# General higher-order breathers and rogue waves in the two-component long-
wave–short-wave resonance-interaction mode

Jiguang Rao1, Boris A. Malomed 2,3, Dumitru Mihalache 4, Jingsong He5,∗
1School of Mathematics and Statistics, Hubei University of Science and
Technology, Xianning, Hubei, 437100, P. R. China;
2 Department of Physical Electronics, School of Electrical Engineering,
Faculty of Engineering, and Center for Light–Matter Interaction, Tel Aviv
University, P.O.B. 39040, Ramat Aviv, Tel Aviv, Israel;
3 Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica,
Chile;
4 Horia Hulubei National Institute of Physics and Nuclear Engineering, P.O.
Box MG–6, Magurele, RO-077125, Romania;
5 Institute for Advanced Study, Shenzhen University, Shenzhen, Guangdong,
518060, P. R. China

###### Abstract.

General higher-order breather and rogue wave (RW) solutions to the two-
component long wave–short wave resonance interaction (2-LSRI) model are
derived via the bilinear Kadomtsev-Petviashvili hierarchy reduction method and
are given in terms of determinants. Under particular parametric conditions,
the breather solutions can reduce to homoclinic orbits, or a mixture of
breathers and homoclinic orbits. There are three families of RW solutions,
which correspond to a simple root, two simple roots, and a double root of an
algebraic equation related to the dimension reduction procedure. The first
family of RW solutions consists of $\frac{N(N+1)}{2}$ bounded fundamental RWs,
the second family is composed of $\frac{N_{1}(N_{1}+1)}{2}$ bounded
fundamental RWs coexisting with another $\frac{N_{2}(N_{2}+1)}{2}$ fundamental
RWs of different bounded state ($N,N_{1},N_{2}$ being positive integers),
while the third one have
${[\widehat{N}_{1}^{2}+\widehat{N}_{2}^{2}-\widehat{N}_{1}(\widehat{N}_{2}-1)]}$
fundamental bounded RWs ($\widehat{N}_{1},\widehat{N}_{2}$ being non-negative
integers). The second family can be regarded as the superpositions of the
first family, while the third family can be the degenerate case of the first
family under particular parameter choices. These diverse RW patterns are
illustrated graphically.

∗ Corresponding author<EMAIL_ADDRESS><EMAIL_ADDRESS>
Keywords: Two-component long wave–short wave resonance interaction model,
Rogue waves, Breathers.

##  1\. Introduction

Resonant three-wave interactions plays a crucially important role in various
setups which occur in Bose-Einstein condensates, fluid mechanics, optics, and
other areas of physics [1, 2, 3, 4, 5]. Theoretical studies of these phenomena
provide an essential contribution to the nonlinear-wave dynamics [6, 7, 8, 9,
10, 11]. Generally, in nonlinear systems with linear dispersion relation
$\omega=\omega(k)$ ($\omega$ and $k$ are, as usual, the frequency and
wavenumber, respectively), the resonant interaction takes place if the
corresponding frequencies and wavenumbers, which form a resonance triad [12],
are mutually locked by conditions

$k_{3}=k_{1}+k_{2},\omega_{3}=\omega_{1}+\omega_{2}.$ (1)

The limit case of the triad amounts to the long-wave–short-wave (LW-SW)
resonance, when one wave is much longer than the other two [12, 13, 14], i.e.,

$k_{3}=k+(\Delta{k})/2,k_{2}=k-(\Delta{k})/2,k_{1}=\Delta{k},|\Delta{k}|\ll|k|.$
(2)

In this special case, condition (1) for the frequency amounts, in the first
approximation, to

${\mathrm{d}}{\omega}/{\mathrm{d}}{k}\approx\omega(\Delta{k})/\Delta{k},$ (3)

i.e., the LW phase velocity must match the SW group velocity, at the special
value of wavenumber $k$. In this case, the nonlinear equation, governing a
slowly varying, complex-valued SW packet envelope ($S$) and a real-valued LW
field ($L$), has been derived by means of the multiple-scale asymptotic
expansion [15, 16, 17]:

$\begin{array}[]{c}iS_{t}-S_{xx}+LS=0,\\\
L_{t}=2(\delta|S|^{2})_{x}.\end{array}$ (4)

The sign of the real nonlinearity coefficient $\delta$ depends on the physical
realization of the system. It is often called the LW–SW resonant interaction
(LSRI) model, or the Yajima–Oikawa (YO) system. It applies to fluid mechanics
and a number of other physical settings.

The LSRI model (4) is completely integrable as it admits a Lax pair, and was
solved by dint of the inverse scattering transform [15]. Additionally, Cheng
had proposed the LSRI model (4) from the so-called $K$-constrained Kadomtsev-
Petviashvili (KP) hierarchy with $K=2$, while $K=1$ corresponds to the
classical nonlinear Schrödinger (NLS) equation [18]. Bright- and dark-soliton
solutions of Eq. (4) were constructed in Ref. [16, 17]. The first-order RW
solutions to the LSRI model (4) were studied by means of the Hirota’s bilinear
method [19] and Darboux transformation [20, 21]. The higher-order breather and
RW solutions were derived with the help pf the bilinear KP hierarchy reduction
method [22, 23]. Homoclinic connections of unstable plane waves were
investigated by means of the Bäcklund transform [24].

RWs are large displacements from an otherwise tranquil (but usually unstable)
background [25, 26, 27, 28, 29, 30, 31, 32, 33], whose most striking features
are unpredictability and localization in time and space [34, 35]. In the past
decade, RWs have attracted a great deal of interest [36, 37, 38, 39, 40, 41,
42, 43] in the experimental and theoretical communities alike. Explicit
solutions of integrable equations for RWs help to understand these phenomena
in the physical systems. Such solutions have been found for many integrable
models, such as the NLS equation and its multicomponent version [44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54], the Davey–Stewartson (DS) [55, 56, 57] and
Ablowitz-Ladik [58, 59] equations, etc. [60, 61, 62, 63, 64, 65, 66, 67, 68,
69, 70]. The RW solutions can be regarded as the limit case of breathers,
which are periodic in time or spatial coordinate. The breathers can also
provide a model for the observation of RWs in experiments [41]. A rogue wave
may be thought of as the consequence of modulation instability (MI), but
conversely not all of MIs necessarily result in rogue wave generation [50,
71]. Besides, Baronio et al. found that rogue wave solutions in several
physical systems exist in the subset of parameters where the MI is present if
and only if the unstable sideband spectrum also contains continuous wave or
zero-frequency perturbations as a limit case [72].

In this paper, we consider the following two-component LSRI (2-LSRI) model
governing the resonance of two SW components with a common LW one:

$\displaystyle iA_{t}-A_{xx}+LA=0,$ $\displaystyle iB_{t}-B_{xx}+LB=0,$ (5)
$\displaystyle L_{t}=2(\delta_{1}|A|^{2}+\delta_{2}|B|^{2})_{x},$

where $A$, $B$ are the SWs and $L$ is the LW, real nonlinear coefficients
$\delta_{1},\delta_{2}$ depend on the precise physical properties of the
system, e.g. the density stratification profile in fluid [1, 73]. In optical
contexts, the two short-wave components $A$ and $B$ are called optical waves,
while the long-wave component $L$ are regarded as the induced optical
rectification [74] or the low–frequency terahertz wave [75].

The 2-LSRI model is an integrable extension of the LSRI model (4), and it
reduces to the LSRI model (4) for $A=B=S$. This system also admits soliton
solutions of bright, dark, and mixed bright–dark types [76, 77, 78]. The
first-order RW solution was first constructed by the Darboux transformation
[79] and later by means of the Hirota’s bilinear method [80]. The connection
between the existence criterion of rogue waves and the onset of baseband MI in
the LSRI model (4) and the 2-LSRI model (5) was confirmed in Ref. [72] and
Ref. [80], respectively. Additionally, Chen et.al. have obtained several RW
solutions from the general rational solutions of the (2+1)-dimensional multi-
component LSRI system [81]. Very recently, Li and Geng have constructed a
family of higher-order bounded RW solutions consisting of $N(N+1)/2$
fundamental RWs with the help of the Darboux transformation [82, 83, 84]. That
family of solutions of the 2-LSRI model (5) shares the same RW patterns with
the higher-order RW of LSRI model [23], and there not exist new RW patterns
for 2-LSRI model in contrast with LSRI model. It is known that the multi-
component NLS equation admits more RW patterns in contrast to the scalar NLS
equation [47, 48, 49, 50, 51, 52], such as the mixed bounded RWs consisting of
RWs of different types, and the degenerate bounded RWs. Very recently, Yang
and Yang derived a family of new bounded RW solutions to the three-wave
resonant interaction system [61], which consists of
${[\widehat{N}_{1}^{2}+\widehat{N}_{2}^{2}-\widehat{N}_{1}(\widehat{N}_{2}-1)]}$
fundamental bounded RWs ($\widehat{N}_{1},\widehat{N}_{2}$ being non-negative
integers). By taking $\widehat{N}_{1}=0$ or $\widehat{N}_{2}=0$, this solution
family amounts to one consisting of the degenerate bounded RW solutions. For
$\widehat{N}_{1}\widehat{N}_{2}\neq 0$, the family contains non-degenerate
bounded RW solutions. Thus, we call this family of RW solutions as degradable
bounded RW solutions in this paper. Up to now, the higher-order mixed bounded
and degradable bounded RWs were not reported for the 2-LSRI model (5), to the
best of our knowledge. A natural motivation is to construct the higher-order
mixed bounded and degradable bounded RWs in the 2-LSRI model (5).
Additionally, the first-order breather solutions were constructed by means of
the Hirota’s bilinear method [80], and the first-order homoclinic orbits were
derived by the Bäcklund transformation [85]. However, higher-order breather
and homoclinic-orbit solutions have not been reported, as yet, for the 2-LSRI
model (5), to the best of our knowledge. Very recently, we constructed general
higher-order breather solutions for the multi-component two-dimensional LSRI
model in terms of determinants via the bilinear KP-hierarchy reduction method
[86], which can also be applied to derive the higher-order breather and
homoclinic-orbit solutions of the 2-LSRI model (5) in the form of
determinants.

The main goal of the present paper is to construct the general higher-order
breather and RW solutions by employing the bilinear KP hierarchy reduction
method. The objectives of our work are as follows:

  * •

The general higher-order breather solutions in terms of determinants will be
constructed. Under particular parametric restrictions, the breather solutions
can reduce to higher-order homoclinic orbits or a mixture of homoclinic orbits
and breathers.

  * •

Three families of RW solutions, corresponding to a simple root, two simple
roots, and a double root of an algebraic equation related to the dimension
reduction procedure, will be derived. The first family is the bounded $N$-th-
order RWs consisting of $N(N+1)/2$ fundamental RWs. The second family is the
mixed bounded $(N_{1},N_{2})$-th order RWs comprising
$N_{1}(N_{1}+1)+N_{2}(N_{2}+1)/2$ fundamental RWs, in which the $N_{1}$-th-
order bounded RWs and $N_{2}$-th bounded RW can be included in different
states. The third one is the degradable bounded
$(\widehat{N}_{1},\widehat{N}_{2})$-th order RWs. Here $N,N_{1},N_{2}$ are
positive integers, and $\widehat{N}_{1},\widehat{N}_{2}$ are non-negative
integers. When $\widehat{N}_{1}=0$ or $\widehat{N}_{2}=0$, the third family
solutions are degenerate RWs.

The paper is organized as follows. In Section 2, we present the general
higher-order breather solutions for the 2-LSRI model in the form of Theorem 1,
and then investigate dynamics of the breathers. In Section 3, we present three
different families of RW solutions by three Theorems (i.e., Theorems 2, 3, 4),
then we exhibit dynamics of these three families of RWs, respectively. In
Section 4, we give the derivation of the breather solutions in Theorem 1 and
RW solutions in Theorems 2, 3, 4, i.e., the proofs of these four Theorems. The
discussion of the obtained results and the conclusions are presented in
Section 5.

##  2\. Dynamics of breathers in the 2-LSRI model

This Section focuses on the dynamics of breathers in the 2-LSRI model (5). For
this purpose, we first present the general breather solutions in forms of
determinants for the 2-LSRI model by the following Theorem. The proof for this
Theorem is postponed in Section 4.

######  Theorem 1.

The 2-LSRI model (5) admits the following breather solutions:

$\displaystyle
A=\rho_{1}e^{i\left(k_{1}x+(\gamma+k_{1}^{2})t\right)}\frac{g}{f},B=\rho_{2}e^{i\left(k_{2}x+(\gamma+k_{2}^{2})t\right)}\frac{h}{f},L=\gamma-2({\rm
log}f)_{xx},$ (6)

where the real function $f$ and the complex functions $g,h$ are given by

$\displaystyle f=\tau_{0,0},g=\tau_{1,0},h=\tau_{0,1}$ (7)

and $\tau_{n,k}$ is defined as the following $N\times N$ determinant:

$\displaystyle\tau_{n,k}=\det_{1<s,j\leq
N}\left(\overline{m}_{s,j}^{(n,k)}\right),$ (8)

with

$\displaystyle\overline{m}_{s,j}^{(n,k)}=$
$\displaystyle\frac{1}{p_{s}^{[1]}+p_{j}^{[1]*}}(-\frac{p_{s}^{[1]}-ik_{1}}{p_{j}^{[1]*}+ik_{1}})^{n}(-\frac{p_{s}^{[1]}-ik_{2}}{p_{j}^{[1]*}+ik_{2}})^{k}e^{\zeta_{s}+\zeta_{j}^{*}}+\frac{1}{p_{s}^{[1]}+p_{j}^{[2]*}}(-\frac{p_{s}^{[1]}-ik_{1}}{p_{j}^{[2]*}+ik_{1}})^{n}$
(9)
$\displaystyle\times(-\frac{p_{s}^{[1]}-ik_{2}}{p_{j}^{[2]*}+ik_{2}})^{k}e^{\zeta_{s}}+\frac{1}{p_{s}^{[2]}+p_{j}^{[1]*}}(-\frac{p_{s}^{[2]}-ik_{1}}{p_{j}^{[1]*}+ik_{1}})^{n}(-\frac{p_{s}^{[2]}-ik_{2}}{p_{j}^{[1]*}+ik_{2}})^{k}e^{\zeta_{j}^{*}}+$
$\displaystyle\frac{1}{p_{s}^{[2]}+p_{j}^{[2]*}}(-\frac{p_{s}^{[2]}-ik_{1}}{p_{j}^{[2]*}+ik_{1}})^{n}(-\frac{p_{s}^{[2]}-ik_{2}}{p_{j}^{[2]*}+ik_{2}})^{k},$

and

$\displaystyle\zeta_{s}=$ $\displaystyle\xi_{s}^{[1]}-\xi_{s}^{[2]},$ (10)
$\displaystyle\xi_{s}^{[\alpha]}=$ $\displaystyle
p_{s}^{[\alpha]}x-ip_{s}^{[\alpha]2}t+\overline{\xi}_{s}^{[\alpha]},\alpha=1,2.$

In the above expressions, $p_{s}^{[\lambda]},\overline{\xi}_{s}^{[\alpha]}$
are arbitrary complex constants,$\rho_{\ell},k_{\ell},\gamma$ are freely real
parameters, and the parameters $p_{s}^{[1]},p_{s}^{[2]}$ have to satisfy the
following constraints:

$\displaystyle\frac{\delta_{1}\rho_{1}^{2}}{(p_{s}^{[1]}-ik_{1})(p_{s}^{[2]}-ik_{1})}+\frac{\delta_{2}\rho_{2}^{2}}{(p_{s}^{[1]}-ik_{2})(p_{s}^{[2]}-ik_{2})}-i(p_{s}^{[1]}+p_{s}^{[2]})=0.$
(11)

######  Remark 1.

If we assume $p_{sR}^{[\alpha]}>0$, then $f$ in Eq. (7) is positive, and the
above breather solutions are nonsingular. Thus hereafter we take
$p_{sR}^{[\alpha]}>0$ to avoid the singularities of the breathers. The
subscripts $R$ and $I$ represent the real and imaginary parts of a given
parameter or a function, respectively.

######  Remark 2.

When $\delta_{1}=\delta_{2},k_{1}=k_{2},\rho_{1}=k\rho_{2}$, the above
breather solution reduces to the breather solutions of the scalar LSRI model
(4) reported in [22]. However, the breather solutions in [22] were given by
$2N\times 2N$ determinants, whereas they are given by $N\times N$ determinants
in Theorem 1.

######  Remark 3.

If $p_{sR}^{[1]2}-p_{sI}^{[1]2}=p_{sR}^{[2]2}-p_{sI}^{[2]2}$, then the
coefficients of $t$ in $\zeta_{sI}$ are zero, and the solutions (6) become the
higher-order homclinic orbits. If
$p_{sR}^{[1]2}-p_{sI}^{[1]2}=p_{sR}^{[2]2}-p_{sI}^{[2]2},p_{jR}^{[1]2}-p_{jI}^{[1]2}\neq
p_{jR}^{[2]2}-p_{jI}^{[2]2}$ ( $1\leq s\neq j\leq N$), the solutions (6) are a
mixture of breathers and homclinic orbits.

By taking $N=1$ in Eq. (8), Theorem 1 yields the first-order breather
solutions of the 2-LSRI model, which can be expressed in terms of hyperbolic
and trigonometric functions as:

$\displaystyle A=$
$\displaystyle\overline{A}\frac{e^{\theta_{0}}\cosh(\zeta_{1R}+\theta_{0}+i\overline{\beta}_{1})+e^{\widehat{\alpha}}\cos(\zeta_{1I}+\widehat{\beta}-i\overline{\alpha}_{1})}{e^{\theta_{0}}\cosh(\zeta_{1R}+\theta_{0})+e^{\widehat{\alpha}}\cos(\zeta_{1I}+\widehat{\beta})},$
(12) $\displaystyle B=$
$\displaystyle\overline{B}\frac{e^{\theta_{0}}\cosh(\zeta_{1R}+\theta_{0}+i\overline{\beta}_{2})+e^{\widehat{\alpha}}\cos(\zeta_{1I}+\widehat{\beta}-i\overline{\alpha}_{2})}{e^{\theta_{0}}\cosh(\zeta_{1R}+\theta_{0})+e^{\widehat{\alpha}}\cos(\zeta_{1I}+\widehat{\beta})},$
$\displaystyle L=$
$\displaystyle\gamma-\frac{y_{2}\cosh(\zeta_{1R}+\theta_{0})\cos(\zeta_{1I}+\widehat{\beta})+y_{1}\sinh(\zeta_{1R}+\theta_{0})\sin(\zeta_{1I}+\widehat{\beta})+y_{0}}{\left(e^{\theta_{0}}\cosh(\zeta_{1R}+\theta_{0})+e^{\widehat{\alpha}}\cos(\zeta_{1I}+\widehat{\beta})\right)^{2}}.$

The auxiliary functions in the above expressions are defined by

$\displaystyle\zeta_{1}=\xi_{1}^{[1]}-\xi_{1}^{[2]}=(p_{1}^{[1]}-p_{1}^{[2]})x-i(p_{1}^{[1]2}-p_{1}^{[2]2})t+\overline{\xi}_{1}^{[1]}-\overline{\xi}_{1}^{[2]},$
(13)
$\displaystyle\overline{A}=-\rho_{1}(p_{1}^{[2]}+p_{1}^{[2]*})\frac{p_{1}^{[2]}+ik_{1}}{p_{1}^{[2]}-ik_{1}}e^{i\left(k_{1}x+(\gamma+k_{1}^{2})t+\overline{\beta}_{1}\right)},$
$\displaystyle\overline{B}=-\rho_{2}(p_{1}^{[2]}+p_{1}^{[2]*})\frac{p_{1}^{[2]}+ik_{2}}{p_{1}^{[2]}-ik_{2}}e^{i\left(k_{2}x+(\gamma+k_{2}^{2})t+\overline{\beta}_{2}\right)},$
$\displaystyle
e^{\overline{\alpha}_{\ell}+i\overline{\beta}_{\ell}}=\frac{p_{1}^{[1]}-ik_{\ell}}{p_{1}^{[2]}-ik_{\ell}},e^{\widehat{\alpha}+i\widehat{\beta}}=\frac{p_{1}^{[2]}+p_{1}^{[2]}}{p_{1}^{[2]}+p_{1}^{[1]}},\ell=1,2,$
$\displaystyle
y_{1}=2e^{\theta_{0}+\widehat{\alpha}}(\overline{c}^{2}-\overline{e}^{2}),y_{2}=4e^{\theta_{0}+\widehat{\alpha}}\overline{c}\overline{e},y_{0}=2(\overline{c}^{2}e^{2\theta_{0}}-\overline{e}^{2}e^{2\widehat{\alpha}}),$
$\displaystyle\overline{c}=\frac{1}{2}\left(p_{1}-p_{2}+p_{1}^{*}-p_{2}^{*}\right),\overline{e}=\frac{1}{2i}\left(p_{1}-p_{2}-(p_{1}^{*}-p_{2}^{*})\right),e^{\theta_{0}}=\sqrt{\frac{p_{2}+p_{2}^{*}}{p_{1}+p_{1}^{*}}},$

and $\xi_{1}^{[1]},\xi_{1}^{[2]}$ are defined in Eq. (10).

From the above expressions of the first-order breather solution (12), we can
obtain that the first-order breather propagates along the line
$\zeta_{1R}+\theta_{0}=0$, and is periodic along the line
$\zeta_{1I}+\widehat{\beta}=0$. Figure 1 shows the first-order breather in the
2-LSRI model (5).

When $p_{1R}^{[1]2}-p_{1I}^{[1]2}=p_{1R}^{[2]2}-p_{1I}^{[2]2}$, the
coefficients of $t$ in $\zeta_{1I}$ are zero, hence the first-order breather
solutions (12) reduce to the first-order homoclinic orbit solutions. Figure 2
shows the first-order homoclinic orbit in the 2-LSRI model (5) with parameters
$p_{1}^{[1]}=1+i,p_{1}^{[2]}=1-i$.

Here we have to note that the first-order breather solutions (12) for the
2-LSRI model (5) have been reported in [80], and were derived by the Hirota’s
direction method. By taking the long wave limit of the obtained first-order
breather solution, the first-order RW solutions were also derived for the
2-LSRI model (5). Additionally, as shown in Figs. 1,2, the LW component $L$
has wave structures similar to those of the SW components $A$ and $B$, so we
only focus on the dynamics of breathers in the SW components $A$ and $B$, and
in what follows we will not show the LW component.

Figure 1. (Colour online) The first-order breather in the 2-LSRI model (5),
which is described by the solutions (12) with parameters
$\delta_{1}=-1,\delta_{2}=1,\rho_{1}=\frac{\sqrt{195}}{2},\rho_{2}=\sqrt{3},k_{1}=0,k_{2}=-1,\gamma=0,p_{1}^{[1]}=2+i,p_{1}^{[2]}=3-2i,\overline{\xi}_{1}^{[1]}=0,\overline{\xi}_{1}^{[2]}=0$.
Figure 2. (Colour online) The first-order homoclinic orbit in the 2-LSRI model
(5), which is described by the solutions (12) with parameters
$\delta_{1}=1,\delta_{2}=-1,\rho_{1}=\sqrt{2},\rho_{2}=\sqrt{5},k_{1}=0,k_{2}=-1,\gamma=0,p_{1}^{[1]}=1+i,p_{1}^{[2]}=1-i,\overline{\xi}_{1}^{[1]}=0,\overline{\xi}_{1}^{[2]}=0$.

By taking $N=2$ in Eq. (8), Theorem 1 generates the second-order breather
solutions of the 2-LSRI model. The determinant forms of the functions $f,g$,
and $h$ of solutions (6) can be explicitly written as

$\displaystyle f=$
$\displaystyle\begin{vmatrix}\overline{m}_{1,1}^{(0,0)}&\overline{m}_{1,2}^{(0,0)}\\\
\overline{m}_{2,1}^{(0,0)}&\overline{m}_{2,2}^{(0,0)}\end{vmatrix},\quad g=$
$\displaystyle\begin{vmatrix}\overline{m}_{1,1}^{(1,0)}&\overline{m}_{1,2}^{(1,0)}\\\
\overline{m}_{2,1}^{(1,0)}&\overline{m}_{2,2}^{(1,0)}\end{vmatrix},\quad h=$
$\displaystyle\begin{vmatrix}\overline{m}_{1,1}^{(0,1)}&\overline{m}_{1,2}^{(0,1)}\\\
\overline{m}_{2,1}^{(0,1)}&\overline{m}_{2,2}^{(0,1)}\end{vmatrix},$ (14)

where $\overline{m}_{s,j}^{(n,k)}$ are given by Eq. (9). Since this second-
order breather is a superposition of first-order breathers, thus these
solutions have three different dynamical behaviours: (i) the second-order
homoclinic orbits for
$p_{sR}^{[1]2}-p_{sI}^{[1]2}=p_{sR}^{[2]2}-p_{sI}^{[2]2},s=1,2$; (ii) the
mixture of the first-order breather and first-order homoclinic orbit for
$p_{sR}^{[1]2}-p_{sI}^{[1]2}=p_{sR}^{[2]2}-p_{sI}^{[2]2}$ and
$p_{3-s\,R}^{[1]2}-p_{3-s\,I}^{[1]2}\neq\,p_{3-s\,R}^{[2]2}-p_{3-s\,I}^{[2]2}$;
and (iii) the second-order breather for $p_{sR}^{[1]2}-p_{sI}^{[1]2}\neq
p_{sR}^{[2]2}-p_{sI}^{[2]2}$. Figure 3 shows these three different types of
periodic waves of the 2-LSRI model (5). In the leftmost panels of Fig. 3,
corresponding to the second-order homoclinic orbits, the two periodic waves
are only periodic in space (i.e., $x$); in the middle panels, corresponding to
a mixture of a first-order breather and a first-order homclinic orbit, one
periodic wave (namely, the one propagating along the $x$-axis) is only
periodic along space $x$ and the other periodic wave is periodic along both
space $x$ and time $t$; in the rightmost panels, both of the two periodic
waves are periodic along space $x$ and time $t$.

For larger $N$ in Theorem 1, the higher-order breathers, or higher-order
homclinic orbits, or a mixture of them to the 2-LSRI model can be obtained,
namely a superposition of $N$ individual first-order solutions given by Eq.
(12).

Figure 3. (Colour online) The leftmost panels: the second-order homoclinic
orbits with parameters
$\delta_{1}=1,\delta_{2}=-1,\rho_{1}=\sqrt{2},\rho_{2}=\sqrt{5},k_{1}=0,k_{2}=-1,\gamma=0,p_{1}^{[1]}=1+i,p_{1}^{[2]}=1-i,p_{2}^{[1]}=1.78521+2i,p_{2}^{[2]}=0.53496-1.048432i$;
The middle panels: a mixture of a first-order breather and a first-order
homoclinic orbit with parameters
$\delta_{1}=1,\delta_{2}=-1,\rho_{1}=\sqrt{2},\rho_{2}=\sqrt{5},k_{1}=0,k_{2}=-1,\gamma=0,p_{1}^{[1]}=1+i,p_{1}^{[2]}=1-i,p_{2}^{[1]}=1+2i,p_{2}^{[2]}=0.6780-1.2539i$;
The rightmost panels: the second-order breather with parameters
$\delta_{1}=-1,\delta_{2}=-1,\rho_{1}=\sqrt{\frac{5}{6}},\rho_{2}=\sqrt{\frac{55}{12}},k_{1}=0,k_{2}=-1,\gamma=0,p_{1}^{[1]}=1+i,p_{1}^{[2]}=1-\frac{1}{2}i,p_{2}^{[1]}=1-\frac{1}{3}i,p_{2}^{[2]}=1.0362+0.7905i$.

##  3\. Dynamics of rogue waves in the 2-LSRI model

In this Section, we study the dynamics of RWs in the 2-LSRI model (5). The RW
solutions, which will be explicitly expressed in this Section, depend on the
root structure of the following algebraic equation

$\displaystyle\frac{\partial\mathcal{Q}}{\partial p}=0,$ (15)

where

$\displaystyle\mathcal{Q}(p)=\frac{\delta_{1}\rho_{1}^{2}}{p-ik_{1}}+\frac{\delta_{2}\rho_{2}^{2}}{p-ik_{2}}+ip^{2}.$
(16)

For different types of roots of Eq. (15), the algebraic expressions of the
corresponding RW solutions to the 2-LSRI model (5) are different. In what
follows, we will mainly discuss the RW solutions associated to a non-imaginary
simple root, two non-imaginary simple roots, and a non-imaginary double root
of Eq. (15), respectively. These three families of solutions are bounded RWs,
mixed bounded RWs, and the degradable bounded RWs.

######  Remark 4.

If $p_{0}$ is a root of Eq. (15), then $-p_{0}^{*}$ is also a root for Eq.
(15), thus hereafter we assume $p_{0R}>0$ without loss of generality.

######  Remark 5.

Eq. (15) is a quintic equation having up to five roots. If $p_{0}$ is a triple
root of Eq. (15), then $-p_{0}^{*}$ is also a triple root of Eq. (15), thus
$p_{0}=-p_{0}^{*}$ must hold, otherwise there are six roots for Eq. (15), so
$p_{0}$ is pure imaginary when it is a triple root of Eq. (15). Besides, there
is a factor $\frac{1}{p_{0}+p_{0}^{*}}$ in RW solutions, hence $p_{0}$ cannot
be pure imaginary. Thus, there do not exist RW solutions when $p_{0}$ is a
triple root of Eq. (15). That is also true for $p_{0}$ being a quadruple or a
quintuple root of Eq. (15), so we only consider the RW solutions associated
with $p_{0}$ being a non-imaginary simple root, two non-imaginary simple
roots, and a non-imaginary double root of Eq. (15).

Below, we give three families of higher-order RW solutions to the 2-LSRI model
(5) and then consider their dynamics. These RW solutions are expressed through
both differential operator forms and Schur polynomials. Before that, we have
to review the definition of these Schur polynomials
$\mathbf{S}_{j}(\mathbf{x})$ with $\mathbf{x}=(x_{1},x_{2},\cdots)$:

$\displaystyle\sum\limits_{j=0}^{\infty}S_{j}(\mathbf{x})\epsilon^{j}=\exp(\sum\limits_{k=1}^{\infty}x_{k}\epsilon^{k}),$
(17)

or more explicitly

$\displaystyle S_{0}(\mathbf{x})=1,$ (18) $\displaystyle
S_{1}(\mathbf{x})=x_{1},$ $\displaystyle
S_{2}(\mathbf{x})=\frac{1}{2}x_{1}^{2}+x_{2},$ $\displaystyle\vdots$
$\displaystyle S_{j}(\mathbf{x})=\sum\limits_{l_{1}+2l_{2}+\cdots
ml_{m}=j}\left(\prod\limits_{k=1}^{m}\frac{x_{k}^{l_{k}}}{l_{k}!}\right).$

###  3.1. The bounded $N$-th-order rogue wave solutions and their dynamics

####  3.1.1. The bounded $N$-th-order rogue wave solutions

If $p_{0}$ is a non-imaginary simple root of Eq. (15), the bounded $N$-th-
order RW solutions of the 2-LSRI model are expressed as the following Theorem.

######  Theorem 2.

The 2-LSRI model (5) admits the following bounded $N$th-order RW solutions

$\displaystyle
A=\rho_{1}e^{i\left(k_{1}x+(\gamma+k_{1}^{2})t\right)}\frac{g}{f},B=\rho_{2}e^{i\left(k_{2}x+(\gamma+k_{2}^{2})t\right)}\frac{h}{f},L=\gamma-2({\rm
log}f)_{xx},$ (19)

where

$\displaystyle f=\tau_{0,0},g=\tau_{1,0},h=\tau_{0,1}$ (20)

and $\tau_{n,k}$ is defined as the following $N\times N$ determinant:

$\displaystyle\tau_{n,k}=\det_{1<s,j\leq
N}\left({m}_{2s-1,2j-1}^{(n,k)}\right),$ (21)

and the matrix elements ${m}_{s,j}^{(n,k)}$ are given either (a) in
differential operator form:

$\displaystyle{m}_{s,j}^{(n,k)}=$
$\displaystyle\left.\frac{[\mathcal{T}(p)\partial_{p}]^{s}}{s!}\frac{[\mathcal{T}^{*}(p)\partial_{p^{*}}]^{j}}{j!}m\right|_{p=p_{0}},$
(22) $\displaystyle m=$
$\displaystyle\frac{1}{p+p^{*}}(-\frac{p-ik_{1}}{p^{*}+ik_{1}})^{n}(-\frac{p-ik_{2}}{p^{*}+ik_{2}})^{k}e^{\xi+\xi^{*}},$

or (b) through Schur polynomials:

$\displaystyle\widehat{m}_{s,j}^{(n,k)}=$
$\displaystyle\sum\limits_{\nu=0}^{\mathrm{min}(i,j)}[\frac{|\lambda_{1}|^{2}}{(p_{0}+p_{0}^{*})^{2}}]^{\nu}\mathbf{S}_{i-\nu}({\mathbf{x}}^{+}(n,k)+\nu{\bf
s})\mathbf{S}_{j-\nu}({\mathbf{x}}^{-}(n,k)+\nu{\bf s}^{*}).$ (23)

Here $N$ is an arbitrary positive integer. The auxiliary functions
$\mathcal{T}(p),\xi$ in Eq. (22) are defined by

$\displaystyle\xi=$ $\displaystyle px-
ip^{2}t+\sum\limits_{r=1}^{\infty}\widehat{a}_{r}\ln^{r}\mathcal{W}(p),$ (24)
$\displaystyle\mathcal{T}(p)=$
$\displaystyle\sqrt{\frac{\mathcal{Q}^{2}(p)-\mathcal{Q}^{2}(p_{0})}{\mathcal{Q}^{{}^{\prime}2}(p)}},$
$\displaystyle\mathcal{W}(p)=$
$\displaystyle\frac{\mathcal{Q}(p)\pm\sqrt{\mathcal{Q}^{2}(p)-\mathcal{Q}(p_{0})}}{\mathcal{Q}(p_{0})}.$

The vectors $\mathbf{x}^{\pm}(n,k)=(x_{1}^{\pm},x_{2}^{\pm},\cdots)$ in Eq.
(23) are defined as:

$\displaystyle x_{r}^{+}(n,k)=$
$\displaystyle\lambda_{r}x-{i}\beta_{r}t+n\theta_{r}^{(1)}+k\theta_{r}^{(2)}+{a}_{r},$
(25) $\displaystyle x_{r}^{-}(n,k)=$
$\displaystyle\lambda_{r}^{*}x+{i}\beta_{r}^{*}t-n\theta_{r}^{(1)}-k\theta_{r}^{(2)}+{a}_{r}^{*},$

where $\alpha_{r},\beta_{r},\theta_{r}^{(1)}$, and $\theta_{r}^{(2)}$ are
coefficients from the expansions

$\displaystyle p(\kappa)-p_{0}=$
$\displaystyle\sum\limits_{r=1}^{\infty}\lambda_{r}\kappa^{r},$ (26)
$\displaystyle p^{2}(\kappa)-p^{2}_{0}=$
$\displaystyle\sum\limits_{r=1}^{\infty}\beta_{r}\kappa^{r},$
$\displaystyle\ln\frac{p(\kappa)-ik_{1}}{p_{0}-ik_{1}}=$
$\displaystyle\sum\limits_{r=1}^{\infty}\theta_{r}^{(1)}\kappa^{r},$
$\displaystyle\ln\frac{p(\kappa)-ik_{2}}{p_{0}-ik_{2}}=$
$\displaystyle\sum\limits_{r=1}^{\infty}\theta_{r}^{(2)}\kappa^{r},$

and the vector $\mathbf{s}=(s_{1},s_{2},s_{3},\cdots)$ is defined by the
expansion:

$\displaystyle\ln\left[\frac{1}{\kappa}(\frac{p_{0}+p_{0}^{*}}{p_{1}})(\frac{p(\kappa)-p_{0}}{p(\kappa)+p_{0}^{*}})\right]=\sum\limits_{r=1}^{\infty}s_{r}\kappa^{r}.$
(27)

####  3.1.2. Dynamics of the bounded rogue waves

We first consider the fundamental (i.e., first-order) RW solutions of the
2-LSRI model (5), which can be regarded as the basic elements of high-order
bounded RW solutions. For this purpose, we take $N=1$ in Theorem 2, the first-
order RW solutions of the 2-LSRI model (5) are explicitly expressed as

$\displaystyle
A=\rho_{1}e^{i\left(k_{1}x+(\gamma+k_{1}^{2})t\right)}\frac{g}{f},B=\rho_{2}e^{i\left(k_{2}x+(\gamma+k_{2}^{2})t\right)}\frac{h}{f},L=\gamma-2({\rm
log}f)_{xx},$ (28)

where

$\displaystyle f=$
$\displaystyle\widehat{m}_{1,1}^{(0,0)},g=\widehat{m}_{1,1}^{(1,0)},h=\widehat{m}_{1,1}^{(0,1)},$
(29) $\displaystyle\widehat{m}_{1,1}^{(n,k)}=$
$\displaystyle(\lambda_{1}x-{i}\beta_{1}t+n\theta_{1}^{(1)}+k\theta_{1}^{(1)})(\lambda_{1}^{*}x+i\beta_{1}^{*}t-n\theta_{1}^{(1)}-k\theta_{1}^{(2)})+\zeta_{0},$

and

$\displaystyle\lambda_{1}=\frac{dp(\kappa)}{d\kappa}|_{\kappa=0},\beta_{1}=2\lambda_{1}p_{0},$
(30)
$\displaystyle\theta_{1}^{(1)}=\frac{\lambda_{1}}{p_{0}-{i}k_{1}},\theta_{1}^{(2)}=\frac{\lambda_{1}}{p_{0}-{i}k_{2}},\zeta_{0}=\frac{|\lambda_{1}|^{2}}{(p_{0}+p_{0}^{*})^{2}},$

Here we have taken $a_{1}=0$ so that the center of the fundamental RW is
located at $x=0,t=0$. After simple algebraic calculations, the above RW
solutions can also be expressed as follows:

$\displaystyle A=$
$\displaystyle\widehat{\rho}_{1}\left[1-\frac{2i(\widehat{a}_{1}\ell_{2}-\widehat{b}_{1}\ell_{1})+(\widehat{a}_{1}^{2}+\widehat{b}_{1}^{2})}{\ell_{1}^{2}+\ell_{2}^{2}+\zeta_{0}}\right],$
(31) $\displaystyle B=$
$\displaystyle\widehat{\rho}_{2}\left[1-\frac{2i(\widehat{a}_{2}\ell_{2}-\widehat{b}_{2}\ell_{1})+(\widehat{a}_{2}^{2}+\widehat{b}_{2}^{2})}{\ell_{1}^{2}+\ell_{2}^{2}+\zeta_{0}}\right],$
$\displaystyle L=$
$\displaystyle\gamma+4\frac{\ell_{1}^{2}-\ell_{2}^{2}-\zeta_{0}}{(\ell_{1}^{2}+\ell_{2}^{2}+\zeta_{0})^{2}},$

where
$\widehat{\rho}_{j}=\rho_{j}e^{i\left(k_{j}x+(\gamma+k_{j}^{2})t\right)},\ell_{1}=x+2p_{0I}t,\ell_{2}=-2p_{0R}t,\zeta_{0}=\frac{1}{4p_{0R}^{2}},\widehat{a}_{j}=\frac{p_{0R}}{p_{0R}^{2}+(p_{0I}-k_{j})^{2}},\widehat{b}_{j}=\frac{k_{j}-p_{0I}}{p_{0R}^{2}+(p_{0I}-k_{j})^{2}}$
for $j=1,2$.

This fundamental RW in the $A$ component is classified into three different
types:

  * •

a bright RW when $(p_{0I}-k_{1})^{2}\leq\frac{1}{3}p_{0R}^{2}$;

  * •

a four–petaled RW when $\frac{1}{3}p_{0R}^{2}<(p_{0I}-k_{1})^{2}<3p_{0R}^{2}$;

  * •

a dark RW when $(p_{0I}-k_{1})^{2}\geq 3p_{0R}^{2}$.

By replacing $k_{1}$ with $k_{2}$ in the corresponding parameter condition,
this classification is also valid for the fundamental RW in the $B$ component.
The LW component $L$ is always a dark RW. Fig. 4 displays these three
different types of fundamental RWs in both $A$ and $B$ components.

These fundamental RWs admit the center amplitudes given below:

$\displaystyle|A^{c}|=$
$\displaystyle|\rho_{1}|\left|1-\frac{4p_{0R}^{2}}{p_{0R}^{2}+(p_{0I}-k_{1})^{2}}\right|,$
(32) $\displaystyle|B^{c}|=$
$\displaystyle|\rho_{2}|\left|1-\frac{4p_{0R}^{2}}{p_{0R}^{2}+(p_{0I}-k_{2})^{2}}\right|,$
$\displaystyle L^{c}=$ $\displaystyle\gamma-16p_{0R}^{2},$

which are calculated at the origin [68, 69]. The peak-to-background ratios of
the fundamental RWs in the two SW components are
$|A^{(c)}|/|\rho_{1}|,|B^{(c)}|/|\rho_{2}|\leq 3$, which indicates that the
fundamental RWs cannot reach a peak amplitude that exceeds three times the
background level, as in the case of fundamental RWs in the 2-NLS equations
[68, 69].

The higher-order bounded RW solutions in Theorem 2 are superpositions of
$N(N+1)/2$ ($N\geq 2$) fundamental RWs (28). The coexistence of these
$N(N+1)/2$ fundamental RWs can generate diverse waveforms. For instance,
taking $N=2$ and $N=3$ in Theorem 2, the second-order and third-order RW
solutions can be derived, respectively. These second-order and third-order RWs
are displayed in Fig. 5. It is seen that the second-order RWs are composed of
three fundamental RWs (the left panels of Fig.5), which form triangle
patterns. The third-order RWs comprising six fundamental RWs exhibit ring
waveforms (the right panels of Fig.5).

Figure 4. (Colour online) The three different types of first-order RW
solutions (31) in the LSRI model. The leftmost panels:
$\delta_{1}=-1,\delta_{2}=-1,\rho_{1}=\frac{5\sqrt{6}}{6},\rho_{2}=\frac{2\sqrt{21}}{3},k_{1}=\frac{3}{2},k_{2}=0,\gamma=0,p_{1}^{[1]}=1+i$;
The middle panels:
$\delta_{1}=-1,\delta_{2}=1,\rho_{1}=\frac{2\sqrt{21}}{3},\rho_{2}=\frac{5\sqrt{6}}{3},k_{1}=0,k_{2}=-1$;
The rightmost panels:
$\delta_{1}=-1,\delta_{2}=-1,\rho_{1}=\frac{5\sqrt{2}}{2},\rho_{2}=\frac{\sqrt{14}}{2},k_{1}=-1,k_{2}=1,\gamma=0,p_{1}^{[1]}=1+i$.
Figure 5. (Colour online) The left panels: the second-order bounded RW
solutions (19) with parameters
$N=2,\delta_{1}=-1,\delta_{2}=-1,\rho_{1}=\frac{5\sqrt{2}}{2},\rho_{2}=\frac{\sqrt{14}}{2},k_{1}=-1,k_{2}=1,\gamma=0,p_{1}^{[1]}=1+i,a_{1}=0,a_{2}=0,a_{3}=100i$.
The right panels: the third-order bounded RW solutions (19) with parameters
$N=3,\delta_{1}=-1,\delta_{2}=-1,\rho_{1}=\frac{5\sqrt{2}}{2},\rho_{2}=\frac{\sqrt{14}}{2},k_{1}=-1,k_{2}=1,\gamma=0,p_{1}^{[1]}=1+i,a_{1}=0,a_{2}=0,a_{3}=0,a_{4}=0,a_{5}=1000-5000i$.

###  3.2. The mixed bounded $(N_{1},N_{2})$-th order rogue wave solutions and
their dynamics

####  3.2.1. The mixed bounded $(N_{1},N_{2})$-th order rogue wave solutions

If $p_{0}^{(1)}$ and $p_{0}^{(2)}$ ($p_{0}^{(1)}\neq p_{0}^{(2)}$) are two
non-imaginary simple roots of Eq. (15), the mixed RW solutions comprising of
bounded $N_{1}$th-order RWs and $N_{2}$th-order RWs are given by the following
Theorem.

######  Theorem 3.

The 2-LSRI model (5) admits the following mixed bounded $(N_{1},N_{2})$th-
order RW solutions

$\displaystyle
A=\rho_{1}e^{i\left(k_{1}x+(\gamma+k_{1}^{2})t\right)}\frac{g}{f},B=\rho_{2}e^{i\left(k_{2}x+(\gamma+k_{2}^{2})t\right)}\frac{h}{f},L=\gamma-2({\rm
log}f)_{xx},$ (33)

where

$\displaystyle f=\tau_{0,0},g=\tau_{1,0},h=\tau_{0,1}$ (34)

and $\tau_{n,k}$ is defined as the following $2\times 2$ block determinant:

$\displaystyle\tau_{n,k}=\det\left(\begin{matrix}\tau_{n,k}^{[1,1]}&\tau_{n,k}^{[1,2]}\\\
\tau_{n,k}^{[2,1]}&\tau_{n,k}^{[2,2]}\end{matrix}\right),$ (35)
$\displaystyle\tau_{n,k}^{[\overline{\alpha},\overline{\beta}]}=\left(m_{2s-1,2j-1}^{(n,k,\overline{\alpha},\overline{\beta})}\right)_{1\leq
s\leq N_{{\overline{\alpha}}},1\leq j\leq N_{\overline{\beta}}},$ (36)

for $\overline{\alpha},\overline{\beta}=1,2$, and the matrix elements
${m}_{s,j}^{(n,k,\overline{\alpha},\overline{\beta})}$ are given either (a) in
differential operator form:

$\displaystyle{m}_{s,j}^{(n,k,\overline{\alpha},\overline{\beta})}=$
$\displaystyle\left.\frac{[\mathcal{T}(p)\partial_{p}]^{s}}{s!}\frac{[\widehat{\mathcal{T}}(\overline{p}^{*})\partial_{\overline{p}^{*}}]^{j}}{j!}m^{(n,k,\overline{\alpha},\overline{\beta})}\right|_{p=p_{0}^{(\overline{\alpha})},\overline{p}=p_{0}^{(\overline{\beta})}},$
(37) $\displaystyle m^{(n,k,\overline{\alpha},\overline{\beta})}=$
$\displaystyle\frac{1}{{p}+\overline{p}^{*}}(-\frac{{p}-ik_{1}}{\overline{p}^{*}+ik_{1}})^{n}(-\frac{{p}-ik_{2}}{\overline{p}^{*}+ik_{2}})^{k}e^{\xi_{\overline{\alpha}}+\overline{\xi}_{\overline{\beta}}^{*}}.$

or (b) through Schur polynomials:

$\displaystyle\widehat{{m}}_{s,j}^{(n,k,\overline{\alpha},\overline{\beta})}=$
$\displaystyle\sum\limits_{\nu=0}^{\mathrm{min}(i,j)}(\frac{1}{p_{0}^{(\overline{\alpha})}+p_{0}^{(\overline{\beta})}})[\frac{|\lambda_{1,\alpha}|^{2}}{(p_{0}^{(\overline{\alpha})}+p_{0}^{(\overline{\beta})})^{2}}]^{\nu}\mathbf{S}_{i-\nu}({\mathbf{x}^{+}_{\overline{\alpha},\overline{\beta}}}(n,k)+\nu{\bf
s_{\overline{\alpha},\overline{\beta}}})\mathbf{S}_{j-\nu}({\mathbf{x}^{-}_{\overline{\alpha},\overline{\beta}}}(n,k)+\nu{\bf
s_{\overline{\beta},\overline{\alpha}}^{*}}),$ (38)

Here $N_{1},N_{2}$ are arbitrary positive integers. The auxiliary functions in
Eq. (37) are defined by

$\displaystyle\mathcal{T}(p)=$
$\displaystyle\sqrt{\frac{\mathcal{Q}^{2}(p)-\mathcal{Q}^{2}(p_{0}^{(\overline{\alpha})})}{\mathcal{Q}^{{}^{\prime}2}(p)}},\widehat{\mathcal{T}}(\overline{p})=\sqrt{\frac{\mathcal{\widehat{Q}}^{2}(\overline{p})-\mathcal{\widehat{Q}}^{2}({p}_{0}^{(\overline{\beta})*})}{\mathcal{\widehat{Q}}^{{}^{\prime}2}(\overline{p})}},$
(39) $\displaystyle\xi_{\overline{\alpha}}=$ $\displaystyle
px-\mathrm{i}p^{2}t+\sum\limits_{r=1}^{\infty}\widehat{a}_{r,\alpha}\ln^{r}\mathcal{W}^{(\overline{\alpha})}(p),\overline{\xi}_{\overline{\beta}}=\overline{p}x-\mathrm{i}\overline{p}^{2}t+\sum\limits_{r=1}^{\infty}\widehat{a}^{*}_{r,\beta}\ln^{r}\mathcal{\widehat{W}^{(\overline{\beta})}}(\overline{p}),$
$\displaystyle\mathcal{W}^{(\overline{\alpha})}(p)=$
$\displaystyle\frac{\mathcal{Q}(p)\pm\sqrt{\mathcal{Q}^{2}(p)-\mathcal{Q}(p^{(\overline{\alpha})}_{0})}}{\mathcal{Q}(p^{(\overline{\alpha})}_{0})},\mathcal{\widehat{W}}^{(\overline{\beta})}(\overline{p})=\frac{\mathcal{\widehat{Q}}(\overline{p})\pm\sqrt{\mathcal{\widehat{Q}}^{2}(\overline{p})-\mathcal{\widehat{Q}}({p}^{(\overline{\beta})}_{0})}}{\mathcal{\widehat{Q}}(p_{0}^{(\overline{\beta})*})},$
$\displaystyle\mathcal{\widehat{Q}}(\overline{p})=$
$\displaystyle\frac{\delta_{1}\rho_{1}^{2}}{\overline{p}+ik_{1}}+\frac{\delta_{2}\rho_{2}^{2}}{\overline{p}+ik_{2}}-i\overline{p}^{2},$

where $\mathcal{Q}$ is given by Eq. (16). The vectors
$\mathbf{x}_{\overline{\alpha},\overline{\beta}}^{\pm}(n,k)=(x_{1,\overline{\alpha},\overline{\beta}}^{\pm},x_{2,\overline{\alpha},\overline{\beta}}^{\pm},\cdots)$
in Eq. (38) are defined as:

$\displaystyle x_{r,\overline{\alpha},\overline{\beta}}^{+}(n,k)=$
$\displaystyle\lambda_{r,\overline{\alpha}}x+{i}\beta_{r,\overline{\alpha}}t+n\theta_{r,\overline{\alpha}}^{(1)}+k\theta_{r,\overline{\alpha}}^{(2)}-b_{r,\overline{\alpha},\overline{\beta}}+{a}_{r,\overline{\alpha}},$
(40) $\displaystyle x_{r,\overline{\alpha},\overline{\beta}}^{-}(n,k)=$
$\displaystyle\lambda_{r,\overline{\beta}}^{*}x-{i}\beta_{r,\overline{\beta}}^{*}t-n\theta_{r,\overline{\beta}}^{(1)}-k\theta_{r,\overline{\beta}}^{(2)}-b_{r,\overline{\beta},\overline{\alpha}}^{*}+{a}_{r,\overline{\beta}}^{*},$

where
$\lambda_{r,\overline{\alpha}},\beta_{r,\overline{\alpha}},\theta_{r,\overline{\alpha}}^{(1)},\theta_{r,\overline{\alpha}}^{(2)}$,
and $b_{r,\overline{\alpha},\overline{\beta}}$ are coefficients from the
expansions

$\displaystyle p(\kappa)-p_{0}^{(\overline{\alpha})}=$
$\displaystyle\sum\limits_{r=1}^{\infty}\lambda_{r,\overline{\alpha}}\kappa^{r},$
(41) $\displaystyle p^{2}(\kappa)-p^{(\overline{\alpha})2}_{0}=$
$\displaystyle\sum\limits_{r=1}^{\infty}\beta_{r,\overline{\alpha}}\kappa^{r},$
$\displaystyle\ln\frac{p(\kappa)-ik_{1}}{p_{0}^{(\overline{\alpha})}-ik_{1}}=$
$\displaystyle\sum\limits_{r=1}^{\infty}\theta_{r,\overline{\alpha}}^{(1)}\kappa^{r},$
$\displaystyle\ln\frac{p(\kappa)-ik_{2}}{p_{0}^{(\overline{\alpha})}-ik_{2}}=$
$\displaystyle\sum\limits_{r=1}^{\infty}\theta_{r,\overline{\alpha}}^{(2)}\kappa^{r},$
$\displaystyle\ln\left[\frac{p^{(\overline{\alpha})}(\kappa)+p_{0}^{(\overline{\beta})}}{p_{0}^{(\overline{\alpha})}+p_{0}^{(\overline{\beta})}}\right]=$
$\displaystyle\sum\limits_{r=1}^{\infty}b_{r,\overline{\alpha},\overline{\beta}}\kappa^{r},$
$\displaystyle\mathcal{Q}[p^{(\overline{\alpha})}(\kappa)]=$
$\displaystyle\mathcal{Q}(p_{0}^{(\overline{\alpha})})\cosh(\kappa),$

and $\mathcal{Q}$ are defined by Eq. (24).

####  3.2.2. The dynamics of the mixed bounded $(N_{1},N_{2})$-th order rogue
waves

This family of RW solutions is a mixture of a bounded $N_{1}$th-order RW and
another different bounded $N_{2}$th-order RW, thus it comprises of
$\frac{N_{1}(N_{1}+1)+N_{2}(N_{2}+1)}{2}$ fundamental RWs. To exhibit the
dynamics of the mixed bounded RWs, here we take the two non-imaginary simple
roots $p_{0}^{(\alpha)}$ ($\alpha=1,2$) of Eq. (15) and the parameters
$\delta_{\alpha},k_{\alpha},\rho_{\alpha}$ as

$\displaystyle p_{0}^{(1)}=\frac{1}{2}+\frac{1}{3}i,p_{0}^{(2)}\approx
0.5466-1.11i,\delta_{1}=1,\delta_{2}=1,$ (42) $\displaystyle
k_{1}=1,k_{2}=-1,\rho_{1}=\frac{25\sqrt{4002}}{1656},\rho_{2}=\frac{73\sqrt{46}}{552},h=0.$

We first consider the simplest RWs in this solution family, which are composed
of two different fundamental RWs. For this purpose, we take $N_{1}=N_{2}=1$ in
Theorem 3. Then, the corresponding mixed bounded RW solution is

$\displaystyle
A=\rho_{1}e^{i\left(k_{1}x+(\gamma+k_{1}^{2})t\right)}\frac{g}{f},B=\rho_{2}e^{i\left(k_{2}x+(\gamma+k_{2}^{2})t\right)}\frac{h}{f},L=\gamma-2({\rm
log}f)_{xx},$ (43)

where

$\displaystyle f=$
$\displaystyle\begin{vmatrix}\widehat{m}_{1,1}^{(0,0,1,1)}&\widehat{m}_{1,1}^{(0,0,1,2)}\\\
\widehat{m}_{1,1}^{(0,0,2,1)}&\widehat{m}_{1,1}^{(0,0,2,2)}\end{vmatrix},\quad
g=$
$\displaystyle\begin{vmatrix}\widehat{m}_{1,1}^{(1,0,1,1)}&\widehat{m}_{1,1}^{(1,0,1,2)}\\\
\widehat{m}_{1,1}^{(1,0,2,1)}&\widehat{m}_{1,1}^{(1,0,2,2)}\end{vmatrix},\quad
h=$
$\displaystyle\begin{vmatrix}\widehat{m}_{1,1}^{(0,1,1,1)}&\widehat{m}_{1,1}^{(0,1,1,2)}\\\
\widehat{m}_{1,1}^{(0,1,2,1)}&\widehat{m}_{1,1}^{(0,1,2,2)}\end{vmatrix},$
(44)

and

$\displaystyle\widehat{m}_{1,1}^{(n,k,\overline{\alpha},\overline{\beta})}=$
$\displaystyle\frac{1}{p_{0}^{(\overline{\alpha})}+p_{0}^{(\overline{\beta})*}}\left[x_{1,\overline{\alpha},\overline{\beta}}^{+}(n,k)x_{1,\overline{\alpha},\overline{\beta}}^{-}(n,k)+\frac{\lambda_{1,\overline{\alpha}}\lambda_{1,\overline{\beta}}^{*}}{(p_{0}^{(\overline{\alpha})}+p_{0}^{(\overline{\beta})*})^{2}}\right],$
(45) $\displaystyle x_{\overline{\alpha},\overline{\beta}}^{+}(n,k)=$
$\displaystyle\alpha_{1,\overline{\alpha}}x+\mathrm{i}\beta_{1,\overline{\alpha}}t+(n+\frac{1}{2})\lambda_{1,\overline{\alpha}}+n_{1}\theta_{1,\overline{\alpha}}^{(1)}+n_{2}\theta_{1,\overline{\alpha}}^{(2)}-b_{1,\overline{\alpha},\overline{\beta}}+a_{1,\overline{\alpha}},$
$\displaystyle x_{\overline{\alpha},\overline{\beta}}^{-}(n,k)=$
$\displaystyle\alpha_{1,\overline{\beta}}^{*}x-\mathrm{i}\beta_{1,\overline{\beta}}^{*}t-(n+\frac{1}{2})\lambda_{1,\overline{\beta}}^{*}-n_{1}\theta_{1,\overline{\beta}}^{(1)*}-n_{2}\theta_{1,\overline{\beta}}^{(2)*}-b_{1,\overline{\beta},\overline{\alpha}}^{*}+a_{1,\overline{\beta}}^{*},$

for $\overline{\alpha},\overline{\beta}=1,2$. Here we have taken
$a_{0}^{[1]}=1$ and $a_{0}^{[2]}=1$ for simplicity. The degree of polynomials
in functions $f,g$, and $h$ is four in both $x$ and $t$.

According to the classifications of the first-order RW solution (31) discussed
previously, since
$\frac{1}{3}p_{0R}^{(1)2}<(p_{0I}^{(1)}-k_{1})^{2}<3p_{0R}^{(1)2},(p_{0I}^{(1)}-k_{1})^{2}>{3}p_{0R}^{(2)2}$
and
$(p_{0I}^{(2)}-k_{1})^{2}<\frac{1}{3}p_{0R}^{(1)2},(p_{0I}^{(2)}-k_{2})^{2}>{3}p_{0R}^{(2)2}$,
thus the $A$ component contains a dark RW and a four–petaled RW, while the $B$
component comprises a bright RW and a dark RW. Fig. 6 shows this mixed RW
solution. It is seen that the component $A$ features a four–petaled RW of
larger shape coexisting with a dark RW of smaller shape, while the component
$B$ displays a dark RW of smaller shape mixing with a bright RW of bigger
shape.

Figure 6. (Colour online) The mixed RW solution (43) comprising of two first-
order RW with parameters given by Eq. (42) and
$N_{1}=1,N_{2}=1,a_{1,1}=0,a_{1,2}=10$.  Figure 7. (Colour online) The
leftmost side panels: the mixed RW solution (33) comprising of a first-order
RW and a bounded second-order RW with parameters given by Eq. (42) and
$N_{1}=1,N_{2}=2,a_{1,1}=0,a_{1,2}=0,a_{2,2}=0,a_{3,2}=400$. The middle
panels: the mixed RW solution (33) comprising of a first-order RW and a
second-order bounded RW with parameters given by Eq. (42) and
$N_{1}=2,N_{2}=1,a_{1,1}=0,a_{2,1}=0,a_{3,1}=200i,a_{1,2}=0$. The rightmost
panels: the mixed RW solution (33) consisting of two second-order bounded RWs
with with parameters (42) and
$N_{1}=2,N_{2}=2,a_{1,1}=0,a_{2,1}=0,a_{3,1}=500i,a_{1,2}=0,a_{2,2}=0,a_{3,2}=200.$

For larger $N_{1}$, or $N_{2}$, or both, the higher-order mixed bounded RWs
can be generated, which are composed of more fundamental RWs. For instance, by
taking $N_{1}=1,N_{2}=2$ or $N_{1}=2,N_{2}=1$ in Theorem 3, the corresponding
solutions consist of a first-order RW and a second-order bounded RW (i.e.,
four first-order RWs), and the degree of the corresponding functions $f,g$,
and $h$ are eight in both $x$ and $t$. The leftmost and middle panels of Fig.
7 display the mixed RWs with $N_{1}=1,N_{2}=2$ and $N_{1}=2,N_{2}=1$,
respectively. It is seen that the component $B$ is composed of three bounded
fundamental RWs of bright type and a fundamental dark RW when
$N_{1}=1,N_{2}=2$ (see the leftmost panels of Fig. 7), while it features three
bounded fundamental RWs of dark type and a fundamental bright RW when
$N_{1}=2,N_{2}=1$ (see the middle panels of Fig. 7 ). With $N_{1}=2,N_{2}=2$,
the corresponding RW solutions (33) are comprised of two bounded second-order
RWs, which are displayed in the rightmost panels of Fig. 7. It is seen that
there are three bounded fundamental bright RWs and three bounded fundamental
dark RWs in the $A$ component, and three bounded fundamental RWs of dark type
mixing with three bounded RWs of four–petaled type in the $A$ component.

It is noted that such mixed RWs were studied for the two component NLS
equations [47, 48, 49, 50, 51, 52], but they have not been reported for the
2-LSRI model before, to the best of our knowledge. Additionally, in Theorem 3,
the parameters $N_{1},N_{2}$ cannot be zero, thus the $2\times 2$ block
determinant in Theorem 3 cannot degenerate into a single block determinant.
Therefore, the corresponding mixed bounded RWs cannot reduce to pure bounded
higher-order RWs, i.e., they are always in mixed bounded states.

###  3.3. The degradable bounded $(\widehat{N}_{1},\widehat{N}_{2})$-th order
rogue wave solutions and their dynamics

####  3.3.1. The degradable bounded $(\widehat{N}_{1},\widehat{N}_{2})$-th
order rogue wave solutions

If $\overline{p}_{0}$ is a non-imaginary double root of Eq. (15), a family of
the degradable bounded RW solutions to the 2-LSRI model (5) can be
constructed, which are given by the following Theorem.

######  Theorem 4.

The 2-LSRI model (5) has the following degradable bounded
$(\widehat{N}_{1},\widehat{N}_{2})$th-order RW solutions

$\displaystyle
A=\rho_{1}e^{i\left(k_{1}x+(\gamma+k_{1}^{2})t\right)}\frac{g}{f},B=\rho_{2}e^{i\left(k_{2}x+(\gamma+k_{2}^{2})t\right)}\frac{h}{f},L=\gamma-2({\rm
log}f)_{xx},$ (46)

where

$\displaystyle f=\tau_{0,0},g=\tau_{1,0},h=\tau_{0,1}$ (47)

and $\tau_{n,k}$ is defined as the following $2\times 2$ block determinant:

$\displaystyle\tau_{n,k}=\det\left(\begin{matrix}\tau_{n,k}^{[1,1]}&\tau_{n,k}^{[1,2]}\\\
\tau_{n,k}^{[2,1]}&\tau_{n,k}^{[2,2]}\end{matrix}\right),$ (48)
$\displaystyle\tau_{n,k}^{[\overline{\alpha},\overline{\beta}]}=\left(m_{3s-\overline{\alpha},3j-\overline{\beta}}^{[n,k,\overline{\alpha},\overline{\beta}]}\right)_{1\leq
s\leq\widehat{N}_{\overline{\alpha}},1\leq
j\leq\widehat{N}_{\overline{\beta}}},$ (49)

for $\overline{\alpha},\overline{\beta}=1,2$, and the matrix elements
${m}_{s,j}^{(n,k,\overline{\alpha},\overline{\beta})}$ are given either (a) in
differential operator form:

$\displaystyle{m}_{s,j}^{(n,k,\overline{\alpha},\overline{\beta})}=$
$\displaystyle\left.\sum\limits_{\mu=1}^{s}\sum\limits_{l=1}^{j}\frac{[\mathcal{T}(p)\partial_{p}]^{s}}{s!}\frac{[{\mathcal{T}}^{*}({p})\partial_{{p}^{*}}]^{j}}{j!}m^{(n,k,\overline{\alpha},\overline{\beta})}\right|_{p=\overline{p}_{0}},$
(50) $\displaystyle m^{(n,k,\overline{\alpha},\overline{\beta})}=$
$\displaystyle\frac{1}{{p}+{p}^{*}}(-\frac{{p}-ik_{1}}{{p}^{*}+ik_{1}})^{n}(-\frac{{p}-ik_{2}}{{p}^{*}+ik_{2}})^{k}e^{\xi_{\overline{\alpha}}+{\xi}^{*}_{\overline{\beta}}}.$

or (b) through Schur polynomials:

$\displaystyle\widehat{{m}}_{s,j}^{(n,k,\overline{\alpha},\overline{\beta})}=$
$\displaystyle\sum\limits_{\nu=0}^{\mathrm{min}(i,j)}[\frac{|\lambda_{1}|^{2}}{(p_{0}+p_{0}^{*})^{2}}]^{\nu}\mathbf{S}_{i-\nu}({\mathbf{x}}_{\overline{\alpha}}^{+}(n,k)+\nu{\bf
s})\mathbf{S}_{j-\nu}({\mathbf{x}}_{\overline{\beta}}^{-}(n,k)+\nu{\bf
s}^{*}).$ (51)

Here $\widehat{N}_{1},\widehat{N}_{2}$ are arbitrary non-negative integers.
The auxiliary functions $\mathcal{T}(p)$ in Eq. (50) is given through
$\mathcal{Q}(p)$

$\displaystyle\left(\mathcal{T}(p)\partial_{p}\right)^{3}\mathcal{Q}(p)=\mathcal{Q}(p),$
(52)

and $\xi_{\overline{\alpha}}$ is defined by

$\displaystyle\xi_{\overline{\alpha}}=$ $\displaystyle px-
ip^{2}t+\sum\limits_{r=1}^{\infty}a_{r,\overline{\alpha}}\ln^{r}\mathcal{W}(p).$
(53)

The vectors
$\mathbf{x}_{\overline{\alpha}}^{\pm}(n,k)=(x_{1,\overline{\alpha}}^{\pm},x_{2,\overline{\alpha}}^{\pm},\cdots)$
in Eq. (51) are defined as:

$\displaystyle x_{r,\overline{\alpha}}^{+}(n,k)=$
$\displaystyle\lambda_{r,\overline{\alpha}}x+{i}\beta_{r,\overline{\alpha}}t+n\theta_{r,\overline{\alpha}}^{(1)}+k\theta_{r,\overline{\alpha}}^{(2)}+{a}_{r,\overline{\alpha}},$
(54) $\displaystyle x_{r,\overline{\alpha}}^{-}(n,k)=$
$\displaystyle\lambda_{r,\overline{\alpha}}^{*}x-{i}\beta_{r,\overline{\alpha}}^{*}t-n\theta_{r,\overline{\alpha}}^{(1)}-k\theta_{r,\overline{\alpha}}^{(2)}+{a}_{r,\overline{\alpha}}^{*},$

where $\alpha_{r},\beta_{r},\theta_{r}^{(1)}$, and $\theta_{r}^{(2)}$ are
defined in Eq. (26) with $p_{0}$ replaced by $\overline{p}_{0}$,
$\mathbf{s}=(s_{1},s_{2},s_{3},\cdots)$ is given by Eq. (27) with $p_{0}$
replaced by $\overline{p}_{0}$, and the function $p(\kappa)$ appearing in Eqs.
(26),(27) are defined by the following equation:

$\displaystyle\mathcal{Q}(p)=\frac{\mathcal{Q}(p_{0})}{3}\left[e^{\kappa}+2e^{-\frac{\kappa}{2}}\cos(\frac{\sqrt{3}}{2}\kappa)\right],$
(55)

where $\mathcal{Q}$ is given by Eq. (16).

######  Remark 6.

By using the same method developed in Ref. [61], one can get that the
polynomial degree of the tau functions $\tau_{n,k}$ in Theorem 2–4 are
$N(N+1)$, $N_{1}(N_{1}+1)+N_{2}(N_{2}+1)$ and
$2[\widehat{N}_{1}^{2}+\widehat{N}_{2}^{2}-\widehat{N}_{1}(\widehat{N}_{2}-1)]$
in both $x$ and $t$ variables, respectively, where $N,N_{1},N_{2}$ are
positive integers and $\widehat{N}_{1},\widehat{N}_{2}$ are non-negative
integers.

####  3.3.2. Dynamics of the degradable bounded rogue waves

Similar to the mixed bounded RWs in Theorem 3, the RW solutions in the above
Theorem are also given by $2\times 2$ block determinants. However, the
$2\times 2$ block determinants in this Theorem can degenerate into single
block determinants if one takes $\widehat{N}_{1}=0$ or $\widehat{N}_{2}=0$.
That is the reason we call this family of solutions degradable RW solutions.
In what follows, we will reveal the dynamics of the degenerate single-block RW
solutions and the non-degenerate $2\times 2$ block solutions, respectively.
For this purpose, we take the following parameters in the solutions (46) of
Theorem 4

$\displaystyle\overline{p}_{0}=\frac{1}{4}(\sqrt{15}+3i),\rho_{1}=\frac{2\sqrt{6}}{3},\rho_{2}=\frac{5\sqrt{3}}{3},\delta_{1}=-1,\delta_{2}=-1,k_{1}=1,k_{2}=-\frac{1}{2}.$
(56)

With this set of parameters, upon the classifications of the first-order RW
solutions (31) discussed previously, the solutions in the components $A$ and
$B$ are dark RWs and bright RWs, respectively.

We first consider the degenerate single-block RW solutions (i.e.,
$\widehat{N}_{1}=0$ or $\widehat{N}_{2}=0$). In the degenerate case of
$\widehat{N}_{1}=0,\widehat{N}_{2}\neq 0$ and the degenerate case of
$\widehat{N}_{1}\neq 0,\widehat{N}_{2}=0$, the dynamical features of the
corresponding RWs are diverse. In order to more clearly show the difference
between the two degenerate RWs, we next exhibit them.

When $\widehat{N}_{1}=0,\widehat{N}_{2}\neq 0$, the tau solution in Eq. (48)
has the following form:

$\displaystyle\tau_{n,k}=\det\left(\tau_{n,k}^{[2,2]}\right)=\det\left(\widehat{m}_{3s-2,3j-2}^{(n,k,2,2)}\right)_{1\leq
s,j\leq\widehat{N}_{2}},$ (57)

with $\widehat{m}_{s,j}^{(n,k,\alpha,\beta)}$ being given by Eq. (51). In this
degenerate case, the parameter $\widehat{N}_{2}$ determines the order of the
degenerate RWs, the corresponding solutions consist of $\widehat{N}_{2}^{2}$
fundamental RWs. Fig. 8 shows the degenerate RWs for $\widehat{N}_{2}=1$ and
$\widehat{N}_{2}=2$. It is seen that there is only one fundamental RW in the
degenerate solutions when $\widehat{N}_{2}=1$ (see the left panels). In this
case, the corresponding degenerate RW solution is equivalent to the
fundamental RW solutions given by Eq. (28). However, in the case of
$\widehat{N}_{2}=2$, there are four fundamental RWs (see the right panels). As
discussed previously, the mixed bounded RWs can also form a RW pattern
consisting of four fundamental RWs (see the leftmost side and middle panels of
Fig. 7). However, the four fundamental RWs in the mixed bounded solutions are
composed of three bounded RWs and a fundamental RW of different state (see the
leftmost side and middle panels of Fig. 7). In the degenerate solution, the
four fundamental RWs are bounded (see the right panels of Fig. 8).

Figure 8. (Colour online) The degenerate RW solutions (46) with
$\widehat{N}_{1}=0$ and parameters given by Eq. (56). The left panels:
$\widehat{N}_{2}=1$ and $a_{1,2}=0$, the corresponding solutions only comprise
a single fundamental RW. The right panels: $\widehat{N}_{2}=2$ and
$a_{1,2}=0,a_{2,2}=0,a_{3,2}=0,a_{4,2}=100$, the corresponding solutions
consist of four fundamental RWs.

When $\widehat{N}_{1}\neq 0,\widehat{N}_{2}=0$, the tau solution in Eq. (48)
is written in the following form:

$\displaystyle\tau_{n,k}=\det\left(\tau_{n,k}^{[1,1]}\right)=\det\left(\widehat{m}_{3s-1,3j-1}^{(n,k,1,1)}\right)_{1\leq
s,j\leq\widehat{N}_{1}},$ (58)

where $\widehat{m}_{s,j}^{(n,k,\alpha,\beta)}$ is given by Eq. (51). In this
case, the parameter $\widehat{N}_{1}$ controls the order of the corresponding
degenerate RW, and there are $(\widehat{N}_{1}^{2}+\widehat{N}_{1})$
fundamental RWs in degenerate RW solutions. The RW solutions (46) with
$\widehat{N}_{1}=1$ and $\widehat{N}_{1}=2$ are displayed in Fig. 9. It is
seen that there are two bounded fundamental RWs in the case
$\widehat{N}_{1}=1$ (see the left panels of Fig. 9) while there are six
bounded fundamental RWs when $\widehat{N}_{1}=2$ (see the right panels of Fig.
9). The solutions shown in Fig. 6 are also composed of two fundamental RWs,
but they are in different states.

Figure 9. (Colour online) The degenerate RW solutions (46) with
$\widehat{N}_{2}=0$ and parameters given by Eq. (56). The left panels:
$\widehat{N}_{1}=1$ and $a_{1,1}=0,a_{2,1}=10$, the corresponding solutions
consist of two bounded fundamental RWs. The right panels: $\widehat{N}_{1}=2$
and $a_{1,1}=0,a_{2,1}=0,a_{3,1}=0,a_{4,1}=0,a_{5,1}=500$, the corresponding
solutions consist of six bounded fundamental RWs.

Then we consider the non-degenerate RWs solutions when
$\widehat{N}_{1}\widehat{N}_{2}\neq 0$, which are expressed by the $2\times 2$
block determinants. To illustrate the dynamics of these non-degenerate RWs, we
consider the case of $\widehat{N}_{1}=2,\widehat{N}_{2}=1$. In this case, the
tau function is explicitly expressed as

$\displaystyle\tau_{n,k}=$
$\displaystyle\det\left(\begin{matrix}\tau_{n,k}^{[1,1]}&\tau_{n,k}^{[1,2]}\\\
\tau_{n,k}^{[2,1]}&\tau_{n,k}^{[2,2]}\end{matrix}\right),$ (59)
$\displaystyle=$
$\displaystyle\det\left(\begin{matrix}\widehat{m}_{2,2}^{(n,k,1,1)}&\widehat{m}_{2,5}^{(n,k,1,1)}&\widehat{m}_{2,1}^{(n,k,1,2)}\\\
\widehat{m}_{5,2}^{(n,k,1,1)}&\widehat{m}_{5,5}^{(n,k,1,1)}&\widehat{m}_{5,1}^{(n,k,1,2)}\\\
\widehat{m}_{1,2}^{(n,k,2,1)}&\widehat{m}_{1,5}^{(n,k,2,1)}&\widehat{m}_{1,1}^{(n,k,2,2)}\end{matrix}\right),$
(60)

where $\widehat{m}_{s,j}^{(n,k,\alpha,\beta)}$ is given in Eq. (51). The
degree of these tau functions is ten in both $x$ and $t$, thus the
corresponding non-degenerate RW should be composed of five fundamental RWs,
which is demonstrated in Fig. 10. Since the solutions in Theorems 2,3 consist
of $\frac{N(N+1)}{2}$ and $\frac{N_{1}(N_{1}+1)+N_{2}(N_{2}+1)}{2}$
fundamental RWs, respectively, which do not contain the solutions comprising
five fundamental RWs, thus the RW solution displayed in Fig. 10 is distinctive
in contrast with the solutions in Theorems 2,3.

Figure 10. (Colour online) The non-degenerate RW solutions (46) with
$\widehat{N}_{1}=2,\widehat{N}_{2}=1$ and parameters (56) and
$a_{1,1}=0,a_{2,1}=0,a_{3,1}=0,a_{4,1}=0,a_{5,1}=500,a_{1,2}=0$, which consist
of five bounded fundamental RWs.

##  4\. Derivation of higher-order breather and RW solutions

In this Section, we construct the general breather in Theorem 1 and RW
solutions in Theorems 2,3,4 to the 2-LSRI model (5) via the bilinear KP
hierarchy reduction method [87, 88, 89].

###  4.1. Tau functions of the 2-LSRI model

The 2-LSRI model (5) is transformed into the bilinear form,

$\displaystyle\left(D_{x}^{2}+2ik_{1}D_{x}-iD_{t}\right)g\cdot f=0,$ (62)
$\displaystyle\left(D_{x}^{2}+2ik_{2}D_{x}-iD_{t}\right)h\cdot f=0,$
$\displaystyle\left(D_{t}D_{x}-2\delta_{1}\rho_{1}^{2}-2\delta_{2}\rho_{2}^{2}\right)f\cdot
f=-2(\delta_{1}\rho_{1}^{2}|g^{2}|+\delta_{2}\rho_{2}^{2}|h|^{2}),$

through the variable transformation,

$\displaystyle
A=\rho_{1}e^{i\left(k_{1}x+(\gamma+k_{1}^{2})t\right)}\frac{g}{f},B=\rho_{2}e^{i\left(k_{2}x+(\gamma+k_{2}^{2})t\right)}\frac{h}{f},L=\gamma-2({\rm
log}f)_{xx},$ (63)

where $f$ is a real function, and $g$, $h$ are complex functions. Here the
operator $D$ is the Hirota’s bilinear differential operator [87] defined by

$\displaystyle P(D_{x},D_{y},D_{t},)F(x,y,t\cdot\cdot\cdot)\cdot
G(x,y,t,\cdot\cdot\cdot)$ $\displaystyle=$ $\displaystyle
P(\partial_{x}-\partial_{x^{{}^{\prime}}},\partial_{y}-\partial_{y^{{}^{\prime}}},\partial_{t}-\partial_{t^{{}^{\prime}}},\cdot\cdot\cdot)F(x,y,t,\cdot\cdot\cdot)G(x^{{}^{\prime}},y^{{}^{\prime}},t^{{}^{\prime}},\cdot\cdot\cdot)|_{x^{{}^{\prime}}=x,y^{{}^{\prime}}=y,t^{{}^{\prime}}=t},$

where $P$ is a polynomial of $D_{x}$,$D_{y}$,$D_{t},\cdot\cdot\cdot$.

Then we start with the general tau functions for the multi-component KP
hierarchy expressed in the forms of Gramian determinants. The following
bilinear equation in the multi-component KP hierarchy [90]

$\displaystyle(D_{x_{1}}^{2}+2\alpha_{1}D_{x_{1}}-D_{x_{2}})\tau_{n+1,k}\cdot\tau_{n,k}=0,$
(64)
$\displaystyle(D_{x_{1}}^{2}+2\alpha_{2}D_{x_{1}}-D_{x_{2}})\tau_{n,k+1}\cdot\tau_{n,k}=0,$
$\displaystyle(D_{x_{1}}D_{r}-2)\tau_{n,k}\cdot\tau_{n,k}=-2\tau_{n+1,k}\tau_{n-1,k},$
$\displaystyle(D_{x_{1}}D_{s}-2)\tau_{n,k}\cdot\tau_{n,k}=-2\tau_{n,k+1}\tau_{n,k-1},$

has the following Gramian determinant tau functions

$\displaystyle\tau_{n,k}=\det\limits_{1\leq s,j\leq N}(m_{s,j}^{(n,k)}).$ (65)

Here the matrix element $m_{ij}^{(n)}$ satisfies

$\displaystyle\partial_{x_{1}}m_{s,j}^{(n,k)}$
$\displaystyle=\psi_{s}^{(n,k)}\phi_{j}^{(n,k)},$ (66)
$\displaystyle\partial_{x_{2}}m_{s,j}^{(n,k)}$
$\displaystyle=\psi_{s}^{(n+1,k)}\phi_{j}^{(n,k)}+\psi_{s}^{(n,k)}\phi_{j}^{(n-1,k)},\partial_{x_{2}}m_{s,j}^{(n)}=\psi_{s}^{(n)}\phi_{j}^{(n,k+1)}+\psi_{s}^{(n,k)}\phi_{j}^{(n,k-1)},$
$\displaystyle m_{s,j}^{(n+1,k)}$
$\displaystyle=m_{s,j}^{(n,k)}+\psi_{s}^{(n,k)}\phi_{j}^{(n+1,k)},m_{s,j}^{(n,k+1)}=m_{s,j}^{(n,k)}+\psi_{s}^{(n,k)}\phi_{j}^{(n,k+1)},$
$\displaystyle\partial_{x_{2}}\psi_{s}^{{(n,k)}}$
$\displaystyle=\partial_{x_{1}}^{2}\psi_{s}^{(n,k)},$
$\displaystyle\psi_{s}^{{(n+1,k)}}$
$\displaystyle=(\partial_{x_{1}}-\alpha_{1})\psi_{s}^{(n,k)},\psi_{s}^{{(n,k+1)}}=(\partial_{x_{1}}-\alpha_{2})\psi_{s}^{(n,k)},$
$\displaystyle\partial_{x_{2}}\phi_{j}^{{(n,k)}}$
$\displaystyle=-\partial_{x_{1}}^{2}\phi_{s}^{(n,k)},$
$\displaystyle\phi_{s}^{{(n+1,k)}}$
$\displaystyle=-(\partial_{x_{1}}+\alpha_{1})\phi_{s}^{(n,k)},\phi_{s}^{{(n,k+1)}}=-(\partial_{x_{1}}+\alpha_{2})\phi_{s}^{(n,k)},$

where $m_{s,j}^{(n,k)},\phi_{s}^{(n,k)}$, and $\phi_{j}^{(n,k)}$ are variables
of $x_{1},x_{2},r$, and $s$.

If one restrict the above tau functions satisfying the following dimension-
reduction condition:

$\displaystyle\mathcal{L}_{0}\tau_{n,k}=(\delta_{1}\rho_{1}^{2}\partial_{s}+\delta_{2}\rho_{2}^{2}\partial_{s}+i\partial_{x_{2}})\tau_{n,k}=C\tau_{n,k},$
(67)

where

$\displaystyle\mathcal{L}_{0}=(\delta_{1}\rho_{1}^{2}\partial_{s}+\delta_{2}\rho_{2}^{2}\partial_{s}+i\partial_{x_{2}}),$
(68)

and $C$ is some constant, then the third and fourth bilinear equations in Eq.
(64) would combine into the following bilinear equation:

$\displaystyle\left(D_{x_{1}}D_{x_{2}}-2\delta_{1}-2\delta_{2}\right)\tau_{n,k}\cdot\tau_{n,k}=-2(\delta_{1}\tau_{n+1,k}\tau_{n-1,k}+\delta_{2}\tau_{n,k+1}\tau_{k-1}).$
(69)

Based on this dimension-reduction condition, the derivatives with respect to
variables $r$ and $s$ are replaced by the derivative with respect to another
variables $x_{1},x_{2}$. Then in the above tau functions, $s$ and $r$ are just
parameters that can be regarded as having any values, thus we take $s=r=0$ for
convenience. Applying the change of independent variables in the tau functions
(65)

$\displaystyle x_{1}=x,x_{2}=-it,\alpha_{1}=ik_{1},\alpha_{2}=ik_{2},$ (70)

if the tau functions $\tau_{n,k}$ satisfy the complex conjugacy condition:

$\displaystyle\tau_{n,k}^{*}(x,t)=\tau_{-n,-k}(x,t),$ (71)

then the bilinear equation (69) and the first and second bilinear equations in
Eq. (64) would become the bilinear equations (62) of the 2-LSRI model for

$f=\tau_{0,0},g=\tau_{1,0},g^{*}=\tau_{-1,0},h=\tau_{0,1},h^{*}=\tau_{0,-1}.$
(72)

In this way, we obtain the tau functions of the 2-LSRI model.

Choosing different forms of the matrix elements $m_{s,j}^{(n,k)}$, we can
construct the breather and RW solutions for the 2-LSRI model (5), the details
of the derivations will be given in the following subsections.

###  4.2. Derivation of breather solutions to the 2-LSRI model

To construct the breather solutions of the 2-LSRI model (5), we take the
matrix elements $m_{s,j}^{(n,k)}$ of tau functions $\tau_{n,k}$ given by Eq.
(65) in the following form:

$\displaystyle m_{s,j}^{(n,k)}=$
$\displaystyle\sum\limits_{\alpha,\beta=1}^{2}\frac{1}{p_{s}^{[\alpha]}+q_{j}^{[\beta]}}(-\frac{p_{s}^{[\alpha]}-\alpha_{1}}{q_{j}^{[\beta]}+\alpha_{1}})^{n}(-\frac{p_{s}^{[\alpha]}-\alpha_{2}}{q_{j}^{[\beta]}+\alpha_{2}})^{k}e^{\xi_{s}^{[\alpha]}+\eta_{j}^{[\beta]}},$
(73) $\displaystyle\psi_{s}^{(n,k)}=$
$\displaystyle\sum\limits_{\alpha=1}^{2}(p_{s}^{[\alpha]}-\alpha_{1})^{n}(p_{s}^{[\alpha]}-\alpha_{2})^{k}e^{\xi_{s}^{[\alpha]}},$
$\displaystyle\phi_{j}^{(n,k)}=$
$\displaystyle\sum\limits_{\beta=1}^{2}(-q_{j}^{[\beta]}-\alpha_{1})^{-n}(-q_{j}^{[\beta]}-\alpha_{2})^{-k}e^{\eta_{j}^{[\beta]}},$

and

$\displaystyle\xi_{s}^{[\alpha]}=$
$\displaystyle\frac{1}{p_{s}^{[\alpha]}-\alpha_{1}}r+\frac{1}{p_{s}^{[\alpha]}-\alpha_{2}}s+p_{s}^{[\alpha]}x_{1}+p_{s}^{[\alpha]2}x_{2}+\overline{\xi}_{s}^{[\alpha]},$
(74) $\displaystyle\eta_{s}^{[\beta]}=$
$\displaystyle-\frac{1}{q_{j}^{[\beta]}+\alpha_{1}}r-\frac{1}{q_{j}^{[\beta]}+\alpha_{2}}s+q_{j}^{[\beta]}x_{1}-q_{j}^{[\beta]2}x_{2}+\overline{\eta}_{j}^{[\beta]},$

where $p_{s}^{[\alpha]},q_{j}^{[\beta]},\overline{\xi}_{s}^{[\alpha]}$, and
$\overline{\eta}_{j}^{[\beta]}$ are arbitrary complex parameters. Then the tau
function in Eq. (65) can be rewritten in the following form:

$\displaystyle\tau_{n,k}=\Lambda\det\limits_{1\leq s,j\leq
N}\left(\overline{m}_{s,j}^{(n,k)}\right),$ (75)

where $\Lambda=\prod\limits_{s=1}^{N}e^{\xi_{s}^{[2]}+\eta_{j}^{[2]}}$ and

$\displaystyle\overline{m}_{s,j}^{(n,k)}=$
$\displaystyle\frac{1}{p_{s}^{[1]}+q_{j}^{[1]}}(-\frac{p_{s}^{[1]}-\alpha_{1}}{q_{j}^{[1]}+\alpha_{1}})^{n}(-\frac{p_{s}^{[1]}-\alpha_{2}}{q_{j}^{[1]}+\alpha_{2}})^{k}e^{\xi_{s}^{[1]}-\xi_{s}^{[2]}+\eta_{j}^{[1]}-\eta_{j}^{[2]}}+\frac{1}{p_{s}^{[1]}+q_{j}^{[2]}}(-\frac{p_{s}^{[1]}-\alpha_{1}}{q_{j}^{[2]}+\alpha_{1}})^{n}$
(76)
$\displaystyle\times(-\frac{p_{s}^{[1]}-\alpha_{2}}{q_{j}^{[2]}+\alpha_{2}})^{k}e^{\xi_{s}^{[1]}-\xi_{s}^{[2]}}+\frac{1}{p_{s}^{[2]}+q_{j}^{[1]}}(-\frac{p_{s}^{[2]}-\alpha_{1}}{q_{j}^{[1]}+\alpha_{1}})^{n}(-\frac{p_{s}^{[2]}-\alpha_{2}}{q_{j}^{[1]}+\alpha_{2}})^{k}e^{\eta_{j}^{[1]}-\eta_{j}^{[2]}}+$
$\displaystyle\frac{1}{p_{s}^{[2]}+q_{j}^{[2]}}(-\frac{p_{s}^{[2]}-\alpha_{1}}{q_{j}^{[2]}+\alpha_{1}})^{n}(-\frac{p_{s}^{[2]}-\alpha_{2}}{q_{j}^{[2]}+\alpha_{2}})^{k}.$

If the parameters $p_{s}^{[\alpha]},q_{j}^{[\beta]}$ satisfy the constraints

$\displaystyle\frac{\delta_{1}\rho_{1}^{2}}{(p_{s}^{[1]}-\alpha_{1})(p_{s}^{[2]}-\alpha_{2})}+\frac{\delta_{2}\rho_{2}^{2}}{(p_{s}^{[1]}-\alpha_{2})(p_{s}^{[2]}-\alpha_{2})}-i(p_{s}^{[1]}+p_{s}^{[2]})=0,$
(77)
$\displaystyle\frac{\delta_{1}\rho_{1}^{2}}{(q_{s}^{[1]}+\alpha_{1})(q_{s}^{[2]}+\alpha_{1})}+\frac{\delta_{2}\rho_{2}^{2}}{(q_{s}^{[1]}+\alpha_{2})(q_{s}^{[2]}+\alpha_{2})}+i(q_{s}^{[1]}+q_{s}^{[2]})=0,$

then from Eqs. (75), (76), we get the tau function in the form of Eq. (75)
satisfying the dimension-reduction condition (67). By taking the variable
transformations (70) into the tau functions (75), and setting the following
complex conjugacy conditions on the parameters:

$\displaystyle
q_{j}^{[\beta]}=p_{j}^{[\beta]*},\overline{\eta}_{j}^{[\beta]}=\overline{\xi}_{s}^{[\alpha]*},$
(78)

then we have

$\displaystyle\eta_{j}^{[1]}-\eta_{j}^{[2]}=\xi_{s}^{[1]*}-\xi_{s}^{[2]*},m_{s,j}^{(n,k)*}=m_{j,s}^{(-n,-k)},$
(79)

which further yields the tau function (75) satisfying the complex conjugacy
condition (71).

Thus, under the parameter constraint in Eq. (77) and the variable
transformations in Eq. (70), the tau functions (75) reduce to the solutions of
bilinear equations (62) of the 2-LSRI model (5) for
$f=\tau_{0,0},g=\tau_{1,0},g^{*}=\tau_{-1,0},h=\tau_{0,1},h^{*}=\tau_{0,-1}$.
The multiplicative factor $\Lambda$ in Eq. (75) can be removed in
$\frac{\tau_{1,0}}{\tau_{0,0}}$ and $\frac{\tau_{0,1}}{\tau_{0,0}}$, and the
variables $s$,$r$ are taken as zero, then we can obtain the breather solutions
of the 2-LSRI model (5). Theorem 1 is then proved.

###  4.3. Derivation of RW solutions to the 2-LSRI model.

In this Section, we construct the rational RW solutions to the 2-LSRI model by
starting with matrix elements $m_{s,j}^{(n,k)}$ of tau functions $\tau_{n,k}$
(65) in the following form:

$\displaystyle m_{s,j}^{(n,k)}=$
$\displaystyle\mathcal{A}_{s}\mathcal{B}_{j}m^{(n,k)},$ (80) $\displaystyle
m^{(n,k)}=$
$\displaystyle\frac{1}{p+q}(-\frac{p-\alpha_{1}}{q+\alpha_{1}})^{n}(-\frac{p-\alpha_{2}}{q+\alpha_{2}})^{k}e^{\xi+\eta},$
$\displaystyle\psi_{s}^{(n,k)}=$
$\displaystyle\mathcal{A}_{s}(p-\alpha_{1})^{n}(p-\alpha_{2})^{k}e^{\xi},$
$\displaystyle\phi_{s}^{(n,k)}=$
$\displaystyle\mathcal{B}_{j}(q+\alpha_{1})^{-n}(q+\alpha_{2})^{-k}e^{\eta},$

where

$\displaystyle\mathcal{A}_{s}=$
$\displaystyle\frac{\left[\mathcal{T}(p)\partial_{p}\right]^{s}}{s!},\mathcal{B}_{j}=\frac{\left[\mathcal{\widehat{T}}(q)\partial_{q}\right]^{j}}{j!},$
(81) $\displaystyle\xi=$
$\displaystyle\frac{1}{p-\alpha_{2}}s+\frac{1}{p-\alpha_{1}}r+px_{1}+p^{2}x_{2}+\xi_{0},$
$\displaystyle\eta=$
$\displaystyle\frac{1}{q+\alpha_{2}}s+\frac{1}{q+\alpha_{1}}r+qx_{1}-q^{2}x_{2}+\eta_{0},$

and $\mathcal{T}(p)$ and $\mathcal{\widehat{T}}(q)$ are arbitrary functions of
$p$ and $q$, respectively. It is easy to see that these functions also satisfy
the differential and difference relations (66), thus
$\tau_{n,k}=\det(m_{s,j}^{(n,k)})$ with (80) satisfies the bilinear equations
(62).

Then, we constrain the tau functions $\tau_{n,k}=\det(m_{s,j}^{(n,k)})$ with
(80) satisfying the dimension reduction condition (67). One can directly
obtain that:

$\displaystyle\mathcal{L}_{0}m_{s,j}^{(n,k)}=$
$\displaystyle\mathcal{A}_{s}\mathcal{B}_{j}m^{(n,k)}=\mathcal{A}_{s}\mathcal{B}_{j}\left(\mathcal{Q}(p)+\mathcal{\widehat{Q}}(q)\right)m^{(n,k)},$
(82)

where

$\displaystyle\mathcal{Q}(p)=\frac{\delta_{1}\rho_{1}^{2}}{p-ik_{1}}+\frac{\delta_{2}\rho_{2}^{2}}{p-ik_{2}}+ip^{2},$
(83)
$\displaystyle\mathcal{\widehat{Q}}(q)=\frac{\delta_{1}\rho_{1}^{2}}{q+ik_{1}}+\frac{\delta_{2}\rho_{2}^{2}}{q+ik_{2}}-iq^{2}.$

Following the works [60, 61], if functions $\mathcal{T}(p)$ and
$\mathcal{\widehat{T}}(q)$ are taken as

$\displaystyle\mathcal{T}(p)=\frac{\mathcal{W}(p)}{\mathcal{W}^{{}^{\prime}}(p)},\mathcal{\widehat{T}}(q)=\frac{\mathcal{\widehat{W}}(q)}{\mathcal{\widehat{W}}^{{}^{\prime}}(q)},$
(84)

we get
$\mathcal{T}(p)\partial_{p}=\partial_{\ln\mathcal{W}},\mathcal{\widehat{T}}(q)\partial_{q}=\partial_{\ln\mathcal{\widehat{W}}}$.
Upon the Leibnitz rules, Eq. (82) is rewritten as

$\displaystyle\mathcal{L}_{0}m_{s,j}^{(n,k)}=$
$\displaystyle\sum\limits_{\mu=0}^{s}\frac{1}{\mu!}\left[(\mathcal{T}(p)\partial_{p})^{\mu}\mathcal{Q}(p)\right]m_{s-\mu,j}^{(n,k)}+\sum\limits_{l=0}^{j}\frac{1}{l!}\left[(\mathcal{\widehat{T}}(q)\partial_{q})^{\ell}\mathcal{\widehat{Q}}(q)\right]m_{s,j-l}^{(n,k)}$
(85) $\displaystyle=$
$\displaystyle\sum\limits_{\mu=0}^{s}\frac{1}{\mu!}\left[\mathcal{W}(p)+(-1)^{\mu}\frac{1}{\mathcal{W}(p)}\right]m_{s-\mu,j}^{(n,k)}+\sum\limits_{l=0}^{j}\frac{1}{l!}\left[\mathcal{\widehat{W}}(q)+(-1)^{l}\frac{1}{\mathcal{\widehat{W}}(q)}\right]m_{s,j-l}^{(n,k)}.$

As in Ref. [61], selecting proper forms of functions
$\mathcal{T}(p),\mathcal{\widehat{T}}(q)$ can give rise to the coefficients of
certain indices on the right hand side of the above equation vanishing at some
values of $p,q$. To this end, we will choose $p=p_{0},q=q_{0}$ as the roots of
the following algebraic equation:

$\displaystyle\frac{\partial\mathcal{Q}}{\partial
p}=0,\frac{\partial\mathcal{\widehat{Q}}}{\partial q}=0,$ (86)

namely,

$\displaystyle\mathcal{Q}^{{}^{\prime}}(p_{0})=0,\mathcal{\widehat{Q}}^{{}^{\prime}}(q_{0})=0.$
(87)

Here $p_{0},q_{0}$ are not pure imaginary, and $p_{0R}>0,q_{0R}>0$.

(1) When $p_{0},q_{0}$ are simple roots of the algebraic equation (86), we
impose functions $\mathcal{T}(p),\mathcal{\widehat{T}}(q)$ meeting the
following condition:

$\displaystyle\left(\mathcal{T}(p)\partial_{p}\right)^{2}\mathcal{Q}(p)=\mathcal{Q}(p),\left(\mathcal{\widehat{T}}(q)\partial_{q}\right)^{2}\mathcal{\widehat{Q}}(q)=\mathcal{\widehat{Q}}(q).$
(88)

From Eq. (84), the above condition is rewritten as

$\displaystyle\partial^{2}_{\ln{\mathcal{W}}}\mathcal{Q}(p)=\mathcal{Q}(p),\partial^{2}_{\ln{\mathcal{\widehat{W}}}}\mathcal{\widehat{Q}}(q)=\mathcal{\widehat{Q}}(q),$
(89)

With Eq. (86) and the scaling
$\mathcal{W}(p_{0})=1,\mathcal{\widehat{W}}(q_{0})=1$, we obtain:

$\displaystyle\mathcal{Q}(p)=\frac{1}{2}Q(p_{0})\left(\mathcal{W}(p)+\frac{1}{\mathcal{W}(p)}\right),\mathcal{\widehat{Q}}(q)=\frac{1}{2}\mathcal{\widehat{Q}}(q_{0})\left(\mathcal{\widehat{W}}(q)+\frac{1}{\mathcal{\widehat{W}}(q)}\right)$
(90)

and

$\displaystyle\mathcal{W}(p)=\frac{\mathcal{Q}(p)\pm\sqrt{\mathcal{Q}^{2}(p)-\mathcal{Q}^{2}(p_{0})}}{\mathcal{Q}(p_{0})},\mathcal{\widehat{W}}(q)=\frac{\mathcal{\widehat{Q}}(q)\pm\sqrt{\mathcal{\widehat{Q}}^{2}(q)-\mathcal{\widehat{Q}}^{2}(q_{0})}}{\mathcal{\widehat{Q}}(q_{0})}.$
(91)

From Eq. (84), the explicit forms of $\mathcal{T}(p),\mathcal{\widehat{T}}(q)$
are given through $\mathcal{Q}(p),\mathcal{\widehat{Q}}(q)$ as:

$\displaystyle\mathcal{T}(p)=$
$\displaystyle\sqrt{\frac{\mathcal{Q}^{2}(p)-\mathcal{Q}^{2}(p_{0})}{\mathcal{Q}^{{}^{\prime}2}(p)}},\mathcal{\widehat{T}}(q)=$
$\displaystyle\sqrt{\frac{\mathcal{\widehat{Q}}^{2}(q)-\mathcal{\widehat{Q}}^{2}(q_{0})}{\mathcal{\widehat{Q}}^{{}^{\prime}2}(q)}}.$
(92)

The Eq. (85) becomes

$\displaystyle\left.\mathcal{L}_{0}m_{s,j}^{(n,k)}\right|_{p=p_{0},q=q_{0}}=$
$\displaystyle\left.\mathcal{Q}(p_{0})\sum\limits_{\mu=0}^{s}\frac{1}{\mu!}m_{s-\mu,j}^{(n,k)}\right|_{p=p_{0},q=q_{0}}+\left.\mathcal{\widehat{Q}}(q_{0})\sum\limits_{l=0}^{j}\frac{1}{l!}m_{s,j-l}^{(n,k)}\right|_{p=p_{0},q=q_{0}}.$
(93)

Then, for the restricted indices of the determinants (65) of the tau
functions:

$\displaystyle\tau_{n,k}=\det\limits_{1\leq s,j\leq
N}\left(m_{2s-1,2j-1}^{(n,k)}\right),$ (94)

upon the relation (85), the above tau functions satisfy the following
relation:

$\displaystyle\mathcal{L}_{0}\tau_{n,k}=\left[\mathcal{Q}(p_{0})+\mathcal{\widehat{Q}}(q_{0})\right]N\tau_{n,k},$
(95)

which is nothing but the dimension reduction condition (67).

(2) When
$p_{0}^{({\overline{\alpha}})},q_{0}^{({\overline{\alpha}})}({\overline{\alpha}}=1,2,p_{0}^{(1)}\neq\pm
p_{0}^{(2)})$ are two simple roots of the algebraic equation (86), then the
tau functions are expressed by the following $2\times 2$ block determinants:

$\displaystyle\tau_{n,k}=\det\left(\begin{matrix}\tau_{n,k}^{[1,1]}&\tau_{n,k}^{[1,2]}\\\
\tau_{n,k}^{[2,1]}&\tau_{n,k}^{[2,2]}\end{matrix}\right),$ (96)

where

$\displaystyle\tau_{n,k}^{[{\overline{\alpha}},{\overline{\beta}}]}=\mathrm{mat}_{1\leq
s\leq N_{\overline{\alpha}},1\leq j\leq
N_{\overline{\beta}}}\left(\left.m_{2s-1,2j-1}^{(n,k)}\right|_{p=p_{0}^{({\overline{\alpha}})},q=q_{0}^{({\overline{\beta}})}}\right),1\leq{\overline{\alpha}},{\overline{\beta}}\leq
2,$ (97)

and $m_{s,j}^{(n,k)}$ are given by Eq. (80) with
$[\mathcal{T}(p),\mathcal{\widehat{T}}(q),\xi_{0},\eta_{0}]$ replaced by
$[\mathcal{T}^{[{\overline{\alpha}}]}(p),\mathcal{\widehat{T}}^{[{\overline{\beta}}]}(q),\xi_{0,{\overline{\alpha}}},\eta_{0,{\overline{\beta}}}]$,
here the functions
$\mathcal{T}^{[{\overline{\alpha}}]}(p),\mathcal{\widehat{T}}^{[{\overline{\beta}}]}(q)$
are provided by (92) with $p_{0},q_{0}$ replaced by
$p_{0}^{({\overline{\alpha}})}$ and $q_{0}^{({\overline{\beta}})}$ for
${\overline{\alpha}},{\overline{\beta}}=1,2$, respectively. These tau
functions (96) also satisfy the bilinear equations (64), which can be proved
by the same method given in Appendix C of Ref. [61], thus the proof is omitted
here.

For this $2\times 2$ block determinant forms of tau functions (96), the
contiguity relation (85) is written as

$\displaystyle\left.\mathcal{L}_{0}m_{s,j}^{(n,k)}\right|_{p=p_{0}^{({\overline{\alpha}})},q=q_{0}^{({\overline{\beta}})}}=$
$\displaystyle\left.\mathcal{Q}(p_{0}^{({\overline{\alpha}})})\sum\limits_{\mu=0}^{s}\frac{1}{\mu!}m_{s-\mu,j}^{(n,k)}\right|_{p=p_{0}^{({\overline{\alpha}})},q=q_{0}^{({\overline{\beta}})}}+\left.\mathcal{\widehat{Q}}(q_{0}^{({\overline{\beta}})})\sum\limits_{l=0}^{j}\frac{1}{l!}m_{s,j-l}^{(n,k)}\right|_{p=p_{0}^{({\overline{\alpha}})},q=q_{0}^{({\overline{\beta}})}}.$
(98)

Similar to the results reported in Ref. [45], this contiguity relation can
further yield:

$\displaystyle\mathcal{L}_{0}\tau_{n,k}=\left\\{\left[\mathcal{Q}(p_{0}^{(1)})+\mathcal{\widehat{Q}}(q_{0}^{(1)})\right]N_{1}+\left[\mathcal{Q}(p_{0}^{(2)})+\mathcal{\widehat{Q}}(q_{0}^{(2)})\right]N_{2}\right\\}\tau_{n,k}$
(99)

namely, the tau functions (96) expressed by $2\times 2$ block determinant also
satisfy the dimension reduction condition (67).

(3) When $p_{0},q_{0}$ are double roots of the algebraic equation (86),
namely,

$\displaystyle\left.\frac{\partial Q}{\partial
p}\right|_{p=p_{0}}=\left.\frac{\partial^{2}Q}{\partial
p^{2}}\right|_{p=p_{0}}=0,\,\left.\frac{\partial\widehat{Q}}{\partial
q}\right|_{q=q_{0}}=\left.\frac{\partial^{2}\widehat{Q}}{\partial
q^{2}}\right|_{q=q_{0}}=0,$ (100)

$\mathcal{T}(p),\mathcal{\widehat{T}}(p)$ have the same forms as in Eq. (84),
but they have to further meet the following condition:

$\displaystyle\left(\mathcal{T}(p)\partial_{p}\right)^{3}\mathcal{Q}(p)=\mathcal{Q}(p),\left(\mathcal{\widehat{T}}(q)\partial_{q}\right)^{3}\mathcal{\widehat{Q}}(p)=\mathcal{\widehat{Q}}(q),$
(101)

namely,

$\displaystyle\partial^{3}_{\ln\mathcal{W}}\mathcal{Q}(p)=\mathcal{Q}(p),\partial^{3}_{\ln\mathcal{\widehat{W}}}\mathcal{\widehat{Q}}(q)=\mathcal{\widehat{Q}}(q).$
(102)

Scaling $\mathcal{W}(p_{0})=\mathcal{\widehat{W}}(q_{0})=1$, a solution to
above equation under conditions (100) is

$\displaystyle\mathcal{Q}(p)=\frac{\mathcal{Q}(p_{0})}{3}\left(\mathcal{W}(p)+\frac{2}{\sqrt{\mathcal{W}(p)}}\cos\left[\frac{\sqrt{3}}{2}\ln\mathcal{W}(p)\right]\right),$
(103)
$\displaystyle\mathcal{\widehat{Q}}(q)=\frac{\mathcal{\widehat{Q}}(q_{0})}{3}\left(\mathcal{\widehat{W}}(q)+\frac{2}{\sqrt{\mathcal{\widehat{W}}(q)}}\cos\left[\frac{\sqrt{3}}{2}\ln\mathcal{\widehat{W}}(q)\right]\right).$

With $\mathcal{Q}(p),\mathcal{\widehat{Q}}(q)$ being of this form, the
following tau functions expressed by block determinants:

$\displaystyle\tau_{n,k}=\det\left(\begin{matrix}\tau_{n,k}^{[1,1]}&\tau_{n,k}^{[1,2]}\\\
\tau_{n,k}^{[2,1]}&\tau_{n,k}^{[2,2]}\end{matrix}\right),$ (104)

where

$\displaystyle\tau_{n,k}^{[\alpha,\beta]}=\mathrm{mat}_{1\leq
s\leq\widehat{N}_{\alpha},1\leq
j\leq\widehat{N}_{\beta}}\left(\left.m_{3s-\alpha,3j-\beta}^{(n,k)}\right|_{p=p_{0}^{(\alpha)},q=q_{0}^{(\beta)}}\right),1\leq\alpha,\beta\leq
2,$ (105)

also satisfy the bilinear equations (64). This result can be proved using the
method given in Appendix C of Ref. [61], thus we omit here its proof. Using
the above relations, the Eq. (85) becomes :

$\displaystyle\left.\mathcal{L}_{0}m_{s,j}^{(n,k)}\right|_{p=p_{0},q=q_{0}}=$
$\displaystyle\left.\mathcal{Q}(p_{0})\sum\limits_{{\scriptsize\begin{matrix}\mu=0\\\
\mu\equiv 0({\bf
mod}3)\end{matrix}}}^{s}\frac{1}{\mu!}m_{s-\mu,j}^{(n,k)}\right|_{p=p_{0},q=q_{0}}+\left.\mathcal{\widehat{Q}}(q_{0})\sum\limits_{{\scriptsize\begin{matrix}l=0\\\
l\equiv 0({\bf
mod}3)\end{matrix}}}^{j}\frac{1}{l!}m_{s,j-l}^{(n,k)}\right|_{p=p_{0},q=q_{0}}.$
(106)

Upon this contiguity relation, then the tau function (96) expressed by
$2\times 2$ block determinant also satisfy the dimension reduction condition
(67).

Finally, we impose the complex conjugacy condition

$\displaystyle\tau_{n,k}^{*}=\tau_{-n,-k}.$ (107)

For this purpose, we first take $\eta_{0}=\xi_{0}^{*}$ in Eq. (94) for a
simple root, and $\eta_{0,\beta}=\xi_{0,\alpha}^{*}$ in Eq. (97) and Eq. (105)
for two simple roots and a double root. Then, by applying the change of
independent variables (70), it is easy to find that $\mathcal{\widehat{Q}}$
will be the conjugate of $\mathcal{Q}$ in Eq. (83) if $q=p^{*}$, thus
$q_{0}^{*}=p_{0}$. When $p_{0},q_{0}$ are single simple roots of Eq. (86),
then

$\displaystyle\left.m_{j,s}^{(-n,-k)}\right|_{p=p_{0},q=p_{0}^{*}}=\left.\left[m_{s,j}^{(n,k)}\right]^{*}\right|_{p=p_{0},q=p_{0}^{*}},$
(108)

thus the complex conjugacy condition (107) is realized. If
$p_{0}^{[\alpha]},q_{0}^{[\beta]}$ are two simple single roots of Eq. (86),
since $q_{0}^{(\alpha)}=p_{0}^{(\alpha)*}$, thus

$\displaystyle\left.m_{j,s}^{(-n,-k)}\right|_{p=p_{0}^{(\alpha)},q=p_{0}^{(\beta)*}}=\left.\left[m_{s,j}^{(n,k)}\right]^{*}\right|_{p=p_{0}^{(\beta)},q=p_{0}^{(\alpha)*}},$
(109)

which can further imply:

$\displaystyle\tau_{n,k}^{[\alpha,\beta]}=\tau_{-n,-k}^{[\beta,\alpha]*},$
(110)

then the complex conjugacy condition (107) is also satisfied. When $p_{0}$ is
a double root of Eq. (86), the complex conjugacy condition (107) can also be
proved in a similar way.

To obtain the solutions in operator differential forms in Theorem 2–4, we take

$\displaystyle\xi_{0}=\sum\limits_{r=1}^{\infty}\widehat{a}_{r}\ln^{r}\mathcal{W}(p),$
(111)

for a simple root $p_{0}$ of Eq. (86), where $\mathcal{W}(p)$ is given by Eq.
(91), and

$\displaystyle\xi_{0,\alpha}=\sum\limits_{r=1}^{\infty}\widehat{a}_{r,\alpha}\ln^{r}\mathcal{W}^{(\alpha)}(p),\alpha=1,2$
(112)

for two simple roots $p_{0}^{(\alpha)}$ and $p_{0}^{(\beta)}$ of Eq. (86),
where $\mathcal{W}^{(\alpha)}(p)$ is defined in Eq. (91) with $p_{0}$ being
replaced by $p_{0}^{(\alpha)}$, and

$\displaystyle\xi_{0,\alpha}=\sum\limits_{r=1}^{\infty}\widehat{a}_{r,\alpha}\ln^{r}\mathcal{W}(p),\alpha=1,2$
(113)

for a double root of Eq. (86), where $\mathcal{W}(p)$ is given by Eq. (103).
Here $\widehat{a}_{r},\widehat{a}_{r,\alpha}$ are arbitrary complex
parameters. Then we can obtain the solutions in differential operator forms in
Theorem 2–4.

The last content of this Section is to convert the solutions given through
differential operator forms to Schur polynomials in Theorems 2, 3, 4. For this
purpose, we use the following generator $\mathcal{G}$ of differential
operators $\left[\mathcal{T}(p)\partial_{p}\right]^{s}$ and
$\left[\mathcal{\widehat{T}}(q)\partial_{q}\right]^{j}$ introduced in Ref.
[45],

$\displaystyle\mathcal{G}=\sum\limits_{s=0}^{\infty}\sum\limits_{j=0}^{\infty}\frac{\kappa^{s}}{s!}\frac{\lambda^{j}}{j!}\left[\mathcal{T}(p)\partial_{p}\right]^{s}\left[\mathcal{\widehat{T}}(q)\partial_{q}\right]^{j}.$
(114)

From the relations between $\mathcal{T},\mathcal{\widehat{T}}$ and
$\mathcal{W},\mathcal{\widehat{W}}$ in Eq. (84), the generator $\mathcal{G}$
can also be written as

$\displaystyle\mathcal{G}=\sum\limits_{s=0}^{\infty}\sum\limits_{j=0}^{\infty}\frac{\kappa^{s}}{s!}\frac{\lambda^{j}}{j!}\left[\partial_{\mathcal{W}(p)}\right]^{s}\left[\partial_{\mathcal{\widehat{W}}(q)}\right]^{j}=\exp\left(\kappa\partial_{\ln\mathcal{W}(p)}+\lambda\partial_{\ln\mathcal{\widehat{W}}(q)}\right),$
(115)

The operator $\mathcal{G}$ acting on an arbitrary function
$\mathcal{F}(\mathcal{W},\mathcal{\widehat{W}})$ can result in the following
relation:

$\displaystyle\mathcal{G}\mathcal{F}(\mathcal{W},\mathcal{\widehat{W}})=\mathcal{F}\left(e^{\kappa}\mathcal{W},e^{\lambda}\mathcal{\widehat{W}}\right).$
(116)

Since the parameter $p$ and $q$ are related to $\mathcal{W}$ and
$\mathcal{\widehat{W}}$, respectively, thus one can regard $p$ as a function
of $\mathcal{W}$, and $q$ as a function of $\mathcal{\widehat{W}}$, namely,
$p=p(\mathcal{W}),q=q(\mathcal{\widehat{W}})$. Furthermore,
$\mathcal{W}(p_{0})=\mathcal{\widehat{W}}(q_{0})=1$.

For $m^{(n,k)}$ in Eq. (22) of Theorem 2,

$\displaystyle\left.\frac{1}{m^{(n,k)}}\mathcal{G}m^{(n,k)}\right|_{p=p_{0},q=q_{0}}$
(117) $\displaystyle=$
$\displaystyle\frac{p_{0}+q_{0}}{p(\kappa)+q(\lambda)}\left(\frac{p(\kappa)-ik_{1}}{p_{0}-ik_{1}}\right)^{n}\left(\frac{q(\lambda)+ik_{1}}{q_{0}+ik_{1}}\right)^{-n}\left(\frac{p(\kappa)-ik_{2}}{p_{0}-ik_{2}}\right)^{k}\left(\frac{q(\lambda)+ik_{2}}{q_{0}+ik_{2}}\right)^{-k}$
$\displaystyle\exp\left(\sum\limits_{r=1}^{\infty}\widehat{a}_{r}\kappa^{r}+\widehat{a}_{r}^{*}\lambda^{r}\right)\times\exp\left[(p(\kappa)-p_{0}+q(\lambda)-q_{0})x-i(p^{2}(\kappa)-p_{0}^{2}-q^{2}(\lambda)+q^{2}_{0})t\right].$

Here we have set $q=p^{*}$ in Eq. (22). After expanding the right side of the
above equation into power series of $\kappa$ and $\lambda$, the first term can
be written into the following form [60, 61]:

$\displaystyle\frac{p_{0}+q_{0}}{p(\kappa)+q(\lambda)}=\sum\limits_{v=0}^{\infty}\left(\frac{|\lambda_{1}|^{2}}{(p_{0}+q_{0})^{2}}\kappa
v\right)^{v}\exp\left(\sum\limits_{r=1}^{\infty}(vs_{r}-b_{r})\kappa^{r}+(vs_{r}^{*}-b_{r}^{*})\lambda^{r}\right),$
(118)

where $\lambda_{1}$ is given by Eq. (26), and $b_{r}$ is given by the Taylor
coefficient of $\kappa^{r}$ in the expansion of

$\displaystyle\ln\left[\frac{p(\kappa)+q_{0}}{p_{0}+q_{0}}\right]=\sum\limits_{r=1}^{\infty}b_{r}\kappa^{r},$
(119)

and $s_{r}$ is defined in Eq. (27). The rest term can be written as

$\displaystyle\exp$
$\displaystyle\left\\{\sum\limits_{r=1}^{\infty}\kappa^{r}\left[\lambda_{r}x-i\beta_{r}t+n\theta_{r}^{(1)}+k\theta_{r}^{(1)}\right]+\sum\limits_{r=1}^{\infty}\lambda^{r}\left[\lambda^{*}_{r}x+i\beta^{*}_{r}t\right.\right.$
(120)
$\displaystyle\left.\left.+n\theta_{r}^{(1)}+k\theta_{r}^{(1)}\right]+\sum\limits_{r=1}^{\infty}\left(\widehat{a}_{r}\kappa^{r}+\widehat{a}_{r}^{*}\kappa^{r}\right)\right\\}.$

Combining these expansions, then Eq. (117) becomes

$\displaystyle\left.\frac{1}{m^{(n,k)}}\mathcal{G}m^{(n,k)}\right|_{p=p_{0},q=q_{0}}$
(121) $\displaystyle=$
$\displaystyle\sum\limits_{v=0}^{\infty}\left(\frac{|\lambda_{1}|^{2}}{(p_{0}+q_{0})^{2}}\kappa\lambda\right)^{v}\exp\left(\sum\limits_{r=1}^{\infty}(x^{+}_{r}+vs_{r})\kappa_{r}+\sum\limits_{r=1}^{\infty}(x_{r}^{-}+vs^{*})\lambda^{r}\right),$

where $x_{r}^{\pm}(n,k)$ are as given by Eq. (25) with

$\displaystyle a_{r}=\widehat{a}_{r}-b_{r}.$ (122)

Taking the coefficients of $\kappa^{s}\lambda^{j}$ on both sides of Eq. (121),
we get

$\displaystyle\frac{m_{s,j}^{(n,k)}}{m^{(n,k)}|_{p=p_{0},q=q_{0}}}=\sum\limits_{v=0}^{\mathrm{min}(s,j)}\left(\frac{|\lambda_{1}|^{2}}{(p_{0}+q_{0})^{2}}\right)^{v}\mathbf{S}_{s-v}(\mathbf{x}^{+}(n,k)+v\mathbf{s})\mathbf{S}_{j-v}(\mathbf{x}^{-}(n,k)+v\mathbf{s}^{*}),$
(123)

where $m_{s,j}^{(n,k)}$ is the matrix element given by Eq. (22). The right
side of the above equation is the $\widehat{m}_{s,j}^{(n,k)}$ defined in Eq.
(23). Furthermore, from Eq. (123) we can obtain the following relation:

$\displaystyle\det\limits_{1\leq s,j\leq
N}\left(m_{2s-1,2j-1}^{(n,k)}\right)=\widehat{H}\det\limits_{1\leq s,j\leq
N}\left(\widehat{m}_{2s-1,2j-1}^{(n,k)}\right)$ (124)

where $\widehat{H}=\left(m^{(n,k)}|_{p=p_{0},q=q_{0}}\right)^{N}$. This
relation indicates that the solutions given through differential operator form
and through Schur polynomials in Theorem 2 are equivalent.

Similarly, we can also transform the solutions expressed by differential
operator into Schur polynomials in this way. This completes the proof of
Theorems 2, 3, 4.

##  5\. Conclusion and discussion

In this paper, we have constructed general higher-order breather and RW
solutions of the 2-LSRI model (5) in the form of determinants, by means of the
bilinear KP-hierarchy reduction method. We first studied the dynamics of the
breather solutions. Under particular restrictions imposef on the parameters,
the breather solutions can become the homoclinic orbits in the 2-LSRI model
(5). It has been shown that the second-order breather solutions have three
different dynamical behaviours: two breathers, second-order homoclinic orbits,
and a mixture of a breather and a first-order homoclinic orbit. We derived
three families of RW solutions to the 2-LSRI model, which correspond to a
simple root, two simple roots, and double roots of the equation (15) related
to the dimension reduction condition. They are bounded RW, mixed bounded RWs,
and degradable bounded RWs. The dynamics of these three families of RWs have
been exhibited. The differences between these three families of RWs can be
summarized as follows:

  * •

The polynomial degree of the tau functions $\tau_{n,k}$ in Theorem 2–4 are
$N(N+1)$, $N_{1}(N_{1}+1)+N_{2}(N_{2}+1)$ and
$2[\widehat{N}_{1}^{2}+\widehat{N}_{2}^{2}-\widehat{N}_{1}(\widehat{N}_{2}-1))$
in both $x$ and $t$ variables, respectively, where $N,N_{1},N_{2}$ are
positive integers and $\widehat{N}_{1},\widehat{N}_{2}$ are non-negative
integers.

  * •

The solutions in Theorem 2–4 comprise of $\frac{N(N+1)}{2}$,
$\frac{N_{1}(N_{1}+1)+N_{2}(N_{2}+1)}{2}$, and
${[\widehat{N}_{1}^{2}+\widehat{N}_{2}^{2}-\widehat{N}_{1}(\widehat{N}_{2}-1)]}$
fundamental RWs, respectively. The RWs in Theorem 2 and Theorem 4 are bounded
states, while they are a mixture of two different bounded states in Theorem 3.

We point out that the breathers and RW solutions to the 2-LSRI model have been
derived earlier by using the bilinear method [80] and the Darboux
transformation [79, 82, 83, 84]. Comparing with those previously reported
results, the main results obtained in this paper can be summarized as follows:

  * •

There were only the first-order breather and first-order RW solutions given in
Ref. [80], while only the first-order RW solutions were studied in Ref. [79].
In the present paper we have constructed the general higher-order breather and
RW solutions in terms of determinants.

  * •

The higher-order RW solutions given in Refs. [82, 83, 84] comprise $N(N+1)/2$
fundamental RWs, which correspond to the family of RW solutions in Theorem 2
of the present paper. The general higher-order RW solutions in Theorems 3 and
4 are new RW solutions to the 2-LSRI model (5), which have not been reported
before, to the best of our knowledge.

## Acknowledgments

The work was supported by the National Natural Science Foundation of China
(Grants 11871446/12071451), and the Guangdong Basic and Applied Basic Research
Foundation (Grant 2022A1515012554), and the Research and Development
Foundation of Hubei University of Science and Technology (Grant BK202302), and
Israel Science Foundation (grant No. 1286/17).

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