1. Introduction
In recent years, the time scales theory has received increasing attention in applied mathematics as a unifying framework for continuous and discrete dynamical systems [1, 2]. The core concept originates from Hilger's 1988 introduction of measure chains, namely arbitrary nonempty closed subsets of the real numbers [3, 4]. For the time scale $\mathbb{T}$ and the space scale $\mathbb{X}$, their respective backward jump operators are denoted by $\sigma:\mathbb{T}\to\mathbb{T}$ and $\rho:\mathbb{X}\to\mathbb{X}$, which map each point to its immediate left neighbor in the corresponding scale, i.e. [5]
$\begin{align} \sigma\left(t\right): = \sup\left\{s\in\mathbb{T}:s \lt t\right\},\qquad \rho\left(x\right): = \sup\left\{s\in\mathbb{X}:s \lt x\right\}.\end{align}$
The associated graininess functions are defined by $\begin{align} \mu\left(t\right): = t-\sigma\left(t\right),\qquad \nu\left(x\right): = x-\rho\left(x\right).\end{align}$
The time-scale calculus provides a unified approach to differential, difference, and quantum calculus, leading to the theory of dynamic equations on time scales [6]. These equations provide a unified description of classical continuous, discrete, and quantum systems, while rigorous results on solution properties, such as uniqueness, stability, and periodicity, have further established the theory [7]. Dynamic equations on time scales also have a wide range of real-world applications. In ecology, they can be used to model insect population dynamics. In addition, time scales provide a unified way to describe both discrete and continuous processes in epidemic transmission models, such as those for West Nile virus [8, 9]. Potential applications also arise in physics and economics [10].Nonlinear evolution equations, especially integrable systems typified by soliton equations, have long been a central topic in mathematical physics [11, 12]. Continuous and discrete soliton equations, as two important classes of integrable systems, share many similarities and close connections [13]. For instance, their integrability is closely related to the construction of Lax pairs [14, 15]. Moreover, continuous systems can often be transformed into discrete ones through suitable discretization rules [16, 17]. To establish a unified framework for continuous and discrete integrable systems, scholars have extended the theory of integrable systems to the time-scale framework. In this direction, Hovhannisyan derived such classical nonlinear models as the sine-Gordon equation and the Burgers equation on time scales by introducing suitable Lax systems in the time-scale setting [18-20]. In addition, with the help of the nabla differential operator on time scales, the nonlinear Schrödinger (NLS) equation and the Gerdjikov-Ivanov equation have also been formulated on time scales [21, 22]. Through these studies, the theory of integrable systems has been conceptually extended and methodologically enriched within the time-scale framework, thereby providing a theoretical basis for the modeling of multiscale nonlinear phenomena.
In many practical problems arising in biochemistry, physics, and mechanics, coupled equations are ubiquitous. Consequently, the study of integrable couplings of soliton equations has attracted considerable attention [23, 24]. Integrable couplings of soliton equations originated from the study of the centerless Virasoro symmetric algebra associated with integrable systems or soliton equations [25]. Their aim is to generalize a single soliton equation to a multi-component coupled system by extending the degrees of freedom or the underlying algebraic structure while preserving integrability. Various approaches have been developed for constructing integrable couplings, including perturbation methods [26-28], methods based on enlarged spectral problems [29], and new Lie-algebraic methods [30].
Integrable couplings have been extensively studied in both continuous and discrete settings. In the continuous case, Ma related the semi-direct sum of Lie algebras to integrable couplings of soliton equations, thereby providing a systematic algebraic framework for their derivation [31]. More recently, Wang and Zhang extended this framework by constructing a multi-component integrable coupling based on a non-semisimple Lie algebra and applied it to both isospectral mKdV and nonisospectral Ablowitz-Kaup-Newell-Segur (AKNS) problems [32]. In addition, integrable couplings arising from three-dimensional unital algebras have also been studied, with applications to the KdV equation [33]. In the discrete case, Ma proposed a construction scheme for nonlinear discretizable couplings, including a Hamiltonian coupling for the Volterra lattice equation [34]. Despite these developments, a unified framework that bridges the continuous and discrete cases is still lacking. The main objective of this paper is to establish such a unified framework for integrable couplings. Within this framework, we construct integrable couplings for the AKNS hierarchy on a space scale by extending the associated spectral problems, and further derive the corresponding integrable couplings for the NLS and mKdV equations on a space scale.
To address this gap, we introduce the notions of time scale and space scale, which together provide a unified framework for continuous and discrete systems. The notion of the time scale was introduced by Hilger in 1988 through measure chains [3, 4], and it provides a unified description of continuous and discrete time evolution. In contrast, the term 'space scale' is used here to indicate that the time-scale framework is imposed on the spatial variable in the study of integrable systems. Under this setting, the spatial part can incorporate both continuous and discrete structures, thereby allowing a unified treatment of continuous and semi-discrete spatial dynamics. In our framework, the time variable remains continuous, whereas the spatial variable is treated within the time-scale setting. This provides a unified theoretical framework for studying integrable systems involving both continuous and semi-discrete structures.
The Darboux transformation is one of the most powerful tools in the theory of integrable systems, serving as a systematic method for generating new exact solutions from known ones [35-37]. A foundational framework for constructing Darboux transformations for integrable couplings based on non-semisimple matrix Lie algebras has been established [38, 39]. In addition, the Darboux method has been extended to more general settings, such as time scales, and applied to equations including the Korteweg-de Vries (mKdV) equation [40] and the complex coupled dispersionless equation [41]. However, to the best of our knowledge, the Darboux transformation for integrable couplings has not yet been developed in the time-scale setting. Motivated by this observation, in this paper we extend the Darboux transformation for integrable couplings to the time-scale setting.
In this paper, we study integrable couplings of the AKNS hierarchy on a space scale and derive the corresponding solutions by means of the Darboux transformation. In our framework, the time variable remains continuous, whereas the spatial variable is treated within the time-scale setting, which distinguishes the space-scale case from the classical one. Based on this setting, we construct the Darboux transformation for integrable couplings on a space scale and obtain soliton solutions for the related systems. The main contribution of this paper is to provide a unified framework for integrable couplings that connects continuous and semi-discrete cases, together with its applications to the NLS and mKdV equations. The paper is organized as follows. Section 2 develops the Darboux transformation for integrable couplings on a space scale and provides rigorous proofs. Section 3 derives integrable couplings of the AKNS hierarchy on a space scale, together with their corresponding Darboux transformations. Section 4 establishes integrable couplings of the NLS and mKdV equations on a space scale; in addition, one-soliton solutions for these coupled systems are obtained via the proposed Darboux transformation. Section 5 concludes the paper with a summary of the main results.
2. Formulation of Darboux transformations of integrable couplings on a space scale
In this section, we present a Darboux transformation for integrable couplings on a space scale. To begin with, we recall a basic product rule from the calculus on time scales. For two functions f and g, the nabla derivative satisfies [6]
$\begin{align*} \nabla_{x}\left(fg\right)& = \left(\nabla_{x}f\,\right)g+f^{\,\rho}\left(\nabla_{x}g\right), \\ \nabla_{t}\left(fg\right)& = \left(\nabla_{t}f\,\right)g+f^{\,\sigma}\left(\nabla_{t}g\right).\end{align*}$
Following [29, 31], the Lax pair associated with integrable couplings can be written in the block form
$\begin{align} \bar{\boldsymbol{U}} = \begin{pmatrix} \boldsymbol{U} & \boldsymbol{U}_1 \\ 0 & \boldsymbol{U} \end{pmatrix},\qquad \bar{\boldsymbol{V}} = \begin{pmatrix} \boldsymbol{V} & \boldsymbol{V}_1 \\ 0 & \boldsymbol{V} \end{pmatrix}.\end{align}$
We then extend the spatial part of the Lax pair to the space-scale setting, which leads to the following enlarged spectral problem:$\begin{align} \nabla_x\boldsymbol{\bar{\Phi}} = \bar{\boldsymbol{U}}\boldsymbol{\bar{\Phi}},\qquad \boldsymbol{\bar{\Phi}}_{t} = \bar{\boldsymbol{V}}\boldsymbol{\bar{\Phi}},\end{align}$
where $\begin{align*} \bar{\boldsymbol{U}} = \begin{pmatrix} \boldsymbol{U} & \boldsymbol{U}_1 \\ 0 & \boldsymbol{U} \end{pmatrix},\qquad \bar{\boldsymbol{V}} = \begin{pmatrix} \boldsymbol{V} & \boldsymbol{V}_1 \\ 0 & \boldsymbol{V} \end{pmatrix},\end{align*}$
with $\begin{align} \begin{cases} \boldsymbol{U} = \lambda \boldsymbol{J}+\boldsymbol{K},\qquad \boldsymbol{U}_1 = \lambda\boldsymbol{J}_1+\boldsymbol{K}_1,\\ \boldsymbol{V} = \sum_{j = 0}^{m}\boldsymbol{V}_j\lambda^{m-j},\qquad \boldsymbol{V}_1 = \sum_{j = 0}^{m}\boldsymbol{V}_{1j}\lambda^{m-j},\qquad m\unicode{x2A7E}0. \end{cases}\end{align}$
Here, J and J1 are constant N × N diagonal matrices, while K and K1 are N × N matrices whose off-diagonal entries depend on the variables and whose diagonal entries vanish. Define $\begin{align} \bar{\boldsymbol{J}} = \begin{pmatrix} \boldsymbol{J} & \boldsymbol{J}_1 \\ 0 & \boldsymbol{J} \end{pmatrix},\qquad \bar{\boldsymbol{K}} = \begin{pmatrix} \boldsymbol{K} & \boldsymbol{K}_1 \\ 0 & \boldsymbol{K} \end{pmatrix},\qquad \bar{\boldsymbol{V}}_j = \begin{pmatrix} \boldsymbol{V}_j & \boldsymbol{V}_{1j} \\ 0 & \boldsymbol{V}_j \end{pmatrix}.\end{align}$
The spectral problem ($\begin{align} \nabla_x\boldsymbol{\bar{\Phi}} = \left(\lambda\bar{\boldsymbol{J}}+\bar{\boldsymbol{K}}\right)\boldsymbol{\bar{\Phi}},\qquad \boldsymbol{\bar{\Phi}}_{t} = \sum_{j = 0}^{m}\bar{\boldsymbol{V}}_j\lambda^{m-j}\boldsymbol{\bar{\Phi}},\qquad m\unicode{x2A7E}0.\end{align}$
Let $\bar{\boldsymbol{D}}$ be a $2N\times 2N$ matrix such that the transformed function $\boldsymbol{\bar{\Phi}}[1] = \bar{\boldsymbol{D}}\boldsymbol{\bar{\Phi}}$ also satisfies a spectral problem of the same form as ($\begin{align} & \nabla_x\boldsymbol{\bar{\Phi}}\left[1\right] = \left(\lambda\bar{\boldsymbol{J}}+\bar{\boldsymbol{K}}\left[1\right]\right)\boldsymbol{\bar{\Phi}}\left[1\right],\quad \boldsymbol{\bar{\Phi}}_{t}\left[1\right] = \sum_{j = 0}^{m}\bar{\boldsymbol{V}}_j\left[1\right]\lambda^{m-j}\boldsymbol{\bar{\Phi}}\left[1\right],\nonumber\\[-4pt] & \qquad m\unicode{x2A7E}0,\end{align}$
where $\bar{\boldsymbol{K}}[1]$ has the same form as $\bar{\boldsymbol{K}}$, and $\bar{\boldsymbol{V}}_j[1]$ has the same form as $\bar{\boldsymbol{V}}_j$.The transformation $\boldsymbol{\bar{\Phi}}[1] = \bar{\boldsymbol{D}}\boldsymbol{\bar{\Phi}}$ is called the Darboux transformation of the spectral problem (7 ). We take $\bar{\boldsymbol{D}}$ in the form
$\begin{equation} \bar{\boldsymbol{D}} = \lambda\bar{\boldsymbol{I}}-\bar{\boldsymbol{S}},\qquad \bar{\boldsymbol{I}} = \mathrm{diag}\left(\boldsymbol{I},\boldsymbol{I}\right),\end{equation}$
where I is the N × N identity matrix, and $\bar{\boldsymbol{S}}$ is a $2N\times 2N$ matrix to be determined.Applying the nabla derivative to both sides of the transformation $\boldsymbol{\bar{\Phi}}[1] = \bar{\boldsymbol{D}}\boldsymbol{\bar{\Phi}}$, we obtain 10 ), we obtain
$\begin{align} \left(\lambda\bar{\boldsymbol{J}}+\bar{\boldsymbol{K}}\left[1\right]\right)\left(\lambda\bar{\boldsymbol{I}}-\bar{\boldsymbol{S}}\right) = \left(\lambda\bar{\boldsymbol{I}}-\bar{\boldsymbol{S}}^{\rho}\right)\left(\lambda\bar{\boldsymbol{J}}+\bar{\boldsymbol{K}}\right) -\nabla_x\bar{\boldsymbol{S}}.\end{align}$
By comparing the coefficients of powers of $\lambda$ in ($\begin{align} \begin{cases} \bar{\boldsymbol{K}}\left[1\right] = \bar{\boldsymbol{K}}+\bar{\boldsymbol{J}}\bar{\boldsymbol{S}}-\bar{\boldsymbol{S}}^{\rho}\bar{\boldsymbol{J}}, \\ \nabla_x\bar{\boldsymbol{S}} = \bar{\boldsymbol{K}}\left[1\right]\bar{\boldsymbol{S}}-\bar{\boldsymbol{S}}^{\rho}\bar{\boldsymbol{K}} \end{cases}.\end{align}$
Differentiating both sides of the transformation $\boldsymbol{\bar{\Phi}}[1] = \bar{\boldsymbol{D}}\boldsymbol{\bar{\Phi}}$ with respect to t, we obtain 12 ), we obtain
$\begin{align} \sum_{i = 0}^{m}\bar{\boldsymbol{V}}_{i}\left[1\right]\lambda^{m-i}\left(\lambda\bar{\boldsymbol{I}}-\bar{\boldsymbol{S}}\right) = \left(\lambda\bar{\boldsymbol{I}}-\bar{\boldsymbol{S}}\right)\sum_{i = 0}^{m}\bar{\boldsymbol{V}}_i\lambda^{m-i}-\bar{\boldsymbol{S}}_t.\end{align}$
By comparing the coefficients of powers of $\lambda$ in ($\left\{\begin{array}{l} \overline{\boldsymbol{V}}_{i+1}[1]=\overline{\boldsymbol{V}}_{i+1}+\overline{\boldsymbol{V}}_{i}[1] \overline{\boldsymbol{S}}-\overline{\boldsymbol{S}} \overline{\boldsymbol{V}}_{i}, \quad 0 \leqslant i \leqslant m-1 \\ \overline{\boldsymbol{S}}_{t}=\overline{\boldsymbol{V}}_{m}[1] \overline{\boldsymbol{S}}-\overline{\boldsymbol{S}} \overline{\boldsymbol{V}}_{m} \end{array}\right.$
Next, we introduce N constant eigenvalues $\lambda$l $(1\unicode{x2A7D} l\unicode{x2A7D} N)$ and their corresponding eigenvectors $\boldsymbol{\bar{\Phi}}^{\,l}$ satisfying
$\begin{align} \nabla_x\boldsymbol{\bar{\Phi}}^{\,l} = \bar{\boldsymbol{U}}^{\,l}\boldsymbol{\bar{\Phi}}^{\,l},\qquad \boldsymbol{\bar{\Phi}}^{\,l}_{t} = \bar{\boldsymbol{V}}^{\,l}\boldsymbol{\bar{\Phi}}^{\,l},\qquad 1\unicode{x2A7D} l\unicode{x2A7D} N,\end{align}$
where $\bar{\boldsymbol{U}}^{\,l} = \bar{\boldsymbol{U}}|_{\lambda = \lambda_l}$ and $\bar{\boldsymbol{V}}^{\,l} = \bar{\boldsymbol{V}}|_{\lambda = \lambda_l}$.For each l, write
$\begin{align} \boldsymbol{\bar{\Phi}}^{\,l} = \left(\left(\Phi_{1}^{\,l}\right)^{\mathrm{T}},\left(\Phi^{\,l}\right)^{\mathrm{T}}\right)^{\mathrm{T}},\end{align}$
where $\Phi_{1}^{\,l}$ and $\Phi^{\,l}$ are N-dimensional column vectors. Define $\begin{align} \bar{\boldsymbol{H}} = \begin{pmatrix} \boldsymbol{H} & \boldsymbol{H}_1 \\ 0 & \boldsymbol{H} \end{pmatrix},\end{align}$
where $\begin{align*} \boldsymbol{H} = \left(\Phi^{1},\Phi^{2},\ldots,\Phi^{N}\right),\qquad \boldsymbol{H}_1 = \left(\Phi_{1}^{1},\Phi_{1}^{2},\ldots,\Phi_{1}^{N}\right).\end{align*}$
Then $\begin{align} \begin{cases} \nabla_x \boldsymbol{H} = \boldsymbol{J}\boldsymbol{H}\Lambda+\boldsymbol{K}\boldsymbol{H}, \\ \nabla_x \boldsymbol{H}_1 = \boldsymbol{J}\boldsymbol{H}_1\Lambda+\boldsymbol{K}\boldsymbol{H}_1+\boldsymbol{J}_1\boldsymbol{H}\Lambda+\boldsymbol{K}_1\boldsymbol{H}, \\ \boldsymbol{H}_t = \sum_{j = 0}^{m}\boldsymbol{V}_j\boldsymbol{H}\Lambda^{m-j}, \\ \boldsymbol{H}_{1t} = \sum_{j = 0}^{m}\boldsymbol{V}_j\boldsymbol{H}_1\Lambda^{m-j}+\sum_{j = 0}^{m}\boldsymbol{V}_{1j}\boldsymbol{H}\Lambda^{m-j}, \end{cases}\end{align}$
where $\begin{align*} \Lambda = \mathrm{diag}\left(\lambda_1,\lambda_2,\dots,\lambda_N\right),\qquad \bar{\Lambda} = \begin{pmatrix} \Lambda & 0\\ 0 & \Lambda \end{pmatrix}.\end{align*}$
Having defined 11 ) and (13 ). It follows that
$\overline{\boldsymbol{S}}=\overline{\boldsymbol{H}} \bar{\Lambda} \overline{\boldsymbol{H}}^{-1},$
we now prove that $\bar{\boldsymbol{S}}$ satisfies ($\begin{align} \begin{split} \bar{\boldsymbol{S}} = \begin{pmatrix} \boldsymbol{S} & \boldsymbol{S}_1 \\ 0 & \boldsymbol{S} \end{pmatrix} & = \begin{pmatrix} \boldsymbol{H} & \boldsymbol{H}_1 \\ 0 & \boldsymbol{H} \end{pmatrix} \begin{pmatrix} \Lambda & 0 \\ 0 & \Lambda \end{pmatrix} \begin{pmatrix} \boldsymbol{H}^{-1} & -\boldsymbol{H}^{-1}\boldsymbol{H}_{1}\boldsymbol{H}^{-1} \\ 0 & \boldsymbol{H}^{-1} \end{pmatrix} \\ & = \begin{pmatrix} \boldsymbol{H}\Lambda \boldsymbol{H}^{-1} & -\boldsymbol{H}\Lambda \boldsymbol{H}^{-1}\boldsymbol{H}_1\boldsymbol{H}^{-1} +\boldsymbol{H}_1\Lambda \boldsymbol{H}^{-1} \\ 0 & \boldsymbol{H}\Lambda \boldsymbol{H}^{-1} \end{pmatrix} \end{split}.\end{align}$
We now verify that the matrix $\bar{\boldsymbol{S}}$ satisfies (11 ) and (13 ). This leads to the following theorem.
The matrix
$\begin{equation*} \bar{\boldsymbol{D}} = \lambda\bar{\boldsymbol{I}}-\bar{\boldsymbol{S}},\end{equation*}$
where $\begin{equation*} \bar{\boldsymbol{S}} = \bar{\boldsymbol{H}}\bar{\Lambda}\bar{\boldsymbol{H}}^{-1},\end{equation*}$
is a Darboux matrix for the spectral problem ($\begin{equation} \bar{\boldsymbol{K}}\left[1\right] = \bar{\boldsymbol{K}}+\bar{\boldsymbol{J}}\bar{\boldsymbol{S}}-\bar{\boldsymbol{S}}^{\rho}\bar{\boldsymbol{J}}.\end{equation}$
By expanding (11 ), we obtain
$\begin{align} \begin{split} \nabla_x\bar{\boldsymbol{S}} & = \bar{\boldsymbol{K}}\bar{\boldsymbol{S}} - \bar{\boldsymbol{S}}^{\rho}\bar{\boldsymbol{K}} + \bar{\boldsymbol{J}}\bar{\boldsymbol{S}}^{2} - \bar{\boldsymbol{S}}^{\rho}\bar{\boldsymbol{J}}\bar{\boldsymbol{S}} \\ & = \begin{pmatrix} \boldsymbol{K} \boldsymbol{S} - \boldsymbol{S}^{\rho} \boldsymbol{K} + \boldsymbol{J}\boldsymbol{S}^{2} - \boldsymbol{S}^{\rho}\boldsymbol{J}\boldsymbol{S} & \begin{array}{c} \boldsymbol{K}\boldsymbol{S}_1 + \boldsymbol{K}_1\boldsymbol{S} - \boldsymbol{K}^{\rho}\boldsymbol{K}_1 - \boldsymbol{S}_1^{\rho}\boldsymbol{K} + \\ \boldsymbol{J}\boldsymbol{S}\boldsymbol{S}_1 + \boldsymbol{J}\boldsymbol{S}_1\boldsymbol{S} + \boldsymbol{J}_1\boldsymbol{S}^{2} - \boldsymbol{S}^{\rho}\boldsymbol{J}\boldsymbol{S}_1 - \boldsymbol{S}^{\rho}\boldsymbol{J}_1\boldsymbol{S} - \boldsymbol{S}_1^{\rho}\boldsymbol{J}\boldsymbol{S} \end{array} \\ 0 & \boldsymbol{K}\boldsymbol{S} - \boldsymbol{S}^{\rho}\boldsymbol{K} + \boldsymbol{J}\boldsymbol{S}^{2} - \boldsymbol{S}^{\rho}\boldsymbol{J}\boldsymbol{S} \end{pmatrix} \end{split}.\end{align}$
Next, we show that the nabla derivatives of the matrices S and S1 satisfy (20 ). Indeed,
$\begin{align} \begin{split} \nabla_x\boldsymbol{S} & = \nabla_x\left(\boldsymbol{H}\Lambda \boldsymbol{H}^{-1}\right) \\ & = \nabla_x\boldsymbol{H}\Lambda\boldsymbol{H}^{-1}+\boldsymbol{H}^{\rho}\Lambda\nabla_x\left(\boldsymbol{H}^{-1}\right) \\ & = \boldsymbol{J}\boldsymbol{S}^{2}+\boldsymbol{K}\boldsymbol{S}-\boldsymbol{S}^{\rho}\boldsymbol{J}\boldsymbol{S}-\boldsymbol{S}^{\rho}\boldsymbol{K} \end{split}.\end{align}$
Moreover, $\begin{align} \begin{split} \nabla_x\boldsymbol{S}_1 & = \nabla_x\left(-\boldsymbol{H}\Lambda \boldsymbol{H}^{-1}\boldsymbol{H}_1\boldsymbol{H}^{-1}+\boldsymbol{H}_1\Lambda \boldsymbol{H}^{-1}\right) \\ & = -\nabla_x\boldsymbol{H}\Lambda\boldsymbol{H}^{-1}-\boldsymbol{H}^{\rho}\Lambda\left(\nabla_x\boldsymbol{H}^{-1}\right)\boldsymbol{H}_1\boldsymbol{H}^{-1}-\boldsymbol{H}^{\rho}\Lambda\boldsymbol{H}^{-1\rho}\left(\nabla_x\boldsymbol{H}\right)\boldsymbol{H}^{-1}\\ &\quad-\boldsymbol{H}^{\rho}\Lambda\boldsymbol{H}^{-1\rho}\boldsymbol{H}_1^{\rho}\left(\nabla_x\boldsymbol{H}_1\right)+\nabla_x\boldsymbol{H}_1\Lambda\boldsymbol{H}+\boldsymbol{H}_1^{\rho}\Lambda\nabla_x\boldsymbol{H}^{-1} \\ & = -\boldsymbol{J}\boldsymbol{H}\Lambda^2\boldsymbol{H}^{-1}\boldsymbol{H}_1\boldsymbol{H}^{-1}-\boldsymbol{K}\boldsymbol{H}\Lambda\boldsymbol{H}^{-1}\boldsymbol{H}_1\boldsymbol{H}^{-1}+\boldsymbol{S}^{\rho}\boldsymbol{J}\boldsymbol{H}\Lambda\boldsymbol{H}^{-1}\boldsymbol{H}_1\boldsymbol{H}^{-1} \\ &\quad-\boldsymbol{S}^{\rho}\boldsymbol{J}\boldsymbol{H}_1\Lambda\boldsymbol{H}^{-1}-\boldsymbol{S}^{\rho}\boldsymbol{J}_1\boldsymbol{S}-\boldsymbol{S}^{\rho}\boldsymbol{K}_1+\boldsymbol{S}^{\rho}\boldsymbol{H}_1^{\rho}\boldsymbol{H}^{\rho-1}\boldsymbol{J}\boldsymbol{S} \\ &\quad+\left(\boldsymbol{H}\Lambda\boldsymbol{H}^{-1}\right)^{\rho}\boldsymbol{H}_1^{\rho}\boldsymbol{H}^{\rho-1}\boldsymbol{K}+\boldsymbol{J}\boldsymbol{H}_1\Lambda^2\boldsymbol{H}^{-1}+\boldsymbol{K}\boldsymbol{H}_1\Lambda\boldsymbol{H}^{-1}+\boldsymbol{J}_1\boldsymbol{H}\Lambda^2\boldsymbol{H}^{-1} \\ &\quad+\boldsymbol{K}_1\boldsymbol{S}-\boldsymbol{H}_1^{\rho}\Lambda\boldsymbol{H}^{\rho-1}\boldsymbol{J}\boldsymbol{S}-\boldsymbol{H}_1^{\rho}\Lambda\boldsymbol{H}^{-1\rho}\boldsymbol{K} \end{split}.\end{align}$
On the other hand,
$\begin{align} \begin{split} &\boldsymbol{K}\boldsymbol{S}_1+\boldsymbol{K}_1\boldsymbol{S}-\boldsymbol{K}^{\rho}\boldsymbol{K}_1-\boldsymbol{S}_1^{\rho}\boldsymbol{K}+\boldsymbol{J}\boldsymbol{S}\boldsymbol{S}_1+\boldsymbol{J}\boldsymbol{S}_1\boldsymbol{S}+\boldsymbol{J}_1\boldsymbol{S}^{2}-\boldsymbol{S}^{\rho}\boldsymbol{J}\boldsymbol{S}_1-\boldsymbol{S}^{\rho}\boldsymbol{J}_1\boldsymbol{S}-\boldsymbol{S}_1^{\rho}\boldsymbol{J}\boldsymbol{S} \\ &\quad = \boldsymbol{K}\left(-\boldsymbol{H}\Lambda \boldsymbol{H}^{-1}\boldsymbol{H}_1\boldsymbol{H}^{-1}+\boldsymbol{H}_1\Lambda \boldsymbol{H}^{-1}\right)+\boldsymbol{K}_1\boldsymbol{S}-\boldsymbol{S}^{\rho}\boldsymbol{K}_1-\left(-\boldsymbol{H}\Lambda \boldsymbol{H}^{-1}\boldsymbol{H}_1\boldsymbol{H}^{-1}+\boldsymbol{H}_1\Lambda \boldsymbol{H}^{-1}\right)^{\rho}\boldsymbol{K} \\ &\qquad +\boldsymbol{J}\boldsymbol{H}\Lambda\boldsymbol{K}^{-1}\left(-\boldsymbol{H}\Lambda \boldsymbol{H}^{-1}\boldsymbol{H}_1\boldsymbol{H}^{-1}+\boldsymbol{H}_1\Lambda \boldsymbol{H}^{-1}\right)+\boldsymbol{J}\left(-\boldsymbol{H}\Lambda \boldsymbol{H}^{-1}\boldsymbol{H}_1\boldsymbol{H}^{-1}+\boldsymbol{H}_1\Lambda \boldsymbol{H}^{-1}\right)\boldsymbol{H}\Lambda\boldsymbol{H}^{-1} \\ &\qquad +\boldsymbol{J}_1\boldsymbol{S}^{2}-\boldsymbol{S}^{\rho}\left(-\boldsymbol{H}\Lambda \boldsymbol{H}^{-1}\boldsymbol{H}_1\boldsymbol{H}^{-1}+\boldsymbol{H}_1\Lambda \boldsymbol{H}^{-1}\right)-\boldsymbol{S}^{\rho}\boldsymbol{J}_1\boldsymbol{S}-\left(-\boldsymbol{H}\Lambda \boldsymbol{H}^{-1}\boldsymbol{H}_1\boldsymbol{H}^{-1}+\boldsymbol{H}_1\Lambda \boldsymbol{H}^{-1}\right)^{\rho}\boldsymbol{J}\boldsymbol{S} \\ &\quad = -\boldsymbol{K}\boldsymbol{H}\Lambda\boldsymbol{H}^{-1}\boldsymbol{H}_1\boldsymbol{H}^{-1}+\boldsymbol{K}\boldsymbol{H}_1\Lambda\boldsymbol{H}^{-1}+\boldsymbol{K}_1\boldsymbol{S}-\boldsymbol{S}^{\rho}\boldsymbol{K}_1+\left(\boldsymbol{H}\Lambda\boldsymbol{H}^{-1}\right)^{\rho}\boldsymbol{H}_1^{\rho}\boldsymbol{H}^{\rho-1}\boldsymbol{K} \\ &\qquad -\boldsymbol{H}_1^{\rho}\Lambda\boldsymbol{H}^{-1\rho}\boldsymbol{K}-\boldsymbol{J}\boldsymbol{H}\Lambda^{2}\boldsymbol{H}^{-1}\boldsymbol{H}_1\boldsymbol{H}^{-1}+\boldsymbol{J}\boldsymbol{H}_1\Lambda^2\boldsymbol{H}^{-1}+\boldsymbol{J}_1\boldsymbol{H}\Lambda^{2}\boldsymbol{H}^{-1}+\boldsymbol{S}^{\rho}\boldsymbol{J}\boldsymbol{H}\Lambda\boldsymbol{H}^{-1}\boldsymbol{H}_1\boldsymbol{H}^{-1} \\ &\qquad -\boldsymbol{S}^{\rho}\boldsymbol{J}\boldsymbol{H}_1\Lambda\boldsymbol{H}^{-1}-\boldsymbol{S}^{\rho}\boldsymbol{J}_1\boldsymbol{S}+\boldsymbol{S}^{\rho}\boldsymbol{H}_1^{\rho}\boldsymbol{H}^{\rho-1}\boldsymbol{J}\boldsymbol{S}-\boldsymbol{H}_1^{\rho}\Lambda\boldsymbol{H}^{\rho-1}\boldsymbol{J}\boldsymbol{S}. \end{split} \end{align}$
Comparing the above expressions, we conclude that the defined $\bar{\boldsymbol{S}}$ satisfies (11 ). Next, from (13 ), we have 13 ) by direct computation. Since
$\begin{align} \bar{\boldsymbol{S}}_t = \left[\sum_{i = 0}^{m}\bar{\boldsymbol{V}}_i\bar{\boldsymbol{S}}^{m-i},\bar{\boldsymbol{S}}\right].\end{align}$
We now verify that $\bar{\boldsymbol{S}}$ satisfies ($\begin{align} \bar{\boldsymbol{S}}^{m-i} = \begin{pmatrix} \boldsymbol{S}^{m-i} & \sum_{k = 1}^{m-i}\boldsymbol{L}_k \\ 0 & \boldsymbol{S}^{m-i} \end{pmatrix},\end{align}$
where $\begin{align} \boldsymbol{L}_l = \underbrace{\boldsymbol{S}, \ldots, \boldsymbol{S}}_{l-1}\boldsymbol{S}_1 \underbrace{\boldsymbol{S}, \ldots, \boldsymbol{S}}_{m-i-l}, \quad 1 \unicode{x2A7D} l \unicode{x2A7D} m-i,\end{align}$
it follows that $\begin{align} \begin{split} \left[\bar{\boldsymbol{V}}_i\bar{\boldsymbol{S}}^{m-i},\bar{\boldsymbol{S}}\right] & = \begin{pmatrix} \boldsymbol{V}_i\boldsymbol{S}^{m-i+1}-\boldsymbol{S}\boldsymbol{V}_i\boldsymbol{S}^{m-i} & \begin{array}{c} \boldsymbol{V}_i\boldsymbol{S}^{m-i}\boldsymbol{S}_1+\boldsymbol{S}_i\sum_{k = 1}^{m-i}\boldsymbol{L}_k\boldsymbol{S}+\boldsymbol{V}_{1i}\boldsymbol{S}^{m-i+1} \\ -\boldsymbol{S}\boldsymbol{V}_i\sum_{k = 1}^{m-i}\boldsymbol{L}_k-\boldsymbol{S}\boldsymbol{V}_{1i}\boldsymbol{S}^{m-i}-\boldsymbol{S}_1\boldsymbol{V}_i\boldsymbol{S}^{m-i} \end{array} \\ 0 & \boldsymbol{V}_i\boldsymbol{S}^{m-i+1}-\boldsymbol{S}\boldsymbol{V}_i\boldsymbol{S}^{m-i} \end{pmatrix} \end{split}.\end{align}$
Moreover, $\begin{align} \begin{split} \sum_{k = 1}^{m-i} \boldsymbol{L}_k & = \sum_{k = 1}^{m-i} \underbrace{\boldsymbol{S}, \ldots, \boldsymbol{S}}_{k-1} \boldsymbol{S}_1 \underbrace{\boldsymbol{S}, \ldots, \boldsymbol{S}}_{m-i-k} \\ & = \sum_{k = 1}^{m-i} \boldsymbol{H} \Lambda^{k-1} \boldsymbol{H}^{-1} \left(-\boldsymbol{H} \Lambda \boldsymbol{H}^{-1} \boldsymbol{H}_1 \boldsymbol{H}^{-1} + \boldsymbol{H}_1 \Lambda \boldsymbol{H}^{-1}\right) \boldsymbol{H} \Lambda^{m-i-k} \boldsymbol{H}^{-1}\\ & = -\sum_{k = 1}^{m-i} \boldsymbol{H} \Lambda^k \boldsymbol{H}^{-1} \boldsymbol{H}_1 \Lambda^{m-i-k} \boldsymbol{H}^{-1} + \sum_{k = 1}^{m-i} \boldsymbol{H} \Lambda^{k-1} \boldsymbol{H}^{-1} \boldsymbol{H}_1 \Lambda^{m-i-k+1} \boldsymbol{H}^{-1}\\ & = -\boldsymbol{H} \Lambda^{m-i} \boldsymbol{H}^{-1} \boldsymbol{H}_1 \boldsymbol{H}^{-1} + \boldsymbol{H}_1 \Lambda^{m-i} \boldsymbol{H}^{-1}.\end{split}\end{align}$
A direct calculation shows that $\begin{equation} \begin{aligned} \boldsymbol{S}_{1t} & = -\boldsymbol{H}_t \Lambda \boldsymbol{H}^{-1} \boldsymbol{H}_1 \boldsymbol{H}^{-1} + \boldsymbol{H} \Lambda \boldsymbol{H}^{-1} \boldsymbol{H}_t \boldsymbol{H}^{-1} \boldsymbol{H}_1 \boldsymbol{H}^{-1} - \boldsymbol{H} \Lambda \boldsymbol{H}^{-1} \boldsymbol{H}_{1t} \boldsymbol{H}^{-1} \\ &\quad + \boldsymbol{H} \Lambda \boldsymbol{H}^{-1} \boldsymbol{H}_1 \boldsymbol{H}^{-1} \boldsymbol{H}_t \boldsymbol{H}^{-1} + \boldsymbol{H}_{1t} \Lambda \boldsymbol{H}^{-1} - \boldsymbol{H}_1 \Lambda \boldsymbol{H}^{-1} \boldsymbol{H}_t \boldsymbol{H}^{-1} \\ & = \sum_{i = 0}^{m} \left[ -\boldsymbol{V}_i \boldsymbol{H} \Lambda^{m-i+1} \boldsymbol{H}^{-1} \boldsymbol{H}_1 \boldsymbol{H}^{-1} + \boldsymbol{H} \Lambda \boldsymbol{H}^{-1} \boldsymbol{V}_i \boldsymbol{H} \Lambda^{m-i} \boldsymbol{H}^{-1} \boldsymbol{H}_1 \boldsymbol{H}^{-1} \right. \\ &\quad \left. - \boldsymbol{H} \Lambda \boldsymbol{H}^{-1} \left(\boldsymbol{V}_i \boldsymbol{H}_1 \Lambda^{m-i} + \boldsymbol{V}_{1i} \boldsymbol{H} \Lambda^{m-i}\right) \boldsymbol{H}^{-1} + \boldsymbol{H} \Lambda \boldsymbol{H}^{-1} \boldsymbol{H}_1 \boldsymbol{H}^{-1} \boldsymbol{V}_i \boldsymbol{H} \Lambda^{m-i} \boldsymbol{H}^{-1} \right. \\ &\quad \left. + \left(\boldsymbol{V}_i \boldsymbol{H}_1 \Lambda^{m-i} + \boldsymbol{V}_i \boldsymbol{H} \Lambda^{m-i}\right) \Lambda^{-1} - \boldsymbol{H}_1 \Lambda \boldsymbol{H}^{-1} \boldsymbol{V}_i \boldsymbol{H} \Lambda^{m-i} \boldsymbol{H}^{-1}] \right] \\ & = \sum_{i = 0}^{m} \left( -\boldsymbol{V}_i \boldsymbol{H} \Lambda^{m-i+1} \boldsymbol{H}^{-1} \boldsymbol{H}_1 \boldsymbol{H}^{-1} + \boldsymbol{H} \Lambda \boldsymbol{H}^{-1} \boldsymbol{V}_i \boldsymbol{H} \Lambda^{m-i} \boldsymbol{H}^{-1} \boldsymbol{H}_1 \boldsymbol{H}^{-1} \right. \\ &\quad \left. - \boldsymbol{H} \Lambda \boldsymbol{H}^{-1} \boldsymbol{V}_i \boldsymbol{H}_1 \Lambda^{m-i} \boldsymbol{H}^{-1} - \boldsymbol{H} \Lambda \boldsymbol{H}^{-1} \boldsymbol{V}_{1i} \boldsymbol{H} \Lambda^{m-i} \boldsymbol{H}^{-1} \right. \\ &\quad \left. + \boldsymbol{H} \Lambda \boldsymbol{H}^{-1} \boldsymbol{H}_1 \boldsymbol{H}^{-1} \boldsymbol{V}_i \boldsymbol{H} \Lambda^{m-i} \boldsymbol{H}^{-1} + \boldsymbol{V}_{1i} \boldsymbol{H} \Lambda^{m-i+1} \boldsymbol{H}^{-1} - \boldsymbol{H}_1 \Lambda \boldsymbol{H}^{-1} \boldsymbol{V}_i \boldsymbol{H} \Lambda^{m-i} \boldsymbol{H}^{-1} \right). \end{aligned}\end{equation}$
Furthermore, $\begin{aligned} {\left[\sum_{i=0}^{m} \bar{V}_{i} \bar{S}^{m-i}, \bar{S}\right]_{12}=} & \sum_{i=0}^{m}\left[V_{i} \sum_{k=1}^{m-i} L_{k} S+V_{1 i} S^{m-i+1}+V_{i} S^{m-i} S_{1}-S V_{i} \sum_{k=1}^{m-i} L_{k}-S V_{1 i} S^{m-i}-S_{1} V_{i} S^{m-i}\right] \\ = & \sum_{i=0}^{m}\left[V_{i}\left(-H \Lambda^{m-i} H^{-1} H_{1} H^{-1}+H_{1} \Lambda^{m-i} H^{-1}\right) H \Lambda H^{-1}+V_{1 i} H \Lambda^{m-i+1} H^{-1}\right. \\ & \left.+V_{i} H \Lambda^{m-i} H^{-1}\left(-H \Lambda H^{-1} H_{1} H^{-1}+H_{1} \Lambda H^{-1}\right)\right] . \end{aligned}$
Comparing the (1, 2)-block, we see that $\boldsymbol{S}_{1t}$ is equal to the (1, 2)-block of 13 ). Combining this with the verification of (11 ), we conclude that $\bar{\boldsymbol{S}}$ satisfies both (11 ) and (13 ). Therefore, $\bar{\boldsymbol{D}} = \lambda\bar{\boldsymbol{I}}-\bar{\boldsymbol{S}}$ is a Darboux matrix of the spectral problem (4 ), and the corresponding Bácklund transformation is given by
$\begin{equation*} \left[\sum_{i = 0}^{m}\bar{\boldsymbol{V}}_i\bar{\boldsymbol{S}}^{m-i},\bar{\boldsymbol{S}}\right].\end{equation*}$
Hence, $\bar{\boldsymbol{S}}$ satisfies ($\begin{align} \bar{\boldsymbol{K}}\left[1\right] = \bar{\boldsymbol{K}}+\bar{\boldsymbol{J}}\bar{\boldsymbol{S}}-\bar{\boldsymbol{S}}^{\rho}\bar{\boldsymbol{J}}.\end{align}$
This completes the proof.3. Integrable couplings of AKNS hierarchy on a space scale
In this section, we proceed to establish a framework for integrable couplings associated with the AKNS hierarchy on a space scale.
3.1. A hierarchy of AKNS integrable couplings on a space scale
We begin with the following spectral problem:
$\begin{align} \nabla_x\boldsymbol{\bar{\Phi}} = \bar{\boldsymbol{U}}\boldsymbol{\bar{\Phi}}.\end{align}$
Here $\begin{align} \bar{\boldsymbol{U}} = \begin{pmatrix} \boldsymbol{U} & \boldsymbol{U}_1 \\ 0 & \boldsymbol{U} \end{pmatrix}, \;\;\, \boldsymbol{U} = \begin{pmatrix} -\lambda & \frac{P+P^{\rho}}{2} \\ \frac{Q+Q^{\rho}}{2} & \lambda \end{pmatrix}, \;\;\, \boldsymbol{U}_1 = \begin{pmatrix} -\lambda & \frac{R+R^{\rho}}{2} \\ \frac{S+S^{\rho}}{2} & \lambda \end{pmatrix},\end{align}$
where P, Q, R, and S are potential functions. For convenience, we set $\begin{align*} p = \frac{P+P^{\rho}}{2},\quad q = \frac{Q+Q^{\rho}}{2},\quad r = \frac{R+R^{\rho}}{2},\quad s = \frac{S+S^{\rho}}{2},\end{align*}$
and write $\begin{align*} \bar{\boldsymbol{\Phi}} = \left(\psi^\mathrm{T},\phi^\mathrm{T}\right)^\mathrm{T},\qquad \boldsymbol{\psi} = \left(\psi_1,\psi_2\right)^\mathrm{T},\qquad \boldsymbol{\phi} = \left(\phi_1,\phi_2\right)^\mathrm{T}.\end{align*}$
Next, we consider the stationary zero-curvature equation in the space-scale setting: $\begin{align} \nabla_x\bar{\boldsymbol{W}} = \bar{\boldsymbol{U}}^{\sigma}\bar{\boldsymbol{W}}-\bar{\boldsymbol{W}}^{\rho}\bar{\boldsymbol{U}}.\end{align}$
Let $\begin{align} \bar{\boldsymbol{W}} = \begin{pmatrix} \boldsymbol{W} & \boldsymbol{W}_1 \\ 0 & \boldsymbol{W} \end{pmatrix}, \quad \boldsymbol{W} = \begin{pmatrix} A & B \\ C &-A \end{pmatrix}, \quad \boldsymbol{W}_1 = \begin{pmatrix} E & F \\ G & -E \end{pmatrix}.\end{align}$
Substituting (35 ) into (34 ), we obtain
$\begin{align} \nabla_x\boldsymbol{W} = \boldsymbol{U}^{\sigma}\boldsymbol{W}-\boldsymbol{W}^{\rho}\boldsymbol{U},\quad \nabla_x\boldsymbol{W}_1 = \boldsymbol{U}_1^{\sigma}\boldsymbol{W}-\boldsymbol{W}^{\rho}\boldsymbol{U}_1+\boldsymbol{U}^{\sigma}\boldsymbol{W}_1-\boldsymbol{W}_1^{\rho}\boldsymbol{U}.\end{align}$
From these relations, one derives $\begin{equation} \begin{cases} \nabla_xA = \lambda\left(A^{\rho}-A\right)+p^{\sigma}C-qB^{\rho}, \\ \nabla_xB = -\lambda\left(B^{\rho}+B\right)-p^{\sigma}A-pA^{\rho}, \\ \nabla_xC = \lambda\left(C+C^{\rho}\right)+q^{\sigma}A+qA^{\rho},\\ \nabla_xA = \lambda\left(A-A^{\rho}\right)+pC^{\rho}-q^{\sigma}B, \end{cases}\end{equation}$
and $\begin{equation} \begin{cases} \nabla_xE = \lambda\left(A^{\rho}-A\right)+\lambda\left(E^{\rho}-E\right)+r^{\sigma}C-sB^{\rho}+p^{\sigma}G-qF^{\rho}, \\ \nabla_xF = -\lambda\left(B^{\rho}+B\right)-\lambda\left(F^{\rho}+F\right)-r^{\sigma}A-rA^{\rho}-p^{\sigma}E-pE^{\rho}, \\ \nabla_xG = -\lambda\left(C^{\rho}+C\right)-\lambda\left(G^{\rho}+G\right)+s^{\sigma}A+sA^{\rho}+q^{\sigma}E+qE^{\rho}, \\ \nabla_xE = \lambda\left(A^{\rho}-A\right)+\lambda\left(E^{\rho}-E\right)+rC^{\rho}-s^{\sigma}B+pG^{\rho}-q^{\sigma}F, \end{cases}\end{equation}$
where $\begin{equation*} p = \frac{P+P^{\rho}}{2},\qquad q = \frac{Q+Q^{\rho}}{2},\qquad r = \frac{R+R^{\rho}}{2},\qquad s = \frac{S+S^{\rho}}{2}.\end{equation*}$
Before proceeding, we specify the inverse operators used below. The operator $\nabla_x^{-1}$ denotes the inverse of the nabla derivative, namely the nabla integral on the space scale $\mathbb{X}$:
$\begin{equation*} \left(\nabla_x^{-1}f\,\right)\left(x\right): = \int_{x_0}^{x} f\left(\xi\right)\,\nabla \xi,\end{equation*}$
where x0 is a fixed initial point, and the integration constants are taken to be zero unless otherwise stated. In addition, we write $\begin{align*} \mathit{G}: = 2-\nu\left(x\right)\nabla_x,\end{align*}$
and denote by $\mathit{G}^{-1} = (2-\nu(x)\nabla_x)^{-1}$ its inverse operator, understood formally through $\begin{align*} \mathit{G}\left(\mathit{G}^{-1}f\,\right) = f.\end{align*}$
In particular, for the continuous case $\nu(x) = 0$, one has $\mathit{G} ^ {-1} = \frac12 I$, while for the semi-discrete case $\nu(x) = 1$, $\mathit{G} = 1+\mathit{E}$, and hence $\mathit{G}^{-1} = (1+\mathit{E})^{-1}$.We expand 39 ) into (37 ) and (38 ), and then comparing the powers of $\lambda$, we deduce
$\begin{equation} \left\{ \begin{aligned} \boldsymbol{W} & = \sum_{i = 0}^{+\infty} W_i \lambda^{-i}, \qquad \boldsymbol{W}_i = \begin{bmatrix} a_i & b_i \\ c_i & -a_i \end{bmatrix}, \\ \boldsymbol{W}_1 & = \sum_{i = 0}^{+\infty} W_{1,i} \lambda^{-i}, \qquad \boldsymbol{W}_{1,i} = \begin{bmatrix} e_i & f_i \\ g_i & -e_i \end{bmatrix}. \end{aligned} \right.\end{equation}$
Substituting ($\begin{align} \begin{cases} a_0 = a_0^0,\qquad b_0 = 0,\qquad c_0 = 0, \\ c_i = -\left(2-\nu\nabla_x\right)^{-1}\left(q^{\sigma}a_{i-1}+qa_{i-1}^{\rho}\right)+\left(2-\nu\nabla_x\right)^{-1}\nabla_xc_{i-1},\qquad i\unicode{x2A7E} 1,\\ b_i = -\left(2-\nu\nabla_x\right)^{-1}\left(p^{\sigma}a_{k-1}+pa_{i-1}^{\rho}\right)-\left(2-\nu\nabla_x\right)^{-1}\nabla_xb_{i-1},\qquad i\unicode{x2A7E} 1,\\ a_i = \frac{1}{2}\nabla_x^{-1}\left[p^{\sigma}c_i+pc_i^{\rho}-qb_i^{\rho}-q^{\sigma}b_i\right]+a_i^{0},\qquad i\unicode{x2A7E} 1, \end{cases}\end{align}$
and $\begin{align} \begin{cases} e_0 = e_0^0,\qquad f_0 = 0,\qquad g_0 = 0, \\ f_i = -b_i-\left(2-\nu\nabla_x\right)^{-1}\nabla_xf_{i-1}, \\ \qquad -\left(2-\nu\nabla_x\right)^{-1}\left(r^{\sigma}+ra_{i-1}^{\rho}+p^{\sigma}e_{i-1}+pe_{i-1}^{\rho}\right), \\ g_i = -c_i+\left(2-\nu\nabla_x\right)^{-1}\nabla_xg_{i-1}, \\ \qquad -\left(2-\nu\mathrm{\nabla}_x\right)^{-1}\left(s^{\sigma}a_{i-1}+sa_{i-1}^{\rho}+q^{\sigma}e_{i-1}+qe_{i-1}\right),\\ e_i = \frac{1}{2}\nabla_x^{-1}\left[r^{\sigma}c_i+rc_i^{\rho}-sb_i^{\rho}-s^{\sigma}b_i + p^{\sigma}g_i+pg_i^{\rho}-qf_i^{\rho}-q^{\sigma}f_i\right]+e_i^{0}. \end{cases} \end{align}$
We choose the constants as $a_0^0 = \alpha$ and $e_0^0 = \beta$, where a and β are constants. For $i\unicode{x2A7E} 1$, we set the integration constants $a_i^0 = 0$ and $e_i^0 = 0$. In the continuous case $\nu(x) = 0$, the first few coefficients are given by
$\begin{align} \begin{cases} a_0 = \alpha,\quad b_0 = c_0 = 0, \\ b_1 = -\alpha P,\quad c_1 = -\alpha Q,\quad a_1 = 0,\\ b_2 = \alpha \mathit{G}^{-1}\nabla_x P, \\ c_2 = -\alpha \mathit{G}^{-1}\nabla_x Q,\quad a_2 = -\frac{\alpha}{2}PQ, \\ b_3 = \frac{\alpha}{4}\mathit{G}^{-1}\left[\mathit{G}P\mathit{G}PQ-4\nabla_x\mathit{G}^{-1}\nabla_xP\right], \\ c_3 = \frac{\alpha}{4}\mathit{G}^{-1}\left[\mathit{G}Q\mathit{G}PQ-4\nabla_x\mathit{G}^{-1}\nabla_xQ\right], \\ a_3 = \frac{\alpha}{4}\nabla_x^{-1}\mathit{G}\left[Q\nabla_x\mathit{G}^{-1}\nabla_xP-P\nabla_x\mathit{G}^{-1}\mathit{G}Q\right], \\ \cdots\,, \end{cases}\end{align}$
and $\begin{align} \begin{cases} e_0 = \beta,\quad f_0 = g_0 = 0,\\ f_1 = \alpha P-\alpha R-\beta P, \\ g_1 = \alpha Q-\alpha S-\beta P,\quad e_1 = 0, \\ f_2 = -\alpha \mathit{G}^{-1}\nabla_x\left(2P-R\right)+\beta \mathit{G}^{-1}\nabla_xP, \\ g_2 = \alpha \mathit{G}^{-1}\nabla_x\left(2Q-S\right)+\beta \mathit{G}^{-1}\nabla_x P, \\ e_2 = -\frac{1}{2}\beta QP+\frac{\alpha}{2}\left(2PQ-QR-SP\right), \\ \cdots\,. \end{cases} \end{align}$
Here $\mathit{G} = 2-\nu\nabla_x$. We now turn to the auxiliary spectral problem
$\begin{align} \boldsymbol{\bar{\Phi}}_t = \bar{\boldsymbol{V}}^{\left[m\right]}\boldsymbol{\bar{\Phi}},\qquad m\unicode{x2A7E}0,\end{align}$
where $\begin{align} \bar{\boldsymbol{V}}^{\left[m\right]} = \begin{pmatrix} \boldsymbol{V}^{\left[m\right]} & \boldsymbol{V}_1^{\left[m\right]} \\ 0 & \boldsymbol{V}^{\left[m\right]} \end{pmatrix}.\end{align}$
Here $\bar{\boldsymbol{V}}^{[m]}$ is a $4\times4$ matrix, whose $2\times2$ block entries are given by $\begin{align} \begin{cases} \boldsymbol{V}^{\left[m\right]} = \begin{pmatrix} A^{\left[m\right]} & B^{\left[m\right]} \\ C^{\left[m\right]} & -A^{\left[m\right]} \end{pmatrix} = \left(\lambda^m \boldsymbol{W}\right)_+ = \sum_{i = 0}^{m} W_i \lambda^{m-i}, \qquad m \unicode{x2A7E} 0, \\ \boldsymbol{V}_1^{\left[m\right]} = \begin{pmatrix} E^{\left[m\right]} &F^{\left[m\right]} \\ G^{\left[m\right]} & -E^{\left[m\right]} \end{pmatrix} = \left(\lambda^m \boldsymbol{W}_1\right)_+ = \sum_{i = 0}^{m} W_{1,i} \lambda^{m-i},\qquad m\unicode{x2A7E}0. \end{cases} \end{align}$
From the compatibility conditions of (32 ) and (44 ), we derive 47 ) is equivalent to
$\begin{align} \bar{\boldsymbol{U}}_t-\nabla_x\bar{\boldsymbol{V}}^{\left[m\right]}+\bar{\boldsymbol{U}}\bar{\boldsymbol{V}}^{\left[m\right]}-\bar{\boldsymbol{V}}^{\left[m\right]\rho}\bar{\boldsymbol{U}} = 0.\end{align}$
Equation ($\begin{align} \begin{cases} \boldsymbol{U}_t-\nabla_x\boldsymbol{V}^{\left[m\right]}+\boldsymbol{U}\boldsymbol{V}^{\left[m\right]}-\boldsymbol{V}^{\left[m\right]\rho}\boldsymbol{U} = 0, \\ \boldsymbol{U}_{1t}-\nabla_x\boldsymbol{V}_{1}^{\left[m\right]}+\boldsymbol{U}_1\boldsymbol{V}^{\left[m\right]}-\boldsymbol{V}^{\left[m\right]\rho}\boldsymbol{U}_1+\boldsymbol{U}\boldsymbol{V}_1^{\left[m\right]}-\boldsymbol{V}_1^{\left[m\right]\rho}\boldsymbol{U} = 0. \end{cases}\end{align}$
Consequently, a hierarchy of AKNS integrable couplings on a space scale is generated: $\begin{align} \begin{cases} p_t = -\left(b_{m+1}+b_{m+1}^{\rho}\right) = -\left(2-\nu\nabla_x\right)b_{m+1}, \\ q_t = c_{m+1}+c_{m+1}^{\rho} = \left(2-\nu\nabla_x\right)b_{m+1}, \\ r_t = -\left(b_{m+1}+b_{m+1}^{\rho}\right)-\left(f_{m+1}+f_{m+1}^{\rho}\right) = -\left(2-\nu\nabla_x\right)\left(b_{m+1}+f_{m+1}\right), \\ s_t = c_{m+1}+c_{m+1}^{\rho}+g_{m+1}+g_{m+1}^{\rho} = \left(2-\nu\nabla_x\right)\left(c_{m+1}+g_{m+1}\right). \end{cases} \end{align}$
By the Tu formula, we obtain
$\begin{align} \begin{pmatrix} p\\ q\\ r\\ s \end{pmatrix}_t = -\mathit{G} \begin{pmatrix} \sigma_3 & 0 \\ 0 & \sigma_3 \end{pmatrix} \bar{\boldsymbol{L}} \begin{pmatrix} b_m\\ c_m\\ f_m+b_m\\ g_m+c_m \end{pmatrix},\end{align}$
where $\begin{equation*} \sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \qquad \mathit{G} = 2-\nu \nabla_{x},\end{equation*}$
and $\bar{\boldsymbol{L}}$ is the $4\times4$ matrix $\begin{align*} \bar{\boldsymbol{L}} = \begin{pmatrix} \boldsymbol{L} & 0\\ \boldsymbol{L}_1-\boldsymbol{L} & \boldsymbol{L} \end{pmatrix},\end{align*}$
with $\begin{align*} \boldsymbol{L} = \begin{pmatrix} \dfrac{1}{2}\mathit{G}^{-1}p\mathit{G}\nabla_x^{-1}q\mathit{G} + \mathit{G}^{-1}\nabla_x & -\dfrac{1}{2}\mathit{G}^{-1}p\mathit{G}\nabla_x^{-1}p\mathit{G} \\[10pt] \dfrac{1}{2}\mathit{G}^{-1}q\mathit{G}\nabla_x^{-1}q\mathit{G} & \mathit{G}\nabla_x - \dfrac{1}{2}\mathit{G}^{-1}q\mathit{G}\nabla_x^{-1}q\mathit{G} \end{pmatrix}\end{align*}$
and $\begin{align*} \boldsymbol{L}_1 = \left( \begin{array}{c|c} \begin{array}{c} \frac{1}{2}\mathit{G}^{-1}r\mathit{G}\nabla_x^{-1}q\mathit{G} \\[5pt] + \frac{1}{2}\mathit{G}^{-1}p\mathit{G}\nabla_x^{-1}s\mathit{G} \end{array} & \begin{array}{c} \frac{1}{2}\mathit{G}^{-1}s\mathit{G}\nabla_x^{-1}q\mathit{G} \\[5pt] - \frac{1}{2}\mathit{G}^{-1}q\mathit{G}\nabla_x^{-1}s\mathit{G} \end{array} \\[9pt] \hline \\[-10pt] \begin{array}{c} -\frac{1}{2}\mathit{G}^{-1}r\mathit{G}\nabla_x^{-1}p\mathit{G} \\[5pt] - \frac{1}{2}\mathit{G}^{-1}p\mathit{G}\nabla_x^{-1}r\mathit{G} \end{array} & \begin{array}{c} -\frac{1}{2}\mathit{G}^{-1}s\mathit{G}\nabla_x^{-1}p\mathit{G} \\[5pt] - \frac{1}{2}\mathit{G}^{-1}q\mathit{G}\nabla_x^{-1}r\mathit{G} \end{array} \end{array} \right).\end{align*}$
Since $\mathit{G} = 2-\nu\nabla_x$, it follows that $\begin{equation} \begin{pmatrix} p\\ q\\ r\\ s \end{pmatrix}_t = -\left(2-\nu\nabla_x\right) \begin{pmatrix} \sigma_3 & 0 \\ 0 & \sigma_3 \end{pmatrix} \bar{\boldsymbol{L}}^{m} \begin{pmatrix} b_1\\ c_1\\ f_1+b_1\\ g_1+c_1 \end{pmatrix}, \qquad m\unicode{x2A7E}0.\end{equation}$
3.2. Construction of the Darboux transformation for AKNS integrable couplings on a space scale
To begin this subsection, we recall the nabla exponential function. For a time scale $\mathbb{T}$ and a given ld-continuous function $p:\mathbb{T}\to\mathbb{R}$, the nabla exponential function is defined by [6]
$\begin{equation*} \hat{\mathrm{e}}_{p}\left(t,s\right): = \exp\left(\int_{s}^{t}\hat{\xi}_{\nu\left(\gamma\right)}\left(p\left(\gamma\right)\right)\,\nabla \gamma\right),\qquad s,t\in\mathbb{T},\end{equation*}$
where the cylinder transformation $\hat{\xi}_h$ is defined by $\begin{equation*} \hat{\xi}_h\left(z\right): = -\frac{1}{h}\log\left(1-zh\right).\end{equation*}$
The nabla exponential function satisfies the following properties: $\begin{equation*} \begin{cases} \left(1\right)\ \hat{\mathrm{e}}_0\left(s_0,s_0\right) = 1, \\ \left(2\right)\ \left(\hat{\mathrm{e}}_{\tau}\left(\cdot,s_0\right)\right)^{\nabla}\left(s\right) = \tau\left(s\right)\hat{\mathrm{e}}_{\tau}\left(s,s_0\right), \\ \left(3\right)\ \hat{\mathrm{e}}_{\tau}^{\rho}\left(s,s_0\right) = \left(1-\nu\left(s\right)\tau\left(s\right)\right)\hat{\mathrm{e}}_{\tau}\left(s,s_0\right),\\ \left(4\right)\ \hat{\mathrm{e}}_{\alpha}\left(s,s_0\right)\hat{\mathrm{e}}_{\tau}\left(s,s_0\right) = \hat{\mathrm{e}}_{\alpha \oplus \tau}\left(s,s_0\right),\\ \left(5\right)\ \left(\alpha \oplus \tau\right)\left(s\right) = \alpha\left(s\right)+\tau\left(s\right)-\nu\left(s\right)\alpha(s)\tau(s). \\ \end{cases} \end{equation*}$
To construct the Darboux transformation for the integrable couplings in (49 ), we apply theorem 1 . In this case,
$\begin{equation*} \boldsymbol{U} = \lambda\boldsymbol{J}+\boldsymbol{K},\qquad \boldsymbol{U}_1 = \lambda\boldsymbol{J}_1+\boldsymbol{K}_1,\end{equation*}$
with $\begin{align*} \boldsymbol{J} &= \boldsymbol{J}_1 = \begin{pmatrix} -1 & 0\\ 0 & 1 \end{pmatrix},\quad \boldsymbol{K} = \begin{pmatrix} 0 & \dfrac{P+P^{\rho}}{2} \\ \dfrac{Q+Q^{\rho}}{2} & 0 \end{pmatrix},\nonumber\\ \boldsymbol{K}_1 &= \begin{pmatrix} 0 & \dfrac{R+R^{\rho}}{2} \\ \dfrac{S+S^{\rho}}{2} & 0 \end{pmatrix}.\end{align*}$
Next, choose two distinct eigenvalues $\lambda$1 and $\lambda$2, and denote the corresponding eigenvectors by
$\begin{align*} \phi_{jk} = \phi_j\left(\lambda_k\right),\quad \psi_{jk} = \psi_j\left(\lambda_k\right),\quad j,k = 1,2.\end{align*}$
Define
$\begin{equation} \Lambda = \begin{pmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{pmatrix},\quad \boldsymbol{H} = \begin{pmatrix} \phi_{11} & \phi_{12} \\ \phi_{21} & \phi_{22} \end{pmatrix},\quad \boldsymbol{H}_1 = \begin{pmatrix} \psi_{11} & \psi_{12} \\ \psi_{21} & \psi_{22} \end{pmatrix}.\end{equation}$
A direct calculation gives $\begin{equation} \boldsymbol{S} = \boldsymbol{H}\Lambda\boldsymbol{H}^{-1},\qquad \boldsymbol{S}_1 = -\boldsymbol{H}\Lambda\boldsymbol{H}^{-1}\boldsymbol{H}_1\boldsymbol{H}^{-1}+\boldsymbol{H}_1\Lambda\boldsymbol{H}^{-1}.\end{equation}$
By theorem 1 , the first-order Darboux transformation takes the form 55 ) is equivalent to
$\begin{align} \bar{\boldsymbol{D}} = \lambda\bar{\boldsymbol{I}}-\bar{\boldsymbol{S}},\,\, \bar{\boldsymbol{I}} = \mathrm{diag}\left(1,1,1,1\right) = \begin{pmatrix} \boldsymbol{I} & 0\\ 0 & \boldsymbol{I} \end{pmatrix},\,\, \bar{\boldsymbol{S}} = \begin{pmatrix} \boldsymbol{S} & \boldsymbol{S}_1 \\ 0 & \boldsymbol{S} \end{pmatrix}.\end{align}$
Therefore, $\begin{align} \boldsymbol{\bar{\Phi}}\left[1\right] = \bar{\boldsymbol{D}}\boldsymbol{\bar{\Phi}}, \quad \bar{\boldsymbol{K}}\left[1\right] = \bar{\boldsymbol{K}}+\bar{\boldsymbol{J}}\bar{\boldsymbol{S}}-\bar{\boldsymbol{S}}^{\rho}\bar{\boldsymbol{J}}.\end{align}$
The second equation in ($\begin{align} \boldsymbol{K}\left[1\right] & = \boldsymbol{K}+\boldsymbol{J}\boldsymbol{S}-\boldsymbol{S}^{\rho}\boldsymbol{J},\nonumber\\ \boldsymbol{K}_1\left[1\right] & = \boldsymbol{K}_1+\boldsymbol{J}\boldsymbol{S}_1+\boldsymbol{J}_1\boldsymbol{S}-\boldsymbol{S}^{\rho}\boldsymbol{J}_1-\boldsymbol{S}^{\rho}_1\boldsymbol{J}.\end{align}$
Here $\boldsymbol{K}[1]$ and $\boldsymbol{K}_1[1]$ have the same forms as K and K1, respectively, i.e. $\begin{align*} \boldsymbol{K}\left[1\right]& = \begin{pmatrix} 0 & \dfrac{P\left[1\right]+P^{\rho}\left[1\right]}{2} \\ \dfrac{Q\left[1\right]+Q^{\rho}\left[1\right]}{2} & 0 \end{pmatrix},\nonumber\\ \boldsymbol{K}_1\left[1\right] &= \begin{pmatrix} 0 & \dfrac{R\left[1\right]+R^{\rho}\left[1\right]}{2} \\ \dfrac{S\left[1\right]+S^{\rho}\left[1\right]}{2} & 0 \end{pmatrix}.\end{align*}$
Write
$\begin{align} \boldsymbol{S} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix},\qquad \boldsymbol{S}_1 = \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}.\end{align}$
Then the corresponding transformations are $\begin{align} \begin{aligned} P\left[1\right]& = P-\left(2-\nu\nabla_x\right)\left(a_{12}+a_{12}^{\rho}\right), \\ Q\left[1\right]& = Q+\left(2-\nu\nabla_x\right)\left(a_{21}+a_{21}^{\rho}\right),\\ R\left[1\right]& = R-\left(2-\nu\nabla_x\right)\left(a_{12}+a_{12}^{\rho}+b_{12}+b_{12}^{\rho}\right), \\ S\left[1\right]& = S+\left(2-\nu\nabla_x\right)\left(a_{21}+a_{21}^{\rho}+b_{21}+b_{21}^{\rho}\right). \end{aligned}\end{align}$
Example 1. The second-order system
Next, we consider the second-order AKNS integrable coupling system on a space scale. According to (17 ), we have
$\begin{equation} \begin{aligned} \bar{\boldsymbol{V}}^{\left[2\right]} = \left(\lambda^2\bar{\boldsymbol{W}}\right)_{+} = \sum_{i = 0}^{2}\bar{\boldsymbol{W}}_{i}\lambda^{2-i} = \sum_{i = 0}^{2}\begin{pmatrix} a_i & b_i & e_i & f_i \\ c_i & -a_i & g_i & -e_i \\ 0 & 0 & a_i & b_i \\ 0 & 0 & c_i & -a_i \end{pmatrix}\lambda^{2-i}, \end{aligned}\end{equation}$
where $\begin{align} \begin{cases} a_0 = \alpha,\qquad b_0 = c_0 = 0,\\ a_1 = 0,\qquad b_1 = \alpha P,\qquad c_1 = -\alpha Q,\\ b_2 = \alpha \mathit{G}^{-1}\nabla_xP,\\ c_2 = -\alpha \mathit{G}^{-1}\nabla_xQ,\\ a_2 = -\dfrac{\alpha}{2}PQ,\\ e_0 = \beta,\qquad f_0 = g_0 = 0,\\ f_1 = \alpha P-\alpha R-\beta P,\\ g_1 = \alpha Q-\alpha S-\beta Q,\qquad e_1 = 0,\\ f_2 = -\alpha \mathit{G}^{-1}\nabla_x\left(2P-R\right)+\beta \mathit{G}^{-1}\nabla_xP,\\ g_2 = \alpha \mathit{G}^{-1}\nabla_x\left(2Q-S\right)-\beta \mathit{G}^{-1}\nabla_xQ,\\ e_2 = -\dfrac{1}{2}\beta QP+\dfrac{\alpha}{2}\left(2PQ-QR-SP\right), \end{cases}\end{align}$
with $\mathit{G} = 2-\nu\nabla_x$.For the zero seed solution, solving the spectral problems (32 ) and (44 ) yields the eigenfunction
$\begin{align} \boldsymbol{\bar{\Phi}}\left(\lambda\right) &= \begin{pmatrix} \psi_1 \\ \psi_2 \\ \phi_1 \\ \phi_2 \end{pmatrix}\nonumber\\ & = \begin{pmatrix} \left(\beta\lambda^2t-\lambda\int_{0}^{x}\hat{\mathrm{e}}_{-\lambda}\left(\tau,\rho\left(\tau\right)\right)\nabla\tau\right)\hat{\mathrm{e}}_{-\lambda}\left(x,0\right)\mathrm{e}^{\alpha\lambda^2t} \\ \left(-\beta\lambda^2t+\lambda\int_{0}^{x}\hat{\mathrm{e}}_{-\lambda}\left(\tau,\rho\left(\tau\right)\right)\nabla\tau\right)\hat{\mathrm{e}}_{\lambda}\left(x,0\right)\mathrm{e}^{-\alpha\lambda^2t}\\ \hat{\mathrm{e}}_{-\lambda}\left(x,0\right)\mathrm{e}^{\alpha\lambda^2t} \\ \hat{\mathrm{e}}_{\lambda}\left(x,0\right)\mathrm{e}^{-\alpha\lambda^2t} \end{pmatrix}.\end{align}$
To obtain explicit solutions via the Darboux transformation, we choose two distinct eigenvalues $\lambda$1 and $\lambda$2 and the corresponding eigenfunctions $\begin{equation} \bar{\boldsymbol{\Phi}}\left(\lambda_1\right) = \begin{pmatrix} \psi_{11} \\ \psi_{21} \\ \phi_{11} \\ \phi_{21} \end{pmatrix},\qquad \boldsymbol{\bar{\Phi}}\left(\lambda_2\right) = \begin{pmatrix} \psi_{12} \\ \psi_{22} \\ \phi_{12} \\ \phi_{22} \end{pmatrix}.\end{equation}$
Example 2. The third-order system
Next, we consider the third-order AKNS integrable coupling system on a space scale. According to (17 ), we have 60 ).
$\begin{equation} \begin{aligned} \bar{\boldsymbol{V}}^{\left[3\right]} = \left(\lambda^3\bar{\boldsymbol{W}}\right)_{+} = \sum_{i = 0}^{3}\bar{\boldsymbol{W}}_{i}\lambda^{3-i} = \sum_{i = 0}^{3}\begin{pmatrix} a_i & b_i & e_i & f_i \\ c_i & -a_i & g_i & -e_i \\ 0 & 0 & a_i & b_i \\ 0 & 0 & c_i & -a_i \end{pmatrix}\lambda^{3-i}, \end{aligned}\end{equation}$
where $\begin{equation} \begin{cases} a_3 = \dfrac{\alpha}{4} \mathit{G}\left[Q\nabla_x\mathit{G}^{-1}\nabla_xP\right], \\ b_3 = \dfrac{\alpha}{4}\mathit{G}^{-1}\left[\mathit{G}P\mathit{G}PQ-4\nabla_x\mathit{G}^{-1}\nabla_xP\right],\\ c_3 = \dfrac{\alpha}{4}\mathit{G}^{-1}\left[\mathit{G}Q\mathit{G}PQ-4\nabla_x\mathit{G}^{-1}\nabla_xQ\right], \end{cases}\end{equation}$
and $\begin{align} \begin{cases} e_3 = &\dfrac{\beta}{2}\nabla_x^{-1}\left[\mathit{G}Q\nabla_x\mathit{G}^{-1}\nabla_xP-\mathit{G}P\nabla_x\mathit{G}^{-1}\nabla_xQ\right] \\ &+\dfrac{\alpha}{2}\nabla_x^{-1}\left[3\mathit{G}P\nabla_x\mathit{G}^{-1}\nabla_xQ-3\mathit{G}Q\nabla_x\mathit{G}^{-1}\nabla_xP\right. \\ &\left. -\mathit{G}R\nabla_x\mathit{G}^{-1}\nabla_xQ+ \mathit{G}Q\nabla_x\mathit{G}^{-1}\nabla_xR\right. \\ &\left. +\mathit{G}S\nabla_x\mathit{G}^{-1}\nabla_xP-\mathit{G}P\nabla_x\mathit{G}^{-1}\nabla_xS\right],\\ f_3 = &\dfrac{\beta}{4}\left[P\mathit{G}QP-4\left[\mathit{G}^{-1}\right]^2Q\right] \\ &+\dfrac{\alpha}{4}\left[R\mathit{G}PQ+P\mathit{G}\left(QR+SP-3PQ\right)\right. \\ &\left. +12\left[\mathit{G}^{-1}\nabla_x\right]^2P-4\left[\mathit{G}^{-1}\nabla_x\right]^2R\right], \\ g_3 = &\dfrac{\beta}{4}\left[Q\mathit{G}QP-4\left[\mathit{G}^{-1}\right]^2P\right] \\ &-\dfrac{\alpha}{4}\left[-S\mathit{G}PQ+Q\mathit{G}\left(3PQ-QR-SP\right)\right. \\ &\left. +12\left[\mathit{G}^{-1}\nabla_x\right]^2Q-4\left[\mathit{G}^{-1}\nabla_x\right]^2S\right]. \end{cases}\end{align}$
Here $\mathit{G} = 2-\nu\nabla_x$, and the remaining coefficients are the same as those given in (For the zero seed solution, solving (32 ) and (44 ) yields the eigenfunction
$\begin{align} \boldsymbol{\bar{\Phi}}\left(\lambda\right)& = \begin{pmatrix} \psi_1 \\ \psi_2 \\ \phi_1 \\ \phi_2 \end{pmatrix}\nonumber\\ &= \begin{pmatrix} \left(\beta\lambda^3t-\lambda\int_{0}^{x}\hat{\mathrm{e}}_{-\lambda}\left(\tau,\rho\left(\tau\right)\right)\nabla\tau\right)\hat{\mathrm{e}}_{-\lambda}\left(x,0\right)\mathrm{e}^{\alpha\lambda^3t} \\ \left(-\beta\lambda^3t+\lambda\int_{0}^{x}\hat{\mathrm{e}}_{-\lambda}\left(\tau,\rho\left(\tau\right)\right)\nabla\tau\right)\hat{\mathrm{e}}_{\lambda}\left(x,0\right)\mathrm{e}^{-\alpha\lambda^3t}\\ \hat{\mathrm{e}}_{-\lambda}\left(x,0\right)\mathrm{e}^{\alpha\lambda^3t} \\ \hat{\mathrm{e}}_{\lambda}\left(x,0\right)\mathrm{e}^{-\alpha\lambda^3t} \end{pmatrix}.\end{align}$
Similarly, by choosing two distinct eigenvalues $\lambda$1 and $\lambda$2, we obtain the corresponding eigenfunctions $\begin{equation} \boldsymbol{\bar{\Phi}}\left(\lambda_1\right) = \begin{pmatrix} \psi_{11} \\ \psi_{21} \\ \phi_{11} \\ \phi_{21} \end{pmatrix},\qquad \boldsymbol{\bar{\Phi}}\left(\lambda_2\right) = \begin{pmatrix} \psi_{12} \\ \psi_{22} \\ \phi_{12} \\ \phi_{22} \end{pmatrix}.\end{equation}$
Thus, the coefficients for the second- and third-order systems are obtained. In the following section, we derive the integrable couplings of the NLS and mKdV equations through an appropriate reduction procedure.
4. Integrable couplings of the NLS equations and mKdV equation on a space scale
In this section, by imposing suitable reduction conditions, we derive the integrable couplings of the NLS and mKdV equations on a space scale and obtain their soliton solutions via the Darboux transformation.
We impose the reduction condition
$\begin{equation} P = -Q^{*},\quad R = -S^{*},\end{equation}$
where $*$ denotes complex conjugation.Thus, if 32 ) and (44 ) corresponding to the eigenvalue $\lambda$1, then
$\begin{equation*} \boldsymbol{\bar{\Phi}} = \left(\psi_1,\psi_2,\phi_1,\phi_2\right)^{\mathrm{T}}\end{equation*}$
is a solution of the spectral problems ($\begin{equation*} \bar{\boldsymbol{\Psi}} = \left(\psi_2^{*},-\psi_1^{*},\phi_2^{*},-\phi_{1}^{*}\right)^{\mathrm{T}}\end{equation*}$
is a solution of the same spectral problems corresponding to the eigenvalue $\lambda_1^{*}$.Example 1. Integrable coupling of the NLS equation on a space scale
Let 49 ), we obtain the integrable coupling of the NLS equation on a space scale:
$\begin{equation} a_0 = \alpha = -2\mathrm{i},\quad e_0 = \beta = 0.\end{equation}$
By setting m = 2 in ($\begin{align} &\mathrm{i}\left[\frac{P+P^{\rho}}{2}\right]_t -\frac{P+P^{\rho}}{2}\left(2-\nu\nabla_x\right)\left|P\right|^2 -2\left(2-\nu\nabla_x\right)^{-1}\nabla_x^2P = 0,\end{align}$
$\begin{array}{l} \mathrm{i}\left[\frac{R+R^{\rho}}{2}\right]_{t}-\frac{R+R^{\rho}}{2}\left(2-\nu \nabla_{x}\right)|P|^{2} \\ \quad+\frac{P+P^{\rho}}{2}\left(2-\nu \nabla_{x}\right)\left(2|P|^{2}-P^{*} R-R^{*} P\right) \\ \quad+2\left(2-\nu \nabla_{x}\right)^{-1}\left(2 \nabla_{x}^{2} P-\nabla_{x}^{2} R\right)=0 . \end{array}$
We next consider two special choices of the space-scale measure.
Setting $\nu(x) = 0$ yields the classical integrable coupling of the nonlinear Schrödinger system:
$\begin{equation} \begin{cases} \mathrm{i}P_t-2\left|P\right|^2P-P_{xx} = 0, \\ \mathrm{i}R_t-R_{xx}+2P_{xx}-4\left|P\right|^2R-2P^2R^{*}+4\left|P\right|^2P = 0. \end{cases}\end{equation}$
This case coincides with that in [38].For the case $\nu(x) = 1$, we obtain an integrable coupling of the generalized semi-discrete NLS equation:
$\begin{equation} \begin{cases} \mathrm{i}\left[\frac{\left(1+\mathit{E}\right)P_n}{2}\right]_t -\frac{\left(1+\mathit{E}\right)P_n}{2}\left(1+\mathit{E}\right)\left|P_n\right|^{2} -2\left(1+\mathit{E}\right)^{-1}\left(1-\mathit{E}\right)^{2}P_n = 0, \\ \mathrm{i}\left[\frac{\left(1+\mathit{E}\right)R_n}{2}\right]_t -\frac{\left(1+\mathit{E}\right)R_n}{2}\left(1+\mathit{E}\right)a_2 -\frac{\left(1+\mathit{E}\right)P_n}{2}\left(1+\mathit{E}\right)e_2 -\left(1-\mathit{E}\right)f_2 = 0, \end{cases}\end{equation}$
where $\begin{equation} \begin{cases} e_2 = \mathrm{i}\left(2\left|P_n\right|^{2}-P_n^{*}R_n-R_n^{*}P_n\right), \\ f_2 = 2\mathrm{i}\left(1+\mathit{E}\right)^{-1}\left(1-\mathit{E}\right)\left(2P_n-R_n\right). \end{cases} \end{equation}$
To obtain exact solutions, we apply the Darboux transformation. Then
$\begin{equation} \bar{\boldsymbol{H}} = \begin{pmatrix} \boldsymbol{H} & \boldsymbol{H}_1 \\ 0 & \boldsymbol{H} \end{pmatrix} = \begin{pmatrix} \phi_{11} & \phi_{21}^{*} & \psi_{11} & \psi_{21}^{*} \\ \phi_{21} & -\phi_{11}^{*} & \psi_{21} & -\psi_{11}^{*} \\ 0 & 0 & \phi_{11} & \phi_{21}^{*} \\ 0 & 0 & \phi_{21} & -\phi_{11}^{*} \end{pmatrix},\end{equation}$
where $\begin{equation} \begin{cases} \phi_{11} = \hat{\mathrm{e}}_{-\lambda_1}\left(x,0\right)\mathrm{e}^{-2\mathrm{i}\lambda_1^2t}, \\ \phi_{21} = \hat{\mathrm{e}}_{\lambda_1}\left(x,0\right)\mathrm{e}^{2\mathrm{i}\lambda_1^2t}, \\ \psi_{11} = -\lambda_1\displaystyle\int_{0}^{x}\hat{\mathrm{e}}_{-\lambda_1}\left(\tau,\rho\left(\tau\right)\right)\,\nabla\tau\, \hat{\mathrm{e}}_{-\lambda_1}\left(x,0\right)\mathrm{e}^{-2\mathrm{i}\lambda_1^2t}, \\ \psi_{21} = \lambda_1\displaystyle\int_{0}^{x}\hat{\mathrm{e}}_{-\lambda_1}\left(\tau,\rho\left(\tau\right)\right)\,\nabla\tau\, \hat{\mathrm{e}}_{\lambda_1}\left(x,0\right)\mathrm{e}^{2\mathrm{i}\lambda_1^2t}. \end{cases} \end{equation}$
Let $\lambda_1 = \xi+\mathrm{i}\eta$. By applying the Bácklund transformation (58 ), we obtain a one-soliton-like solution of the NLS equation on a space scale:
$\begin{align} P\left[1\right] &= -4\left(2-\nu\nabla_x\right)^{-1}\xi\left[ \frac{\hat{\mathrm{e}}_{M_1}\left(x,0\right)\mathrm{e}^{-4\mathrm{i}\left(\xi^2-\eta^2\right)t}} {\hat{\mathrm{e}}_{M_2}\left(x,0\right)\mathrm{e}^{8\xi\eta t}+\hat{\mathrm{e}}_{M_3}\left(x,0\right)\mathrm{e}^{-8\xi\eta t}} \right. \nonumber\\ &\quad \left. +\frac{\left(1-\nu M_1\right)\hat{\mathrm{e}}_{M_1}\left(x,0\right)\mathrm{e}^{-4\mathrm{i}\left(\xi^2-\eta^2\right)t}} {\left(1-\nu M_2\right)\hat{\mathrm{e}}_{M_2}\left(x,0\right)\mathrm{e}^{8\xi\eta t}+\left(1-\nu M_3\right)\hat{\mathrm{e}}_{M_3}\left(x,0\right)\mathrm{e}^{-8\xi\eta t}} \right],\end{align}$
$\begin{align} R\left[1\right] &= -2\left(2-\nu\nabla_x\right)^{-1}\left(a_{12}+a^{\rho}_{12}+b_{12}+b^{\rho}_{12}\right).\end{align}$
Here $\begin{align} \begin{cases} a_{12} = 2\xi\displaystyle\frac{\hat{\mathrm{e}}_{M_1}(x,0)\mathrm{e}^{-4\mathrm{i}(\xi^2-\eta^2)t}} {\hat{\mathrm{e}}_{M_2}(x,0)\mathrm{e}^{8\xi\eta t}+\hat{\mathrm{e}}_{M_3}(x,0)\mathrm{e}^{-8\xi\eta t}}, \\ a^{\rho}_{12} = 2\xi\displaystyle\frac{(1-\nu M_1)\hat{\mathrm{e}}_{M_1}(x,0)\mathrm{e}^{-4\mathrm{i}(\xi^2-\eta^2)t}} {(1-\nu M_2)\hat{\mathrm{e}}_{M_2}(x,0)\mathrm{e}^{8\xi\eta t}+(1-\nu M_3)\hat{\mathrm{e}}_{M_3}(x,0)\mathrm{e}^{-8\xi\eta t}}, \\ b_{12} = \displaystyle\frac{2\xi}{\bigl(\hat{\mathrm{e}}_{M_2}(x,0)\mathrm{e}^{8\xi\eta t}+\hat{\mathrm{e}}_{M_3}(x,0)\mathrm{e}^{-8\xi\eta t}\bigr)^{2}} \\ \quad\times\left[ (\xi-\mathrm{i}\eta)\hat{\mathrm{e}}_{M_4}(x,0)\mathrm{e}^{M_6t} \left(\int_{0}^{x}\hat{\mathrm{e}}_{\xi-\mathrm{i}\eta}(\tau,\rho(\tau))\,\nabla\tau +\int_{0}^{x}\hat{\mathrm{e}}_{-\xi+\mathrm{i}\eta}(\tau,\rho(\tau))\,\nabla\tau\right)\right.\\ \qquad\left. -(\xi+\mathrm{i}\eta)\hat{\mathrm{e}}_{M_5}(x,0)\mathrm{e}^{M_7t} \left(\int_{0}^{x}\hat{\mathrm{e}}_{-\xi-\mathrm{i}\eta}(\tau,\rho(\tau))\,\nabla\tau +\int_{0}^{x}\hat{\mathrm{e}}_{\xi+\mathrm{i}\eta}(\tau,\rho(\tau))\,\nabla\tau\right) \right], \\ b_{12}^{\rho} = \displaystyle\frac{2\xi}{\Bigl[(1-\nu M_2)\hat{\mathrm{e}}_{M_2}(x,0)\mathrm{e}^{8\xi\eta t}+(1-\nu M_3)\hat{\mathrm{e}}_{M_3}(x,0)\mathrm{e}^{-8\xi\eta t}\Bigr]^{2}} \\ \qquad\times\left[ (\xi-\mathrm{i}\eta)(1-\nu M_4)\hat{\mathrm{e}}_{M_4}(x,0)\mathrm{e}^{M_6t} \left(\int_{0}^{x}\hat{\mathrm{e}}_{\xi-\mathrm{i}\eta}(\tau,\rho(\tau))\,\nabla\tau +\int_{0}^{x}\hat{\mathrm{e}}_{-\xi+\mathrm{i}\eta}(\tau,\rho(\tau))\,\nabla\tau\right)\right.\\ \qquad\left. -(\xi+\mathrm{i}\eta)(1-\nu M_5)\hat{\mathrm{e}}_{M_5}(x,0)\mathrm{e}^{M_7t} \left(\int_{0}^{x}\hat{\mathrm{e}}_{-\xi-\mathrm{i}\eta}(\tau,\rho(\tau))\,\nabla\tau +\int_{0}^{x}\hat{\mathrm{e}}_{\xi+\mathrm{i}\eta}(\tau,\rho(\tau))\,\nabla\tau\right) \right]. \end{cases}\end{align}$
where $\begin{equation} \begin{cases} M_1 = -2\mathrm{i}\eta+\nu\left(\xi^2+\eta^{2}\right), \\ M_2 = -2\xi-\nu\left(\xi^2+\eta^2\right), \\ M_3 = 2\xi-\nu\left(\xi^2+\eta^2\right),\\ M_4 = -2\xi-2\mathrm{i}\eta+\nu\left(\xi^4+\eta^4+2\xi^2\eta^2\right),\\ M_5 = 2\xi-2\mathrm{i}\eta+\nu\left(\xi^4+\eta^4+2\xi^2\eta^2\right), \\ M_6 = -4\mathrm{i}\left(\xi^2-\eta^2+2\mathrm{i}\xi\eta\right), \\ M_7 = -4\mathrm{i}\left(\xi^2-\eta^2-2\mathrm{i}\xi\eta\right). \end{cases} \end{equation}$
When $\nu(x) = 0$, this reduces to the classical one-soliton-like solution:
$\begin{align} \begin{split} P\left[1\right] &= -2\xi \mathrm{e}^{-2\eta \mathrm{i}x-4\mathrm{i}\left(\xi^2-\eta^2\right)t}\,\mathrm{sech}\left(2\xi x-8\xi\eta t\right), \\ R\left[1\right] &= \xi\,\mathrm{sech}^2\left(2\xi x-8\xi\eta t\right)\mathrm{e}^{-2\eta \mathrm{i}x-4\mathrm{i}\left(\xi^2-\eta^2\right)t} \\ &\quad \times\left[\left(2\xi x+2\mathrm{i}\eta x-1\right)\mathrm{e}^{2\xi x-8\xi\eta t} +\left(2\mathrm{i}\eta x-2\xi x-1\right)\mathrm{e}^{-2\xi x+8\xi \eta t}\right]. \end{split}\end{align}$
Example 2. Integrable couplings of the mKdV Equations on a space scale
Let
$\begin{equation} a_0 = \alpha = -2,\quad e_0 = \beta = 0.\end{equation}$
By setting m = 3 in (49 ), we obtain the integrable coupling of the mKdV equation on a space scale:
$\begin{align} &\left[\frac{P+P^{\rho}}{2}\right]_t -\frac{P+P^{\rho}}{4}\mathit{G} \nabla_x^{-1}\mathit{G}\left[P^{*}\mathit{G}^{-1}\nabla_x^{2}P -P\mathit{G}^{-1}\nabla_x^2P^{*}\right] \nonumber\\ &\quad -\frac{1}{2}\nabla_x\mathit{G}^{-1}\left[4\mathit{G}^{-1}\nabla_x^2P +\mathit{G}P\nabla_x\left|P\right|^2 \right] = 0, \end{align}$
and $\begin{align} &\left[\frac{R+R^{\rho}}{2}\right]_t -\frac{R+R^{\rho}}{4}\mathit{G}\nabla_x^{-1}\mathit{G} \left[P^{*}\mathit{G}^{-1}\nabla_x^{2}P -P\mathit{G}^{-1}\nabla_x^2P^{*}\right]\nonumber\\ &\quad -\frac{P+P^{\rho}}{4}\mathit{G}\nabla_x^{-1}\left[3\mathit{G}P\mathit{G}^{-1}\nabla_x^2P^{*} +3\left(P^*+P^{*\rho}\right)\mathit{G}^{-1}\nabla_x^2P-\left(R+R^{\rho}\right)\mathit{G}^{-1}\nabla_x^2P^* \right. \nonumber\\ &\quad\left. +\left(P^*+P^{*\rho}\right)\mathit{G}^{-1}\nabla_x^2R +\left(R^*+R^{*\rho}\right)\mathit{G}^{-1}\nabla_x^2P -\left(P+P^{\rho}\right)\mathit{G}^{-1}\nabla_x^2R^* \right] \nonumber\\ &\quad -\frac{1}{2}\nabla_x\left[R\mathit{G} \left|P\right|^2-12\mathit{G}^{-2}\nabla_x^2P +4\mathit{G}^{-2}\nabla_x^2R+P\mathit{G}\left(R^*R+R^*P-3\left|P\right|^2\right) \right] = 0, \end{align}$
where $\mathit{G} = 2-\nu\nabla_{x}$. We next consider two special choices of the space scale measure.When $\nu(x) = 0$, the classical integrable coupling of the mKdV equation is obtained:
$\begin{equation} \begin{cases} P_t-\dfrac{1}{2}P_{xxx}-3\left|P\right|^2P_x = 0, \\[5pt] R_t+\dfrac{3}{2}P_{xxx}-\dfrac{1}{2}R_{xxx}+9\left|P\right|^2P_x-3\left|P\right|^2R_x-3P^*RP_x-3PR^*P_x = 0. \end{cases}\end{equation}$
This case coincides with that in [38].When $\nu(x) = 1$, an integrable coupling of the generalized semi-discrete mKdV equation is obtained:
$\begin{equation} \begin{cases} \dfrac{1+\mathit{E}}{2}P_{nt}-\dfrac{\left(1+\mathit{E}\right)P_n}{2}\left(1+\mathit{E}\right)a_3-\left(1-\mathit{E}\right)b_3 = 0, \\ \dfrac{1+\mathit{E}}{2}R_{nt}-\dfrac{\left(1+\mathit{E}\right)R_n}{2}\left(1+\mathit{E}\right)a_3-\dfrac{\left(1+\mathit{E}\right)P_n}{2}\left(1+\mathit{E}\right)e_3-\left(1-\mathit{E}\right)f_3 = 0. \end{cases}\end{equation}$
where $\begin{equation} \begin{cases} a_3 = \dfrac{1}{2}\left(1-\mathit{E}\right)^{-1}\left(1+\mathit{E}\right)\left[P^{*}_n\left(1+\mathit{E}\right)^{-1}\left(1-\mathit{E}\right)^2P_n-P_n\left(1+\mathit{E}\right)^{-1}\left(1-\mathit{E}\right)^2P^{*}_n\right], \\ b_3 = \dfrac{1}{2}\left(1+\mathit{E}\right)^{-1}\left[\left(1+\mathit{E}\right)P_n\left(1+\mathit{E}\right)\left|P_n\right|^2-P_n\left(1+\mathit{E}\right)^{-1}\left(1-\mathit{E}\right)^2P^{*}_n\right], \\ e_3 = \dfrac{1}{2}\left[3\left(1+\mathit{E}\right)P_n\left(1+\mathit{E}\right)^{-1}\left(1-\mathit{E}\right)^2P^{*}_n+3\left(1+\mathit{E}\right)P^*_n\left(1+\mathit{E}\right)^{-1}\left(1-\mathit{E}\right)^2P_n\right. \\ \qquad\left. -\left(1+\mathit{E}\right)R_n\left(1+\mathit{E}\right)^{-1}\left(1-\mathit{E}\right)^2P^{*}_n+\left(1+\mathit{E}\right)P^*_n\left(1+\mathit{E}\right)^{-1}\left(1-\mathit{E}\right)^2R_n\right. \\ \qquad\left. +\left(1+\mathit{E}\right)R^*_n(1+\mathit{E})^{-1}(1-\mathit{E})^2P_n-(1+\mathit{E})P_n(1+\mathit{E})^{-1}(1-\mathit{E})^2R^{*}_n\right], \\ f_3 = \dfrac{1}{2}\left[R_n(1+\mathit{E})\left|P_n\right|^2+P_n(1+\mathit{E})(P^*_nR_n+R^*_nP_n-3\left|P_n\right|^2)\right. \\ \qquad\left. +12(1+\mathit{E})(1-\mathit{E})^2P_n-4(1+\mathit{E})(1-\mathit{E})^2R_n\right]. \end{cases} \end{equation}$
To obtain exact solutions, we apply the Darboux transformation. Then
$\begin{equation} \bar{\boldsymbol{H}} = \begin{pmatrix} \boldsymbol{H} & \boldsymbol{H}_1 \\ 0 & \boldsymbol{H} \end{pmatrix} = \begin{pmatrix} \phi_{11} & \phi_{21}^{*} & \psi_{11} & \psi_{21}^{*} \\ \phi_{21} & -\phi_{11}^{*} & \psi_{21} & -\psi_{11}^{*} \\ 0 & 0 & \phi_{11} & \phi_{21}^{*} \\ 0 & 0 & \phi_{21} & -\phi_{11}^{*} \end{pmatrix},\end{equation}$
where $\begin{align} \begin{cases} \phi_{11} = \hat{\mathrm{e}}_{-\lambda_1}\left(x,0\right)\mathrm{e}^{-2\lambda_1^3t}, \\ \phi_{21} = \hat{\mathrm{e}}_{\lambda_1}\left(x,0\right)\mathrm{e}^{2\lambda_1^3t}, \\ \psi_{11} = -\lambda_1\displaystyle\int_{0}^{x}\hat{\mathrm{e}}_{-\lambda_1}\left(\tau,\rho\left(\tau\right)\right)\,\nabla\tau\, \hat{\mathrm{e}}_{-\lambda_1}\left(x,0\right)\mathrm{e}^{-2\lambda_1^3t}, \\ \psi_{21} = \lambda_1\displaystyle\int_{0}^{x}\hat{\mathrm{e}}_{-\lambda_1}\left(\tau,\rho\left(\tau\right)\right)\,\nabla\tau\, \hat{\mathrm{e}}_{\lambda_1}\left(x,0\right)\mathrm{e}^{2\lambda_1^3t}. \end{cases} \end{align}$
Let $\lambda_1 = \xi+\mathrm{i}\eta$. By applying the Bácklund transformation (58 ), we obtain a one-soliton-like solution of the mKdV equation on a space scale: 80 ).
$\begin{align} P\left[1\right] & = -4\left(2-\nu\nabla_x\right)^{-1}\xi\left[ \frac{\hat{\mathrm{e}}_{M_1}\left(x,0\right)\mathrm{e}^{4\left(\eta^3-3\xi^2\eta\right)t}} {\hat{\mathrm{e}}_{M_2}\left(x,0\right)\mathrm{e}^{-4\left(\xi^3-3\xi\eta^2\right)t}+\hat{\mathrm{e}}_{M_3}\left(x,0\right)\mathrm{e}^{4\left(\xi^3-3\xi\eta^2\right)t}} \right. \nonumber\\ &\quad \left. +\frac{\left(1-\nu M_1\right)\hat{\mathrm{e}}_{M_1}\left(x,0\right)\mathrm{e}^{4\left(\eta^3-3\xi^2\eta\right)t}} {\left(1-\nu M_2\right)\hat{\mathrm{e}}_{M_2}\left(x,0\right)\mathrm{e}^{-4\left(\xi^3-3\xi\eta^2\right)t}+\left(1-\nu M_3\right)\hat{\mathrm{e}}_{M_3}\left(x,0\right)\mathrm{e}^{4\left(\xi^3-3\xi\eta^2\right)t}} \right], \end{align}$
$\begin{align} R\left[1\right] & = -2\left(2-\nu\nabla_x\right)^{-1}\left(a_{12}+a^{\rho}_{12}+b_{12}+b^{\rho}_{12}\right),\end{align}$
where $\begin{align} \begin{cases} a_{12} = 2\xi\displaystyle\frac{\hat{\mathrm{e}}_{M_1}(x,0)\mathrm{e}^{4(\eta^3-3\xi^2\eta)t}} {\hat{\mathrm{e}}_{M_2}(x,0)\mathrm{e}^{-4(\xi^3-3\xi\eta^2)t}+\hat{\mathrm{e}}_{M_3}(x,0)\mathrm{e}^{4(\xi^3-3\xi\eta^2)t}}, \\ a^{\rho}_{12} = 2\xi\displaystyle\frac{(1-\nu M_1)\hat{\mathrm{e}}_{M_1}(x,0)\mathrm{e}^{4(\eta^3-3\xi^2\eta)t}} {(1-\nu M_2)\hat{\mathrm{e}}_{M_2}(x,0)\mathrm{e}^{-4(\xi^3-3\xi\eta^2)t}+(1-\nu M_3)\hat{\mathrm{e}}_{M_3}(x,0)\mathrm{e}^{4(\xi^3-3\xi\eta^2)t}}, \\ b_{12} = \dfrac{2\xi}{\bigl(\hat{\mathrm{e}}_{M_2}(x,0)\mathrm{e}^{-4(\xi^3-3\xi\eta^2)t}+\hat{\mathrm{e}}_{M_3}(x,0)\mathrm{e}^{4(\xi^3-3\xi\eta^2)t}\bigr)^{2}} \\ \qquad\times\left[ (\xi-\mathrm{i}\eta)\hat{\mathrm{e}}_{M_4}(x,0)\mathrm{e}^{M_8t} \left(\int_{0}^{x}\hat{\mathrm{e}}_{\xi-\mathrm{i}\eta}(\tau,\rho(\tau))\,\nabla\tau +\int_{0}^{x}\hat{\mathrm{e}}_{-\xi+\mathrm{i}\eta}(\tau,\rho(\tau))\,\nabla\tau\right)\right.\\ \qquad\left. -(\xi+\mathrm{i}\eta)\hat{\mathrm{e}}_{M_5}(x,0)\mathrm{e}^{M_9t} \left(\int_{0}^{x}\hat{\mathrm{e}}_{-\xi-\mathrm{i}\eta}(\tau,\rho(\tau))\,\nabla\tau +\int_{0}^{x}\hat{\mathrm{e}}_{\xi+\mathrm{i}\eta}(\tau,\rho(\tau))\,\nabla\tau\right) \right], \\ b_{12}^{\rho} = \dfrac{2\xi}{\Bigl[(1-\nu M_2)\hat{\mathrm{e}}_{M_2}(x,0)\mathrm{e}^{-4(\xi^3-3\xi\eta^2)t}+(1-\nu M_3)\hat{\mathrm{e}}_{M_3}(x,0)\mathrm{e}^{4(\xi^3-3\xi\eta^2)t}\Bigr]^2} \\ \qquad\times\left[ (\xi-\mathrm{i}\eta)(1-\nu M_4)\hat{\mathrm{e}}_{M_4}(x,0)\mathrm{e}^{M_8t} \left(\int_{0}^{x}\hat{\mathrm{e}}_{\xi-\mathrm{i}\eta}(\tau,\rho(\tau))\,\nabla\tau +\int_{0}^{x}\hat{\mathrm{e}}_{-\xi+\mathrm{i}\eta}(\tau,\rho(\tau))\,\nabla\tau\right)\right.\\ \qquad\left. -(\xi+\mathrm{i}\eta)(1-\nu M_5)\hat{\mathrm{e}}_{M_5}(x,0)\mathrm{e}^{M_9t} \left(\int_{0}^{x}\hat{\mathrm{e}}_{-\xi-\mathrm{i}\eta}(\tau,\rho(\tau))\,\nabla\tau +\int_{0}^{x}\hat{\mathrm{e}}_{\xi+\mathrm{i}\eta}(\tau,\rho(\tau))\,\nabla\tau\right) \right]. \end{cases}\end{align}$
where $\begin{align} \begin{cases} M_8 = -4\left(\xi^3-\mathrm{i}\eta^3+3\mathrm{i}\xi^2\eta-3\xi\eta^2\right), \\ M_9 = 4\left(\xi^3+\mathrm{i}\eta^3-3\mathrm{i}\xi^2\eta-3\xi\eta^2\right), \end{cases}\end{align}$
and $M_1,\dots,M_7$ are given in (When $\nu(x) = 0$, this reduces to the classical one-soliton-like solution:
$\begin{align} \begin{split} P\left[1\right] &= -2\xi \mathrm{e}^{-2\mathrm{i}\eta x+4\left(\eta^3-3\xi^2\eta\right)t}\,\mathrm{sech}\left(2\xi x+4\left(\xi^3-3\xi\eta^2\right)t\right), \\ R\left[1\right] &= \xi \mathrm{e}^{-2\mathrm{i}\eta x+4\xi^3 t-12\xi\eta^2 t}\,\mathrm{sech}^2\left(2\xi x+4\xi^3 t-12\xi\eta^2 t\right) \\ &\quad \times\left[\left(2\xi x+2\mathrm{i}\eta x-1\right)\mathrm{e}^{2\xi x+4\left(\xi^3-3\xi\eta^2\right)t}\right.\nonumber\\ &\quad \left.+\left(2\mathrm{i}\eta x-2\xi x-1\right)\mathrm{e}^{-2\xi x-4\left(\xi^3-3\xi\eta^2\right)t}\right]. \end{split} \end{align}$
5. Conclusions
This study establishes a Darboux transformation formulation for integrable couplings on a space scale. By extending the spectral problem and the associated Lax pair, we derive integrable couplings of the AKNS hierarchy within this framework. The integrable couplings of the AKNS hierarchy on a space scale overcome the limitation that classical constructions usually treat continuous and semi-discrete cases separately. By introducing the space-scale setting, the coupling relations are formulated along the spatial direction of scale transformations, which provides a unified framework for both continuous and generalized semi-discrete systems. In particular, through suitable reductions, we derive integrable couplings related to the NLS and mKdV equations on a space scale. Moreover, by choosing appropriate space scales, the proposed method unifies the continuous and generalized semi-discrete versions of these coupled systems. Finally, soliton solutions for these systems are obtained via the Darboux transformation.