frac{p_{i}q_{j}}{(p_{i}+q_{j})^{2}}\kappa\lambda\right)^{\nu}\exp\left(\sum_{r=1}^{\infty}\left(x^{+}_{r,i,j}(n,k)+\nu
s_{r,i,j}\right)\kappa^{r}+\sum_{r=1}^{\infty}\left(x^{-}_{r,j,i}(n,k)+\nu
s^{*}_{r,j,i}\right)\lambda^{r}\right),$

where $x^{\pm}_{r,i,j}(n,k)$ and $s_{r,i,j}$ are as defined in Theorem 2.
Taking the coefficients of $\kappa^{i}\lambda^{j}$ on both sides of the above
equation, we get

$\frac{m_{i,j}^{(n,k)}}{(p_{i}+p_{j}^{*})\widetilde{m}_{i,j}^{(n,k)}}=\sum_{\nu=0}^{\min(n_{i},n_{j})}\left(\frac{1}{p_{i}+p_{j}^{*}}\right)\left[\frac{p_{i}p_{j}^{*}}{(p_{i}+p_{j}^{*})^{2}}\right]^{\nu}\hskip
1.70709ptS_{n_{i}-\nu}[\textbf{\emph{x}}^{+}_{i,j}(n,k)+\nu\textbf{\emph{s}}_{i,j}]\hskip
1.70709ptS_{n_{j}-\nu}[\textbf{\emph{x}}^{-}_{j,i}(n,k)+\nu\textbf{\emph{s}}^{*}_{j,i}],$
(63)

where $m_{i,j}^{(n,k)}$ is the matrix element defined in Eq. (52). It is easy
to see that the function $\tau_{n,k}$, with its matrix element
$m_{i,j}^{(n,k)}$ scaled as the left side of the above equation, still
satisfies the bilinear equations (48), because the scaling factor is an
exponential of a linear function in $(z_{1},z_{2},z_{3})$. Thus, the scaled
$\tau_{n,k}$ function, with its matrix elements given by the right side of the
above equation, i.e., by Eq. (14) of Theorem 2, still satisfies the bilinear
equations (48) of the three-wave system. This completes the proof of Theorem
2.

###  VI.3 Proof of Theorem 3

To derive semi-rational solutions in the three-wave system (1), we choose the
matrix element $m_{i,j}^{(n,k)}$ of the $\tau_{n,k}$ function as in Eq. (37),
which is the same as that in Eq. (52) plus a complex constant $\pi_{i,j}$. It
is easy to see that such a $\tau_{n,k}$ function still satisfies the bilinear
system (56) for reasons explained in Ref. YangYang3wave . Under the condition
$\pi_{i,j}=\pi^{*}_{j,i}$, the corresponding $\tau_{n,k}$ function satisfies
the complex-conjugation condition (60) as well. Thus, this $\tau_{n,k}$
function with the matrix element $m_{i,j}^{(n,k)}$ as given in Eq. (37)
satisfies the bilinear equations (48) of the three-wave system. To derive the
more explicit expression (V.2) of $m_{i,j}^{(n,k)}$ in Theorem 3, we just need
to repeat the calculations in the proof of Theorem 2, except that the
$\pi_{i,j}$ term in $m_{i,j}^{(n,k)}$, after the scaling of
$(p_{i}+p_{j}^{*})\widetilde{m}_{i,j}^{(n,k)}$ in Eq. (63), gives rise to the
exponential terms in Eq. (V.2) of Theorem 3. This theorem is then proved.

##  VII Summary and Discussion

In this paper, we have investigated rogue waves in the (2+1)-dimensional
three-wave system (1) by the bilinear method. General rogue waves arising from
a constant background, from a lump-soliton background and from a dark-soliton
background have been derived, and their dynamics illustrated. We have shown
that for rogue waves arising from a constant background, fundamental rogue
waves are line-shaped, and multi-rogue waves exhibit multiple intersecting
lines. In addition, higher-order rogue waves could also be line-shaped, but
would exhibit multiple parallel lines. For rogue waves arising from a lump-
soliton background, they could exhibit distinctive patterns due to their
interaction with the lump soliton. For rogue waves arising from a dark-soliton
background, their intensity pattern could feature half-line shapes or lump
shapes, which are very novel.

In this article, we have only illustrated the simplest rogue wave solutions.
It is known that as the order $N$ increases, rogue waves could exhibit more
striking patterns YangYang2021b . Whether rogue waves in the (2+1)-dimensional
three-wave system (1) also display novel patterns at large $N$ values is an
interesting question which merits further study.

## Acknowledgment

The work of J.Y. was supported in part by the National Science Foundation
under award number DMS-1616122.

## References

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