Soliton Solutions to Sasa–Satsuma-Type Modified Korteweg–De Vries Equations by Binary Darboux Transformations

Abstract

Sasa–Satsuma (SS)-type integrable matrix modified Korteweg–de Vries (mKdV) equations are derived from two group constraints, involving the replacement of the spectral matrix in the Ablowitz–Kaup–Newell–Segur matrix eigenproblems with its matrix transpose and its Hermitian transpose. Using the Lax pairs and dual Lax pairs of matrix eigenproblems as a foundation, binary Darboux transformations are constructed. These transformations, initiated with a zero seed solution, facilitate the generation of soliton solutions for the SS-type integrable matrix mKdV equations presented.

1. Introduction

Soliton theory represents a dynamic area within mathematical physics, offering diverse methods for solving nonlinear integrable equations. Key approaches include the Hirota direct approach, the inverse scattering technique, Darboux transformations (DT), the Riemann–Hilbert approach, Lie symmetry analysis, and Painlevé singularity confinement analysis [1,2,3]. Integral to these methods is the concept of an integrable equation, which is linked to two matrix eigenproblems known as a Lax pair. A binary Darboux transformation (DT) is derived from both a Lax pair and its adjoint, termed the dual Lax pair, which are equivalent representations of the same integrable equation [4,5]. Matrix eigenproblems play a crucial role not only in constructing DTs and binary DTs, but also in designing inverse scattering transforms and Riemann–Hilbert problems [1,2,3].

In the framework of (1+1)-dimensional integrable equations, where t and x are independent variables and $p=p(t,x)$ represents a column vector of unknown variables, spatial and temporal matrix eigenproblems are defined as follows:

$$\begin{array}{c}{\displaystyle -i{\varphi }_x=\mathcal{E}\varphi =\mathcal{E}(p,z)\varphi ,}\hfill \end{array}$$

(1)

and

$${\displaystyle -i{\varphi }_t=\mathcal{F}\varphi =\mathcal{F}(p,z)\varphi ,}$$

(2)

where i denotes the imaginary unit, $\varphi$ is a column eigenfunction, and $\mathcal{E}$ and $\mathcal{F}$ are square spectral matrices depending on p and the spectral parameter z. The consistency condition of these eigenproblems yields an integrable evolution equation

$$p_t=X\left(p\right),$$

(3)

through the flatness condition

$${\mathcal{E}}_t-{\mathcal{F}}_x+i[\mathcal{E},\mathcal{F}]=0,$$

(4)

where $[·,·]$ denotes the matrix bracket. Flatness equations are known for their elegant algebraic structures, ensuring the existence of a limitless array of symmetries for the considered nonlinear integrable model [6]. Obviously, the dual Lax pair of the matrix eigenproblems, defined as

$$i{\tilde{\varphi }}_x=\tilde{\varphi }\mathcal{E}=\tilde{\varphi }\mathcal{E}(p,z),$$

(5)

and

$$i{\tilde{\varphi }}_t=\tilde{\varphi }\mathcal{F}=\tilde{\varphi }\mathcal{F}(p,z),$$

(6)

where $\tilde{\varphi }$ is a row eigenfunction, results in an identical flatness condition (4) without generating any extra conditions. All of the Lax pair of eigenproblems and the dual Lax pair eigenproblems are utilized in our discussion to formulate binary DTs.

Based on a Lax pair and its adjoint in eigenproblems, a binary DT is constructed for an integrable equation under examination, expressed as

$${\varphi }^{\prime }=D^{+}\varphi =D^{+}(p,z)\varphi ,{\tilde{\varphi }}^{\prime }=\tilde{\varphi }{D}^{-}=\tilde{\varphi }{D}^{-}(p,z),$$

(7)

where $D^{-}={\left(D^{+}\right)}^{-1}$. This transformation ensures that ${\varphi }^{\prime }$ and ${\tilde{\varphi }}^{\prime }$ solve new matrix eigenproblems as follows:

$$-i{\varphi }_x^{\prime }={\mathcal{E}}^{\prime }{\varphi }^{\prime },-i{\varphi }_t^{\prime }={\mathcal{F}}^{\prime }{\varphi }^{\prime },$$

(8)

and their adjoint counterparts as follows:

$$i{\tilde{\varphi }}_x^{\prime }={\tilde{\varphi }}^{\prime }{\mathcal{E}}^{\prime },i{\tilde{\varphi }}_t^{\prime }={\tilde{\varphi }}^{\prime }{\mathcal{F}}^{\prime },$$

(9)

in which the updated Lax pair matrices read as follows:

$${\mathcal{E}}^{\prime }=\mathcal{E}(p^{\prime },z),{\mathcal{F}}^{\prime }=\mathcal{F}(p^{\prime },z),$$

(10)

and

$$p^{\prime }=S\left(p\right),$$

(11)

defines a Bäcklund transformation. For the binary DT formulation, the Darboux matrices $D^{+}$ and $D^{-}$ must satisfy

$$-i{D}_x^{+}{D}^{-}+D^{+}\mathcal{E}{D}^{-}={\mathcal{E}}^{\prime }=\mathcal{E}(S\left(p\right),z),$$

(12)

and

$$-i{D}_t^{+}{D}^{-}+D^{+}\mathcal{F}{D}^{-}={\mathcal{F}}^{\prime }=\mathcal{F}(S\left(p\right),z).$$

(13)

The Equations (8) and (9) ensure that the novel Lax pair, ${\mathcal{E}}^{\prime }$ and ${\mathcal{F}}^{\prime }$, satisfy the identical flatness condition (4), where p is replaced by $p^{\prime }$. Moreover, $p^{\prime }$ represents a novel solution to the original integrable model if p does. Hence, (11) provides a Bäcklund transformation of the given integrable model. It is well documented that Darboux transformations and binary DTs have been extensively studied for single- and multi-component integrable equations [4,7,8,9,10,11,12,13,14]. However, research in non-commutative cases, including matrix integrable equations, remains relatively sparse despite notable contributions [15,16,17].

It is widely recognized that integrable reductions can generate reduced integrable equations under certain conditions, particularly when specific group constraints are applied to the spectral matrix $\mathcal{E}$. One common class of such reductions is expressed as follows:

$${\left(\mathcal{E}(t,x,z^{*})\right)}^{†}=\mathsf{\Pi }\mathcal{E}(t,x,z){\mathsf{\Pi }}^{-1},$$

(14)

where † denotes the Hermitian transpose, * represents the complex conjugate, and $\mathsf{\Pi }$ is a constant Hermitian non-singular matrix (see, e.g., [18,19,20]). This reduction involves replacing the parameter z with $z^{*}$ and is effective for either nonlinear Schrödinger (NLS) or modified Korteweg–de Vries (mKdV)-type integrable models. A second class of integrable reductions specific to mKdV-type equations involves replacing z with its negative counterpart $-z$ in given matrix eigenproblems as follows:

$${\left(\mathcal{E}(t,x,-z)\right)}^T=-\mathsf{\Pi }\mathcal{E}(t,x,z){\mathsf{\Pi }}^{-1},$$

(15)

where T denotes the transpose of a matrix and $\mathsf{\Pi }$ is a constant symmetric non-singular matrix. This reduction is effective for mKdV-type equations but is not applicable to NLS-type equations [20]. Other substitutions such as $z\to -z^{*}$ and $z\to z$ lead to nonlocal integrable reductions, often accompanied by reflections of the independent variables $t\to -t$, $x\to -x$, or $(t,x)\to (-t,-x)$. These transformations are notable for their role in generating distinct forms of integrable reductions across different classes of nonlinear equations.

In this paper, we aim to explore two kinds of integrable reductions, (14) and (15), simultaneously applied to construct Sasa–Satsuma (SS)-type integrable matrix mKdV equations. Our approach will utilize matrix Ablowitz–Kaup–Newell–Segur (AKNS) eigenproblems of an arbitrary order as the foundational framework. The primary objective is to derive binary DTs for the resultant integrable matrix mKdV models. These binary DTs will demonstrate an N-fold decomposition property, particularly when the eigenvalues and dual eigenvalues differ from each other in the regular case. Starting from the zero seed solution, the derived binary DTs will be employed to compute soliton solutions specific to the SS-type integrable matrix mKdV models. The paper will conclude with a summary in the Section 5, accompanied by pertinent concluding remarks on the significance and implications of the findings.

2. SS-Type Integrable Matrix mKdV Models

2.1. The Integrable Matrix AKNS Hierarchy Revisited

Let us revisit the procedure for constructing the hierarchy of integrable matrix AKNS equations. Assume $m,n\ge 1$ are two integers, z represents the spectral parameter, $I_k$ denotes the $k\times k$ identity matrix ($k\in \mathbb{N}$), and $r,s$ are two matrix potentials:

$$r=r(t,x)={\left(r_{jl}\right)}_{m\times n},s=s(t,x)={\left(s_{lj}\right)}_{n\times m}.$$

(16)

The local integrable matrix AKNS hierarchy originates from the matrix AKNS eigenproblems characterized by the matrix potentials

$$-i{\varphi }_x=\mathcal{E}\varphi =\mathcal{E}(p,z)\varphi ,-i{\varphi }_t={\mathcal{F}}^{\left[k\right]}\varphi ={\mathcal{F}}^{\left[k\right]}(p,z)\varphi ,k\ge 0,$$

(17)

whose Lax pair reads

$$\mathcal{E}=z\Sigma +A\left(p\right),{\mathcal{F}}^{\left[k\right]}=z^k\Delta +B^{\left[k\right]}\left(p\right),$$

(18)

with $\Sigma ,\Delta$ defined by

$$\begin{array}{cc}& \Sigma =\mathrm{diag}({\gamma }_1{I}_m,{\gamma }_2{I}_n),\end{array}$$

(19)

$$\begin{array}{cc}& \Delta =\mathrm{diag}({\delta }_1{I}_m,{\delta }_2{I}_n).\end{array}$$

(20)

Here, ${\gamma }_1,{\gamma }_2$, and ${\delta }_1,{\delta }_2$ are pairs of distinct arbitrary real numbers. In addition, the other two $m+n$th-order matrices are determined as follows:

$$A\left(p\right)=\left[\begin{array}{cc}0& r\\ s& 0\end{array}\right],$$

(21)

referred to as the potential matrix, and

$$B^{\left[k\right]}\left(p\right)=\sum _{l=0}^{k-1}{z}^l\left[\begin{array}{cc}{e}^{\left[s\right]}& f^{\left[s\right]}\\ g^{\left[s\right]}& h^{\left[s\right]}\end{array}\right],$$

(22)

where four sequences of differential polynomial matrices, $e^{\left[s\right]},f^{\left[s\right]},g^{\left[s\right]}$, and $h^{\left[s\right]}$, are determined in a recursive fashion by

$$\begin{array}{c}{f}^{\left[0\right]}=0,g^{\left[0\right]}=0,e^{\left[0\right]}={\delta }_1{I}_m,h^{\left[0\right]}={\delta }_2{I}_n,\end{array}$$

(23a)

$$\begin{array}{c}{\displaystyle f^{[l+1]}=\frac{1}{\gamma }(-i{g}_x^{\left[l\right]}-r{h}^{\left[l\right]}+e^{\left[l\right]}r),l\ge 0,}\end{array}$$

(23b)

$$\begin{array}{c}{\displaystyle g^{[l+1]}=\frac{1}{\gamma }(i{g}_x^{\left[k\right]}+s{e}^{\left[l\right]}-h^{\left[l\right]}s),l\ge 0,}\end{array}$$

(23c)

$$\begin{array}{c}{e}_x^{\left[l\right]}=i(r{g}^{\left[l\right]}-f^{\left[l\right]}s),h_x^{\left[l\right]}=i(s{f}^{\left[l\right]}-g^{\left[l\right]}r),l\ge 1,\end{array}$$

(23d)

with $\gamma ={\gamma }_1-{\gamma }_2,\delta ={\delta }_1-{\delta }_2.$ It is required that the constants of integration vanish, i.e., $e^{\left[l\right]}{{|}_{r=0,s=0}=0,h^{\left[l\right]}|}_{r=0,s=0}=0,l\ge 1$), in order to uniquely determine the differential polynomial matrices $B^{\left[l\right]},l\ge 1.$ For instance, using this approach, one can present

$$B^{\left[1\right]}={\displaystyle \frac{\delta }{\gamma }}\left[\begin{array}{cc}0& r\\ s& 0\end{array}\right],B^{\left[2\right]}={\displaystyle \frac{\delta }{\gamma }}z\left[\begin{array}{cc}0& r\\ s& 0\end{array}\right]-{\displaystyle \frac{\delta }{{\gamma }^2}}\left[\begin{array}{cc}rs& i{r}_x\\ -i{s}_x& -sr\end{array}\right],$$

(24)

and

$$B^{\left[3\right]}={\displaystyle \frac{\delta }{\gamma }}{z}^2\left[\begin{array}{cc}0& r\\ s& 0\end{array}\right]-{\displaystyle \frac{\delta }{{\gamma }^2}}z\left[\begin{array}{cc}rs& i{r}_x\\ -i{s}_x& -sr\end{array}\right]-{\displaystyle \frac{\delta }{{\gamma }^3}}\left[\begin{array}{cc}i(r{s}_x-r_{x}s)& r_{xx}+2rsr\\ s_{xx}+2srs& i(s{r}_x-s_{x}r)\end{array}\right].$$

(25)

It is evident that for $m=1$, the matrix eigenproblems described in (17) simplify to the case of multiple components. Specifically, when a set of potentials $r_{1l}$ and $s_{l1}$ are nonzero for $1\le l\le n,$ the matrix eigenproblems in (17) transform into the standard AKNS eigenproblems outlined in [21].

For any given pair of natural numbers m and n, the consistency conditions of the matrix eigenproblems defined in (17), represented by the flatness conditions

$${\mathcal{E}}_t-{\mathcal{F}}_x^{\left[k\right]}+i[\mathcal{E},{\mathcal{F}}^{\left[k\right]}]=0,k\ge 0,$$

(26)

generate a local hierarchy of integrable matrix AKNS equations

$$r_t=i\gamma f^{[k+1]},s_t=-i\gamma g^{[k+1]},k\ge 0.$$

(27)

Based on the trace variational identity [22] and the Lax operator algebra formulation, it can be readily demonstrated that (27) forms a series of local commuting models. Every member in the hierarchy possesses a bi-Hamiltonian formulation, indicating the presence of infinitely many commuting conserved quantities.

2.2. AKNS Integrable Matrix mKdV Models

The AKNS integrable matrix mKdV equation models can be expressed as

$${\displaystyle r_t=-\frac{\delta }{{\gamma }^3}(r_{xxx}+3rs{r}_x+3r_{x}sr),s_t=-\frac{\delta }{{\gamma }^3}(s_{xxx}+3s_{x}rs+3sr{s}_x),}$$

(28)

in which the two matrix potentials, r and s, are given by (16). These equations correspond to the following spectral matrices:

$$\mathcal{E}=z\Sigma +A,{\mathcal{F}}^{\left[3\right]}=z^3\Delta +B^{\left[3\right]},$$

(29)

where $\Sigma$ and $\Delta$ are given by (19) and (20), A is defined by (21), and $B^{\left[3\right]}$ is expressed in terms of A as

$$B^{\left[3\right]}=B^{\left[3\right]}(p,z)={\displaystyle \frac{\delta }{\gamma }}{z}^{2}A-{\displaystyle \frac{\delta }{{\gamma }^2}}z{I}_{m,n}(A^2+i{A}_x)-{\displaystyle \frac{\delta }{{\gamma }^3}}(i[A,A_x]+A_{xx}+2A^3),$$

(30)

with $I_{m,n}=\mathrm{diag}(I_m,-I_n)$.

If $m=1$ and $n=1$, the AKNS integrable matrix mKdV equations reduce to

$$\left\{\begin{array}{c}{\displaystyle r_{11,t}=r_{11,xxx}+6r_{11}{s}_{11}{r}_{11,x},}\hfill \\ s_{11,t}=s_{11,xxx}+6r_{11}{s}_{11}{s}_{11,x}.\hfill \end{array}\right.$$

(31)

For $m=2$ and $n=1$, the equations become

$$\left\{\begin{array}{c}{\displaystyle r_{j1,t}=r_{j1,xxx}+3(r_{11,x}{s}_{11}+r_{21,x}{s}_{12})r_{j1}+3(r_{11}{s}_{11}+r_{21}{s}_{12})r_{j1,x},}\hfill \\ {\displaystyle s_{1j,t}=s_{1j,xxx}+3(r_{11}{s}_{11,x}+r_{21}{s}_{12,x})s_{1j}+3(r_{11}{s}_{11}+r_{21}{s}_{12})s_{1j,x},}\hfill \end{array}\right.$$

(32)

with $1\le j\le 2$. And for $m=2$ and $n=2$, the equations are

$$\left\{\begin{array}{c}{\displaystyle r_{jl,t}=r_{jl,xxx}+3\sum _{p,q=1}^2{r}_{jp}{s}_{pq}{r}_{ql,x}+3\sum _{p,q=1}^2{r}_{jp,x}{s}_{pq}{r}_{ql},}\hfill \\ {\displaystyle s_{lj,t}=s_{lj,xxx}+3\sum _{p,q=1}^2{s}_{lp,x}{r}_{pq}{s}_{qj}+3\sum _{p,q=1}^2{s}_{lp}{r}_{pq}{s}_{qj,x},}\hfill \end{array}\right.$$

(33)

with $1\le j,l\le 2$. These equations are all Liouville integrable and exhibit symmetry under the reflection about the origin $(x\to -x,t\to -t)$.

2.3. SS-Type Integrable Matrix AKNS Equations

Let us now construct integrable reductions from the general integrable matrix AKNS Equation (27).

We choose two constant non-singular Hermitian matrices ${\mathsf{\Theta }}_1,{\mathsf{\Theta }}_2$ and another two constant non-singular symmetric matrices ${\mathsf{\Xi }}_1,{\mathsf{\Xi }}_2$, and we then define two local reductions for the initial spectral matrix $\mathcal{E}$ given in (29) as follows:

$${\left(\mathcal{E}(t,x,z^{*})\right)}^{†}=\mathsf{\Theta }\mathcal{E}(t,x,z){\mathsf{\Theta }}^{-1},$$

(34)

and

$${\left(\mathcal{E}(t,x,-z)\right)}^T=-\mathsf{\Xi }\mathcal{E}(t,x,z){\mathsf{\Xi }}^{-1},$$

(35)

in which two constant square matrices, $\mathsf{\Theta }$ and $\mathsf{\Xi }$, are determined by

$$\mathsf{\Theta }=\left[\begin{array}{cc}{\mathsf{\Theta }}_1& 0\\ 0& {\mathsf{\Theta }}_2\end{array}\right],\mathrm{where}{\mathsf{\Theta }}_j^{†}={\mathsf{\Theta }}_j,j=1,2,$$

(36)

and

$$\mathsf{\Xi }=\left[\begin{array}{cc}{\mathsf{\Xi }}_1& 0\\ 0& {\mathsf{\Xi }}_2\end{array}\right],\mathrm{where}{\mathsf{\Xi }}_j^T={\mathsf{\Xi }}_j,j=1,2.$$

(37)

These two group constraints precisely require

$$A(t,x)={\mathsf{\Theta }}^{-1}{A}^{†}(t,x)\mathsf{\Theta },$$

(38)

and

$$A(t,x)=-{\mathsf{\Xi }}^{-1}{A}^T(t,x)\mathsf{\Xi },$$

(39)

which lead equivalently to the following potential reductions:

$$r(t,x)={\mathsf{\Theta }}_1^{-1}{s}^{†}(t,x){\mathsf{\Theta }}_2,$$

(40)

and

$$r(t,x)=-{\mathsf{\Xi }}_1^{-1}{s}^T(t,x){\mathsf{\Xi }}_2,$$

(41)

respectively. Therefore, we need a restriction for r as follows:

$${\mathsf{\Theta }}_2^{-1}{r}^{†}(t,x){\mathsf{\Theta }}_1=-{\mathsf{\Xi }}_2^{-1}{r}^T(t,x){\mathsf{\Xi }}_1,$$

(42)

due to taking a pair of group constraints simultaneously.

We observe that the reductions in Equations (34) and (35) lead to the following transformations:

$$\left\{\begin{array}{c}{\displaystyle {\left({\mathcal{F}}^{[2l+1]}(t,x,z^{*})\right)}^{†}=\mathsf{\Theta }{\mathcal{F}}^{[2l+1]}(t,x,z){\mathsf{\Theta }}^{-1},}\hfill \\ {\displaystyle {\left({\mathcal{F}}^{[2l+1]}(t,x,-z)\right)}^T=-\mathsf{\Xi }{\mathcal{F}}^{[2l+1]}(t,x,z){\mathsf{\Xi }}^{-1},}\hfill \end{array}\right.$$

(43)

and

$$\left\{\begin{array}{c}{\displaystyle {\left(B^{[2l+1]}(t,x,z^{*})\right)}^{†}=\mathsf{\Theta }{B}^{[2l+1]}(t,x,z){\mathsf{\Theta }}^{-1},}\hfill \\ {\displaystyle {\left(B^{[2l+1]}(t,x,-z)\right)}^T=-\mathsf{\Xi }{B}^{[2l+1]}(t,x,z){\mathsf{\Xi }}^{-1},}\hfill \end{array}\right.$$

(44)

where $l\ge 0$. Here, ${\mathcal{F}}^{[2l+1]}$ is defined as in (18) and $B^{[2l+1]}$ is determined through (22).

As a result of reductions (40) and (41), the integrable matrix AKNS equations described in (27) with $k=2l+1,l\ge 0,$ lead to a series of SS-type integrable matrix AKNS equations:

$$r_t=i\gamma f^{[2l+2]}{|}_{s={\mathsf{\Theta }}_2^{-1}{r}^{†}{\mathsf{\Theta }}_1=-{\mathsf{\Xi }}_2^{-1}{r}^T{\mathsf{\Xi }}_1},l\ge 0,$$

(45)

where $r={\left(r_{jl}\right)}_{m\times n}$ satisfies (42). Here, ${\mathsf{\Theta }}_1$ and ${\mathsf{\Theta }}_2$ are arbitrarily non-singular Hermitian matrices of sizes m and n, respectively, and ${\mathsf{\Xi }}_1$ and ${\mathsf{\Xi }}_2$ are arbitrarily non-singular symmetric matrices of sizes m and n. Each member in the hierarchy (45) possesses a Lax pair derived from the reduced matrix eigenproblems defined in (17) with $k=2l+1(l\ge 0)$. Furthermore, these equations exhibit infinitely many local commuting conserved quantities and symmetries, inherited from those of the integrable matrix AKNS equations defined in (27) with $k=2l+1(l\ge 0)$ under the two reductions.

2.4. SS-Type Integrable mKdV Equations

Let us focus on the case where $k=3$, which corresponds to $l=1$. In this context, the reduced integrable matrix AKNS equation from (45) specifically becomes

$$\begin{array}{cc}& {\displaystyle r_t=-\frac{\delta }{{\gamma }^3}(r_{xxx}+3r{\mathsf{\Theta }}_2^{-1}{r}^{†}{\mathsf{\Theta }}_1{r}_x+3r_x{\mathsf{\Theta }}_2^{-1}{r}^{†}{\mathsf{\Theta }}_{1}r)}\\ & {\displaystyle =-\frac{\delta }{{\gamma }^3}(r_{xxx}-3r{\mathsf{\Xi }}_2^{-1}{r}^T{\mathsf{\Xi }}_1{r}_x-3r_x{\mathsf{\Theta }}_2^{-1}{r}^T{\mathsf{\Xi }}_{1}r),}\end{array}$$

(46)

where the $m\times n$ matrix potential r needs to satisfy (42). This equation is known as the SS-type integrable matrix mKdV equation.

Based on the provided choices and calculations, let us illustrate the derivation and the resulting SS-type integrable mKdV equation.

If we take $m=1$ and $n=2$, and choose

$${\mathsf{\Theta }}_1=1,{\mathsf{\Theta }}_2^{-1}=\left[\begin{array}{cc}\theta & 0\\ 0& \theta \end{array}\right],{\mathsf{\Xi }}_1=1,{\mathsf{\Xi }}_2^{-1}=\left[\begin{array}{cc}0& \xi \\ \xi & 0\end{array}\right],$$

(47)

with two real constants $\theta$ and $\xi$ subject to ${\theta }^2={\xi }^2=1$. Then, the potential restriction (42) yields

$$r_2=-\theta \xi r_1^{*},$$

(48)

and, furthermore, the potential matrix A satisfying (42) becomes

$$A=\left[\begin{array}{ccc}0& r_1& -\theta \xi r_1^{*}\\ \theta r_1^{*}& 0& 0\\ -\xi r_1& 0& 0\end{array}\right].$$

(49)

Thus, the corresponding SS-type integrable mKdV equation reads

$$r_{1,t}=-{\displaystyle \frac{\delta }{{\gamma }^3}}[r_{1,xxx}+6\theta |r_1{|}^2{r}_{1,x}+3\theta r_1(|r_1{|}^2{)}_x],$$

(50)

with $\theta =\pm 1$. Obviously, for $\theta =1$, this equation precisely matches the SS mKdV equation, as referenced in the literature [23]. The equation allows for higher-order soliton solutions through its generalized Darboux transformation, as discussed in further studies [24].

If we still take $m=1$ and $n=2$, but choose

$${\mathsf{\Theta }}_1=1,{\mathsf{\Theta }}_2^{-1}=\left[\begin{array}{cc}0& \theta \\ \theta & 0\end{array}\right],{\mathsf{\Xi }}_1=1,{\mathsf{\Xi }}_2^{-1}=\left[\begin{array}{cc}\xi & 0\\ 0& \xi \end{array}\right],$$

(51)

in which $\theta$ and $\xi$ are two real constanst subject to the condition ${\theta }^2={\xi }^2=1$, then, the potential restriction (42) still leads to the relation (48), but the potential matrix A satisfying (42) becomes

$$A=\left[\begin{array}{ccc}0& r_1& -\theta \xi r_1^{*}\\ -\xi r_1& 0& 0\\ \theta r_1^{*}& 0& 0\end{array}\right].$$

(52)

Therefore, the corresponding SS-type integrable mKdV equation gives

$$r_{1,t}=-{\displaystyle \frac{\delta }{{\gamma }^3}}[r_{1,xxx}-6\xi r_1^2{r}_{1,x}-3\xi r_1^{*}(|r_1{|}^2{)}_x],$$

(53)

with $\xi =\pm 1$. Clearly, these mathematical forms are different from the ones previously presented in (50).

Now, if we take $m=1$ and $n=4$, and choose

$${\mathsf{\Theta }}_1=1,{\mathsf{\Theta }}_2^{-1}=\left[\begin{array}{cccc}{\theta }_1& 0& 0& 0\\ 0& {\theta }_1& 0& 0\\ 0& 0& {\theta }_2& 0\\ 0& 0& 0& {\theta }_2\end{array}\right],{\mathsf{\Xi }}_1=1,{\mathsf{\Xi }}_2^{-1}=\left[\begin{array}{cccc}0& {\xi }_1& 0& 0\\ {\xi }_1& 0& 0& 0\\ 0& 0& 0& {\xi }_2\\ 0& 0& {\xi }_2& 0\end{array}\right],$$

(54)

in which ${\theta }_j$ and ${\xi }_j$ are real constants subject to the condition ${\theta }_j^2={\xi }_j^2=1,j=1,2$, then the potential restriction (42) engenders

$$r_2=-{\theta }_1{\xi }_1{r}_1^{*},r_4=-{\theta }_2{\xi }_2{r}_3^{*},$$

(55)

and, furthermore, the potential matrix A satisfying (42) becomes

$$A=\left[\begin{array}{ccccc}0& r_1& -{\theta }_1{\xi }_1{r}_1^{*}& r_3& -{\theta }_2{\xi }_2{r}_3^{*}\\ {\theta }_1{r}_1^{*}& 0& 0& 0& 0\\ -{\xi }_1{r}_1& 0& 0& 0& 0\\ {\theta }_2{r}_3^{*}& 0& 0& 0& 0\\ -{\xi }_2{r}_3& 0& 0& 0& 0\end{array}\right].$$

(56)

This allows us to generate the following two-component SS-type integrable mKdV equation:

$$\left\{\begin{array}{c}{\displaystyle r_{1,t}=-\frac{\delta }{{\gamma }^3}[r_{1,xxx}+6({\theta }_1|r_1{|}^2+{\theta }_2|r_3{|}^2)r_{1,x}+3({\theta }_1|r_1{|}^2+{\theta }_2|r_3{|}^2{)}_x{r}_1],}\hfill \\ {\displaystyle r_{3,t}=-\frac{\delta }{{\gamma }^3}[r_{3,xxx}+6({\theta }_1|r_1{|}^2+{\theta }_2|r_3{|}^2)r_{3,x}+3({\theta }_1|r_1{|}^2+{\theta }_2|r_3{|}^2{)}_x{r}_3],}\hfill \end{array}\right.$$

(57)

with the real constants ${\theta }_j$ subject to ${\theta }_j^2=1,j=1,2$. The long-time asymptotics of the equation associated with ${\theta }_j=1,j=1,2$ have been studied via the nonlinear Deift–Zhou steepest descent technique in [25,26].

Following the same approach as in the previous example, we can derive another two-component SS-type integrable mKdV equation as follows:

$$\left\{\begin{array}{c}{\displaystyle r_{1,t}=-\frac{\delta }{{\gamma }^3}\{r_{1,xxx}-3{\xi }_1[2r_1^2{r}_{1,x}+r_1^{*}(|r_1{|}^2{)}_x]-3{\xi }_2[r_3{(r_1{r}_3)}_x+r_3^{*}{(r_1{r}_3^{*})}_x]\},}\hfill \\ {\displaystyle r_{3,t}=-\frac{\delta }{{\gamma }^3}\{r_{3,xxx}-3{\xi }_1[r_1{(r_1{r}_3)}_x+r_1^{*}{(r_1^{*}{r}_3)}_x]-3{\xi }_2[2r_3^2{r}_{3,x}+r_3^{*}(|r_3{|}^2{)}_x]\},}\hfill \end{array}\right.$$

(58)

with the real constants ${\xi }_j$ subject to ${\xi }_j^2=1,j=1,2$. This example is different from the previous one in (57).

Similarly, we can derive two N-component SS-type integrable mKdV equations. The equations for the three-component case are as follows:

$$\left\{\begin{array}{c}{\displaystyle r_{1,t}=-\frac{\delta }{{\gamma }^3}[r_{1,xxx}+6({\theta }_1|r_1{|}^2+{\theta }_2|r_3{|}^2+{\theta }_3|r_5{|}^2)r_{1,x}+3({\theta }_1|r_1{|}^2+{\theta }_2|r_3{|}^2+{\theta }_3|r_5{|}^2{)}_x{r}_1],}\hfill \\ {\displaystyle r_{3,t}=-\frac{\delta }{{\gamma }^3}[r_{3,xxx}+6({\theta }_1|r_1{|}^2+{\theta }_2|r_3{|}^2+{\theta }_3|r_5{|}^2)r_{3,x}+3({\theta }_1|r_1{|}^2+{\theta }_2|r_3{|}^2+{\theta }_3|r_5{|}^2{)}_x{r}_3],}\hfill \\ {\displaystyle r_{5,t}=-\frac{\delta }{{\gamma }^3}[r_{5,xxx}+6({\theta }_1|r_1{|}^2+{\theta }_2|r_3{|}^2+{\theta }_3|r_5{|}^2)r_{5,x}+3({\theta }_1|r_1{|}^2+{\theta }_2|r_3{|}^2+{\theta }_3|r_5{|}^2{)}_x{r}_5],}\hfill \end{array}\right.$$

(59)

with the real constants ${\theta }_j$ subject to ${\theta }_j^2=1,j=1,2,3$, and

$$\left\{\begin{array}{c}{\displaystyle r_{1,t}=-\frac{\delta }{{\gamma }^3}\{r_{1,xxx}-3{\xi }_1[2r_1^2{r}_{1,x}+r_1^{*}{(r_1{r}_1^{*})}_x]-3{\xi }_2[r_3{(r_1{r}_3)}_x+r_3^{*}{(r_1{r}_3^{*})}_x]}\hfill \\ {\displaystyle -3{\xi }_3[r_5{(r_1{r}_5)}_x+r_5^{*}{(r_1{r}_5^{*})}_x]\},}\hfill \\ {\displaystyle r_{3,t}=-\frac{\delta }{{\gamma }^3}\{r_{3,xxx}-3{\xi }_1[r_1{(r_1{r}_3)}_x+r_1^{*}{(r_1^{*}{r}_3)}_x]-3{\xi }_2[2r_3^2{r}_{3,x}+r_3^{*}(|r_3{|}^2{)}_x]}\hfill \\ {\displaystyle -3{\xi }_3[r_5{(r_3{r}_5)}_x+r_5^{*}{(r_3{r}_5^{*})}_x]\},}\hfill \\ {\displaystyle r_{5,t}=-\frac{\delta }{{\gamma }^3}\{r_{5,xxx}-3{\xi }_1[r_1{(r_1{r}_5)}_x+r_1^{*}{(r_1^{*}{r}_5)}_x]-3{\xi }_2[r_3{(r_3{r}_5)}_x+r_3^{*}{(r_3^{*}{r}_5)}_x]}\hfill \\ {\displaystyle -3{\xi }_3[2r_5^2{r}_{5,x}+r_5^{*}(|r_5{|}^2{)}_x\left]\right\},}\hfill \end{array}\right.$$

(60)

with the real constants ${\xi }_j$ subject to ${\xi }_j^2=1,j=1,2,3$, The equation defined by (59) with ${\theta }_j=1,j=1,2,3$, has been investigated by the Riemann–Hilbert method in [27]. Additionally, the equations defined by (60) are entirely novel.

Some of the aforementioned examples have also been addressed and solved using corresponding Riemann–Hilbert problems in [28]. Certainly, there are numerous other intriguing examples, such as those discussed in [13,29], where different choices of non-singular Hermitian matrices ${\mathsf{\Theta }}_1,{\mathsf{\Theta }}_2$ and non-singular symmetric matrices ${\mathsf{\Xi }}_1,{\mathsf{\Xi }}_2$ were taken.

3. Binary Darboux Transformations

3.1. Distribution of Eigenvalues

Observe that the local reduction in (34) (or (35)) ensures that z is an eigenvalue of the matrix eigenproblems in (17) if $\widehat{z}:=z^{*}$ (or $\widehat{z}:=-z$) is a dual eigenvalue. Specifically, it solves the adjoint counterparts of the matrix eigenproblems:

$$i{\tilde{\varphi }}_x=\mathcal{E}\tilde{\varphi }=\mathcal{E}(p,\widehat{z})\tilde{\varphi },i{\tilde{\varphi }}_t={\mathcal{F}}^{\left[k\right]}\tilde{\varphi }={\mathcal{F}}^{\left[k\right]}(p,\widehat{z})\tilde{\varphi },$$

where $k=2l+1$ with $l\ge 0$. Therefore, eigenvalues exhibit the following patterns: $z:\eta ,-{\eta }^{*},i\zeta$, and dual eigenvalues have a corresponding pattern: $z^{*}:{\eta }^{*},-\eta ,-i\zeta$ (or $-z:-\eta ,{\eta }^{*},-i\zeta$), where $\eta \notin i\mathbb{R}$ and $\zeta \in \mathbb{R}$.

3.2. Darboux Matrices

We aim to establish a general framework for Darboux matrices, where the chosen eigenvalues and their adjoints are potentially equal to each other.

Assume that $N_1,N_2\ge 0$ are two natural numbers such that $N=2N_1+N_2\ge 1$. First, we take a set of eigenvalues as follows:

$$\{z_k|1\le k\le N\}=\{{\eta }_k,-{\eta }_k^{*},1\le k\le N_1;i{\zeta }_k,1\le k\le N_2\}$$

(61)

and another set of dual eigenvalues as follows:

$$\{{\widehat{z}}_k|1\le k\le N\}=\{{\eta }_k^{*},-{\eta }_k,1\le k\le N_1;-i{\zeta }_k,1\le k\le N_2\},$$

(62)

where ${\eta }_k\notin i\mathbb{R},1\le k\le N_1$ and ${\zeta }_k\in \mathbb{R},1\le k\le N_2$. Two groups of the corresponding eigenfunctions and dual eigenfunctions are determined by

$$-i{\phi }_{k,x}=\mathcal{E}(r,s;z_k){\phi }_k,-i{\phi }_{k,t}={\mathcal{F}}^{\left[3\right]}(r,s;z_k){\phi }_k,1\le k\le N,$$

(63)

and

$$i{\widehat{\phi }}_{k,x}={\widehat{\phi }}_k\mathcal{E}(r,s;{\widehat{z}}_k),i{\widehat{\phi }}_{k,t}={\widehat{\phi }}_k{\mathcal{F}}^{\left[3\right]}(r,s;{\widehat{z}}_k),1\le k\le N.$$

(64)

For simplicity, we introduce

$$\phi =({\phi }_1,\cdots ,{\phi }_N),\widehat{\phi }={({\widehat{\phi }}_1^T,\cdots ,{\widehat{\phi }}_N^T)}^T,$$

(65)

and

$$\mathsf{\Lambda }=\mathrm{diag}(z_1,\cdots ,z_N),\widehat{\mathsf{\Lambda }}=\mathrm{diag}({\widehat{z}}_1,\cdots ,{\widehat{z}}_N).$$

(66)

Then, the equations for the eigenfunctions read

$$-i{\phi }_x=\Sigma \phi \mathsf{\Lambda }+A\phi ,i{\widehat{\phi }}_x=\widehat{\mathsf{\Lambda }}\widehat{\phi }\Sigma +\widehat{\phi }A,$$

(67)

and

$$\left\{\begin{array}{c}-i{\phi }_t=\Delta \phi {\mathsf{\Lambda }}^3+(B^{\left[3\right]}\left(z_1\right){\phi }_1,\cdots ,B^{\left[3\right]}\left(z_N\right){\phi }_N),\hfill \\ i{\widehat{\phi }}_t={\widehat{\mathsf{\Lambda }}}^3\widehat{\phi }\Delta +({\widehat{\phi }}_1{B}^{\left[3\right]}\left({\widehat{z}}_1\right),\cdots ,{\widehat{\phi }}_N{B}^{\left[3\right]}\left({\widehat{z}}_N\right)),\hfill \end{array}\right.$$

(68)

where the four square matrices $\Sigma ,\Delta ,A$, and $B^{\left[3\right]}$ are given by (19), (20), (21), and (30), respectively.

To establish a general framework for Darboux matrices, where the chosen eigenvalues and their adjoints may coincide, we introduce an $N\times N$ matrix $\mathsf{\Omega }={\left({\omega }_{kl}\right)}_{N\times N}$, whose elements are given by

$${\omega }_{kl}=\left\{\begin{array}{cc}{\displaystyle \frac{{\widehat{\phi }}_k{\phi }_l}{z_l-{\widehat{z}}_k},}& \mathrm{if}{z}_l\ne {\widehat{z}}_k,\hfill \\ {\omega }_{kl}^c(t,x),& \mathrm{if}{z}_l={\widehat{z}}_k,\hfill \end{array}\right.\mathrm{where}1\le k,l\le N.$$

(69)

Here, the matrix $\mathsf{\Omega }$ includes a novel type of elements ${\omega }_{kl}^c$ in the case of $z_l={\widehat{z}}_k$ for a pair $1\le k,l\le N$, which will be specified later. This generalization extends beyond the traditional cases found in the literature (see, e.g., [3,30,31]). Such novel matrices $\mathsf{\Omega }$ arise particularly in the formulation of soliton solutions for nonlocal integrable equations.

If $\mathsf{\Omega }$ is non-singular, let us define the following two Darboux matrices:

$$\left\{\begin{array}{c}{\displaystyle D^{+}=D^{+}\left(z\right)=I_{m+n}-\sum _{k,l=1}^N\frac{{\phi }_k{\left({\mathsf{\Omega }}^{-1}\right)}_{kl}{\widehat{\phi }}_l}{z-{\widehat{z}}_l},}\hfill \\ {\displaystyle D^{-}=D^{-}\left(z\right)=I_{m+n}+\sum _{k,l=1}^N\frac{{\phi }_k{\left({\mathsf{\Omega }}^{-1}\right)}_{kl}{\widehat{\phi }}_l}{z-z_k}.}\hfill \end{array}\right.$$

(70)

Through partial fractional decomposition, these two Darboux matrices can be expressed concisely. It is important to note that the partial fractional decomposition yields

$${\displaystyle D^{+}=I_{m+n}-\sum _{l=1}^N\frac{{\phi }_l^{\mathsf{\Omega }}{\widehat{\phi }}_l}{z-{\widehat{z}}_l},D^{-}=I_{m+n}+\sum _{k=1}^N\frac{{\phi }_k{\widehat{\phi }}_k^{\mathsf{\Omega }}}{z-z_k},}$$

(71)

where one assumes

$${\displaystyle ({\phi }_1^{\mathsf{\Omega }},\cdots ,{\phi }_N^{\mathsf{\Omega }})=({\phi }_1,\cdots ,{\phi }_N){\mathsf{\Omega }}^{-1},}$$

(72)

and

$${\displaystyle {({\left({\widehat{\phi }}_1^{\mathsf{\Omega }}\right)}^T,\cdots ,{\left({\widehat{\phi }}_N^{\mathsf{\Omega }}\right)}^T)}^T={\mathsf{\Omega }}^{-1}{({\widehat{\phi }}_1^T,\cdots ,{\widehat{\phi }}_N^T)}^T.}$$

(73)

Therefore, the two Darboux matrices can be expressed concisely as follows:

$$D^{+}=I_{m+n}-\phi {\mathsf{\Omega }}^{-1}\widehat{R}\widehat{\phi },D^{-}=I_{m+n}+\phi R{\mathsf{\Omega }}^{-1}\widehat{\phi },$$

(74)

where we denote

$${\displaystyle R=\left[\begin{array}{ccc}\frac{1}{z-z_1}& & 0\\ & \ddots & \\ 0& & \frac{1}{z-z_N}\end{array}\right],\widehat{R}=\left[\begin{array}{ccc}\frac{1}{z-{\widehat{z}}_1}& & 0\\ & \ddots & \\ 0& & \frac{1}{z-{\widehat{z}}_N}\end{array}\right].}$$

(75)

Now, we introduce

$$D_1^{\pm }=\underset{z\to \infty }{lim}\left[z(D^{\pm }\left(z\right)-I_{m+n})\right].$$

(76)

It then follows that

$$D_1^{+}=-\phi {\mathsf{\Omega }}^{-1}\widehat{\phi },D_1^{-}=\phi {\mathsf{\Omega }}^{-1}\widehat{\phi },$$

(77)

which implies that

$$D_1^{+}=-D_1^{-}.$$

Finally, we can derive two fundamental properties for the resulting Darboux matrices $D^{+}$ and $D^{-}$ as follows:

(i)

A spectral characteristic identity

$$\left[\prod _{l=1}^N(z-{\widehat{z}}_l)D^{+}\left(z\right)\right]{|}_{z=z_k}{\phi }_k=0,{\widehat{\phi }}_k\left[\prod _{l=1}^N(z-z_l)D^{-}\left(z\right)\right]{|}_{z={\widehat{z}}_k}=0,1\le k\le N,$$

(78)

is satisfied.

(ii)

When an orthogonality

$${\widehat{\phi }}_k{\phi }_l=0\mathrm{when}{z}_l={\widehat{z}}_k,\mathrm{where}1\le k,l\le N,$$

(79)

holds, we achieve $\widehat{R}\widehat{\phi }\phi R=\mathsf{\Omega }R-\widehat{R}\mathsf{\Omega },$ and so, $D^{+}$ and $D^{-}$ solve

$$D^{+}\left(z\right)D^{-}\left(z\right)=I_{m+n}.$$

(80)

This implies that when all eigenvalues $z_k$, $1\le k\le N$, are situated in the upper right quadrant of the complex plane, $D^{+}$ and $D^{-}$ yield a solution to a reflectionless Riemann–Hilbert problem.

3.3. Binary Darboux Transformations

To construct binary DTs, it is essential to examine the dependence of the $\mathsf{\Omega }$-matrix on t and x. Note that the condition

$${\omega }_{kl,x}^c=i{\widehat{\phi }}_k{\displaystyle \frac{\mathcal{E}(r,s;z_l)-\mathcal{E}(r,s;{\widehat{z}}_k)}{z_l-{\widehat{z}}_k}}{\phi }_l=i{\widehat{\phi }}_k\Sigma {\phi }_l\mathrm{when}{z}_l={\widehat{z}}_k,$$

(81)

with $1\le k,l\le N,$ guarantees the following spatial derivative formula:

$${\mathsf{\Omega }}_x=i\widehat{\phi }\Sigma \phi ;$$

(82)

and, moreover, the condition

$${\omega }_{kl,t}^c=i{\widehat{\phi }}_k{\displaystyle \frac{{\mathcal{F}}^{\left[3\right]}(r,s;z_l)-{\mathcal{F}}^{\left[3\right]}(r,s;{\widehat{z}}_k)}{z_l-{\widehat{z}}_k}}{\phi }_l=i{\widehat{\phi }}_k{\Delta }_{[k,l]}{\phi }_l\mathrm{when}{z}_l={\widehat{z}}_k,$$

(83)

with $1\le k,l\le N,$ and ${\Delta }_{[k,l]}$ being worked out as follows:

$${\Delta }_{[k,l]}=({\widehat{z}}_k^2+{\widehat{z}}_k{z}_l+z_l^2)\Delta +{\displaystyle \frac{\delta }{\gamma }}({\widehat{z}}_k+z_l)A-{\displaystyle \frac{\delta }{{\gamma }^2}}{I}_{m,n}(A^2+i{A}_x),1\le k,l\le N,$$

(84)

which guarantees the following temporal derivative formula:

$${\mathsf{\Omega }}_t=i[\widehat{\phi }{\widehat{\mathsf{\Lambda }}}^2\Delta \phi +\widehat{\phi }\widehat{\mathsf{\Lambda }}\Delta \mathsf{\Lambda }\phi +\widehat{\phi }\Delta {\mathsf{\Lambda }}^2\phi +{\displaystyle \frac{\delta }{\gamma }}(\widehat{\phi }\widehat{\mathsf{\Lambda }}A\phi +\widehat{\phi }A\mathsf{\Lambda }\phi )-{\displaystyle \frac{\delta }{{\gamma }^2}}\widehat{\phi }{I}_{m,n}(A^2+i{A}_x)\phi ].$$

(85)

Given the defined pattern of eigenvalues and dual eigenvalues in (61) and (62), let us consider

$${\widehat{\phi }}_k={\phi }_k^{†}\mathsf{\Theta }={\phi }_{N_1+k}^T\mathsf{\Xi },{\widehat{\phi }}_{N_1+k}={\phi }_{N_1+k}^{†}\mathsf{\Theta }={\phi }_k^T\mathsf{\Xi },1\le k\le N_1,$$

(86)

and

$${\widehat{\phi }}_k={\phi }_k^{†}\mathsf{\Theta }={\phi }_k^T\mathsf{\Xi },2N_1+1\le k\le N.$$

(87)

In this way, it becomes evident that $D_1^{+}$ satisfies the required involution properties as follows:

$${\left(D_1^{+}(t,x)\right)}^{†}=-\mathsf{\Theta }{D}_1^{+}(t,x){\mathsf{\Theta }}^{-1},{\left(D_1^{+}(t,x)\right)}^T=\mathsf{\Xi }{D}_1^{+}(t,x){\mathsf{\Xi }}^{-1},$$

(88)

where $\mathsf{\Theta }$ and $\mathsf{\Xi }$ are defined by (36) and (37), and $A^{\prime }=[D_1^{+},\Sigma ]$ satisfies the two group constraints in (34) and (35).

All of the analyses above enable us to formulate a comprehensive framework for binary DTs as outlined below.

Theorem 1.

Let Σ, Δ, and ${\Delta }_{[k,l]}$ be given by (19), (20), and (84), and let the dual eigenvalues $\{{\widehat{z}}_k|1\le k\le N\}$ be chosen as in (62) and the corresponding dual eigenfunctions $\{{\widehat{\phi }}_k|1\le k\le N\}$ be specified through (86) and (87). Assume that $D^{\pm }$ and $D_1^{\pm }$ are presented by (70) and (76). Then, the conditions in (79), (81), and (83) guarantee a binary DT as follows:

$${\varphi }^{\prime }=D^{+}\varphi ,{\tilde{\varphi }}^{\prime }=\tilde{\varphi }{D}^{-},$$

(89)

with a Bäcklund transformation

$$A^{\prime }=A+[D_1^{+},\Sigma ],$$

(90)

for the SS-type integrable matrix mKdV equation (46).

We note that if we choose

$${\omega }_{kl}^c=0\mathrm{when}{z}_l={\widehat{z}}_k,$$

(91)

the three conditions in (79), (81), and (83) can be satisfied by requiring

$${\phi }_k^{†}\mathsf{\Theta }{\phi }_l={\phi }_k^{†}\mathsf{\Theta }\Sigma {\phi }_l={\phi }_k^{†}\mathsf{\Theta }{\Delta }_{[k,l]}{\phi }_l=0\mathrm{when}{z}_l={\widehat{z}}_k,$$

(92)

with $1\le k,l\le N.$ By utilizing (86) and (87), those orthogonal conditions can be expressed equivalently as

$${\phi }_{N_1+k}^T\mathsf{\Xi }{\phi }_l={\phi }_{N_1+k}^T\mathsf{\Xi }\Sigma {\phi }_l={\phi }_{N_1+k}^T\mathsf{\Xi }{\Delta }_{[k,l]}{\phi }_l=0,\mathrm{when}{z}_l={\widehat{z}}_k,1\le k\le N_1,$$

(93)

$${\phi }_{k-N_1}^T\mathsf{\Xi }{\phi }_l={\phi }_{k-N_1}^T\mathsf{\Xi }\Sigma {\phi }_l={\phi }_{k-N_1}^T\mathsf{\Xi }{\Delta }_{[k,l]}{\phi }_l=0,\mathrm{when}{z}_l={\widehat{z}}_k,N_1+1\le k\le 2N_1,$$

(94)

and

$${\phi }_k^T\mathsf{\Xi }{\phi }_l={\phi }_k^T\mathsf{\Xi }\Sigma {\phi }_l={\phi }_k^T\mathsf{\Xi }{\Delta }_{[k,l]}{\phi }_l=0,\mathrm{when}{z}_l={\widehat{z}}_k,2N_1+1\le k\le N,$$

(95)

with $1\le l\le N.$

3.4. N-fold Decomposition Feature

Next, we would like to demonstrate an N-fold decomposition property for the established binary DT in the traditional case, focusing from the fact that the intersection of $\{z_k|1\le k\le N\}$ and $\{{\widehat{z}}_k|1\le k\le N\}$ is empty.

To proceed, let us establish two groups of novel binary Darboux matrices using a single pair of eigenvalues and dual eigenvalues recursively as follows:

$$\left\{\begin{array}{c}{\displaystyle D^{+}\left\{k\right\}=D^{+}\left\{k\right\}\left(z\right)=I_{m+n}-\frac{z_k-{\widehat{z}}_k}{z-{\widehat{z}}_k}\frac{{\phi }_k^{\prime }{\widehat{\phi }}_k^{\prime }}{{\widehat{\phi }}_k^{\prime }{\phi }_k^{\prime }},1\le k\le N,}\hfill \\ {\displaystyle D^{-}\left\{k\right\}=D^{-}\left\{k\right\}\left(z\right)=I_{m+n}+\frac{z_k-{\widehat{z}}_k}{z-z_k}\frac{{\phi }_k^{\prime }{\widehat{\phi }}_k^{\prime }}{{\widehat{\phi }}_k^{\prime }{\phi }_k^{\prime }},1\le k\le N.}\hfill \end{array}\right.$$

(96)

Here, N pairs of new eigenfunctions and dual eigenfunctions are given by

$${\phi }_k^{\prime }=D^{+}\left[[k-1]\right]\left(z_k\right){\phi }_k,{\widehat{\phi }}_k^{\prime }={\widehat{\phi }}_k{D}^{-}\left[[k-1]\right]\left({\widehat{z}}_k\right),1\le k\le N,$$

(97)

with

$$\left\{\begin{array}{c}{D}^{+}\left[\left[0\right]\right]=D^{-}\left[\left[0\right]\right]=I_{m+n},\hfill \\ D^{+}\left[\left[k\right]\right]=D^{+}\left\{k\right\}\cdots D^{+}\left\{2\right\}{D}^{+}\left\{1\right\},1\le k\le N,\hfill \\ D^{-}\left[\left[k\right]\right]=D^{-}\left\{1\right\}{D}^{-}\left\{2\right\}\cdots D^{-}\left\{k\right\},1\le k\le N.\hfill \end{array}\right.$$

(98)

At this moment, a straightforward computation can confirm the following N-fold decomposition:

$$D^{+}=D^{+}\left\{N\right\}{D}^{+}\{N-1\}\cdots D^{+}\left\{1\right\},D^{-}=D^{-}\left\{1\right\}\cdots D^{-}\{N-1\}{D}^{-}\left\{N\right\},$$

(99)

where $D^{+}\left\{k\right\}$ and $D^{-}\left\{k\right\}$, $1\le k\le N$, are defined by (96).

4. Soliton Solutions

We focus on the dual eigenvalues $\{{\widehat{z}}_k|1\le k\le N\}$ defined previously as in (62). By considering the zero seed solution $r=0$, which implies the zero potential matrix $A=0$ due to the two group constraints, we can readily determine the corresponding eigenfunctions and dual eigenfunctions

$$\begin{array}{ccc}& & {\phi }_k(t,x)={\mathrm{e}}^{i{z}_k\Sigma x+i{z}_k^3\Delta t}{\chi }_k,1\le k\le N,\hfill \end{array}$$

(100)

$$\begin{array}{ccc}& & {\widehat{\phi }}_k(t,x)={\chi }_k^{†}{\mathrm{e}}^{-i{\widehat{z}}_k\Sigma x-i{\widehat{z}}_k^3\Delta t}\mathsf{\Theta },1\le k\le N,\hfill \end{array}$$

(101)

where ${\chi }_k$, $1\le k\le N$, denote constant column vectors. Those choices in (86) and (87) result in the conditions on the constant vectors ${\chi }_k,1\le k\le N$ as follows:

$$\left\{\begin{array}{c}{\displaystyle {\chi }_k^T(\mathsf{\Xi }{\mathsf{\Theta }}^{-1}-{\mathsf{\Theta }}^{*}{\mathsf{\Xi }}^{*-1})=0,1\le k\le N_1,}\hfill \\ {\displaystyle {\chi }_k={\mathsf{\Xi }}^{-1}{\mathsf{\Theta }}^{*}{\chi }_{k-N_1}^{*},N_1+1\le k\le 2N_1,}\hfill \\ {\displaystyle {\chi }_k^{†}\mathsf{\Theta }={\chi }_k^T\mathsf{\Xi },2N_1+1\le k\le N,}\hfill \end{array}\right.$$

(102)

where * stands for the complex conjugate of a matrix. t is important to note that the purpose of all these conditions is to satisfy the group constraints in (38) and (39).

The three orthogonal conditions in (92) can be expressed as follows:

$${\chi }_k^{†}\mathsf{\Theta }{\chi }_l={\chi }_k^{†}\mathsf{\Theta }\Sigma {\chi }_l=({\widehat{z}}_k^2+{\widehat{z}}_k{z}_l+z_l^2){\chi }_k^{†}\mathsf{\Theta }\Delta {\chi }_l=0\mathrm{when}{z}_l={\widehat{z}}_k,$$

(103)

with $1\le k,l\le N,$ and $\Sigma$, $\Delta$, and $\mathsf{\Theta }$ being defined by (19), (20), and (36), respectively.

It is worth noting that the case where $z_k={\widehat{z}}_k$ occurs only when $z_k=0$ for $2N_1+1\le k\le N$. Given that ${\gamma }_1$ and ${\delta }_1$ are different from ${\gamma }_2$ and ${\delta }_2$, respectively, the three conditions outlined in (103) equivalently yield

$${\left({\chi }_k^1\right)}^{†}{\mathsf{\Theta }}_1{\chi }_l^1=0,{\left({\chi }_k^2\right)}^{†}{\mathsf{\Theta }}_2{\chi }_l^2=0\mathrm{when}{z}_l={\widehat{z}}_k,\mathrm{where}1\le k,l\le N,$$

(104)

where we assume ${\chi }_k={({\left({\chi }_k^1\right)}^T,{\left({\chi }_k^2\right)}^T)}^T$. Here, ${\chi }_k^1$ and ${\chi }_k^2$ denote column vectors of dimensions m and n, respectively. These conditions ensure the orthogonality restrictions, which can also be represented using the non-singular matrix $\mathsf{\Xi }$.

Now, following the binary DT theory presented in the previous theorem, we define a new potential matrix as follows:

$$A^{\prime }=[D_1^{+},\Sigma ],D_1^{+}=-\phi {\mathsf{\Omega }}^{-1}\widehat{\phi }=-\sum _{k,l=1}^N{\phi }_k{\left({\mathsf{\Omega }}^{-1}\right)}_{kl}{\widehat{\phi }}_l.$$

(105)

This formula leads to a class of soliton solutions for the following SS-type integrable matrix mKdV Equation (46):

$${\displaystyle r=\gamma \sum _{k,l=1}^N{\phi }_k^1{\left({\mathsf{\Omega }}^{-1}\right)}_{kl}{\widehat{\phi }}_l^2,}$$

(106)

where we split ${\phi }_k={({\left({\phi }_k^1\right)}^T,{\left({\phi }_k^2\right)}^T)}^T$, as we did for ${\chi }_k$ before, and ${\widehat{\phi }}_k=({\widehat{\phi }}_k^1,{\widehat{\phi }}_k^2)$. Here, ${\widehat{\phi }}_k^1$ and ${\widehat{\phi }}_k^2$ denote row vectors of dimensions m and n, respectively.

Finally, we conclude that by selecting ${\chi }_k$ as in (102) and satisfying the conditions in (104), Formula (106), along with (69), (100), and (101), yields a matrix potential r satisfying (42). Consequently, these provide the required soliton solutions for the SS-type integrable matrix mKdV equations (46). Such soliton solutions complement those obtained in [32].

5. Concluding Remarks

The paper explores Sasa–Satsuma (SS)-type integrable matrix mKdV equations through two local group constraints applied to the matrix AKNS eigenvalue problem of general order. It establishes a general framework for binary Darboux transformations (DTs) applicable to the derived SS-type integrable matrix mKdV equations, leveraging the associated Lax pair and dual Lax pair of matrix eigenproblems. These binary DTs are then employed to formuate soliton solutions for the SS-type integrable matrix mKdV equations, expanding upon the binary DT theory developed for reduced integrable mKdV equations [32,33].

The crucial aspect of our analysis involves applying both local group constraints simultaneously to derive reduced integrable equations, which forms the foundation for the SS mKdV equation. In constructing binary DTs, we utilize a generalized $\mathsf{\Omega }$-matrix where adopted eigenvalues and dual eigenvalues can coincide. This extension of the $\mathsf{\Omega }$-matrix is inspired by a comprehensive exploration of Riemann–Hilbert problems in the context of nonlocal integrable equations. The framework for binary DTs presented here is applicable to both local and nonlocal integrable equations (see, for example, [34,35,36,37] for nonlocal cases). It is noteworthy that Darboux matrices involving higher order singularities can be generated by introducing repeated eigenvalues or dual eigenvalues, while generalized DTs can be constructed by differentiating with respect to eigenvalues or dual eigenvalues.

We emphasize that the discussed group symmetric reductions for matrix AKNS eigenproblems, involving transformations $z\to z^{*}$ and $z\to -z$, constitute two fundamental classes producing reduced local integrable equations. Exploring the simultaneous adoption of these two reductions for other matrix eigenproblems could yield diverse forms of reduced local integrable equations, which presents an intriguing avenue for future research. In the realm of DTs, there are numerous intriguing challenges. For instance, how can DTs be effectively employed to generate additional types of exact and explicit solutions such as for instance, breather and rogue wave solutions and lump wave solutions? Another important question is the formulation of binary DTs for reduced integrable couplings linked to non-semisimple Lie algebras. Furthermore, it is crucial to explore the connections between binary DT theories and other robust solution techniques, including the inverse scattering approach, the Hirota direct method, and the Riemann–Hilbert technique (see, e.g., [38,39,40]). Understanding these connections could lead to deeper insights and broader applications in the field of integrability research, particularly on multi-compoenent integrable models (see, e.g., [41,42]).