General rogue wave solutions under SU(2)
transformation in the vector Chen-Lee-Liu nonlinear
Schrödinger equation

Changchang Pan, Lili Bu, Shihua Chen, Wen-Xing Yang, Dumitru Mihalache, Philippe Grelu, Fabio Baronio

► To cite this version:

Changchang Pan, Lili Bu, Shihua Chen, Wen-Xing Yang, Dumitru Mihalache, et al. General rogue wave solutions under SU(2) transformation in the vector Chen-Lee-Liu nonlinear Schrödinger equation. Physica D: Nonlinear Phenomena, 2022, 434, 10.1016/j.physd.2022.133204. hal-03980344

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General rogue wave solutions under SU(2) transformation in the vector Chen–Lee–Liu
nonlinear Schrödinger equation

Changchang Panª, Lili Buª, Shihua Chenª,b,*, Wen-Xing Yangc,*, Dumitru Mihalache d, Philippe Grelu e, Fabio Baroniof,.*

ªSchool of Physics and Frontiers Science Center for Mobile Information Communication and Security, Southeast University, Nanjing 211189, China

ᵇek Mountain Laboratories, Nanjing 211111, China

cSchool of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China

dHoria Hulubei National Institute for Physics and Nuclear Engineering, Department of Theoretical Physics, Magurele-Bucharest, RO-077125, Romania

Laboratoire ICB, U.M.R. 6303 C.N.R.S., Université Bourgogne Franche-Comté, 9 avenue A. Savary, F-21078 Dijon, France

fINO CNR and Dipartimento di Ingegneria dell'Informazione, Università di Brescia, Via Branze 38, 25123 Brescia, Italy

Abstract

We obtain analytically the exact explicit rogue wave solutions up to the second order of the vector Chen–Lee–Liu nonlinear Schrödinger equation, using the generalized non-recursive Darboux transformation method. In terms of these solutions, we demonstrate the fundamental Peregrine solitons as well as their doublet, triplet, quartet, and sextet counterparts on the general periodic backgrounds caused by SU(2) transformation. We numerically confirm that, although the Peregrine solitons sitting on SU(2) periodic backgrounds may suffer from larger disturbances than what they experience on SO(2) periodic backgrounds, such rational solitons developed on either kind of backgrounds can manifest clearly in spite of strong non-integrable perturbations. Other rogue wave topics such as rogue wave coexistence and the related parametric conditions are also discussed.

Keywords: rogue wave, Peregrine soliton, vector Chen–Lee–Liu equation

PACS: 05.45.Yv, 02.30.Ik, 47.20.Ky, 47.54.-r

1. Introduction

Rogue waves, alias *solitons on a finite background* [1], are a fascinating multidisciplinary subject [2, 3] as has been enjoyed by ordinary solitons [4, 5], and have attracted a surge of research activities on such areas as oceanography [6], hydrodynamics [7, 8], nonlinear optics [9, 10], magnetics [11], 30 acoustics [12], plasma physics [13], Bose-Einstein condensation [14, 15], economics [16], and even artificial intelligence [17]. One important reason for such a boom lies in that rogue waves can be mathematically constructed based on the integrability of a class of nonlinear partial differential equations [18–23], whereby the otherwise unpredictable extreme wave events observed in practical environments can be understood on an analytical level [24–30]. This in turn brings flourishing development of different analytical methods such as inverse scattering transform [31], Darboux transformation [32–34], and Hirota bi-linearization approach [35, 36], to name a few.

Peregrine soliton is a typical example of such deterministic
analytical solutions, which features a doubly-localized peak and
two side troughs on a finite background [37]. Mathematically,
it is the limiting case of both the Akhmediev breather and the
Kuznetsov–Ma soliton, where the breathing period of the lat-
ter two approaches infinity [24, 25, 38, 39]. This bizarre spa-
tiotemporal structure is particularly suitable for explaining the

fleeting nature of realistic rogue waves [40] and thus has gen-
erally been thought of as the prototype of rogue waves [41]. To
date, the single Peregrine soliton event has been observed in a
water-wave tank [24], in plasmas [13], in optical fibers [25, 42],
and in irregular oceanic sea states [43].

In vector or multi-component nonlinear systems, the Pere-
grine soliton solutions exist as well [44–47], exhibiting some
features that are unavailable to scalar integrable models. As ex-
amples, the Peregrine soliton states formed in coupled systems
can manifest as a dark structure on some (or even all) of their
components [27, 45, 46, 48], occur in defocusing (or equiva-
lently, normally dispersive) nonlinear media [49, 50], coexist
with other cousins on the same background [51, 52], or entail
a strong peak enhancement in the presence of self-steepening
effect [53, 54]. Besides, there also exist Peregrine soliton so-
lutions on the periodic backgrounds that are caused by the in-
terference between continuous-wave fields [55, 56]. Such a pe-
riodicity is of trigonometrical origin and is therefore different
from the cnoidal-wave periodicity occurring in scalar nonlinear
systems [57–60], where the doubly-periodic Jacobian or Weier-
strass elliptic functions are involved [61]. Generally, the two-
component Peregrine soliton solutions on the periodic back-
grounds can be easily obtained by means of a special unitary
group operation, namely the SU(2) transformation [52].

Meanwhile, much attention has been paid to the higher-order
rogue wave solutions, including super rogue waves [62–64] and
multiple rogue waves [50, 65–68], of the integrable nonlinear
systems. There are actual needs for such investigations, among

*Corresponding author
Email addresses: cshua@seu.edu.cn (Shihua Chen),
wenxingyang2@126.com (Wen-Xing Yang), fabio.baronio@unibs.it
(Fabio Baronio)
which are the fundamental scientific interest involved in obtain-
ing a hierarchy of rogue wave solutions [18], and, more impor-
tantly, the emergence of some complicated extreme wave phe-
nomena that can not be well explained by the simple Peregrine
solitons mentioned above [2, 3, 62].

In this article, we try to find, in a systematic way, the ex-
act explicit fundamental and higher-order rogue wave solutions,¹¹⁰
particularly those on periodic backgrounds caused by SU(2)
transformation, of the vector Chen-Lee-Liu (CLL) nonlinear
Schrödinger (NLS) equation [52, 69], using a generalized non-
recursive Darboux transformation method [63, 70]. In terms
of these solutions, the intriguing dynamics developed on the ¹¹⁵
general SU(2) periodic backgrounds for the fundamental Pere-
grine solitons as well as their doublet, triplet, quartet, and sex-
tet counterparts are demonstrated. As a comparison, the Pere-
grine solitons sitting on the SO(2) periodic backgrounds are
also shown. We numerically confirm that the Peregrine solitons
sitting on either SO(2) or SU(2) periodic backgrounds are rela-
tively stable against the perturbations that violate the integrabili-
ty of the governing model, but the latter may suffer from larger
disturbances arising from the modulation instability (MI). In ¹²⁰
addition, we establish a bridge between the complex roots of
two different quartic equations, by which the parameter condi-
tions for rogue wave coexistence can be easily obtained.

2. The vector model and basic procedures for derivation of
general rogue wave solutions

For our studies, we consider the vector CLL–NLS model
that, for instance, governs the mixing of two fundamental-
frequency (FF) pulses in a quadratic nonlinear crystal via a type
II highly phase-mismatched second-harmonic generation pro-
cess [71–73]. In dimensionless form, it can be written as

$$
iu_{1z} + \frac{\sigma}{2} u_{1tt} + (|u_1|^2 + \kappa|u_2|^2)(u_1 + i\gamma u_{1t}) = 0, \quad (1)
$$

$$
iu_{2z} + \frac{\sigma}{2} u_{2tt} + (|u_1|^2 + \kappa |u_2|^2)(u_2 + i y u_{2t}) = 0, \quad (2)
$$

where $u_{1,2}(z,t)$ are the normalized complex envelopes of the
two FF components, with $z$ and $t$ being the propagation dis-
tance and retarded time, respectively. The subscripts $z$ and $t$
represent the partial derivatives. In cases of anomalous and
normal dispersion, the parameter $\sigma$ can be normalized to 1 and
$-1$, respectively. The third term on the left-hand side of Eqs.
(1) and (2) include the Kerr nonlinear effect that results from
the intensity-dependent refractive index [74] and the concomi-
tant self-steepening effect that becomes significant for optical
pulses with possible shock formation [75–77]. To maintain the
integrability of the coupled equations, we will give the same
weight or importance to the self- and cross-phase modulation
contributions (i.e., $\kappa = 1$), whereas assign a free parameter $\gamma$
for the self-steepening, in view of the latter having such addi-
tional applications as supercontinuum generation [78, 79] and
formation of anomalous extreme wave events [53, 54]. As a
special case, when $\gamma = 0$, the above vector CLL model can be
reduced to the well-known Manakov system, which has been
widely used to model the propagation of optical pulses in ran-
domly birefringent fibers [74], the formation of spatial solitons

in planar waveguides [80], or the crossing sea states occurring
in the open ocean [81]. Naturally, this gives us a lot of space to
imagine the possible application of the vector CLL model into
the aforementioned areas, with higher accuracy.

Because of the integrability [69], this vector CLL-NLS equa-
tion can be solved by the non-recursive Darboux transformation
method [32, 63, 70]. Most recently, we have reported its funda-
mental Peregrine soliton solutions on either continuous or peri-
odic backgrounds in Ref. [52], based on the SO(2) group oper-
ation. In the following, we will derive the exact explicit rogue
wave solutions up to the second-order, using the more general
SU(2) transformation. Particularly, we will perform numerical
simulations to investigate the stability discrepancy between the
Peregrine soliton solutions involving SO(2) symmetry and the
ones involving SU(2) symmetry.

For this end, let us first cast the coupled system of Eqs. (1)
and (2) into a 3 × 3 linear eigenvalue problem [52]:

$$
\mathbf{R}_t = \mathbf{U}\mathbf{R}, \quad \mathbf{R}_z = \mathbf{V}\mathbf{R}, \tag{3}
$$

where **R** = [r, s, w]ᵀ is the eigenfunction (*T* signifies the matrix transpose, and *r*, *s*, and *w* are functions of *z*, *t* and the complex spectral parameter *λ*), and the matrices **U** and **V** are given by

$$
\mathbf{U} = - \frac{i(\lambda - \sigma)\sigma_3}{2\sigma\gamma} + \frac{\sqrt{\lambda}}{\sigma}\mathbf{Q} + \frac{i\gamma}{\sigma}\mathbf{K}\mathbf{Q}^2, \quad (4)
$$

$$
\mathbf{V} = \frac{(\lambda - \sigma)}{2\gamma}\mathbf{U} - \frac{i(\lambda - \sigma)}{2\sigma}\mathbf{K}\mathbf{Q}^2 - \frac{\gamma}{2}\mathbf{K}(\mathbf{Q}\mathbf{Q}_t - \mathbf{Q}_t\mathbf{Q}) \\
\qquad + \frac{i\gamma^2}{2\sigma}\mathbf{K}\mathbf{Q}^4 - \frac{\sqrt{\lambda}}{2\sigma}(i\sqrt{\lambda}\sigma_3\mathbf{Q}^2 - \gamma\mathbf{Q}^3 - i\sigma\sigma_3\mathbf{Q}_t)(5)
$$

with $\mathbf{K} = \mathrm{diag}(0, 1, 1)$, $\sigma_3 = \mathrm{diag}(1, -1, -1)$, and

$$
\mathbf{Q} = \begin{bmatrix} 0 & u_1 & u_2 \\ -u_1^* & 0 & 0 \\ -u_2^* & 0 & 0 \end{bmatrix}.
$$

Here ‘diag’ means the diagonal matrix and the asterisk denotes
the complex conjugation. Obviously, for any spectral parameter
$\lambda$, Eqs. (1) and (2) can be reproduced from the compatibility
condition $\mathbf{R}_{1z} = \mathbf{R}_{z1}$, i.e., $\mathbf{U}_z - \mathbf{V}_t + \mathbf{U}\mathbf{V} - \mathbf{V}\mathbf{U} = 0$.

Subsequently, we need to solve the Lax pair (3), using the ini-
tial potentials $u_j = u_{j0}$ ($j = 1, 2$) that should be simple enough.
Depending on the kind of localized solutions one seeks, these
initial potentials can be made zeros (for soliton solutions) or be
finite-background plane waves (for rogue wave or Akhmediev
breather solutions). For our present purposes, we use the plane
wave solutions of Eqs. (1) and (2) as initial potentials, which
can be expressed as

$$
u_{j0} = a_j \exp(i\omega_j t - ik_j z), \quad (j = 1, 2), \qquad (6)
$$

with $a_j$, $k_j$, and $\omega_j$ being the amplitudes, wavenumbers, and
frequencies, respectively, but related by

$$
k_j = A v_j + \frac{1}{2} \sigma \omega_j^2, \quad v_j = \gamma \omega_j - 1, \quad A = a_1^2 + a_2^2. \quad (7)
$$

Inserting these plane-wave solutions into Eqs. (3), one can read-
ily find the expression of **R**(*λ*), given by

$$
\mathbf{R}(\lambda) = \mathbf{G}\Phi(\lambda), \qquad (8)
$$
$$ \Phi(\lambda) = \Gamma_1 \begin{bmatrix} 1 \\ c_{11} \\ c_{21} \end{bmatrix} e^{i\varphi_1} + \Gamma_2 \begin{bmatrix} 1 \\ c_{12} \\ c_{22} \end{bmatrix} e^{i\varphi_2} + \Gamma_3 \begin{bmatrix} 1 \\ c_{13} \\ c_{23} \end{bmatrix} e^{i\varphi_3}, \quad (9) $$

where $G = \text{diag}(1, u_{10}^*/a_1, u_{20}^*/a_2)$, $\varphi_j = \mu_j t + \nu_j z$ ($j = 1, 2, 3$, the same below), $\Gamma_j$ are arbitrary complex constants, and

$$ c_{lj} = - \frac{i a_j \gamma (2 \sigma \gamma \mu_j - \lambda - \sigma)}{\sqrt{\lambda} (2 \sigma \gamma \mu_j - \lambda - \sigma - 2 \sigma \nu_l)}, \quad (l = 1, 2), \quad (10) $$

$$ \nu_j = \frac{2A\gamma^2\lambda - (\lambda - \sigma)^2}{4\sigma\gamma^2} + \frac{i\sqrt{\lambda}}{2}(\omega_1 a_1 c_{1j} + \omega_2 a_2 c_{2j}) + \frac{i\sqrt{\lambda}}{2\gamma\sigma}(A\gamma^2 - \lambda + \sigma)(a_1 c_{1j} + a_2 c_{2j}). \quad (11) $$

The parameters $\mu_j$ in Eqs. (10) and (11) are the three roots of the cubic equation of $\mu$,

$$ (\mu - \frac{\beta}{2} + \frac{A\gamma}{2\sigma} - \frac{\kappa}{4})^3 - 3\zeta(\mu - \frac{\beta}{2} + \frac{A\gamma}{2\sigma} - \frac{\kappa}{4}) + 2\varrho = 0, \quad (12) $$

where the parameters $\beta$, $\zeta$, and $\varrho$ are defined by

$$ \beta = \frac{A\gamma^2 + \lambda - \sigma}{3\sigma\gamma} + \frac{\kappa}{6}, \quad (13) $$

$$ \zeta = \beta^2 + \frac{\delta^2}{12} - \frac{\iota}{3\sigma}, \quad \varrho = \beta^3 - \frac{\beta(2\iota + \sigma\delta^2)}{4\sigma} + \frac{\delta\varpi}{4\sigma}, \quad (14) $$

with

$$ \kappa = \omega_1 + \omega_2, \quad \delta = \omega_1 - \omega_2, \quad (15) $$

$$ \iota = a_1^2 \nu_1 + a_2^2 \nu_2, \quad \varpi = a_1^2 \nu_1 - a_2^2 \nu_2. \quad (16) $$

Particular attention should be paid to the cubic equation (12),¹³⁵ which could allow a double root, say, $\mu_1 = \mu_2 = \mu_0$, or a triple root $\mu_1 = \mu_2 = \mu_3 = \mu_0$, when the spectral parameter $\lambda$ satisfies the discriminant condition

$$ \Delta = \zeta^3 - \varrho^2 = 0. \quad (17) $$

Actually, the triple root can be found directly using $\zeta = \varrho = 0$, which is a special case of Eq. (17). The seeking of this double or triple root $\mu_0$ is very crucial, as it gets involved in constructing a hierarchy of rational rogue wave solutions. It turns out that only the spectral parameter $\lambda_0$ that fulfils Eq. (17) and must be a complex number (with nonzero imaginary part) does admit such kind of rational solutions.

Interestingly, as one can verify, the complex roots $\lambda_0$ of the real-coefficient quartic Eq. (17) can be related to the complex roots $\Lambda$ of the much simpler, symmetric quartic equation

$$ \sigma - \frac{a_1^2 \nu_1}{(\Lambda + \omega_1)^2} - \frac{a_2^2 \nu_2}{(\Lambda + \omega_2)^2} = 0. \quad (18) $$

There is a one-to-one correspondence between the complex roots of these two different quartic equations. In other words, if $\Lambda = m + i\ell$ ($\ell \neq 0$) is a complex root of Eq. (18), then

$$ \lambda_0 = \sigma - A\gamma^2 + \sigma\gamma(3m + \kappa) + \frac{4\gamma\sigma\ell^2(2i\ell - 2m - \kappa)}{4\ell^2 + (\kappa + 2m)^2 - \delta^2}, \quad (19) $$

will definitely be the complex root of Eq. (17). Meanwhile, as $\lambda_0$ satisfies Eq. (17), the double or triple root $\mu_0$ of the cubic Eq. (12) can also be expressed in terms of $m$ and $\ell$, viz.,

$$ \mu_0 = \frac{1}{2}(m + \kappa - A\gamma/\sigma) - i\ell + \frac{2\ell^2(2i\ell - 2m - \kappa)}{4\ell^2 + (\kappa + 2m)^2 - \delta^2}. \quad (20) $$

One of the attracting traits of these $\lambda_0$ and $\mu_0$ formulas is that the real and imaginary parts have been well separated, which would facilitate the simplification of analytical derivations. Hence, in some cases, it is convenient to write $\lambda_0 = \lambda_r + i\lambda_m$, with

$$ \lambda_r = \sigma - A\gamma^2 + \sigma\gamma(3m + \kappa) - \frac{4\gamma\sigma\ell^2(2m + \kappa)}{4\ell^2 + (\kappa + 2m)^2 - \delta^2}, \quad (21) $$

$$ \lambda_m = \frac{8\gamma\sigma\ell^3}{4\ell^2 + (\kappa + 2m)^2 - \delta^2}. \quad (22) $$

It then follows that, for this special spectral parameter $\lambda = \lambda_0$, which would lead to $\mu_1 = \mu_2 = \mu_0$, the factored eigenfunction $\Phi(\lambda)$, given by Eq. (9), will vanish under an appropriate choice of $\Gamma_j$, e.g., when $\Gamma_1 = -\Gamma_2$ and $\Gamma_3 = 0$. However, the two ratios of its components $s/r$ and $w/r$ will not vanish and indeed possess rational function forms, which can be obtained according to l'Hôpital's rule. Routinely, substitution of these rational polynomials into the Darboux transformation formulas could yield the rogue wave solutions as required [34]. As a matter of fact, there is a more convenient way to find the rational polynomials. Specifically, one can let $\lambda = \lambda_0 + (\lambda_0 - \lambda_0^*)\epsilon^D$ ($D = 2, 3$), and then expand $\Phi(\lambda)$ into a Taylor series

$$ \Phi(\lambda) = \Phi^{(0)} + \Phi^{(1)}\epsilon^D + \Phi^{(2)}\epsilon^{2D} + \cdots + O(\epsilon^{nD}), \quad (23) $$

where $\epsilon$ is a complex small quantity and $O$ means the infinitesimal of the same order. This series expansion in powers of $\epsilon^D$ can only be achieved when an appropriate set of $\Gamma_j$ is adopted, namely,

$$ \begin{align} \Gamma_1 &= \frac{1}{2} \sum_{j=1}^{n} \left( \gamma_{2j-1} + \frac{\gamma_{2j}}{\epsilon} \right) \epsilon^{2(j-1)}, \notag \\ \Gamma_2 &= \frac{1}{2} \sum_{j=1}^{n} \left( \gamma_{2j-1} - \frac{\gamma_{2j}}{\epsilon} \right) \epsilon^{2(j-1)}, \tag{24} \\ \Gamma_3 &= 0, \notag \end{align} $$

for $D = 2$ (the double-root case) and

$$ \begin{align} \Gamma_1 &= \frac{1}{3} \sum_{j=1}^{n} \left( \gamma_{3j-2} - \frac{\gamma_{3j-1}\phi_2}{\epsilon} + \frac{\gamma_{3j}\phi_1}{\epsilon^2} \right) \epsilon^{3(j-1)}, \notag \\ \Gamma_2 &= \frac{1}{3} \sum_{j=1}^{n} \left( \gamma_{3j-2} - \frac{\gamma_{3j-1}\phi_1}{\epsilon} + \frac{\gamma_{3j}\phi_2}{\epsilon^2} \right) \epsilon^{3(j-1)}, \tag{25} \\ \Gamma_3 &= \frac{1}{3} \sum_{j=1}^{n} \left( \gamma_{3j-2} + \frac{\gamma_{3j-1}}{\epsilon} - \frac{\gamma_{3j}}{\epsilon^2} \right) \epsilon^{3(j-1)}, \notag \end{align} $$
for $D = 3$ (the triple-root case). Here $\phi_1 = 1/2 + i\sqrt{3}/2$, $\phi_2 = 1/2 - i\sqrt{3}/2$, and $\gamma_s$ are arbitrary complex constants termed structural parameters.

Once $\Phi(\lambda)$ is expanded as in Eq. (23), one can construct the nth-order rogue wave solutions on continuous backgrounds as

$$u_{1,cw}^{[n]} = u_{10} \left( 1 - \frac{i}{|u_{10}|} \mathbf{Y}_1 \mathbf{M}^{-1} \mathbf{Y}_2^{\dagger} \right) \left( \frac{\det(\mathbf{M})}{\det(\mathbf{M}^{\dagger})} \right), \quad (26)$$

$$u_{2,cw}^{[n]} = u_{20} \left( 1 - \frac{i}{|u_{20}|} \mathbf{Y}_1 \mathbf{M}^{-1} \mathbf{Y}_3^{\dagger} \right) \left( \frac{\det(\mathbf{M})}{\det(\mathbf{M}^{\dagger})} \right), \quad (27)$$

where $\mathbf{Y}_j$ ($j = 1, 2, 3$) are $1 \times n$ row vectors, determined by

$$\begin{bmatrix} \mathbf{Y}_1 \\ \mathbf{Y}_2 \\ \mathbf{Y}_3 \end{bmatrix} = [\Phi^{(0)}, \Phi^{(1)}, \Phi^{(2)}, \dots, \Phi^{(n-1)}], \quad (28)$$

and $\mathbf{M}$ is an $n \times n$ square matrix, whose elements $M_{jk}$ can be found through the Taylor expansion

$$\frac{\Phi^{\dagger} \mathbf{X} \Phi}{\lambda - \lambda^{*}} = \sum_{j=1}^{n} \sum_{k=1}^{n} M_{jk} \epsilon^{*(j-1)D} \epsilon^{(k-1)D} + O(|\epsilon|^{2nD}), \quad (29)$$

with $\mathbf{X} = \text{diag}(\sqrt{\lambda^*}, \sqrt{\lambda}, \sqrt{\lambda})$. Here the symbols $\dagger$ and “det” stand for the complex conjugate transpose and the determinant of a matrix, respectively.

As the last step, by performing an SU(2) group operation on Eqs. (26) and (27), we can obtain the more general rogue wave solutions developed on the periodic backgrounds:

$$\left( \begin{matrix} u_1^{[n]} \\ u_2^{[n]} \end{matrix} \right) = \left[ \begin{matrix} \cos \phi - in_3 \sin \phi & -(in_1 + n_2) \sin \phi \\ -(in_1 + n_2) \sin \phi & \cos \phi + in_3 \sin \phi \end{matrix} \right] \left( \begin{matrix} u_{1,cw}^{[n]} \\ u_{2,cw}^{[n]} \end{matrix} \right), \quad (30)$$

where $\phi$ is a free rotation angle, and $n_1, n_2$, and $n_3$ are the directional cosines satisfying $n_1^2 + n_2^2 + n_3^2 = 1$. It should be noted that the complex $2 \times 2$ matrix on the right-hand side of Eq. (30) has a determinant of positive unity and thus belongs to the SU(2) group. Of course, when $n_1 = n_3 = 0$ and $n_2 = 1$, this matrix belongs to the SO(2) group, whose elements become all real. By the SU(2) group transformation, the rogue wave solutions (30) involve three additional free parameters $\phi$ and $n_{1,2}$, as compared to the ones given by Eqs. (26) and (27). As a consequence, the backgrounds of these rogue waves would exhibit interference fringes that depend not only on the temporal beat frequency $\delta$, but also on the three free parameters introduced by SU(2) transformation. In our previous work [52], we have mentioned the above SU(2)-transformed solutions, but only discussed their reduced SO(2) solution forms in detail. As a companion work, we will investigate here the rogue wave solutions (30) in greater depth, with their explicit solution forms being extended to up to the second order that were not reported before.

### 3. Peregrine soliton solutions and their triplet counterparts in the double root ($D = 2$) case

In this section, we will derive the explicit fundamental Peregrine soliton and triplet rogue wave solutions, which correspond to the double root ($D = 2$) case in the procedures outlined

in Sec. 2. In this case, only the preceding two series terms of $\Phi(\lambda)$ in Eq. (23) need to be found, with the help of Eqs. (24). We now provide the resulting formulas of $\mathbf{\Phi}^{(0)}$ and $\mathbf{\Phi}^{(1)}$ as

$$\mathbf{\Phi}^{(0)} = \mathbf{T} \begin{bmatrix} R_0 \\ R_1 \\ R_2 \end{bmatrix} e^{i\varphi_0}, \quad \mathbf{\Phi}^{(1)} = \mathbf{T} \begin{bmatrix} S_0 \\ S_1 \\ S_2 \end{bmatrix} e^{i\varphi_0}, \quad (31)$$

where $\mathbf{T} = \text{diag}(1, c_{10}, c_{20})$, $\varphi_0 = \mu_0 t + \nu_0 z$, $c_{10}$ ($l = 1, 2$) and $\nu_0$ are given by Eqs. (10) and (11) (just letting $j = 0$ therein), respectively, and

$$R_0 = \gamma_1 + i\gamma_2\chi_0\vartheta, \quad R_j = \gamma_1 + i\gamma_2\chi_0\theta_j, \quad (32)$$

$$S_0 = \gamma_1 p + \gamma_2 q + \gamma_3 + i\gamma_4\chi_0\vartheta, \quad (33)$$

$$S_j = \gamma_1 \left( p + \frac{i\chi_0^2\theta_j}{\alpha_j} + \frac{\varphi_j}{\alpha_j} \right) + \gamma_2 \left( q - \frac{g_j}{\alpha_j} \right) + \gamma_3 + i\gamma_4\chi_0\theta_j. \quad (34)$$

with $j = 1, 2$ and $\gamma_s$ ($s = 1, \dots, 4$) being four arbitrary complex parameters. The other parameters or polynomials in Eqs. (32)–(34) are defined by

$$\psi = -2\sigma(1 + m\gamma + il\gamma), \quad (35)$$

$$\vartheta = t - \frac{\gamma A z}{2} + \frac{(\lambda_0 - \sigma)z}{2\gamma} - 2\beta_2\gamma^2 z, \quad (36)$$

$$\theta_j = \vartheta - \frac{i}{\alpha_j}, \quad \alpha_j = -\frac{\psi(\psi - 2\sigma v_j)}{4\gamma v_j}, \quad (37)$$

$$\beta_j = \frac{a_1^2 \omega_1 v_1}{(\psi - 2\sigma v_1)^j} + \frac{a_2^2 \omega_2 v_2}{(\psi - 2\sigma v_2)^j}, \quad (j = 2, 3, 4), \quad (38)$$

$$\chi_0 = \sqrt{\frac{2i\lambda_m K}{3} - \frac{4i\lambda_m (\iota - \delta^2\sigma)}{3\sigma\gamma^2 P}}, \quad (39)$$

$$\chi_1 = \frac{i\lambda_m(2K + P) - \chi_0^2}{\gamma\sigma P}, \quad (40)$$

$$\chi_2 = \frac{(5\chi_0^2 - 2i\lambda_m K)(\chi_0^2 - 2i\lambda_m K) - 4i\lambda_m \chi_0^2 P}{2\gamma^2 \chi_0 P^2}, \quad (41)$$

$$K = \frac{\sigma(\gamma\kappa - 2) - \psi}{\gamma^2}, \quad (42)$$

$$P = 2A - \frac{2\sigma(\kappa - 3\mu_0)}{\gamma} - \frac{\lambda_0 - \sigma}{\gamma^2}, \quad (43)$$

$$\varphi_j = -\frac{2\chi_0^2\sigma\gamma}{\psi} + \chi_1 - i\lambda_m \left(\frac{\alpha_j}{\lambda_0} + \frac{1}{\gamma\sigma}\right), \quad (44)$$

$$\varepsilon = -4\chi_0^2\sigma\gamma^3\beta_3 - 2i\lambda_m\beta_2\gamma\sigma - \frac{i\lambda_m(A\gamma + 2\mu_0\sigma)}{2\sigma\gamma}, \quad (45)$$

$$\rho = -\frac{\lambda_m}{\gamma} - 8i\gamma^2\beta_3(i\lambda_m - \chi_1\sigma\gamma) - 8i\gamma^4\beta_4\chi_0^2, \quad (46)$$
$$p = -\frac{1}{2}\chi_0^2\theta^2 + i\chi_1\theta - i\epsilon\zeta, \quad (47)$$

$$q = i\chi_0\theta p + \rho\chi_0z + i\chi_2\theta + \frac{i}{3}\theta^3\chi_0^3, \quad (48)$$

$$g_j = i\chi_0\epsilon z + \frac{1}{2}\chi_0^3\theta_j^2 - i\chi_0(\chi_1 + \varphi_j)\theta_j - \chi_2 + \frac{i\lambda_m\chi_0}{\alpha_j\sigma\gamma} - \frac{2\chi_0^3\gamma^2}{\psi^2} - \frac{2\chi_0^3\gamma^2}{(\psi - 2\sigma\upsilon_j)^2} + \frac{4\chi_0}{\psi}(\chi_1\sigma\gamma - i\lambda_m). \quad (49)$$

Here the values of $m$ and $\ell$ ($\ell \neq 0$) can be solved for from Eq. (18), and then the values of $\lambda_0$ and $\mu_0$ can be given by Eqs. (19) and (20), respectively.

Quite straightforwardly, one can substitute Eq. (31) into Eqs. (28) and (29) to yield the expressions for $\mathbf{Y}_j$ and $\mathbf{M}$, respectively, and then substitute the obtained $\mathbf{Y}_j$ and $\mathbf{M}$ formulas into Eqs. (26) and (27) to give the resultant rogue wave solutions up to the second order, of which the fundamental rogue wave solutions can be written as

$$u_{1,cw}^{[1]} = u_{10} \left[ 1 + \frac{2i\lambda_m \psi^* R_0 R_1^*}{(\psi^* - 2\sigma\upsilon_1) m_{11}} \right] \left( \frac{\sqrt{\lambda_0} m_{11}}{\sqrt{\lambda_0^*} m_{11}^*} \right), \quad (50)$$

$$u_{2,cw}^{[1]} = u_{20} \left[ 1 + \frac{2i\lambda_m \psi^* R_0 R_2^*}{(\psi^* - 2\sigma\upsilon_2) m_{11}} \right] \left( \frac{\sqrt{\lambda_0} m_{11}}{\sqrt{\lambda_0^*} m_{11}^*} \right), \quad (51)$$

where

$$m_{11} = \lambda_0^* |R_0|^2 + \iota_1 |R_1|^2 + \iota_2 |R_2|^2, \quad (52)$$

$$\iota_j = \frac{a_j^2 [\ell^2 \gamma^2 + (\gamma m + 1)^2]}{\ell^2 + (m + \omega_j)^2}, \quad (j = 1, 2). \quad (53)$$

Meanwhile, the second-order rogue wave solutions can be read

$$u_{1,cw}^{[2]} = u_{10} \left[ 1 + \frac{2i\lambda_m \psi^*(R_1^* X - S_1^* Y)}{(\psi^* - 2\sigma\upsilon_1)W} \right] \left( \frac{\lambda_0 W}{\lambda_0^* W^*} \right), \quad (54)$$

$$u_{2,cw}^{[2]} = u_{20} \left[ 1 + \frac{2i\lambda_m \psi^*(R_2^* X - S_2^* Y)}{(\psi^* - 2\sigma\upsilon_2)W} \right] \left( \frac{\lambda_0 W}{\lambda_0^* W^*} \right), \quad (55)$$

where

$$m_{12} = \lambda_0^* R_0^* S_0 + \iota_1 R_1^* S_1 + \iota_2 R_2^* S_2 - \frac{i \lambda_m \lambda_0^* |R_0|^2}{\lambda_0} - \frac{\lambda_r m_{11}}{\lambda_0}, \quad (56)$$

$$m_{21} = \lambda_0^* R_0 S_0^* + \iota_1 R_1 S_1^* + \iota_2 R_2 S_2^* - i \lambda_m |R_0|^2 - m_{11}, \quad (57)$$

$$m_{22} = \lambda_0^* |S_0|^2 + \iota_1 |S_1|^2 + \iota_2 |S_2|^2 - i\lambda_m(R_0 S_0^* + S_0 R_0^*) - m_{12} - \frac{\lambda_m^2 R_0 (R_0^* + 2S_0^*)}{\lambda_0} - \frac{\lambda_r m_{21} - i\lambda_m m_{11}}{\lambda_0} \quad (58)$$

$$X = R_0 m_{22} - S_0 m_{21}, \quad Y = R_0 m_{12} - S_0 m_{11}, \quad (59)$$

$$W = m_{11}m_{22} - m_{12}m_{21}. \quad (60)$$

It should be noted that in the above formulas, we have made a change of variables $m_{ij} = (\lambda_0 - \lambda_0^*) \sqrt{\lambda_0^*} M_{ij}$, where $M_{ij}$ are the elements of $\mathbf{M}$ defined in Eq. (29). As seen, these $m_{ij}$ formulas are all complex and more complicated than those presented for the Manakov system in which $m_{11}$ and $m_{22}$ are always real [46]. It therefore follows that these solutions obtained will have no singularity problem, as their denominators will never be zero, of course under the parameter condition $\ell \neq 0$.

### 3.1. Peregrine solitons on periodic backgrounds

In view of the fact that the spatiotemporal structures of the first-order solutions (50) and (51) do not depend on the values of $\gamma_1$ and $\gamma_2$, we can further simplify them by simply setting $\gamma_1 = 0$ and $\gamma_2 = 1$ [63, 64]. After cumbersome manipulations, the fundamental Peregrine soliton solutions can be simplified as

$$u_{1,cw}^{[1]} = u_{10} \left( 1 + \frac{iG_1}{M - iN} \right), \quad u_{2,cw}^{[1]} = u_{20} \left( 1 + \frac{iG_2}{M - iN} \right), \quad (61)$$

where $G_{1,2}$, $M$, and $N$ are polynomials of $z$ and $t$, given by

$$M = \theta^2 + \sigma^2 \ell^2 z^2 + \frac{\eta^2}{4\alpha \ell^2}, \quad N = \frac{\gamma}{\alpha}(\sigma\gamma \ell^2 z - \eta\theta) \quad (62)$$

$$G_j = \frac{2[(m + \omega_j)\theta - \sigma\ell^2 z]}{(m + \omega_j)^2 + \ell^2} + \frac{2\gamma(\sigma\gamma\ell^2 z - \eta\theta)}{\alpha} - \frac{\eta\nu_j}{\alpha[(m + \omega_j)^2 + \ell^2]}, \quad (63)$$

with $\theta = t + (\sigma m - A\gamma)z$, $\eta = \gamma m + 1$ and $\alpha = \gamma^2 \ell^2 + \eta^2$. We point out that these solutions have been well translated on the plane $(z, t)$ so that their peaks exactly locate on the origin. Hence the central magnitude of each rogue wave component relative to its background can easily be found to be

$$f_j = 1 + \frac{4\nu_j \ell^2}{\eta [(m + \omega_j)^2 + \ell^2]}. \quad (64)$$

Hence, according to Eqs. (30), the general Peregrine soliton solutions on periodic backgrounds can be expressed as

$$u_1^{[1]} = u_{1,cw}^{[1]}(\cos \phi - i n_3 \sin \phi) - u_{2,cw}^{[1]}(i n_1 + n_2) \sin \phi, \quad (65)$$

$$u_2^{[1]} = u_{1,cw}^{[1]}(-i n_1 + n_2) \sin \phi + u_{2,cw}^{[1]}(\cos \phi + i n_3 \sin \phi). \quad (66)$$

It is easy to find that the background fields will exhibit interference fringes, with the average background height $d_1 = [a_1^2 - (a_1^2 - a_2^2)(n_1^2 + n_2^2)\sin^2\phi]^{1/2}$ for the $u_1$ component and $d_2 = [a_2^2 + (a_1^2 - a_2^2)(n_1^2 + n_2^2)\sin^2\phi]^{1/2}$ for the $u_2$ component. Accordingly, the enhancement factors for two Peregrine soliton components, which define the ratio of the peak amplitude to the respective average background height, can be given by

$$f_{u_1} = \frac{|a_1 f_1(\cos \phi - i n_3 \sin \phi) - a_2 f_2(i n_1 + n_2) \sin \phi|}{\sqrt{a_1^2 - (a_1^2 - a_2^2)(n_1^2 + n_2^2) \sin^2 \phi}}, \quad (67)$$
<figure><img src="image_2.png" /><figcaption>Figure 1: 3D surface and contour plots of the Peregrine soliton states on the periodic backgrounds in the anomalous dispersion regime ($\sigma = 1$), obtained with the initial potential parameters $\gamma = 1$, $\phi = \pi/18$, $\omega_1 = -1/8$, $\omega_2 = -7/8$, $a_1 = \sqrt{2}/2$, and $a_2 = \sqrt{30}/10$, which bring about $m = 1/2$ and $\ell = 3\sqrt{3}/8$. (a)–(d): $n_1 = n_3 = 0$ and $n_2 = 1$ [SO(2) transformation]; (e)–(h): $n_1 = n_2 = n_3 = \sqrt{3}/3$ [SU(2) transformation].</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2: Peregrine soliton states on the periodic backgrounds in the normal dispersion regime ($\sigma = -1$), obtained with another set of initial parameters $\gamma = 1$, $\phi = \pi/18$, $\omega_1 = 19/10$, $\omega_2 = 13/10$, $a_1 = \sqrt{10}/5$ and $a_2 = \sqrt{30}/5$ (yielding $m = -8/5$ and $\ell = 3\sqrt{3}/10$). (a)–(d): $n_1 = n_3 = 0$ and $n_2 = 1$ [SO(2) transformation]; (e)–(h): $n_1 = n_2 = n_3 = \sqrt{3}/3$ [SU(2) transformation].</figcaption></figure>

$$f_{u_2} = \frac{|a_1 f_1(-in_1 + n_2) \sin \phi + a_2 f_2(\cos \phi + in_3 \sin \phi)|}{\sqrt{a_2^2 + (a_1^2 - a_2^2)(n_1^2 + n_2^2) \sin^2 \phi}} \quad (68)$$

Basically, by virtue of Eqs. (18), (64), (67) and (68), one can determine the general parameter conditions under which different Peregrine rogue wave states can be identified.

Figure 1 shows the Peregrine soliton states on the periodic backgrounds in the anomalous dispersion regime ($\sigma = 1$), for²⁰⁵ given set of initial potential parameters $\gamma = 1$, $\omega_1 = -1/8$, $\omega_2 = -7/8$, $a_1 = \sqrt{2}/2$ and $a_2 = \sqrt{30}/10$. With this set of parameters, one can find easily from Eq. (18) that $m = 1/2$ and $\ell = \pm 3\sqrt{3}/8$. In what follows, we will drop the “±” from the $\ell$ formulas, as the rogue wave solutions depend on²¹⁰ $\ell^2$ only. To demonstrate the subtleties of the periodic backgrounds that support the rogue waves, we used two different sets of directional cosine parameters, that is, $n_1 = n_3 = 0$, $n_2 = 1$ for the SO(2)-transformed solutions [see Figs. 1(a)–1(d)] and $n_1 = n_2 = n_3 = \sqrt{3}/3$ for the SU(2) transformed so-²¹⁵ lutions [see Figs. 1(e)–1(h)], with the rotation angle $\phi = \pi/18$

kept the same. It is clear that under the SO(2) transformation, the background fields generate regular interference fringes so that the Peregrine solitons formed can exhibit antisymmetric patterns with respect to the origin, whereas, under the SU(2) transformation, the background fields own an additional phase that causes the fringes an offset and as a result make the Peregrine solitons asymmetric around the origin. This is also true in the normal dispersion regime, as indicated in Fig. 2, where we used another set of initial parameters $\gamma = 1$, $\sigma = -1$, $\phi = \pi/18$, $\omega_1 = 19/10$, $\omega_2 = 13/10$, $a_1 = \sqrt{10}/5$ and $a_2 = \sqrt{30}/5$. Additionally, in Fig. 2 we also demonstrate that the Peregrine solitons obtained with SU(2) transformation are still endowed with an anomalous peak amplitude feature, that is, one Peregrine soliton component can grow beyond the three times average height at the expense of the abrupt descent of the other wave component in the central position, as occurred for the Peregrine solitons built on the continuous backgrounds [53, 54].

It is evident from Figs. 1 and 2 that, within the framework of integrable model, the SU(2) transformed rogue wave solutions, although involving two more independent variables, exhibit rogue wave signatures that seem not to be significantly dif-
<figure><img src="image_4.png" /><figcaption>Figure 3: Numerical simulations of Peregrine solitons on the periodic backgrounds caused by SO(2) transformation, whose analytical solutions are shown in Figs. 2(a)–2(d), under non-integrable perturbations. (a), (b): $κ = 1$ (no perturbation); (c), (d): $κ = 1.05$; (e), (f): $κ = 1.1$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 4: Numerical simulations of Peregrine solitons on the periodic backgrounds caused by SU(2) transformation that are demonstrated in Figs. 2(e)–2(h), under non-integrable perturbations. (a), (b): $κ = 1$ (no perturbation); (c), (d): $κ = 1.05$; (e), (f): $κ = 1.1$.</figcaption></figure>

ferent from those of the SO(2) transformed ones, as pointed out in Ref. [52]. However, one may wonder what would happen, or precisely which one would be more stable, when imposing non-integrable perturbations on these two kinds of rogue waves. Here by *non-integrable perturbation* we mean a perturbation that can violate the integrability of the vector CLL model, e.g., letting the weight factor $κ$ in Eqs. (1) and (2) take a value other than one. This is actually a sort of quite strong perturbation, from the perspective of numerical simulations.

To clarify the issue mentioned above, we numerically inspect whether the analytical Peregrine soliton solutions shown in Fig. 2 are stable and to what extent they are robust against the non-integrable perturbations. Specifically, we set $κ = 1, 1.05$, and $1.1$ in the vector CLL model, respectively, and numerically integrate this model equation from $z = -4$ to $8$ using the analytical solutions at $z = -4$ as initial profiles, by means of a split-step Fourier method [53, 54]. The simulation results are shown in Figs. 3 and 4, corresponding to the SO(2) transformed solutions and SU(2) transformed ones, respectively. For these two types of Peregrine solitons, our numerical code can perfectly reproduce them when no perturbations are exerted (i.e., $κ = 1$), as seen from the contour plots (a) and (b) in both Figs. 3 and 5.

4. However, when putting non-integrable perturbations of same amount on them, the SU(2) transformed Peregrine solitons will experience slightly stronger disturbances than the SO(2) transformed ones, as suggested by the comparison between Figs. 3(c) and 3(d) and Figs. 4(c) and 4(d), where $κ = 1.05$ is used. Despite this stability discrepancy, both types of Peregrine solitons can still manifest themselves as predicted by analytical solutions, even when 10% amount of non-integrable perturbations (i.e., $κ = 1.1$) is added, as seen from the bottom row of contour plots in Figs. 3 and 4.

We need to point out that in both Figs. 1 and 2 we used the special sets of initial parameters that satisfy

$$a_1 = \sqrt{-\frac{\sigma\delta^2}{v_1}}, \quad a_2 = \sqrt{-\frac{\sigma\delta^2}{v_2}}, \quad (\sigma v_{1,2} < 0). \quad (69)$$

Under the amplitude conditions (69), Eq. (18) will admit two pairs of equal complex roots, with $m = -κ/2$ and $ℓ = \sqrt{3}δ/2$, which lead to one family of Peregrine soliton states as shown in Figs. 1 and 2. In fact, in a majority of cases, Eq. (18) could admit two pairs of different complex roots or two different $(m, ℓ)$ pairs, resulting in the coexistence of two families of Peregrine soliton states on the same background [52]. Figure 5
<figure><img src="image_6.png" /><figcaption>Figure 5: Coexistence of two different families of Peregrine solitons on the same periodic background ($\phi = \pi/18$ and $n_1 = n_2 = n_3 = \sqrt{3}/3$) in the normal dispersion regime ($\sigma = -1$), with $\gamma = 1$, $\omega_1 = 19/10$, $\omega_2 = 11/10$, $a_1 = 2\sqrt{10}/15$ and $a_2 = 2\sqrt{10}/5$. (a)–(d): $m = -3/2 + 12^{1/4}/5$ and $\ell = 12^{1/4}/5$; (e)–(h): $m = -3/2 - 12^{1/4}/5$ and $\ell = 12^{1/4}/5$.</figcaption></figure>

shows such rogue wave coexistence for the SU(2) transformation situation ($n_1 = n_2 = n_3 = \sqrt{3}/3$) in the normal dispersion regime, by choosing a typical set of parameters $\gamma = 1$, $\sigma = -1$, $\phi = \pi/18$, $\omega_1 = 19/10$, $\omega_2 = 11/10$, $a_1 = 2\sqrt{10}/15$, and $a_2 = 2\sqrt{10}/5$. It is easy to find, from Eq. (18), that this set of parameters will give rise to $(m, \ell) = (-3/2+12^{1/4}/5, 12^{1/4}/5)$ or $(-3/2 - 12^{1/4}/5, 12^{1/4}/5)$, and hence two different rogue wave families. It is seen that the two rogue wave components ($u_1, u_2$) may feature a bright-bright structure [see Figs. 5(a) and 5(b)] or a bright-dark structure [see Figs. 5(e) and 5(f)], either of which is possible to be excited from the MI of the background fields, for given set of system parameters [18, 51, 52].

We further point out that in the normal dispersion regime, there would occur dark-dark Peregrine soliton structures as well [18, 27, 50], within a broad range of parameters that can be uniquely determined by Eqs. (18), (64), (67) and (68). Figure 6 shows such dark-dark rogue wave states on either the continuous backgrounds [see Figs. 6(a)–6(d)] or the periodic backgrounds [see Figs. 6(e)–6(h)], using a special set of initial parameters $\sigma = -1$ (normal dispersion), $\gamma = 1$, $\omega_1 = 7/10$, $\omega_2 = -6/5$, and $a_1 = a_2 = 1$. The values of $m$ and $\ell$ can be calculated

<figure><img src="image_7.png" /><figcaption>Figure 6: Dark-dark Peregrine soliton structures formed in the normal dispersion regime ($\sigma = -1$), with $\gamma = 1$, $\omega_1 = 7/10$, $\omega_2 = -6/5$, and $a_1 = a_2 = 1$. (a)–(d) on the continuous backgrounds ($\phi = 0$); (e)–(h) on the periodic backgrounds ($\phi = \pi/18$, $n_1 = n_3 = 0$, $n_2 = 1$). The values of $m$ and $\ell$ are calculated from Eq. (18), which are now given by $m \approx -0.1661$ and $\ell \approx 0.4354$.</figcaption></figure>

from Eq. (18), yielding $m \approx -0.1661$ and $\ell \approx 0.4354$ (truncated to four decimal places). However, different from the bright-bright or bright-dark case shown in Fig. 5, the dark-dark rogue waves formed in the normal dispersion regime will generally not admit coexisting structures, as only one pair of valid $(m, \ell)$ values can be obtained for given set of parameters.

### 3.2. Triplet rogue waves on periodic backgrounds

Similarly, the general second-order solutions built on the periodic backgrounds can be rewritten as

$$u_1^{[2]} = u_{1,cw}^{[2]}(\cos \phi - in_3 \sin \phi) - u_{2,cw}^{[2]}(in_1 + n_2) \sin \phi, \quad (70)$$

$$u_2^{[2]} = u_{1,cw}^{[2]}(-in_1 + n_2) \sin \phi + u_{2,cw}^{[2]}(\cos \phi + in_3 \sin \phi), \quad (71)$$

where $u_{1,2,cw}^{[2]}$ are given by Eqs. (54) and (55). Depending on the choice of four structural parameters $\lambda_s$, these solutions can describe either the super rogue wave dynamics, by which we mean a single rogue wave event that exhibits a high peak of utmost extent on a finite background [62–64], or the triplet rogue waves,
<figure><img src="image_8.png" /><figcaption>Figure 7: Coexistence of two different families of triplet rogue waves on the same periodic background ($\phi = \pi/18$ and $n_1 = n_2 = n_3 = \sqrt{3}/3$), under the same parameter condition as in Fig. 5. (a)–(d): $m = -3/2 + 12^{1/4}/5$ and $\ell = 12^{1/4}/5$; (e)–(h): $m = -3/2 - 12^{1/4}/5$ and $\ell = 12^{1/4}/5$.</figcaption></figure>

which refer to the ones whose pattern is composed of three Peregrine solitons well separated from each other [18, 50, 67]. Strictly speaking, the fundamental Peregrine solitons discussed in the last subsection are the lowest order super rogue waves [63]. In the current subsection, we do not discuss the second-order super rogue waves, but the triplet rogue waves, which exist with tremendous possibilities in parameter space. As is well known, the tales of “three-sister” rogue waves have existed for many years, but only in the past decade have the hydrodynamical rogue wave triplets and their optical equivalents been observed in practical settings [30, 67]. This also provides grounds for an investigation of triplet rogue waves that share an SU(2) periodic background discussed here.

Despite the complexity of the triplet rogue wave solutions, their existence can still be predicted by the complex roots of the simple quartic Eq. (18). When this algebraic equation allows one pair of complex roots or two pairs of equal complex roots, there will be only one family of triplet rogue waves allowed by Eqs. (70) and (71). However, if two pairs of different complex roots can be found, our solutions (70) and (71) would exhibit two different families of triplet rogue waves that could coexist on either continuous or periodic backgrounds.

We demonstrate in Fig. 7 the coexistence of two different families of triplet rogue waves on the periodic backgrounds in the normal dispersion regime, using the same parameter conditions as in Fig. 5. The extra structural parameters are specified by $\gamma_1 = 3i$, $\gamma_2 = 1$, $\gamma_3 = 0$, and $\gamma_4 = 20$. It is clearly seen that each field component could exhibit two different triplet rogue wave patterns, each of which is composed of three well-separated Peregrine solitons. In either triplet rogue wave pattern, the layout of three Peregrine solitons is highly alterable, which depends strongly on the choice of four structural parameters $\gamma_s$. Theoretically, when these structural parameters are appropriately selected, the three Peregrine solitons can be well superposed to give birth to a super rogue wave state [63, 64].

### 4. Doublet, quartet, and sextet rogue waves in the triple root ($D = 3$) case

In fact, about the parameter conditions (69), there is more to the story. When these conditions hold, the cubic Eq. (12) will have a triple root $\mu_{1,2,3} = \mu_0 = -\gamma A/(2\sigma) + \kappa/4 + i\sqrt{3}\delta/4$, and this will correspond to the triple root ($D = 3$) case mentioned in Sec. 2. Under these circumstances, the vector CLL model will admit even-number multiple rogue wave solutions as well. In this section, let us follow the procedures in Sec. 2 to derive the explicit analytical solutions for doublet, quartet, and sextet rogue waves, of course on the periodic backgrounds caused by SU(2) transformation.

Obviously, under the parameter conditions (69), one can get from Eq. (18) that $m = -\kappa/2$ and $\ell = \sqrt{3}\delta/2$, and correspondingly from Eqs. (21) and (22) that $\lambda_r = -A\gamma^2 - \sigma\gamma\kappa/2 + \sigma$ and $\lambda_m = 3\sqrt{3}\sigma\gamma\delta/2$. With these inputs and appropriate $\Gamma_j$ formulas (25), the factored eigenfunction $\Phi(\lambda)$ in Eq. (9) can be exactly expanded into Eq. (23) with $D = 3$. The proceeding two series expansion terms $\Phi^{(0)}$ and $\Phi^{(1)}$ can still be expressed by Eq. (31), but now the $R_{0,1,2}$ and $S_{0,1,2}$ polynomials are expressed as follows

$$R_0 = \gamma_1 - \frac{i\gamma_2}{2}\tau + \frac{i\gamma_3}{4}(\tau^2 + 2\tau + 3i\xi), \quad (72)$$

$$R_j = R_0 - \frac{\sqrt{3}\delta}{2\alpha_j} [\gamma_2 - \gamma_3(\tau - \phi_j)], \quad (j = 1, 2), \quad (73)$$

$$S_0 = \gamma_1 b_0 + \gamma_2 e_0 + \gamma_3 h_0 + \gamma_4 - \frac{i\gamma_5}{2}\tau + \frac{i\gamma_6}{4}(\tau^2 + 2\tau + 3i\xi), \quad (74)$$

$$S_j = S_0 - \frac{\sqrt{3}\delta}{2\alpha_j} [\gamma_1 b_j + \gamma_2 e_j + \gamma_3 h_j + \gamma_5 - \gamma_6(\tau - \phi_j)]. \quad (75)$$

where $\tau = \sqrt{3}\delta\theta$, $\xi = \delta^2\sigma z$, and $\gamma_s$ ($s = 1, \dots, 6$) are six complex structural parameters. The polynomial functions $b_{0,1,2}$, $e_{0,1,2}$ and $h_{0,1,2}$ in Eqs. (74) and (75) are given by

$$b_0 = \frac{1}{6}\tau(\tau^2 + 6\tau - 3) + \frac{3i}{2}(\tau - 1)\xi, \quad (76)$$
$$b_1 = -3\xi + i\tau^2 + \frac{2i(\tau - 1)}{\phi_1} + \frac{6i\sigma\delta^2\gamma}{\psi\alpha_2}, \quad (77)$$

$$b_2 = -3\xi + i\tau^2 + \frac{2i(\tau - 1)}{\phi_2} + \frac{6i\sigma\delta^2\gamma}{\psi\alpha_1}, \quad (78)$$

$$e_0 = \frac{3}{16}(3i\xi + 2\tau^2 + 12)\xi - \frac{i}{48}\tau(\tau^3 + 12\tau^2 + 16), \quad (79)$$

$$e_1 = \frac{1}{6}\tau^3 - \frac{3}{2}\tau + \frac{2}{3} + \frac{3i}{2}\xi\tau - \frac{i\sqrt{3}}{2}\left(\tau^2\phi_1 + \frac{3i\xi}{\phi_1}\right) - \frac{3i\gamma(\sqrt{3}\delta + i\tau\alpha_1)\nu_2\delta^3}{2\alpha_1^3\nu_1^2}, \quad (80)$$

$$e_2 = \frac{1}{6}\tau^3 - \frac{3}{2}\tau + \frac{2}{3} + \frac{3i}{2}\xi\tau + \frac{i\sqrt{3}}{2}\left(\tau^2\phi_2 + \frac{3i\xi}{\phi_2}\right) + \frac{3i\gamma(\sqrt{3}\delta + i\tau\alpha_2)\nu_1\delta^3}{2\alpha_2^3\nu_2^2}, \quad (81)$$

$$h_0 = -\frac{9i}{16}(\tau - 2)\xi^2 - \frac{1}{8}(\tau^3 + 3\tau^2 + 9\tau + 20)\xi + \frac{i}{24}\tau\left(\frac{\tau^4}{10} + 2\tau^3 + 3\tau^2 + 2\tau + 4\right), \quad (82)$$

$$h_1 = \frac{9\xi^2}{8} + \frac{3i\phi_1\xi\tau}{2} - \frac{3i\xi\tau^2}{4} - \frac{\tau^4}{24} - \frac{\tau(5\tau^2 + 13)}{12} + \frac{i\sqrt{3}(\tau^2 - 9)\tau}{12} + \frac{\sqrt{3}(6\xi + i\tau^2)}{2\phi_1} - \frac{3\sqrt{3}i\gamma\nu_2\delta^4\phi_1}{2\alpha_1^3\nu_1^2} + \frac{\phi_1}{3} - \frac{3\gamma\nu_2(\tau^2 + 3i\xi)\delta^3}{4\alpha_1^2\nu_1^2} - \frac{3\gamma\nu_2\delta^4\tau(\delta\gamma + \nu_1)}{2\alpha_1^3\nu_1^3\phi_1} \quad (83)$$

$$h_2 = \frac{9\xi^2}{8} + \frac{3i\phi_2\xi\tau}{2} - \frac{3i\xi\tau^2}{4} - \frac{\tau^4}{24} - \frac{\tau(5\tau^2 + 13)}{12} - \frac{i\sqrt{3}(\tau^2 - 9)\tau}{12} - \frac{\sqrt{3}(6\xi + i\tau^2)}{2\phi_2} + \frac{3\sqrt{3}i\gamma\nu_1\delta^4\phi_2}{2\alpha_2^3\nu_2^2} + \frac{\phi_2}{3} + \frac{3\gamma\nu_1(\tau^2 + 3i\xi)\delta^3}{4\alpha_2^2\nu_2^2} + \frac{3\gamma\nu_1\delta^4\tau(\delta\gamma - \nu_2)}{2\alpha_2^3\nu_2^3\phi_2} \quad (84)$$

The function $\theta$ and the parameters $\alpha_j$, $\psi$, and $\phi_{1,2}$ are all kept the same as defined in Secs. 2 and 3.

In a similar fashion, substituting these expansion terms into Eqs. (26) and (27), and then into Eq. (30), one can obtain the general rogue wave solutions up to the second order on the periodic backgrounds. The resultant solution forms are still expressed by Eqs. (50)–(60), (70), and (71), and thus we do not repeat them here. But now the $R_{0,1,2}$ and $S_{0,1,2}$ polynomials are given by Eqs. (72)–(75), involving six structural parameters $\gamma_s$. Generally, leaving the matter of backgrounds aside, when $\gamma_3 \neq 0$, the first-order solutions (50) and (51) can describe the doublet rogue waves, but the second-order solutions (54) and (55) stand for the sextet rogue waves. When $\gamma_3 = 0$ yet $\gamma_2 \neq 0$,

<figure><img src="image_9.png" /><figcaption>Figure 8: 3D surface (top) and contour (bottom) plots of the doublet, quartet, and sextet rogue wave structures on the continuous wave backgrounds ($\phi = 0$) in the normal dispersion regime ($\sigma = -1$), with $\gamma = 1$, $\omega_1 = 19/10$, $\omega_2 = 13/10$, $a_1 = \sqrt{10}/5$ and $a_2 = \sqrt{30}/5$, but three different sets of structural parameters. (a), (b): $\gamma_1 = 1$, $\gamma_5 = 50$, $\gamma_2 = \gamma_3 = \gamma_4 = \gamma_6 = 0$ (doublets); (c), (d): $\gamma_1 = i$, $\gamma_2 = 1$, $\gamma_5 = 300$, $\gamma_3 = \gamma_4 = \gamma_6 = 0$ (quartets); (e), (f): $\gamma_1 = 5i$, $\gamma_2 = 4$, $\gamma_3 = 1$, $\gamma_5 = 400$, $\gamma_4 = \gamma_6 = 0$ (sextets).</figcaption></figure>

these first- and second-order solutions will describe the Peregrine solitons (like those shown in Figs. 1 and 2) and the quartet rogue waves, respectively. After a close inspection of Eqs. (72)–(84), one can find that the first-order solutions (50) and (51) are actually included in the second-order solutions (54) and (55), and can be obtained from the latter by setting $\gamma_2 = \gamma_3 = 0$. Hence, for this reason, we will use the second-order solutions (54) and (55) to describe the doublet, quartet, and sextet rogue waves altogether in the following example.

Figure 8 demonstrates such even-number multiple rogue waves on the continuous wave backgrounds ($\phi = 0$) in the normal dispersion regime, where we adopt the same set of parameters as in Fig. 2, but with different structural parameters that have been specified in the caption. It is seen that under different sets of structural parameters, there would appear the doublet rogue waves [see Figs. 8(a) and 8(b)], the quartet rogue waves [see Figs. 8(c) and 8(d)], and the sextet rogue waves [see Figs. 8(e) and 8(f)]. But, in contrast to the coexisting rogue wave case illustrated in Fig. 7, every constituent in these multiple rogue wave states is actually the same, despite that some may
<figure><img src="image_10.png" /><figcaption>Figure 9: Contour plots of the [(a), (b)] doublet, [(c), (d)] quartet, and [(e), (f)] sextet rogue waves on the periodic backgrounds in the normal dispersion regime, using the same parameter conditions as in Fig. 8.</figcaption></figure>

have a tilted peak orientation caused by the strong interaction$^{425}$ with the nearby constituents. The same is the case with these even-number rogue waves sitting on the periodic backgrounds, as seen in Fig. 9, where we used the same parameters as in Fig. 8 but with $\phi = \pi/18$ and $n_1 = n_2 = n_3 = \sqrt{3}/3$. This is not surprising as Eq. (18) only yields one pair of $(m, \ell)$ values under the parameter conditions (69) and thus no coexisting rogue wave phenomena occur for this special case.

Finally, let us make a remark on the rule governing the number of fundamental Peregrine solitons involving in multiple rogue wave states. Basically, for the *n*th-order rogue wave solutions denoted by Eqs. (26) and (27), the number of the rogue wave constituents is equal to half of the degree of the polynomial, det(*M*), that appears in the solution denominator. Our preliminary studies show that, for given set of parameters that allows the characteristic equation (12) to have a double root, the *n*th-order multiple rogue wave state will consist of *n*(n + 1)/2 fundamental Peregrine solitons, as occurred in the scalar systems [63, 82, 83]. However, for those allowing the cubic equation (12) to have a triple root, or equivalently, satisfying the parameter condition (69), this *n*th-order rogue wave state will exhibit more complicated patterns, consisting of a maximum of *n*(n + 1) (even number) Peregrine solitons. Hence, returning to

the specific cases shown in Figs. 8 and 9, one can see at most 2 and 6 fundamental Peregrine solitons that constitute the first-order and second-order rogue wave states, respectively.

## 5. Conclusion

In conclusion, the rogue wave dynamics describing the fundamental Peregrine solitons as well as their doublet, triplet, quartet, and sextet counterparts developed on periodic backgrounds are analytically investigated, within the framework of the vector CLL–NLS equation, which can govern the interaction of two FF optical waves in a quadratic nonlinear medium. We provided basic procedures to derive the exact explicit solutions for them and established a bridge between the complex roots of two different quartic equations, by which the parameter condition for the existence of rogue waves can be unveiled. Particularly, we numerically confirmed that the Peregrine solitons sitting on either SO(2) or SU(2) periodic backgrounds are quite stable against non-integrable perturbations, but the latter may suffer from slightly stronger disturbances arising from the spontaneous MI. We also demonstrated the manifestation of rogue wave coexistence that is inherent to vector systems and an anomalous peak enhancement that is attributed to the self-steepening effect. We expect that these results and the related method formulated here will spark interest in applying them to other similar multi-component systems [18, 49, 54, 70], particularly to those possessing SU(N) symmetry (N ≥ 2) [84].

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11974075, 11774054, 12075036) and by the Progetti di Ricerca di Interesse Nazionale (PRIN) financed by the Italian Ministry of Education, Universities and Research (MIUR) (Project No. 2020X4T57A). Ph.G. acknowledges support from the Engineering and Innovation through Physical Sciences, High-technologies, and Cross-disciplinary research (EUR EIPHI) Graduate School (Contract No. ANR-17-EURE-0004).

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