Higher-order matrix nonlinear Schrödinger equation with the negative coherent coupling: binary Darboux transformation, vector solitons, breathers and rogue waves

Zhong Du; Yao Nie; Qian Guo

doi:10.1364/OE.506566

1. Introduction

Nonlinear Schrödinger (NLS)-type equations are extensively utilized as universal models to describe the propagation of diverse nonlinear waves in various fields, including nonlinear optics, fluid mechanics, plasma physics, and Bose-Einstein condensates. [1–8]. In optical fibers, solitons, among the different types of nonlinear waves, form through the interaction of group velocity dispersion and self-phase modulation nonlinearity [9,10]. They are particularly valuable in the long-distance, large capacity and high-bit-rate communication systems [9,10]. Breathers, as a special class of solitons, exhibit periodicity either along the temporal dimension or the spatial direction [11]. Currently, breathers are classified into three distinct types: Akhmediev breathers, Kuznetsov-Ma breathers, and Spatio-temporal breathers. Akhmediev breathers are localized in the temporal dimension and exhibit periodic propagation along the spatial direction [12]. On the other hand, Kuznetsov-Ma breathers are localized in the spatial direction and display periodic propagation along the temporal dimension [13,14]. Spatio-temporal breathers propagate periodically in both the temporal dimension and the propagation direction [15].

When the periods of breathers extend to infinity, they transform into a fractional form known as the rational solution, specifically the Peregrine soliton, which is used in the description and characterization of rogue waves [16]. Rogue waves, as the special types of solitons on the non-vanishing background, possess an amplitude that is at least four times higher compared to the surrounding waves in the ocean [17]. Breathers and rogue waves have been theoretically predicted in various other mediums such as optical fibers, water-wave tanks, and plasmas that contain negative ions [18–22]. Rogue waves are generated through the nonlinear superposition of the nonlinear waves, including solitons and breathers, as well as modulation instability in nonlinear dispersive systems [21–24]. Moreover, breathers and rogue waves have been observed in optical fiber experiments for NLS-type equations [25,26].

With the development in optical-fiber technology, researchers have introduced the matrix NLS equation with negative coherent coupling, which are utilized to govern the dynamics of orthogonally polarized nonlinear waves in the isotropic medium [27–31]. The classic matrix NLS equation with negative coherent coupling can be written as [27–31]

(1)$$i{\bf P}_t+{\bf P}_{xx}+2{\bf P}{\bf P}^{\dagger} {\bf P}=0,$$

where ${\bf P}=\left ( \begin {array}{cc} q_1 & q_2 \\ -q_2 & q_1 \end {array} \right )$, $q_1$ and $q_2$ are the slowly varying envelopes of two interacting optical modes, “ ${\dagger }$ ” is the Hermitian conjugate, the variables $x$ and $t$ denote the spatial and temporal coordinates, respectively, and the subscripts represent the partial derivatives. Substituting ${\bf P}=\left ( \begin {array}{cc} q_1 & q_2 \\ -q_2 & q_1 \end {array} \right )$ into Eq. (1), we derive the coupled NLS system with the negative coherent coupling as

\begin{aligned} &iq_{1,t}+q_{1,xx}+2\left(|q_1|^2+2|q_2|^2\right)q_1-2q_1^*q_2^2=0,\\ & iq_{2,t}+q_{2,xx}+2\left(|q_2|^2+2|q_1|^2\right)q_2-2q_2^*q_1^2=0, \end{aligned}

where the terms $|q_1|^2q_1$ and $|q_2|^2q_2$ denote the self-phase modulation, $|q_2|^2q_1$ and $|q_1|^2q_2$ represent the cross-phase modulation, $q_1^*q_2^2$ and $q_2^*q_1^2$ are referred to as the coherent coupling terms [27–31]. When considering the transmission of ultra-short pulses in optical fiber, it is necessary to take into account certain higher-order effects which are not included in Eq. (1). Under these circumstances, it is essential to incorporate higher-order effects, such as self-steepening, higher-order dispersion, and self-frequency, into the nonlinear system [32,33].

Therefore, this paper is dedicated to investigating the higher-order matrix nonlinear Schrödinger equation with negative coherent coupling in a birefringent fiber [34,35]:

(2)$$\begin{aligned} i{\bf P}_t&+{\bf P}_{xx}+2{\bf P}{\bf P}^{\dagger} {\bf P}+\gamma[{\bf P}_{xxxx}+8{\bf P}{\bf P}^{\dagger} {\bf P}_{xx}\\ &+2{\bf P}^2{\bf P}_{xx}^{\dagger}+6{\bf P}^{\dagger} {\bf P}^2_x+4{\bf P}_x{\bf P}^{\dagger}_x{\bf P}+6({\bf P}{\bf P}^{\dagger})^2{\bf P}] = 0, \end{aligned}$$

where $\gamma$ denotes the coefficient of higher-order dispersion and nonlinear terms. Several studies have been conducted on Eq. (2): Properties of vector rogue waves, as well as the conversion of rogue waves to solitons and modulation instability, have been investigated via the Darboux dressing transformation (DDT) [34]; Riemann-Hilbert problem with zero boundary conditions has been utilized to study inverse scattering transformation and soliton solutions [35].

Both the Darboux transformation (DT) and DDT are sets of recurrence formulas for keeping the invariance of the eigenvalue problem associated with integrable nonlinear equations [36–38]. However, the binary DT takes it a step further by not only keeping the eigenvalue problem but also maintaining the invariance of the corresponding adjoint eigenvalue problem [39–41]. Therefore, the binary DT has been utilized to derive degenerate nonlinear wave solutions, which are nonlinear wave solutions that involve the merging of eigenvalues [41–43]. In the context of inverse scattering transformation, eigenvalues can be understood as the poles of the reflection coefficient associated with the corresponding inverse proble [44,45]. Hence, the solutions obtained through the binary DT can also be referred to as double-pole solutions.

However, so far, the binary DT, degenerate solitons, breathers, and beak-type rogue waves has not been constructed. In Section 2, we will establish the binary DT Eq. (2) by employing the corresponding Lax pair and symmetry condition. In Section 3, using this binary DT, we will derive and discuss the vector degenerate and exponential solitons from the vanishing seed solutions. In Section 4, by utilizing the non-vanishing seed solutions and the binary DT constructed in Section 2, we will obtain and analyze vector exponential and degenerate breathers, as well as the beak-type rogue waves. Our conclusions will be presented in Section 5.

2. Binary DT for Eq. (2)

In this section, we will construct the binary DT for Eq. (2), which enable us to obtain analytical solutions for certain nonlinear waves. We start with the Lax pair of Eq. (2) [34,35]:

(3)$$\begin{aligned} &\Phi_x=U(\lambda;Q)\Phi=(i\sigma\lambda + Q)\Phi,\\ &\Phi_t=V(\lambda;Q)\Phi=\left[-(8\lambda^4\gamma-2\lambda^2)i\sigma-8\lambda^3\gamma Q+\widehat{Q}\right]\Phi, \end{aligned}$$

with

\begin{aligned} &\sigma = \begin{pmatrix} -{\bf I}_{2\times 2} & {\bf O}_{2\times 2}\\ {\bf O}_{2\times} & {\bf I}_{2\times 2}\end{pmatrix}, \qquad Q= \begin{pmatrix} {\bf O}_{2\times 2} & {\bf P}\\- {\bf P}^{\dagger} & {\bf O}_{2\times 2} \end{pmatrix}\\ &\widehat{Q}={-}4\lambda^2\left(\gamma Q^2+\gamma Q_x\right)i\sigma+\lambda\left(2 \gamma[Q,Q_x]+2Q+ 2 \gamma Q_{xx}-4 \gamma Q^3\right)\\ &\qquad-\left(3\gamma Q^4-Q^2+\gamma Q_x^2-\gamma Q_{xx}Q-\gamma Q Q_{xx}-Q_x+6\gamma Q^2Q_x-\gamma Q_{xxx}\right)i\sigma, \end{aligned}

where $\Phi (x,t,\lambda )$ is a $4\times 2$ matrix eigenfunction, $\lambda$ is a complex eigenvalue, ${\bf I}_{2\times 2}$ is a $2\times 2$ identity matrix, ${\bf O}_{2\times 2}$ is a $2 \times 2$ zero matrix, and $[\cdot \,,\cdot ]$ is the matrix communicator. After calculating, we discover that the zero curvature equation $U_t-V_x+[U,V]=0$ can yield Eq. (2). Lax Pair (3) shows that the matrices $U(\lambda ;Q)$ and $V(\lambda ;Q)$ satisfy the symmetry condition

(4a)$$U(\lambda;Q)=S_1^{{-}1}U(\lambda;Q)S_1,\qquad V(\lambda;Q)=S_1^{{-}1}V(\lambda;Q)S_1,$$

(4b)$$U^*(\lambda;Q)=S_2^{{-}1}U(\lambda^*;Q)S_2,\qquad V^*(\lambda;Q)=S_2^{{-}1}V(\lambda^*;Q)S_2,$$

with

S_1=\left( \begin{array}{cccc} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ \end{array} \right),\qquad S_2=\left( \begin{array}{cccc} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ \end{array} \right),

where the superscript “ $-1$ ” indicates the inverse of a matrix.

Set that $q_1$ and $q_2$ are the seed solutions of Eq. (2), the vector $\phi _1=(\varphi _1,\varphi _2,\varphi _3,\varphi _4)^T$ is an eigenfunction of Lax Pair (3) with the complex eigenvalue $\lambda _1$, where $\varphi _d$’s ($d=1,2,3,4$) are the functions with respect to $x$ and $t$, and the superscript ‘ T ’ denotes the transpose of a vector/matrix. Via Symmetry Condition (4), we can deduce that vector $S_1\phi _1=(-\varphi _2,\varphi _1,-\varphi _4,\varphi _3)^T$ is an eigenfunction of Lax Pair (3) with $\lambda _1$. Similarly, the vector $S_2\phi _1^*=(-\varphi _3^*,\varphi _4^*,\varphi _1^*,-\varphi _2^*)^T$ is also an eigenfunction of Lax Pair (3) with $\lambda _1^*$. By means of the dimensional reduction [39,40], we construct the binary DT $\left (\phi _1,{\bf P}\right )\xrightarrow {\phi _1,\lambda _1}(\widetilde {\phi _1},\widetilde {{\bf P}})$ of Lax Pair (3):

(5a)$$\widetilde{\phi_1}=\phi_1-\Upsilon_1\Omega^{{-}1}\omega,$$

(5b)$$\widetilde{{\bf P}}={\bf P}-2iW_{12},$$

where $W_d$’s are the $2\times 2$ matrices determined by $\begin {pmatrix}W_{11} & W_{12}\\W_{13} & W_{14} \end {pmatrix}=\Upsilon _1\Omega ^{-1}\Upsilon _1^{\dagger }$,

(6)$$\Upsilon_1=\left( \begin{array}{c} \Phi_1 \\ \Phi_2 \\ \Phi_3 \\ \Phi_4 \end{array} \right)=\left( \begin{array}{cccc} \varphi_1 & -\varphi_2 & -\varphi_3^* & \varphi_4^* \\ \varphi_2 & \varphi_1 & \varphi_4^* & \varphi_3^* \\ \varphi_3 & -\varphi_4 & \varphi_1^* & -\varphi_2^* \\ \varphi_4 & \varphi_3 & -\varphi_2^* & -\varphi_1^* \end{array} \right),$$

the vector $\widetilde {\phi _1}=\left (\widetilde {\varphi }_1,\widetilde {\varphi }_2,\widetilde {\varphi }_3,\widetilde {\varphi }_4\right )^T$ satisfies Lax Pair (3) with the matrix $\widetilde {\bf P}=\begin {pmatrix}q_1[1] & q_2[1]\\-q_2[1] & q_1[1] \end {pmatrix}$, $\widetilde {\varphi }_d$’s are the functions with respect to $x$ and $t$, $q_1[1]$ and $q_2[1]$ respectively represent the first iteration of $q_1$ and $q_2$, $W_{ij}$’s $(i,j=1,2)$ are the $2\times 2$ matrices, and the $4\times 4$ matrix $\Omega$ and $4\times 1$ vector $\omega$ are determined by the algebraic equations

(7a)$$\Omega \Lambda_1-\Lambda_1^{\dagger}\Omega=\Upsilon_1^{\dagger}\Upsilon_1,$$

(7b)$$\lambda \omega-\Lambda_1^{\dagger}\omega=\Upsilon_1^{\dagger}\phi_1,$$

and $\Lambda _1=\begin {pmatrix} \lambda _1 {\bf I}_{2\times 2} & {\bf O}_{2\times 2}\\ {\bf O}_{2\times 2} & \lambda _1^* {\bf I}_{2\times 2} \end {pmatrix}$. Substituting Expression (6) into binary DT (5b), we can obtain the one-fold binary DT formulas for Eq. (2)

(8a)$$\begin{aligned}& q_1[1]=q_1+2i\frac{\det\begin{pmatrix} \Omega & \Phi_3^{\dagger}\\ \Phi_1 & 0 \end{pmatrix}}{\det \Omega}, \end{aligned}$$

(8b)$$\begin{aligned}& q_2[1]=q_2+2i\frac{\det\begin{pmatrix} \Omega & \Phi_4^{\dagger}\\ \Phi_1 & 0 \end{pmatrix}}{\det \Omega}. \end{aligned}$$

To derive the explicit expression of $\Omega$ in Eq. (7a), we use the method mentioned in Refs. [39,40]. By assuming that

(9)$$\Upsilon_1=(\Theta,S_2\Theta^*),\qquad \Omega=\begin{pmatrix} F_{11} & F_{12}\\ F_{21} & F_{22} \end{pmatrix},$$

and substituting Expressions (9) into Eq. (7a), we derive that

(10)$$\begin{pmatrix}(\lambda_1-\lambda_1^*)F_{11} & (\lambda_1^*-\lambda_1^*)F_{12}\\ (\lambda_1-\lambda_1)F_{21} & (\lambda_1^*-\lambda_1)F_{22}\end{pmatrix} = \begin{pmatrix} \Theta^{\dagger}\Theta & \Theta^{\dagger} S_2 \Theta^*\\ -\Theta^T S_2 \Theta & \Theta^T\Theta^* \end{pmatrix},$$

where $\Theta =\begin {pmatrix} \varphi _1 & \varphi _2 & \varphi _3 & \varphi _4 \\ -\varphi _2 & \varphi _1 & -\varphi _4 & \varphi _3 \end {pmatrix}^T$, and $F_{ij}$’s are the $2\times 2$ matrices. Obviously, $(\lambda _1^*-\lambda _1^*)F_{12}=(\lambda _1-\lambda _1)F_{21}=0$ and $\Theta ^{\dagger } S_2 \Theta ^*=-\Theta ^T S_2 \Theta =0$, thus, we can not directly derive $F_{12}$ and $F_{22}$ from Expression (10). To address this problem, we take the limit approach [41] and assume that

(11)$$\Omega = \begin{pmatrix} \frac{\Theta^{\dagger}\Theta}{\lambda_1-\lambda_1^*} & -\lim_{\substack{\lambda \to \lambda_1}} \frac{\Phi(\lambda)^{\dagger} S_2 \Theta^*}{\lambda^*-\lambda_1^*}-K^*\\ \lim_{\substack{\lambda \to \lambda_1}} \frac{\Phi(\lambda)^T S \Theta}{\lambda-\lambda_1}+K & \frac{\Theta^T\Theta^*}{\lambda_1^*-\lambda_1} \end{pmatrix}=\begin{pmatrix}F_{11} & -F_{21}^*\\ F_{21} & F_{11}^*\end{pmatrix},$$

where $\Phi (\lambda )=\Theta \left |{\substack {\lambda _1=\lambda }}\right.$, $K=\begin {pmatrix}k_1 & k_2\\k_2 & -k_1\end {pmatrix}$, $F_{11}$ and $F_{21}$ are $2\times 2$ matrices written as

(12a)$$\begin{aligned}&F_{11}=\frac{\Theta^{\dagger}\Theta}{\lambda_1-\lambda_1^*}=\begin{pmatrix} g_{1} & g_{2} \\ -g_{2} & g_{1} \end{pmatrix}, \end{aligned}$$

(12b)$$\begin{aligned}& F_{21}=\lim_{\substack{\lambda \to \lambda_1}} \frac{\Phi(\lambda)^T S \Theta}{\lambda-\lambda_1}+K =\begin{pmatrix} h_{1}+k_1 & h_{2}+k_2 \\ h_{2}+k_2 & -h_{1}-k_1 \end{pmatrix}, \end{aligned}$$

with $h_1$ and $h_2$ as the complex functions of $x$ and $t$, $k_1$ and $k_2$ being the complex constants, and

\begin{aligned} & g_1=\frac{1} {\lambda_1-\lambda_1^*}\left(|\varphi_1|^2+|\varphi_2|^2+|\varphi_3|^2+|\varphi_4|^2\right),\\ & g_2=\frac{1}{\lambda_1-\lambda_1^*}\left(\varphi_1\varphi_2^*-\varphi_1^*\varphi_2+\varphi_3\varphi_4^*-\varphi_3^*\varphi_4 \right). \end{aligned}

In what follows, we will employ binary DT formulas (Eq. 8(a) and (b)) to present different kinds of analytic solutions, including degenerate and exponential soliton, breather solutions, and rogue wave solutions, for a high-order matrix nonlinear Schrödinger equation, i.e., Eq. (2), from vanishing and non-vanishing seed solutions.

3. Vector soliton solutions for Eq. (2)

Vector soliton solutions will be worked out via the vanishing seed solutions. With $q_1=q_2=0$ of Eq. (2), Lax Pair (3) is simplified into

(13)$$\Phi_x=i\lambda\sigma\Phi, \qquad \Phi_t={-}i(8\lambda^4\gamma-2\lambda^2)\sigma\Phi.$$

The fundamental solutions of Eq. (13) corresponding to $\lambda _1$ can be expressed as

(14)$$\phi_1=(\varphi_1,\varphi_2,\varphi_3,\varphi_4)^T=\left(l_1e^{{-}iY},l_2e^{{-}iY},l_3e^{iY},l_4e^{iY}\right)^T,$$

where $Y=\lambda _1\left (x+2\lambda _1 t-8\gamma \lambda _1^3t\right )$, $l_1$, $l_2$, $l_3$ and $l_4$ are four arbitrary complex constants. Substituting Expression (14) into Expression (12), we obtain that

(15)$$\begin{aligned} &h_1=2i(l_1l_3-l_2l_4)\left[x+4t\lambda_1\left(1-8\gamma\lambda_1^2\right)\right],\\ &h_2={-}2i(l_1l_4+l_2l_3)\left[x+4t\lambda_1\left(1-8\gamma\lambda_1^2\right)\right],\\ &g_1=\frac{1}{\lambda_1-\lambda_1^*}\left[e^{{-}i(Y-Y^*)}\left(|l_1|^2+|l_2|^2\right)+e^{ i(Y-Y^*)}\left(|l_3|^2+|l_4|^2\right)\right],\\ &g_2=\frac{1}{\lambda_1-\lambda_1^*}\left[e^{{-}i(Y-Y^*)}\left(l_1l_2^*-l_1^*l_2\right)+e^{ i(Y-Y^*)}\left(l_3l_4^*-l_3^*l_4\right)\right]. \end{aligned}$$

Substituting $q_1=q_2=0$, Expressions (14) and (15) into Binary DT Formulas (8), we can derive the semi-rational soliton solutions for Eq. (2) as

(16a)$$\begin{aligned}q_{1,semi-soliton}=&{-}2i\left( \begin{array}{cccc} l_1 e^{{-}i Y} & -l_2e^{{-}i Y} & -l_3^* e^{{-}i Y^*} & l_4^* e^{{-}iY^*} \end{array} \right)\\ &\left( \begin{array}{cccc} g_1 & g_2 & -h_1^*-k_1^* & -h_2^*-k_2^* \\ -g_2 & g_1 & -h_2^*-k_2^* & h_1^*+k_1^* \\ h_1+k_1 & h_2+k_2 & -g_1 & -g_2 \\ h_2+k_2 & -h_1-k_1 & g_2 & -g_1 \end{array} \right)^{{-}1}\left( \begin{array}{c} l_3^* e^{{-}i Y^*} \\ -l_4^* e^{{-}i Y^*} \\ l_1 e^{{-}i Y} \\ -l_2 e^{{-}i Y} \end{array} \right),\\ \end{aligned}$$

(16b)$$\begin{aligned}q_{2,semi-soliton}=&{-}2i\left( \begin{array}{cccc} l_1 e^{{-}i Y} & -l_2e^{{-}i Y} & -l_3^* e^{{-}i Y^*} & l_4^* e^{{-}iY^*} \end{array} \right) \\ &\left( \begin{array}{cccc} g_1 & g_2 & -h_1^*-k_1^* & -h_2^*-k_2^* \\ -g_2 & g_1 & -h_2^*-k_2^* & h_1^*+k_1^* \\ h_1+k_1 & h_2+k_2 & -g_1 & -g_2 \\ h_2+k_2 & -h_1-k_1 & g_2 & -g_1 \end{array} \right)^{{-}1}\left( \begin{array}{c} l_4^* e^{{-}i Y^*} \\ l_3^*e^{{-}i Y^*}\\ -l_2e^{{-}i Y} \\ -l_1e^{{-}i Y} \end{array} \right). \end{aligned}$$

Obviously, Solutions (16) comprise exponential and polynomial functions of $x$ and $t$. The constants $l_1$, $l_2$, $l_3$ and $l_4$ have contributions to both the exponential part and the rational part in Solutions (16). By adjusting these constants, we can investigate different types of degenerate solitons and exponential solitons. For instance, when $l_1=l_3=i$ and $l_2=l_4=0$, Fig. 1(a₁) describes a degenerate soliton consisting of two soliton-type hump and a Peregrine hump in the $q_1$ component, while Fig. 1(a₂) shows two Peregrine humps in the $q_2$ component. As the value of $l_4$ increases, the amplitude of the degenerate soliton in the $q_1$ component also increases. Simultaneously, in the $q_2$ component, the soliton-type hump becomes more prominent and transforms into the double-hump structure, as illustrated in Figs. 1(b₁-b₂). When $l_4$ increases up to $1$, we observe the emergence of elastic interaction between the counter-propagating wave and the degenerate soliton. As displayed in Figs. 1(c₁-c₂), the counter-propagating wave and the degenerate soliton correspond to the same eigenvalue and interact with each other. This interaction occurs only once. As a result, counter-propagating waves are generated in both components.

Fig. 1. Degenerate solitons via Solutions (16) with $\lambda _1=\frac {3}{2}i$, $l_1=l_3=i$, $l_2=0$, $k_1=0$, $k_2=\frac {1}{2}$, $\gamma =\frac {1}{25}$, (a₁-a₂) $l_4=0$; (b₁-b₂) $l_4=\frac {1}{4}$; (c₁-c₂) $l_4=1$.

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When $l_1=-il_2$ and $l_3=il_4$, the polynomial functions in Expressions (15) vanish, and Solutions (16) demonstrate vector exponential solitons. Without loss of generality, setting that $l_1=l_3=\frac {\sqrt {2\lambda _{1,I}}}{2}$ and $l_2=-l_4=i\frac {\sqrt {2\lambda _{1,I}}}{2}$, we find that the expressions of the vector solitons can be written as

(17a)$$q_{1,soliton}={-}i\lambda_{1,I}\left[e^{{-}2iY_{R}+i \delta_I} {\rm sech} (2Y_I+\delta_R)+e^{{-}2iY_R-i\epsilon_I}{\rm sech}(2Y_I-\epsilon_R)\right],$$

(17b)$$q_{2,soliton}=\lambda_{1,I}\left[e^{{-}2iY_{R}+i \delta_I} {\rm sech} (2Y_I+\delta_R)-e^{{-}2iY_R-i\epsilon_I}{\rm sech}(2Y_I-\epsilon_R)\right],$$

where the subscripts $R$ and $I$ represent the real and imaginary parts, respectively, $\delta$ and $\epsilon$ are the complex constants, $k_1=\frac {e^{\delta }+e^{\epsilon }}{2}$, $k_2=\frac {e^{\delta }-e^{\epsilon }}{2}i$, $Y_I=\lambda _{1,I}\left \{x+4\lambda _{1,R}\left [1+8\gamma \left (\lambda _{1,I}^2-\lambda _{1,R}^2\right )\right ]t\right \}$, and $Y_R=\lambda _{1,R}x-2\left [\lambda _{1,I}^2-\lambda _{1,R}^2+4\gamma (\lambda _{1,I}^4-6 \lambda _{1,I}^2 \lambda _{1,R}^2+\lambda _{1,R}^4)\right ]t$.

Based on Solutions (17), we can deduce that the vector solitons in the $q_1$ and $q_2$ components have velocities of $-4\lambda _{1,R}$. Additionally, the centre of the vector solitons is located on the line $Y_{I}+\frac {\delta _R-\epsilon _R}{4}=0$. By choosing appropriate values for $\delta$ and $\epsilon$, we can obtain certain types of solitons. For instance, with $\delta =\epsilon =\frac {1}{2}$, we can observe a single-hump bright soliton in the $q_1$ component and a double-hump bright soliton in the $q_2$ component, as shown in Figs. 2(a₁-a₂). Furthermore, when $\delta =\epsilon =\frac {1}{2}$, the soliton in the $q_1$ component becomes a flat-top type, while the distance between the two humps of the double-hump soliton increases in the $q_2$ component, as displayed in Figs. 2(b₁-b₂). Based on Solutions (17), we can deduce that the distances between the two humps in the $q_1$ and $q_2$ components are given by $\frac {|\delta _R+\epsilon _R|}{2\lambda _{1,I}\sqrt {1+16\lambda _{1,R}^2\left [1+8\gamma \left (\lambda _{1,I}^2-\lambda _{1,R}^2\right )\right ]^2}}$. Thus, as the value of $|\delta _R+\epsilon _R|$ increases, the two humps in the vector solitons can separate completely, as illustrated in Figs. 2(c₁-c₂).

Fig. 2. Vector solitons via Solutions (17) with $\lambda _1=\frac {1}{2}i$, $\gamma =\frac {1}{25}$, (a₁-a₂) $\delta =\epsilon =\frac {1}{2}$; (b₁-b₂) $\delta =\epsilon =1$; (c₁-c₂) $\delta =\epsilon =3$.

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In what follows, to observe vector solitons presented in Figs. 1 and 2, we briefly discuss possible experimental realization of solitons in the context of nonlinear optics. Inspired by Ref. [46], we could consider studying dynamics of vector solitons in a biased bulk photorefractive crystal. To achieve this, we can refer to the experimental setup described in Ref. [46]. In the experiment, a collimated continuous wave argon-ion laser beam is split using a polarization beam splitter. Furthermore, the phenomenon of incoherently coupled photorefractive spatial solitons has been studied in Ref. [47]. Additionally, Ref. [48] presents observations of incoherently coupled photorefractive spatial solitons. Moreover, Ref. [49] experimentally confirms the existence of optical vector solitons with bimodal structures in optical Kerr media.

4. Vector breather and rouge wave solutions for Eq. (2)

4.1 Vector breather solutions for Eq. (2)

In this section, vector breather and rogue wave solutions will be derived via Binary DT Formulas (8) with non-vanishing seed solutions. Taking the non-vanishing seed solutions as

(18)$$\begin{aligned} q_1=&a_1e^{i\tau},\qquad q_2=a_2e^{i\tau},\\ \tau=&bx+\left\{2\left(a_1^2 + a_2^2\right)-b^2\right.\\ &\left. +\left[ 6 \left(a_1^2 + a_2^2\right)^2 -12\left(a_1^2+a_2^2\right)b^2+ b^4\right] \gamma\right\}t,\end{aligned}$$

and substituting seed Solutions (18) into Lax Pair (3), we derive that the fundamental solutions of Lax Pair (3) with $\lambda =\lambda _1$ as

(19)$$\begin{pmatrix} \varphi_{1,br}\\ \varphi_{2,br}\\ \varphi_{3,br}\\ \varphi_{4,br} \end{pmatrix}=G \begin{pmatrix} l_1\left(z_1-\lambda_1-\frac{b}{2}\right)e^{i\rho_1}+l_3\left(z_2-\lambda_1-\frac{b}{2}\right)e^{i\rho_2}\\ l_2\left(z_1-\lambda_1-\frac{b}{2}\right)e^{i\rho_1}+l_4\left(z_2-\lambda_1-\frac{b}{2}\right)e^{i\rho_2}\\ i\left(l_1a_1-l_2a_2\right)e^{i\rho_1}+i\left(l_3a_1-l_4a_2\right)e^{i\rho_2}\\ i\left(l_1a_2+l_2a_1\right)e^{i\rho_1}+i\left(l_3a_2+l_4a_1\right)e^{i\rho_2} \end{pmatrix},$$

where $G={\rm diag}\left (e^{i\frac {\tau }{2}},e^{i\frac {\tau }{2}},e^{-i\frac {\tau }{2}},e^{-i\frac {\tau }{2}}\right )$, $a_1$, $a_2$ and $b$ are all the real constants,

(20a)$$\begin{aligned}\rho_1=&z_1x +\left\{{-}8\gamma\lambda_1^3 + 2\left(2 a_1^2 + 2 a_2^2-b^2\right)\gamma \lambda_1 \right.\\ &\left. + 4 b \gamma\lambda_1^2 + \left[b^3 - 6 \left(a_1^2 + a_2^2\right)b\right]\gamma+2\lambda_1 - b\right\} z_1 t,\\ \end{aligned}$$

(20b)$$\begin{aligned}\rho_2=&z_2x +\left\{{-}8\gamma\lambda_1^3 + 2\left(2 a_1^2 + 2 a_2^2-b^2\right)\gamma \lambda_1 \right. \\ &\left. + 4 b \gamma\lambda_1^2 + \left[b^3 - 6 \left(a_1^2 + a_2^2\right)b\right]\gamma+2\lambda_1 - b\right\} z_2 t,\end{aligned}$$

and $z_1$ and $z_2$ are two different roots of the quadratic equation

(21)$$z^2-\left(\lambda+\frac{b}{2}\right)^2-\left(a_1^2+a_2^2\right)=0.$$

Solving Eq. (21), we obtain that

(22)$$z_1={-}z_2={-}\sqrt{\left(\lambda + \frac{b}{2}\right)^2 + a_1^2 + a_2^2}.$$

Substituting Expressions (19) into Expressions (12), we correspondingly work out

(23)$$\begin{aligned} g_{1,br}=&2(a_1^2+a_2^2) \left[\frac{|l_1|^2+|l_2|^2}{z_1-z_1^*+\lambda_1-\lambda_1^*}e^{i(\rho_1-\rho_1^*)}+\frac{l_1l_3^*+l_2l_4^*}{z_1-z_2^*+\lambda_1-\lambda_1^*}e^{i(\rho_1-\rho_2^*)}\right.\\ &\left.+\frac{l_1^*l_3+l_2^*l_4}{z_2-z_1^*+\lambda_1-\lambda_1^*}e^{i(\rho_2-\rho_1^*)}+\frac{|l_3|^2+|l_4|^2}{z_2-z_2^*+\lambda_1-\lambda_1^*}e^{i(\rho_2-\rho_2^*)} \right],\\ g_{2,br}=&{-}2(a_1^2+a_2^2) \left[\frac{l_1^*l_2-l_1l_2^*}{z_1-z_1^*+\lambda_1-\lambda_1^*}e^{i(\rho_1-\rho_1^*)}+\frac{l_2l_3^*-l_1l_4^*}{z_1-z_2^*+\lambda_1-\lambda_1^*}e^{i(\rho_1-\rho_2^*)}\right.\\ &\left.+\frac{l_1^*l_4-l_2^*l_3}{z_2-z_1^*+\lambda_1-\lambda_1^*}e^{i(\rho_2-\rho_1^*)}+\frac{l_3^*l_4-l_3l_4^*}{z_2-z_2^*+\lambda_1-\lambda_1^*}e^{i(\rho_2-\rho_2^*)} \right],\end{aligned}$$

and

(24)$$\begin{aligned} h_{1,br}=&i \left[2 a_2 l_1l_2-a_1 \left(l_1^2-l_2^2\right)\right] \alpha+i \left[2 a_2 l_3l_4-a_1 \left(l_3^2-l_4^2\right)\right]\beta\\ &+i \left[a_1 (l_2l_4-l_1l_3)+a_2 (l_1l_4+l_2l_3)\right]\zeta,\\ h_{2,br}=&i \left[2 a_1 l_1l_2+a_2 \left(l_1^2-l_2^2\right)\right]\alpha+i \left[2 a_1l_3l_4+a_2 \left(l_3^2-l_4^2\right)\right]\beta\\ &+i \left[a_1 (l_1 l_4+l_2l_3)+a_2 (l_1l_3-l_2l_4)\right]\zeta,\end{aligned}$$

with

\begin{aligned} \alpha=&\left(\frac{\lambda_1+\frac{b}{2}}{z_1}-1\right)e^{2i\rho_1},\quad \beta=\left(\frac{\lambda_1+\frac{b}{2}}{z_2}-1\right)e^{2i\rho_2},\\ \zeta=&{-}2+2i\left[\left(2\lambda_1+b\right)x+4\left(2\lambda_1^2+b\lambda_1+a_1^2+a_2^2\right)t\right]\\ &-16 i \gamma \left[4 \lambda_1^3 (2\lambda_1+b)+\left(a_1^2+a_2^2\right) \left(4\lambda_1^2-2 b\lambda_1 +b^2\right)-\left(a_1^2+a_2^2\right)^2\right]t. \end{aligned}

By substituting seed Solutions (18), Expressions (19), (23) and (24) into binary DT formulas (8), we have the semi-rational soliton solutions as follows

(25a)$$\begin{aligned}q_{1,semi-br}=&a_1 e^{i\tau} -2i\left( \begin{array}{cccc} \varphi_{1,br} & -\varphi_{2,br} & -\varphi^*_{3,br} & \varphi^*_{4,br} \end{array} \right)\\ &\left( \begin{array}{cccc} g_{1,br} & g_{2,br} & -h_{1,br}^*-k_1^* & -h_{2,br}^*-k_2^* \\ -g_{2,br} & g_{1,br} & -h_{2,br}^*-k_2^* & h_{1,br}^*+k_1^* \\ h_{1,br}+k_1 & h_{2,br}+k_2 & -g_{1,br} & -g_{2,br} \\ h_{2,br}+k_2 & -h_{1,br}-k_1 & g_{2,br} & -g_{1,br} \end{array} \right)^{{-}1}\left( \begin{array}{c} \varphi_{3,br}^* \\ -\varphi_{4,br}^*\\ \varphi_{1,br}\\ -\varphi_{2,br} \end{array} \right),\\ \end{aligned}$$

(25b)$$\begin{aligned}q_{2,semi-br}=&a_2 e^{i\tau}-2i\left( \begin{array}{cccc} \varphi_{1,br} & -\varphi_{2,br} & -\varphi^*_{3,br} & \varphi^*_{4,br} \end{array} \right) \\ &\left( \begin{array}{cccc} g_{1,br} & g_{2,br} & -h_{1,br}^*-k_1^* & -h_{2,br}^*-k_2^* \\ -g_{2,br} & g_{1,br} & -h_{2,br}^*-k_2^* & h_{1,br}^*+k_1^* \\ h_{1,br}+k_1 & h_{2,br}+k_2 & -g_{1,br} & -g_{2,br} \\ h_{2,br}+k_2 & -h_{1,br}-k_1 & g_{2,br} & -g_{1,br} \end{array} \right)^{{-}1}\left( \begin{array}{c} \varphi_{4,br}^* \\ \varphi_{3,br}^*\\ -\varphi_{2,br}\\ -\varphi_{1,br} \end{array} \right). \end{aligned}$$

Here, taking $\lambda =\lambda _1=-\frac {b}{2}+i\eta$ in Expressions (22), we obtain that $z_1=-z_2=-\sqrt {a_1^2+a_2^2-\eta ^2}$.

When $l_1l_3-l_2l_4=l_1l_4+l_2l_3=0$, the polynomial functions in Solutions (25) vanish, resulting in the generation of vector breathers for Eq. (2). To simplify the calculations, we can assume, without loss of generality, that $l_2=il_1$ and $l_3=il_4$.

When $k_1=\pm ik_2$, we can generate vector beak-type breathers based on Solutions (25), as shown in Figs. 3. If both $z_1$ and $z_2$ are real, i.e. $\eta ^2<a_1^2+a_2^2$, Figs. 3(a₁-a₂) display a vector beak-type Akhmediev breather, which is periodic in spatial direction and localized in temporal dimension. It is worth noting that each hump in the breather of the $q_1$ component corresponds to two valleys in the breather of the $q_2$ component. Furthermore, the background amplitudes in Figs. 3(a₁-a₂) change before and after the periodic humps appear. This change reflects the energy transition between the two components. If both $z_1$ and $z_2$ are purely imaginary, i.e. $\eta ^2>a_1^2+a_2^2$, Figs. 3(b₁-b₂) present a vector beak-type Kuznetsov-Ma breather, which is periodic in temporal dimension and localized in spatial direction. As the value of $\gamma$ increases, the period of the vector beak-type Kuznetsov-Ma breather decreases, as illustrated in Figs. 3(c₁-c₂).

Fig. 3. Beak-type breathers via Solutions (25) with $b=0$, $a_1=\frac {3}{5}$, $a_2=\frac {4}{5}$, $l_1=l_4=1$, $l_2=l_3=i$, $k_1=1$, $k_2=i$, (a₁-a₂) $\eta =\frac {3}{4}$, $\gamma =\frac {1}{25}$; (b₁-b₂) $\eta =\frac {\sqrt {23}}{4}$, $\gamma =\frac {1}{25}$; (c₁-c₂) $\eta =\frac {\sqrt {23}}{4}$, $\gamma =\frac {1}{5}$.

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When $k_1\neq \pm ik_2$, we can derive the vector double-hump beak-type breathers for Eq. (2) using Solutions (25), as depicted in Figs. 4. If $\eta ^2<a_1^2+a_2^2$, Figs. 4(a₁-a₂) exhibit a vector double-hump beak-type Akhmediev breather. It can be noted that the background amplitudes in the $q_1$ and $q_2$ components keep invariant before and after the periodic humps appear. If $\eta ^2>a_1^2+a_2^2$, Figs. 4(b₁-b₂) display a vector double-hump beak-type Kuznetsov-Ma breather. By choosing suitable values for $m_1$ and $m_2$, we can observe a complete separation of the two humps in the vector double-hump beak-type Kuznetsov-Ma breather in Figs. 4(c₁-c₂).

Fig. 4. Double-hump beak-type breathers via Solutions (25) with $b=0$, $a_1=\frac {3}{5}$, $a_2=\frac {4}{5}$, $l_1=l_4=1$, $l_2=l_3=i$, $\gamma =\frac {1}{25}$, (a₁-a₂) $\eta =\frac {3}{4}$, $k_1=1$, $k_2=0$; (b₁-b₂) $\eta =\frac {\sqrt {23}}{4}$, $k_1=1$, $k_2=0$; (c₁-c₂) $\eta =\frac {\sqrt {23}}{4}$, $k_1=\frac {e^8+e^5}{2}$, $k_2=-\frac {e^8-e^5}{2}i$.

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When the polynomial functions in Solutions (25) do not vanish, Solutions (25) are semi-rational functions, which can describe the degenerate breathers consisting of two coincident breathers with the same eigenvalue. For example, we set that $l_1=l_4=1$ and $l_2=l_3=0$: Figs. 5(a₁-a₂) and 5(b₁-b₂) present a vector degenerate beak-type Akhmediev breather when both $z_1$ and $z_2$ are real; Figs. 5(c₁-c₂) illustrate a vector degenerate beak-type Kuznetsov-Ma breather when both $z_1$ and $z_2$ are purely imaginary. The profiles of degenerate beak-type breathers can be seen as the combination of two lines of beak-shaped peaks upon a rogue wave.

Fig. 5. Degenerate beak-type breathers via Solutions (25) with $b=0$, $a_1=\frac {3}{5}$, $a_2=\frac {4}{5}$, $l_1=l_4=1$, $l_2=l_3=0$, $\gamma =\frac {1}{25}$, $k_1=1$, $k_2=i$, (a₁-a₂) and (b₁-b₂) $\eta =\frac {3}{4}$; (c₁-c₂) $\eta =\frac {\sqrt {23}}{4}$.

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4.2 Vector rogue wave solutions for Eq. (2)

When both $z_1$ and $z_2$ tend towards zero, i.e., $\lambda _1\to -\frac {b}{2}\pm i\sqrt {a_1^2+a_2^2}$, both $h_{1,br}$ and $h_{2,br}$ become singular. In this situation, to avoid singularities of $h_{1,br}$ and $h_{2,br}$, $l_1$, $l_2$, $l_3$ and $l_4$ should satisfy the following conditions:

(26a)$$2a_2l_1l_2-a_1\left(l_1^2-l_2^2\right)=2a_2l_3l_4-a_1\left(l_3^2-l_4^2\right),$$

(26b)$$ 2 a_1l_1l_2+a_2\left(l_1^2-l_2^2\right)=2a_1l_3l_4+a_2\left(l_3^2-l_4^2\right).$$

Under Conditions (26), taking the limit $\lambda _1\to -\frac {b}{2}\pm i\sqrt {a_1^2+a_2^2}$ in Expressions (24), we derive that

(27a)$$\begin{aligned} h_{1,rw}=&\lim_{\lambda_1\to-\frac{b}{2}\pm i\sqrt{a_1^2+a_2^2}}h_{1,br}\\ =&i \left[2 a_2 l_1l_2-a_1 \left(l_1^2-l_2^2\right)+a_1 (l_2l_4-l_1l_3)+a_2 (l_1l_4+l_2l_3)\right]\zeta,\\ \end{aligned}$$

(27b)$$\begin{aligned} h_{2,rw}=&\lim_{\lambda_1\to-\frac{b}{2}\pm i\sqrt{a_1^2+a_2^2} }h_{2,br}\\=&i \left[2 a_1 l_1l_2+a_2 \left(l_1^2-l_2^2\right)+a_1 (l_1 l_4+l_2l_3)+a_2 (l_1l_3-l_2l_4)\right]\zeta.\end{aligned}$$

When $\lambda _{1,I}\neq 0$, taking the limit $\lambda _1\to -\frac {b}{2}\pm i\sqrt {a_1^2+a_2^2}$ in Expressions (19) and (23), we have that

(28)$$\begin{pmatrix} \varphi_{1,rw}\\ \varphi_{2,rw}\\ \varphi_{3,rw}\\ \varphi_{4,rw} \end{pmatrix}=G \begin{pmatrix} l_1\left(-\lambda_1-\frac{b}{2}\right)+l_3\left(-\lambda_1-\frac{b}{2}\right)\\ l_2\left(-\lambda_1-\frac{b}{2}\right)+l_4\left(-\lambda_1-\frac{b}{2}\right)\\ i\left(l_1a_1-l_2a_2\right)+i\left(l_3a_1-l_4a_2\right)\\ i\left(l_1a_2+l_2a_1\right)+i\left(l_3a_2+l_4a_1\right) \end{pmatrix},$$

and

(29)$$\begin{aligned} &g_{1,rw}=2(a_1^2+a_2^2) \left[\frac{|l_1|^2+|l_2|^2}{\lambda_1-\lambda_1^*}+\frac{l_1l_3^*+l_2l_4^*}{\lambda_1-\lambda_1^*}+\frac{l_1^*l_3+l_2^*l_4}{\lambda_1-\lambda_1^*}+\frac{|l_3|^2+|l_4|^2}{\lambda_1-\lambda_1^*} \right],\\ &g_{2,rw}={-}2(a_1^2+a_2^2) \left[\frac{l_1^*l_2-l_1l_2^*}{\lambda_1-\lambda_1^*}+\frac{l_2l_3^*-l_1l_4^*}{\lambda_1-\lambda_1^*}+\frac{l_1^*l_4-l_2^*l_3}{\lambda_1-\lambda_1^*}+\frac{l_3^*l_4-l_3l_4^*}{\lambda_1-\lambda_1^*} \right].\end{aligned}$$

By substituting seed Solutions (18), Expressions (27), (28) and (29) into binary DT formulas (8), we derive the rogue wave solutions for Eq. (2) as

(30a)$$\begin{aligned}q_{1,rw}=&a_1 e^{i\tau} -2i\left( \begin{array}{cccc} \varphi_{1,rw} & -\varphi_{2,rw} & -\varphi^*_{3,rw} & \varphi^*_{4,rw} \end{array} \right)\\ &\left( \begin{array}{cccc} g_{1,rw} & g_{2,rw} & -h_{1,rw}^*-k_1^* & -h_{2,rw}^*-k_2^* \\ -g_{2,rw} & g_{1,rw} & -h_{2,rw}^*-k_2^* & h_{1,rw}^*+k_1^* \\ h_{1,rw}+k_1 & h_{2,rw}+k_2 & -g_{1,rw} & -g_{2,rw} \\ h_{2,rw}+k_2 & -h_{1,rw}-k_1 & g_{2,rw} & -g_{1,rw} \end{array} \right)^{{-}1}\left( \begin{array}{c} \varphi_{3,rw}^* \\ -\varphi_{4,rw}^*\\ \varphi_{1,rw}\\ -\varphi_{2,rw} \end{array} \right), \end{aligned}$$

(30b)$$\begin{aligned}q_{2,rw}=&a_2 e^{i\tau}-2i\left( \begin{array}{cccc} \varphi_{1,rw} & -\varphi_{2,rw} & -\varphi^*_{3,rw} & \varphi^*_{4,rw} \end{array} \right)\\ &\left( \begin{array}{cccc} g_{1,rw} & g_{2,rw} & -h_{1,rw}^*-k_1^* & -h_{2,rw}^*-k_2^* \\ -g_{2,rw} & g_{1,rw} & -h_{2,rw}^*-k_2^* & h_{1,rw}^*+k_1^* \\ h_{1,rw}+k_1 & h_{2,rw}+k_2 & -g_{1,rw} & -g_{2,rw} \\ h_{2,rw}+k_2 & -h_{1,rw}-k_1 & g_{2,rw} & -g_{1,rw} \end{array} \right)^{{-}1}\left( \begin{array}{c} \varphi_{4,rw}^* \\ \varphi_{3,rw}^*\\ -\varphi_{2,rw}\\ -\varphi_{1,rw} \end{array} \right). \end{aligned}$$

To simplify the analysis, setting that $l_3=-il_2$, $l_4=il_1$ or $l_1=l_3$, $l_2=l_4$ satisfies Conditions (26), we can generate various rogue waves by means of Solutions (30).

When $l_3=-il_2$, $l_4=il_1$, we obtain two distinct types of vector rogue waves, as presented in Fig. 6. When $c_1\neq 0$, Figs. 6(a₁-a₂) show a vector beak-type rogue wave, where the one hump and two valleys are not aligned on a single line. The presence of a hump in the $q_1$ component corresponding to two valleys in the $q_2$ component reflects the dynamics of energy transition, which is governed by the negative coherent coupling terms in Eq. (2). When $c_1=0$, Figs. 6(b₁-b₂) and 6(c₁-c₂) display a bright rogue wave with one hump and two grooves in the $q_1$ component, along with a dark rogue wave with two valleys and no humps in the $q_2$ component.

Fig. 6. Vector rogue waves via Solutions (30) with $b=0$, $a_2=\frac {4}{5}$, $\gamma =\frac {1}{25}$, $l_1=l_2=1$, $l_3=-i$, $l_4=i$, $k_1=1$, $k_2=i$, (a₁-a₂) $a_1=\frac {3}{5}$, $\lambda _1=i$, (b₁-b₂) and (c₁-c₂) $a_1=0$, $\lambda _1=\frac {4}{5}i$.

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When $l_1=l_3$, $l_2=l_4$, $k_1=1$, $k_2=i$ and $a_1\neq 0$, we observe a vector bright rogue wave with eye-shaped distribution in Figs. 7(a₁-a₂). As $|k_1|$ increases, the vector bright rogue wave splits into two vector beak-type rogue waves, as shown in Figs. 7(b₁-b₂). The distance between the two beak-type rogue waves also increases with the value of $|k_1|$.

Fig. 7. Vector rogue waves via Solutions (30) with $b=0$, $a_1=\frac {3}{5}$, $a_2=\frac {4}{5}$, $l_1=l_2=l_3=l_4=1$, $\gamma =\frac {1}{25}$, $\lambda _1=i$, $k_2=i$, (a₁-a₂) $k_1=1$; (b₁-b₂) $k_1=30$.

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When $l_1=l_3$, $l_2=l_4$, $k_2=ik_1$ and $a_1=0$, Figs. 8(a₁-a₂) exhibit a bright rogue wave without valleys in the $q_1$ component, along with an eye-shaped rogue wave in the $q_2$ component. As $|k_1|$ increases, the bright rogue wave in the $q_1$ component splits into two bright rogue waves; Similarly, the eye-shaped rogue wave in the $q_2$ component splits into two beak-type rogue waves, as depicted in Figs. 8(b₁-b₂).

Fig. 8. Vector rogue waves via Solutions (30) with $b=0$, $a_1=0$, $a_2=\frac {4}{5}$, $l_1=l_2=l_3=l_4=1$, $\gamma =\frac {1}{25}$, $\lambda _1=\frac {4}{5}i$, $k_2=ik_1$, (a₁-a₂) $k_1=1$; (b₁-b₂) $k_1=30$.

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5. Conclusions

In summary, we have investigated the higher-order matrix nonlinear Schrödinger equation with the negative coherent coupling in a birefringent fiber, i.e., Eq. (2). For $q_1$ and $q_2$, the slowly varying envelopes of two interacting optical modes, we have constructed Binary DT Formulas (8) based on Lax Pair (3).

Using the vanishing seed solutions and binary DT formulas (8), we have obtained Solutions (16) to examine various types of degenerate solitons and exponential solitons. In Fig. 1, we have shown three types of degenerate solitons that depend on the increase of $l_4$. By setting $l_1=l_3=\frac {\sqrt {2\lambda _{1,I}}}{2}$ and $l_2=-l_4=i\frac {\sqrt {2\lambda _{1,I}}}{2}$, Solutions (16) have been simplified to Solutions (17), which describe double-hump bright solitons capable of separating into two individual humps, as shown in Fig. 2.

Using the non-vanishing seed solutions and Binary DT Formulas (8), we have derived Solutions (25) to demonstrate various types of vector breathers, including vector single-hump beak-type Akhmediev breather and Kuznetsov-Ma breathers, double-hump beak-type Akhmediev breather and Kuznetsov-Ma breathers, as well as vector degenerate beak-type breathers. These breathers are illustrated in Figs. 3, 4, and 5, respectively. It is worth noting that the double-hump breathers can also exhibit separated humps. Based on the breather solutions, by taking the limit $\lambda _1\to -\frac {b}{2}\pm i\sqrt {a_1^2+a_2^2}$ in Expressions (24), we have obtained rogue wave solutions, i.e., Solutions (30). Using these solutions, Fig. 6 display the vector beak-type rogue wave and bright-dark rogue wave, while Fig. 7 show the splitting of the eye-shaped rogue wave into two vector beak-type rogue waves. Additionally, Fig. 8 illustrate the splitting of the bright rogue wave without valleys in the $q_1$ component into two bright rogue waves, and the splitting of the eye-shaped rogue wave in the $q_2$ component into two beak-type rogue waves.

The results obtained in this work may potentially provide valuable insights for experimentally manipulating the separation of two-hump solitons, breathers, and rogue waves in optical fibers.

Funding

National Natural Science Foundation of China (12305001); Natural Science Foundation of Hebei Province (A2021502003); Fundamental Research Funds for the Central Universities (2023MS163).

Acknowledgments

We express our sincere thanks to all the members of our discussion group for their valuable comments.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Higher-order matrix nonlinear Schrödinger equation with the negative coherent coupling: binary Darboux transformation, vector solitons, breathers and rogue waves

Abstract

1. Introduction

2. Binary DT for Eq. (2)

3. Vector soliton solutions for Eq. (2)

4. Vector breather and rouge wave solutions for Eq. (2)

4.1 Vector breather solutions for Eq. (2)

4.2 Vector rogue wave solutions for Eq. (2)

5. Conclusions

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Equations (47)

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