Abstract We present a matrix coupled dispersionless (CD) system. A Lax pair for the matrix CD system is proposed and Darboux transformation is constructed on the solutions of the matrix CD system and the associated Lax pair. We express an N soliton formula for the solutions of the matrix CD system in terms of quasideterminants. By using properties of the quasideterminants, we obtain some exact solutions, including bright and dark-type solitons, rogue wave and breather solutions of the matrix CD system. Furthermore, it has been shown that the solutions of the matrix CD system are expressed in terms of solutions to the usual CD system, sine-Gordon equation and Maxwell–Bloch system. Keywords：Matrix coupled dispersionless system;Darboux transformation;solitons
PDF (4387KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article H Wajahat A Riaz. Darboux transformation and exact multisolitons for a matrix coupled dispersionless system. Communications in Theoretical Physics, 2020, 72(7): 075001- doi:10.1088/1572-9494/ab8a16
1. Introduction
Soliton equations play an important role in various fields of science, such as nonlinear optics, fluid dynamics, magnetics and others [1, 2]. Such equations admit various localized solutions, such as solitons, rogue wave and breather solutions. Unlike a rogue wave, which is spatiotemporally localized, solitons and breathers are partially localized. In addition, a soliton is a localized wave that occurs in a medium due to the balance between nonlinearity and dispersion. This nice balance causes the pulse to propagate with constant speed and undeformed shape on both vanishing and non-vanishing backgrounds [3]. On the other hand, a breather is a localized wave with periodicity in a certain direction, i.e. time-periodic breather (also known as a Kuznetsov–Ma breather) [4–7], spatial periodic (Akhmeidiev breather) [5–7] and spatiotemporally periodic (general breather) [7, 8]. Rogue waves, also known as extreme and freak waves [9], are unexpectedly high-amplitude single waves that appear from nowhere and disappear without a trace [10]. Rogue waves are known to appear both in coastal areas and the open ocean [11]. In various nonlinear phenomena breather appears along with soliton or rogue wave. Therefore, it is significant to investigate such types of localized solutions, i.e. soliton, breather and rogue waves.
For the most part, rogue waves have been observed and have become one of the most attractive topics for researchers. Mathematically, soliton equations have been studied by using various mathematical methods. Methods to derive the soliton solutions have been propose: e.g. the inverse scattering method [12], Painlevé analysis technique [12], Bäcklund transformation [13], algebraic geometry method [14], Darboux transformation (DT) [15] and Hirota's bilinear method [16]. Among the list, DT is one of the methods used to construct various exact solutions, including bright and dark soliton solutions, periodic solutions, breather solutions and rogue wave solutions [17–22].
Dispersionless integrable systems have attracted a great deal of interest due to their emergence in various areas of theoretical physics [23–27]. Many of these integrable systems arise as semi-classical limits of ordinary integrable systems with a dispersion term. In the semi-classical limit the dispersion term of an ordinary integrable system is eliminated resulting in a dispersionless integrable system with no dispersion term. The coupled dispersionless (CD) integrable system is an important example of an integrable nonlinear dynamical system which has various applications in diverse areas, including theoretical physics, mathematics and engineering sciences [28–33].
In this work, we consider the matrix CD system given by$ \begin{eqnarray}\begin{array}{rcl} & & {\partial }_{t}{q}_{+}+{\partial }_{x}(| {r}_{+}{| }^{2}+| {r}_{0}{| }^{2})=0,\\ & & {\partial }_{t}{q}_{0}+{\partial }_{x}({r}_{+}{r}_{0}^{* }+{r}_{0}{r}_{-}^{* })=0,\\ & & {\partial }_{t}{q}_{-}+{\partial }_{x}(| {r}_{-}{| }^{2}+| {r}_{0}{| }^{2})=0,\\ & & {\partial }_{x}{\partial }_{t}{r}_{+}-2({q}_{+}{r}_{+}+{q}_{0}{r}_{0})=0,\\ & & {\partial }_{x}{\partial }_{t}{r}_{0}-({q}_{+}{r}_{0}+{q}_{0}{r}_{-}+{r}_{+}{q}_{0}^{* }+{r}_{0}{q}_{-})=0,\\ & & {\partial }_{x}{\partial }_{t}{r}_{-}-2({q}_{-}{r}_{-}+{q}_{0}^{* }{r}_{0})=0.\end{array}\end{eqnarray}$The system (1.1) can be regarded as a matrix CD system as the solutions of the above-mentioned system can be expressed in terms of solutions to the well-studied equations.
(I) For ${r}_{+}={r}_{0}={r}_{-}=r/2$ and ${q}_{+}={q}_{0}={q}_{-}={q}_{x}/2$, one obtains$ \begin{eqnarray}{\partial }_{x}{\partial }_{t}q+{\partial }_{x}{{rr}}^{* }+r{\partial }_{x}{r}^{* }=0,\,{\partial }_{x}{\partial }_{t}r-2{q}_{x}r=0,\end{eqnarray}$with the conserved quantity ${q}_{x}^{2}+| {r}_{x}{| }^{2}=c$, where c is constant. Equation (1.2) is the CD system originally introduced in [30]. If r is real, then the system (1.2) reads$ \begin{eqnarray}{\partial }_{x}{\partial }_{t}q+2r{\partial }_{x}r=0,\,{\partial }_{x}{\partial }_{t}r-2{q}_{x}r=0,\end{eqnarray}$with the conserved quantity ${q}_{x}^{2}+{r}_{x}^{2}=c$.
(II) Further substitution ${q}_{x}=\cos \varphi ,\,r=\tfrac{1}{2}{\partial }_{t}\varphi $ in (1.3) brings us to the sine-Gordon equation$ \begin{eqnarray}{\partial }_{x}{\partial }_{t}\varphi =2\sin \varphi .\end{eqnarray}$
(III) Furthermore, substitution ${r}_{+}={r}_{-}={r}_{0}\,=\tfrac{1}{4}E,{r}_{+x}\,={r}_{-,x}={r}_{0,x}=\tfrac{1}{4}P$ and ${q}_{-}={q}_{+}={q}_{0}=\tfrac{1}{4}N$ in (1.1) gives$ \begin{eqnarray}\begin{array}{rcl}{E}_{x} & = & P,\,{\partial }_{t}N=-\displaystyle \frac{1}{2}({{EP}}^{* }+{{PE}}^{* }),\\ {\partial }_{t}P & = & {NE},\,{N}^{2}+P\bar{P}=1,\end{array}\end{eqnarray}$which coincided with the Maxwell–Bloch system [34].
As we reduced the matrix CD system (1.1) into the usual CD system (1.2)–(1.3), sine-Gordon equation (1.4) and Maxwell–Bloch system (1.5), therefore, the matrix CD system can be regarded as a generalization of these equations. The Darboux/Bäcklund transformations for these equations have been studied in various references, such as [19–34]. The present work deals with the matrix CD system, DT and some exact solutions. We introduce a Lax pair for the matrix CD system and derive the DT on the solutions of the matrix CD system and the associated Lax pair. The solutions are expressed in terms of quasideterminants. Further, we show that the matrix CD system exhibits various exact solutions, such as bright and dark soliton, rogue wave and breather solutions.
2. Lax pair and DT
The Lax pair for the matrix CD system is given by$ \begin{eqnarray}{\partial }_{x}\psi ={\lambda }^{-1}A\psi ,\end{eqnarray}$$ \begin{eqnarray}{\partial }_{t}\psi =(\lambda J+B)\psi ,\end{eqnarray}$where$ \begin{eqnarray}\begin{array}{rcl}J & = & \left(\begin{array}{cc}\tfrac{{\rm{i}}}{2}{I}_{2\times 2} & {0}_{2\times 2}\\ {0}_{2\times 2} & -\tfrac{{\rm{i}}}{2}{I}_{2\times 2}\end{array}\right),\,A=-{\rm{i}}\left(\begin{array}{cc}Q & {\partial }_{x}R\\ -{\partial }_{x}{R}^{\dagger } & -{Q}^{* }\end{array}\right),\\ B & = & \left(\begin{array}{cc}{0}_{2\times 2} & -R\\ {R}^{\dagger } & {0}_{2\times 2}\end{array}\right),\end{array}\end{eqnarray}$where $J,\,A$ and B are the matrices of the order 4×4, $\psi =\psi (x,\,t,\,\lambda )$ is also an eigenfunction matrix of the order 4×4 , and λ is the spectral parameter (real or complex). The symbol † in the superscript of R stands for Hermitian conjugation, while * in the superscript of Q denotes complex conjugation. The compatibility condition ${\partial }_{x}{\partial }_{t}\psi ={\partial }_{t}{\partial }_{x}\psi $ yields the zero curvature condition given by$ \begin{eqnarray}{\partial }_{t}Q+{\partial }_{x}({{RR}}^{\dagger })=0,\quad {\partial }_{x}{\partial }_{t}R-{QR}-{{RQ}}^{* }=0,\end{eqnarray}$which coincides with the system (1.1) if the matrices Q and R take the form$ \begin{eqnarray}Q=\left(\begin{array}{cc}{q}_{+} & {q}_{0}\\ {q}_{0} & {q}_{-}\end{array}\right)\quad R=\left(\begin{array}{cc}{r}_{+} & {r}_{0}\\ {r}_{0} & {r}_{-}\end{array}\right).\end{eqnarray}$Note that in (2.5) ${q}_{+}={q}_{+}^{* }$ and ${q}_{-}={q}_{-}^{* }$.
In what follows, we apply DT on the matrix CD system and obtain multisoliton solutions within the framework of quasideterminants. We define a Darboux matrix $D=D(x,\,t;\,\lambda )$ of size 4×4 that brings the trivial solution ψ of the matrix CD system into the non-trivial solution ${\psi }^{\left[1\right]}$. The new solution ${\psi }^{\left[1\right]}$ reads$ \begin{eqnarray}{\psi }^{\left[1\right]}=D\psi ,\end{eqnarray}$which satisfies the same Lax pair (2.1) and (2.2), but with the new solutions$ \begin{eqnarray}{\partial }_{x}{\psi }^{\left[1\right]}={\lambda }^{-1}{A}^{\left[1\right]}{\psi }^{\left[1\right]},\end{eqnarray}$$ \begin{eqnarray}{\partial }_{t}{\psi }^{\left[1\right]}=(\lambda {J}^{\left[1\right]}+{B}^{\left[1\right]}){\psi }^{\left[1\right]}.\end{eqnarray}$Now, we wish to find the DT on the solutions ${J}^{\left[1\right]},\,{A}^{\left[1\right]}$ and ${B}^{\left[1\right]}$. For this, we make the following ansatz for the Darboux matrix D:$ \begin{eqnarray}D=\lambda I-M,\end{eqnarray}$where M is an auxiliary matrix of size 4×4 to be determined, I is an identity matrix of size 4×4 and λ is the spectral parameter. Since D in equation (2.9) is in the first order of the polynomial of λ, it can be referred to as a one-fold Darboux matrix. It should be noted that the matrix M can be constructed from the particular matrix solutions to the Lax pair (2.1) and (2.2). For this, we define an auxiliary matrix M: $M={\rm{\Xi }}{ \mathcal L }{{\rm{\Xi }}}^{-1}$, where Ξ is the matrix solution of the Lax pair (2.1) and (2.2) at a particular eigenvalue matrix ${ \mathcal L }=\mathrm{diag}\left({\lambda }_{1},...,\,{\lambda }_{k}\right)$. The matrix Ξ can be defined as$ \begin{eqnarray}{\rm{\Xi }}\equiv \left(\psi ({\lambda }_{1})\left|1\right\rangle ,...,\,\psi ({\lambda }_{k})\left|k\right\rangle \right)=\left({{\boldsymbol{\Xi }}}_{1},...,\,{{\boldsymbol{\Xi }}}_{k}\right),\end{eqnarray}$where ${{\boldsymbol{\Xi }}}_{l},\,l=1,...,\,k$ are the particular eigenvector solutions to the Lax pair equations (2.1) and (2.2). The matrix Ξ in equation (2.10) satisfies the same Lax pair equations but with the eigenvalue matrix ${ \mathcal L }$:$ \begin{eqnarray}{\partial }_{x}{\rm{\Xi }}=A{\rm{\Xi }}{{ \mathcal L }}^{-1},\quad {\partial }_{t}{\rm{\Xi }}=J{\rm{\Xi }}{ \mathcal L }+B{\rm{\Xi }}.\end{eqnarray}$For $M={\rm{\Xi }}{ \mathcal L }{{\rm{\Xi }}}^{-1}$, we have the following propositions.
Equation (2.7) remains invariant under the DT (2.6), provided the new solution ${A}^{\left[1\right]}$:$ \begin{eqnarray}{A}^{\left[1\right]}=A-{\partial }_{x}M,\end{eqnarray}$with the covariance condition$ \begin{eqnarray}{\partial }_{x}M=\left[A,\,M\right]{M}^{-1}.\end{eqnarray}$
The relation between the new potential ${A}^{\left[1\right]}$ and the old potential A is built and given in (2.12). Now, we need to show that the matrix $M={\rm{\Xi }}{ \mathcal L }{{\rm{\Xi }}}^{-1}$ satisfies the condition (2.13). For this, we have$ \begin{eqnarray}\begin{array}{rcl}{\partial }_{x}M & = & {\partial }_{x}({\rm{\Xi }}{ \mathcal L }{{\rm{\Xi }}}^{-1})=({\partial }_{x}{\rm{\Xi }}){ \mathcal L }{{\rm{\Xi }}}^{-1}-{\rm{\Xi }}{ \mathcal L }{{\rm{\Xi }}}^{-1}({\partial }_{x}{\rm{\Xi }}){{\rm{\Xi }}}^{-1},\\ & = & (A{\rm{\Xi }}{{ \mathcal L }}^{-1}){ \mathcal L }{{\rm{\Xi }}}^{-1}-{\rm{\Xi }}{ \mathcal L }{{\rm{\Xi }}}^{-1}(A{\rm{\Xi }}{{ \mathcal L }}^{-1}){{\rm{\Xi }}}^{-1},\\ & = & A-{{MAM}}^{-1}=[A,\,M]{M}^{-1},\end{array}\end{eqnarray}$which is (2.13). This completes the proof.
Equation (2.8) remains invariant under the DT (2.6), provided the new solutions ${B}^{\left[1\right]}$ and ${J}^{\left[1\right]}$:$ \begin{eqnarray}{B}^{\left[1\right]}=B+[J,\,M],\quad {J}^{\left[1\right]}=J,\end{eqnarray}$with the covariance condition$ \begin{eqnarray}{\partial }_{t}M=[B,\,M]+[J,\,M]M.\end{eqnarray}$
The relation between the new potential ${B}^{\left[1\right]}$ and the old potential B is built and given in (2.15). Now, we need to show that the matrix $M={\rm{\Xi }}{ \mathcal L }{{\rm{\Xi }}}^{-1}$ satisfies the condition (2.16). For this, we take$ \begin{eqnarray}\begin{array}{rcl}{\partial }_{t}M & = & {\partial }_{t}({\rm{\Xi }}{ \mathcal L }{{\rm{\Xi }}}^{-1})=({\partial }_{t}{\rm{\Xi }}){ \mathcal L }{{\rm{\Xi }}}^{-1}-{\rm{\Xi }}{ \mathcal L }{{\rm{\Xi }}}^{-1}({\partial }_{t}{\rm{\Xi }}){{\rm{\Xi }}}^{-1},\\ & = & (J{\rm{\Xi }}{ \mathcal L }+B{\rm{\Xi }}){ \mathcal L }{{\rm{\Xi }}}^{-1}-{\rm{\Xi }}{ \mathcal L }{{\rm{\Xi }}}^{-1}(J{\rm{\Xi }}{ \mathcal L }+B{\rm{\Xi }}){{\rm{\Xi }}}^{-1},\\ & = & [J,\,M]M+[B,\,M],\end{array}\end{eqnarray}$which is (2.16). This completes the proof.
Now, we iterate DT (2.6) N-times and present the N-fold solutions.
Suppose that ${{\rm{\Xi }}}_{1},\,{{\rm{\Xi }}}_{2},...,\,{{\rm{\Xi }}}_{N}$ are particular matrix solutions of the Lax pair (2.1)–(2.2) evaluated at ${{ \mathcal L }}_{1},\,{{ \mathcal L }}_{2},....,\,{{ \mathcal L }}_{N}$, respectively. Then, $N$-fold iteration of DT in terms of quasideterminants1(1 In this paper we will consider only quasideterminants that are expanded about an $m\times m$ matrix. The quasideterminant of the $J\times J$ matrix expanded about the $m\times m$ matrix $D$ is defined as$ \begin{eqnarray}\left|\begin{array}{l}A\quad B\\ C\quad \boxed{D}\end{array}\right|=D-{{CA}}^{-1}B.\end{eqnarray}$The noncommutative Jacobi identity is given by$ \begin{eqnarray}\begin{array}{l}\left|\begin{array}{c}A\quad B\quad C\\ D\quad f\quad g\\ E\quad h\quad \boxed{i}\end{array}\right|=\left|\begin{array}{c}A\quad C\\ E\quad \boxed{i}\end{array}\right|\\ \quad -\,\left|\begin{array}{c}A\quad B\\ E\quad \boxed{h}\end{array}\right|{\left|\begin{array}{c}AB\\ D\boxed{f}\end{array}\right|}^{-1}\left|\begin{array}{c}A\quad C\\ D\quad \boxed{g}\end{array}\right|.\end{array}\end{eqnarray}$For more details see [35] and the references therein.) is expressed as$ \begin{eqnarray}{\psi }^{\left[N\right]}=\left|\begin{array}{cc}\widetilde{{\rm{\Xi }}} & {{\rm{\Pi }}}^{(N-1)}\\ \widehat{{\rm{\Xi }}} & \boxed{{\lambda }^{N}I}\end{array}\right|\psi ,\end{eqnarray}$where$ \begin{eqnarray}\begin{array}{rcl}\widetilde{{\rm{\Xi }}} & = & \left(\begin{array}{cccc}{{\rm{\Xi }}}_{1} & {{\rm{\Xi }}}_{2} & \ldots & {{\rm{\Xi }}}_{N}\\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1} & {{\rm{\Xi }}}_{2}{{ \mathcal L }}_{2} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}\\ \vdots & \vdots & \ddots & \vdots \\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N-1} & {{\rm{\Xi }}}_{2}{{ \mathcal L }}_{2}^{N-1} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N-1}\end{array}\right),\\ \widehat{{\rm{\Xi }}} & = & {\left(\begin{array}{c}{{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N}\\ {{\rm{\Xi }}}_{2}{{ \mathcal L }}_{2}^{N}\\ \vdots \\ {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N}\end{array}\right)}^{{\rm{T}}},\,{{\rm{\Pi }}}^{(N-1)}=\left(\begin{array}{c}I\\ \lambda I\\ \vdots \\ {\lambda }^{N-1}I\end{array}\right),\end{array}\end{eqnarray}$are the matrices of sizes $4N\times 4N,\,4\times 4N$ and $4N\times 4$, respectively.
The proof of (2.20) is by induction on N as follows.
The expression is true for N=1 and gives (2.6), since$ \begin{eqnarray}{\psi }^{\left[1\right]}=\left|\begin{array}{cc}{{\rm{\Xi }}}_{1} & I\\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1} & \boxed{\lambda I}\end{array}\right|\psi =(\lambda I-{{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}{{\rm{\Xi }}}_{1}^{-1})\psi .\end{eqnarray}$Now, assume that (2.20) is true for some fixed $N\geqslant 1$. We prove that the expression is also true for $N+1$, i.e. $ \begin{eqnarray*}\begin{array}{rcl}{\psi }^{\left[N+1\right]} & = & \left|\begin{array}{cccc}{{\rm{\Xi }}}_{1} & \ldots & {{\rm{\Xi }}}_{N} & \lambda I\\ \vdots & \ddots & \vdots & \vdots \\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N-1} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N-1} & {\lambda }^{N}I\\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N} & \boxed{{\lambda }^{N+1}I}\end{array}\right|\psi \\ & & -\,\left|\begin{array}{cccc}{{\rm{\Xi }}}_{1} & \ldots & {{\rm{\Xi }}}_{N} & {{\rm{\Xi }}}_{N+1}\\ \vdots & \ddots & \vdots & \vdots \\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N-1} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N-1} & {{\rm{\Xi }}}_{N+1}{{ \mathcal L }}_{N+1}^{N-1}\\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N} & \boxed{{{\rm{\Xi }}}_{N+1}{{ \mathcal L }}_{N+1}^{N}}\end{array}\right|\\ & & \times \,{{ \mathcal L }}_{N+1}{\left|\begin{array}{cccc}{{\rm{\Xi }}}_{1} & \ldots & {{\rm{\Xi }}}_{N} & {{\rm{\Xi }}}_{N+1}\\ \vdots & \ddots & \vdots & \vdots \\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N-1} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N-1} & {{\rm{\Xi }}}_{N+1}{{ \mathcal L }}_{N+1}^{N-1}\\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N} & \boxed{{{\rm{\Xi }}}_{N+1}{{ \mathcal L }}_{N+1}^{N}}\end{array}\right|}^{-1}\\ & & \times \,\left|\begin{array}{cccc}{{\rm{\Xi }}}_{1} & \ldots & {{\rm{\Xi }}}_{N} & I\\ \vdots & \ddots & \vdots & \vdots \\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N-1} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N-1} & {\lambda }^{N-1}I\\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N} & \boxed{{\lambda }^{N}I}\end{array}\right|\psi ,\\ {\psi }^{\left[N+1\right]} & & =\left|\begin{array}{cccc}{{\rm{\Xi }}}_{1} & \ldots & {{\rm{\Xi }}}_{N} & \lambda I\\ \vdots & \ddots & \vdots & \vdots \\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N-1} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N-1} & {\lambda }^{N}I\\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N} & \boxed{{\lambda }^{N+1}I}\end{array}\right|\psi \\ & & -\,\left|\begin{array}{cccc}{{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N} & {{\rm{\Xi }}}_{N+1}{{ \mathcal L }}_{N+1}\\ \vdots & \ddots & \vdots & \vdots \\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N} & {{\rm{\Xi }}}_{N+1}{{ \mathcal L }}_{N+1}^{N}\\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N+1} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N+1} & \boxed{{{\rm{\Xi }}}_{N+1}{{ \mathcal L }}_{N+1}^{N+1}}\end{array}\right|\end{array}\end{eqnarray*}$$ \begin{eqnarray}\begin{array}{rcl} & & \times \,{\left|\begin{array}{cccc}{{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N+1} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N+1} & {{\rm{\Xi }}}_{N+1}{{ \mathcal L }}_{N+1}^{N+1}\\ \vdots & \ddots & \vdots & \vdots \\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N} & {{\rm{\Xi }}}_{N+1}{{ \mathcal L }}_{N+1}\\ {{\rm{\Xi }}}_{1} & \ldots & {{\rm{\Xi }}}_{N} & \boxed{{{\rm{\Xi }}}_{N+1}}\end{array}\right|}^{-1}\\ & & \times \,\left|\begin{array}{cccc}{{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N} & \lambda I\\ \vdots & \ddots & \vdots & \vdots \\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N} & {\lambda }^{N}I\\ {{\rm{\Xi }}}_{1} & \ldots & {{\rm{\Xi }}}_{N} & \boxed{I}\end{array}\right|\psi ,\end{array}\end{eqnarray}$$ \begin{eqnarray}{\psi }^{\left[N+1\right]}=\left|\begin{array}{cccc}{{\rm{\Xi }}}_{1} & \ldots & {{\rm{\Xi }}}_{N+1} & I\\ \vdots & \ddots & \vdots & \vdots \\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N+1}^{N+1} & {\lambda }^{N}I\\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N+1} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{1}^{N+1} & \boxed{{\lambda }^{N+1}I}\end{array}\right|\psi ,\end{eqnarray}$by the noncommutative Jacobi identity (2.19). This completes the proof.
The N-times iteration of DT for the fields A and B can be given by the following theorem.
Suppose that ${{\rm{\Xi }}}_{1},\,{{\rm{\Xi }}}_{2},...,\,{{\rm{\Xi }}}_{N}$ are particular matrix solutions of the Lax pair (2.1)–(2.2) corresponding to ${{ \mathcal L }}_{1},\,{{ \mathcal L }}_{2},....,\,{{ \mathcal L }}_{N}$, respectively. Then, the $N$-fold DT for the solutions of the matrix CD system in terms of quasideterminants is expressed as$ \begin{eqnarray}\begin{array}{rcl}{A}^{\left[N\right]} & = & A+{\partial }_{x}\left|\begin{array}{cc}\widetilde{{\rm{\Xi }}} & { \mathcal I }\\ \widehat{{\rm{\Xi }}} & \boxed{O}\end{array}\right|,\\ {B}^{N} & = & B-\left[J,\,\left|\begin{array}{cc}\widetilde{{\rm{\Xi }}} & { \mathcal I }\\ \widehat{{\rm{\Xi }}} & \boxed{O}\end{array}\right|\right],\end{array}\end{eqnarray}$where $\widetilde{{\rm{\Xi }}},\,\widehat{{\rm{\Xi }}}$ are given in (2.21), and ${ \mathcal I }={(O,O,...,I)}^{{\rm{T}}}$ is the matrix of size $4N\times 4$.
The proof of (2.25) is by induction on N as follows.
The expression is true for N=1 and gives (2.12) and (2.15), since$ \begin{eqnarray}\begin{array}{rcl}{A}^{\left[1\right]} & = & A+{\partial }_{x}\left|\begin{array}{cc}{{\rm{\Xi }}}_{1} & I\\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1} & \boxed{O}\end{array}\right|=A-{\partial }_{x}({{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}{{\rm{\Xi }}}^{-1}),\\ {B}^{\left[1\right]} & = & B-\left[J,\,\left|\begin{array}{cc}{{\rm{\Xi }}}_{1} & I\\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1} & \boxed{O}\end{array}\right|\right]=B+[J,\,{\rm{\Xi }}{ \mathcal L }{{\rm{\Xi }}}^{-1}].\end{array}\end{eqnarray}$Now, assume that (2.25) is true for some fixed $N\geqslant 1$. We prove that the expression is also true for $N+1$, i.e. $ \begin{eqnarray*}\begin{array}{rcl}{B}^{\left[N+1\right]} & = & B-\left[J,\,\left|\begin{array}{cccc}{{\rm{\Xi }}}_{1} & \ldots & {{\rm{\Xi }}}_{N} & O\\ \vdots & \ddots & \vdots & \vdots \\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N-1} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N-1} & I\\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N} & \boxed{O}\end{array}\right|\right]\\ & & +\left[J,\left|\begin{array}{cccc}{{\rm{\Xi }}}_{1} & \ldots & {{\rm{\Xi }}}_{N} & {{\rm{\Xi }}}_{N+1}\\ \vdots & \ddots & \vdots & \vdots \\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N-1} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N-1} & {{\rm{\Xi }}}_{N+1}{{ \mathcal L }}_{N+1}^{N-1}\\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N} & \boxed{{{\rm{\Xi }}}_{N+1}{{ \mathcal L }}_{N+1}^{N}}\end{array}\right|\right.\end{array}\end{eqnarray*}$ $ \begin{eqnarray*}\begin{array}{rcl} & & \left.\times \,{{ \mathcal L }}_{N+1}{\left|\begin{array}{cccc}{{\rm{\Xi }}}_{1} & \ldots & {{\rm{\Xi }}}_{N} & {{\rm{\Xi }}}_{N+1}\\ \vdots & \ddots & \vdots & \vdots \\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N-1} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N-1} & {{\rm{\Xi }}}_{N+1}{{ \mathcal L }}_{N+1}^{N-1}\\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N} & \boxed{{{\rm{\Xi }}}_{N+1}{{ \mathcal L }}_{N+1}^{N}}\end{array}\right|}^{-1}\right],\end{array}\end{eqnarray*}$ $ \begin{eqnarray*}\begin{array}{rcl}{B}^{\left[N+1\right]} & = & B-\left[J,\,\left|\begin{array}{cccc}{{\rm{\Xi }}}_{1} & \ldots & {{\rm{\Xi }}}_{N} & O\\ \vdots & \ddots & \vdots & \vdots \\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N-1} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N-1} & I\\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N} & \boxed{O}\end{array}\right|\right]\\ & & +\,\left[\Space{0ex}{6.70ex}{0ex}J,\right.\,\left|\begin{array}{cccc}{{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N} & {{\rm{\Xi }}}_{N+1}{{ \mathcal L }}_{N+1}\\ \vdots & \ddots & \vdots & \vdots \\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N} & {{\rm{\Xi }}}_{N+1}{{ \mathcal L }}_{N+1}^{N}\\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N+1} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N+1} & \boxed{{{\rm{\Xi }}}_{N+1}{{ \mathcal L }}_{N+1}^{N+1}}\end{array}\right|\\ & & \times \,{\left|\begin{array}{cccc}{{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N+1} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N+1} & {{\rm{\Xi }}}_{N+1}{{ \mathcal L }}_{N+1}^{N+1}\\ \vdots & \ddots & \vdots & \vdots \\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N} & {{\rm{\Xi }}}_{N+1}{{ \mathcal L }}_{N+1}\\ {{\rm{\Xi }}}_{1} & \ldots & {{\rm{\Xi }}}_{N} & \boxed{{{\rm{\Xi }}}_{N+1}}\end{array}\right|}^{-1}\\ & & \left.\left|\begin{array}{cccc}{{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N+1} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N+1} & O\\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N}^{N} & I\\ \vdots & \ddots & \vdots & \vdots \\ {{\rm{\Xi }}}_{1} & \ldots & {{\rm{\Xi }}}_{N} & \boxed{O}\end{array}\right|\right],\end{array}\end{eqnarray*}$$ \begin{eqnarray}\begin{array}{rcl}{B}^{\left[N+1\right]} & = & B-\left[J,\,\left|\begin{array}{cccc}{{\rm{\Xi }}}_{1} & \ldots & {{\rm{\Xi }}}_{N+1} & O\\ \vdots & \ddots & \vdots & \vdots \\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N} & \ldots & {{\rm{\Xi }}}_{N}{{ \mathcal L }}_{N+1}^{N} & I\\ {{\rm{\Xi }}}_{1}{{ \mathcal L }}_{1}^{N+1} & \ldots & {{\rm{\Xi }}}_{N+1}{{ \mathcal L }}_{N+1}^{N+1} & \boxed{O}\end{array}\right|\right],\end{array}\end{eqnarray}$by the noncommutative Jacobi identity (2.19). This completes the proof. And the proof for ${A}^{\left[N+1\right]}$ is similar to ${B}^{\left[N+1\right]}$.
By using the expression (2.25), one obtains DT on the potentials ${q}_{\pm },\,{q}_{0}$ and ${r}_{\pm },\,{r}_{0}$ in quasideterminant form$ \begin{eqnarray}\begin{array}{rcl}{q}_{+}^{\left[N\right]} & = & {q}_{+}+{\rm{i}}{\partial }_{x}\left|\begin{array}{cc}\widetilde{{\rm{\Xi }}} & {{ \mathcal I }}_{4N-3}\\ {\widehat{{\rm{\Xi }}}}_{4N-3} & \boxed{0}\end{array}\right|,\\ {q}_{-}^{\left[N\right]} & = & {q}_{-}+{\rm{i}}{\partial }_{x}\left|\begin{array}{cc}\widetilde{{\rm{\Xi }}} & {{ \mathcal I }}_{4N-2}\\ {\widehat{{\rm{\Xi }}}}_{4N-2} & \boxed{0}\end{array}\right|,\end{array}\end{eqnarray}$$ \begin{eqnarray}\begin{array}{rcl}{q}_{0}^{\left[N\right]} & = & {q}_{0}+{\rm{i}}{\partial }_{x}\left|\begin{array}{cc}\widetilde{{\rm{\Xi }}} & {{ \mathcal I }}_{4N-2}\\ {\widehat{{\rm{\Xi }}}}_{4N-3} & \boxed{0}\end{array}\right|,\\ {r}_{0}^{\left[N\right]} & = & {r}_{0}+{\rm{i}}\left|\begin{array}{cc}\widetilde{{\rm{\Xi }}} & {{ \mathcal I }}_{4N}\\ {\widehat{{\rm{\Xi }}}}_{4N-3} & \boxed{0}\end{array}\right|,\end{array}\end{eqnarray}$$ \begin{eqnarray}\begin{array}{rcl}{r}_{+}^{\left[N\right]} & = & {r}_{+}+{\rm{i}}\left|\begin{array}{cc}\widetilde{{\rm{\Xi }}} & {{ \mathcal I }}_{4N-1}\\ {\widehat{{\rm{\Xi }}}}_{4N-3} & \boxed{0}\end{array}\right|,\\ {r}_{-}^{\left[N\right]} & = & {r}_{-}+{\rm{i}}\left|\begin{array}{cc}\widetilde{{\rm{\Xi }}} & {{ \mathcal I }}_{4N}\\ {\widehat{{\rm{\Xi }}}}_{4N-2} & \boxed{0}\end{array}\right|,\end{array}\end{eqnarray}$where ${{ \mathcal I }}_{4N-3},\,{{ \mathcal I }}_{4N-2},\,{{ \mathcal I }}_{4N-1}$ and ${{ \mathcal I }}_{4N}$ are the column vectors with entry 1 in the $4N-3,\,4N-2,\,4N-1$ and $4N$row-th vector, respectively and zeros elsewhere, whereas ${\widehat{{\rm{\Xi }}}}_{4N-3}$ and ${\widehat{{\rm{\Xi }}}}_{4N-2}$ are, respectively, the $4N-3$ and $4N-2$row vectors of $\widetilde{{\rm{\Xi }}}$.
3. Explicit solutions
In this section, we shall investigate the effect of DT $D=\lambda I-{\rm{\Xi }}{ \mathcal L }{{\rm{\Xi }}}^{-1}$ explicitly on the potentials ${q}_{+},\,{q}_{-},\,{q}_{0}$ and ${r}_{+},\,{r}_{-},\,{r}_{0}$ of the matrix CD system (1.1). The solution ψ of the Lax pair (2.1)–(2.2) for the seed solutions (i.e. $R={0}_{2\times 2},\,Q=\mathrm{diag}(1,\,-1)$) is given by$ \begin{eqnarray}\psi =\mathrm{diag}\left({{\rm{e}}}^{\alpha (\lambda )},\,{{\rm{e}}}^{\beta (\lambda )},\,{{\rm{e}}}^{\bar{\alpha }(\lambda )},\,{{\rm{e}}}^{\bar{\beta }(\lambda )}\right),\end{eqnarray}$where $\alpha (\lambda )=-{\rm{i}}{\lambda }^{-1}x+\tfrac{{\rm{i}}\lambda t}{2},\,\beta (\lambda )={\rm{i}}\lambda x+\tfrac{{\rm{i}}\lambda t}{2}$.
To find the soliton solutions, we need to have the matrix Ξ in an explicit form. This matrix Ξ can be constructed from the particular vector solutions to the Lax pair (2.1)–(2.2). We now define the vectors satisfying the Lax pair equations for the particular value of λ, so that, for example, ${\xi }_{1}=\left({\epsilon }_{1}{{\rm{e}}}^{\alpha ({\lambda }_{1})}\right.$, ${\left.{\epsilon }_{2}{{\rm{e}}}^{\beta ({\lambda }_{1})},-{{\rm{e}}}^{\bar{\alpha }({\lambda }_{1})},0\right)}^{{\rm{T}}}$, ${\xi }_{2}=\left({\bar{\epsilon }}_{2}{{\rm{e}}}^{\alpha ({\lambda }_{1})}\right.$, ${\left.-{\bar{\epsilon }}_{1}{{\rm{e}}}^{\beta ({\lambda }_{1})},0,{{\rm{e}}}^{\bar{\beta }({\lambda }_{1})}\right)}^{{\rm{T}}}$ are independent column vector solutions of the system (2.1)–(2.2) evaluated at $\lambda ={\lambda }_{1}$, and ${\xi }_{3}\,=$${\left({{\rm{e}}}^{\alpha ({\bar{\lambda }}_{1})},0,{\bar{\epsilon }}_{1}{{\rm{e}}}^{\bar{\alpha }({\bar{\lambda }}_{1})},-{\epsilon }_{2}{{\rm{e}}}^{\bar{\beta }({\bar{\lambda }}_{1})}\right)}^{{\rm{T}}}$, ${\xi }_{4}={\left(0,{{\rm{e}}}^{\beta ({\bar{\lambda }}_{1})},{\bar{\epsilon }}_{2}{{\rm{e}}}^{\bar{\alpha }({\bar{\lambda }}_{1})},{\epsilon }_{1}{{\rm{e}}}^{\bar{\beta }({\bar{\lambda }}_{1})}\right)}^{{\rm{T}}}$ are independent column solutions of the system (2.1)–(2.2) evaluated at $\lambda ={\bar{\lambda }}_{1}$, so that the matrix Ξ with the particular eigenvalue matrix ${ \mathcal L }$ reads$ \begin{eqnarray}\begin{array}{rcl}{\rm{\Xi }} & = & \left({\xi }_{1},{\xi }_{2},{\xi }_{3},{\xi }_{4}\right),\\ { \mathcal L } & = & \mathrm{diag}\left({\lambda }_{1},{\lambda }_{1},{\bar{\lambda }}_{1},{\bar{\lambda }}_{1}\right).\end{array}\end{eqnarray}$By substituting (3.2) into (2.28)–(2.30), we obtain one-fold solutions given by$ \begin{eqnarray}\begin{array}{rcl}{q}_{\pm }^{\left[1\right]} & = & \pm 1-{\partial }_{x}[{\rm{i}}\displaystyle \frac{b}{A}\left({\epsilon }^{2}{A}_{+}{A}_{-}\pm | {\epsilon }_{1}{| }^{2}\right.\\ & & \times \,\left.({A}_{+}{B}_{-}-{B}_{+}{A}_{-})\mp {B}_{+}{B}_{-}\right)],\end{array}\end{eqnarray}$$ \begin{eqnarray}\begin{array}{rcl}{r}_{+}^{\left[1\right]} & = & {\rm{i}}\displaystyle \frac{b}{A}{\epsilon }_{1}{\phi }_{+}{\bar{\chi }}_{+}(\epsilon {A}_{-}+{B}_{-}),\\ {r}_{-}^{\left[1\right]} & = & {\rm{i}}\displaystyle \frac{b}{A}{\bar{\epsilon }}_{1}{\phi }_{-}{\bar{\chi }}_{-}(\epsilon {A}_{+}+{B}_{+}),\end{array}\end{eqnarray}$$ \begin{eqnarray}\begin{array}{rcl}{q}_{0}^{\left[1\right]} & = & {\rm{i}}{\epsilon }_{1}{\bar{\epsilon }}_{2}{\partial }_{x}[\displaystyle \frac{b}{A}{\phi }_{+}{\bar{\chi }}_{+}({\chi }_{+}{\bar{\phi }}_{-}-{\chi }_{-}{\bar{\phi }}_{+})],\\ {r}_{0}^{\left[1\right]} & = & -{\rm{i}}\displaystyle \frac{b}{A}{\bar{\epsilon }}_{2}{\phi }_{+}{\bar{\chi }}_{+}(\epsilon {\phi }_{-}{\bar{\phi }}_{+}+{\chi }_{+}{\bar{\chi }}_{-}),\end{array}\end{eqnarray}$where $A=[{\epsilon }^{2}{A}_{+}{A}_{-}+| {\epsilon }_{1}{| }^{2}({A}_{+}{B}_{-}+{B}_{+}{A}_{-})+{B}_{+}{B}_{-}+2| {\epsilon }_{2}{| }^{2}]$, $\epsilon =\left(| {\epsilon }_{1}{| }^{2}+| {\epsilon }_{1}{| }^{2}\right)$, ${A}_{\pm }=| {\phi }_{\pm }{| }^{2},\,{B}_{\pm }=| {\chi }_{\pm }{| }^{2}$ and ${\phi }_{\pm }\,={{\rm{e}}}^{\mp {\rm{i}}{\lambda }_{1}^{-1}x+\tfrac{{\rm{i}}{\lambda }_{1}t}{2}},\,{\chi }_{\pm }={{\rm{e}}}^{\pm {\rm{i}}{\lambda }_{1}^{-1}x-\tfrac{{\rm{i}}{\lambda }_{1}t}{2}}$ and $b={\mathfrak{I}}({\lambda }_{1})$. Equations (3.3) and (3.4) represent dark-type or bright-type soliton solutions, while equation (3.5) represents rogue wave solutions. The evolution plots of the rogue wave solutions ${q}_{0}^{[j]}$ and ${r}_{0}^{[j]}$ governed by the spectral parameters ${\lambda }_{j}={a}_{j}+{\rm{i}}{b}_{j},j=1,\,2,\,3$ are shown in figures 1 and 2.
Figure 1.
New window|Download| PPT slide Figure 1.(a), (b) and (c) are, respectively, the profiles of ${q}_{0}^{[j]}$ (2.29) for $j=1,\,2$ and 3 with parameters ${\lambda }_{1}=0.2+2{\rm{i}},\,{\lambda }_{2}=0.3-2{\rm{i}},\,{\lambda }_{3}\,=2{\rm{i}},\,{\epsilon }_{1}=1-2{\rm{i}}$ and ${\epsilon }_{2}=1-{\rm{i}}$.
Figure 2.
New window|Download| PPT slide Figure 2.(a), (b) and (c) are, respectively, the profiles of ${r}_{0}^{[j]}$ (2.29) for $j=1,\,2$ and 3 with parameters ${\lambda }_{1}=0.2+2{\rm{i}},\,{\lambda }_{2}=0.3-2{\rm{i}},\,{\lambda }_{3}\,=2{\rm{i}},\,{\epsilon }_{1}=1-2{\rm{i}}$ and ${\epsilon }_{2}=1-{\rm{i}}$.
Here, an explicit form of ${q}_{0}^{[2]},\,{q}_{0}^{[3]}$ and ${r}_{0}^{[2]},\,{r}_{0}^{[3]}$ is not given as it is rather cumbersome.
The evolution plots of the solutions ${q}_{0}^{\left[1\right]}$ and ${r}_{0}^{\left[1\right]}$ under the seed solutions $R={0}_{2\times 2},\,Q=\mathrm{diag}(\cos x,\,-\cos x)$ is depicted in figure 3.
Figure 3.
New window|Download| PPT slide Figure 3.(a), (b) are, respectively, the profiles of ${q}_{0}^{\left[1\right]}$ and ${r}_{0}^{\left[1\right]}$ (3.5) with parameters ${\lambda }_{1}=-0.1+{\rm{i}},\,{\epsilon }_{1}=1-2{\rm{i}}$ and ${\epsilon }_{2}=1-0.1{\rm{i}}$.
Figure 3 shows the breather solution of the matrix CD system.
4. Concluding remarks
In this paper, we have derived a DT for the matrix CD system and constructed some exact solutions. We have constructed a quasideterminant Darboux matrix and obtained N soliton solutions. By using properties of the quasideterminants and symbolic computation, we have computed first-, second- and third-order rogue wave solutions. Moreover, breather solutions are also presented. The solutions of the matrix CD system have been shown to give solutions of the usual CD system, sine-Gordon equation and Maxwell–Bloch system. These results can be further extended to the case of supersymmetric generalization of the system. The matrix superfield Lax representation can be used to obtain superfield Riccati equations, which may result in superfield Bäcklund transformation of the supersymmetric matrix CD system. It would also be interesting to construct an integrable discretization of the system in order to explore numerical simulation by using various numerical schemes.
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11 871 471, 11 331 008 and 11 931 017), Foreign Experts Scientific Cooperation Fund. The author thanks the editor and anonymous referees for their useful comments and suggestions, which helped to improve the paper.
KivsharY SAgrawalG P2003Optical Solitons: From Fibers to Photonic Crystals New York Academic [Cited within: 1]
Cuevas-MaraverJKevrekidisP GWilliamsF2014The sine-Gordon Model and its Applications: From Pendula and Josephson Junctions to Gravity and High Energy Physics Berlin Springer [Cited within: 1]
ShenJGengXXueB2019 Exact solutions of a negative matrix AKNS system with a Hermitian symmetric space Appl. Math. Lett. 90 8 DOI:10.1016/j.aml.2018.10.006
RiazH W AHassanM2018 A discrete generalized coupled dispersionless integrable system and its multisoliton solutions J. Math. Anal. Appl. 458 1639 DOI:10.1016/j.jmaa.2017.10.046 [Cited within: 1]
RiazH W AHassanM2018 Integrability of a multicomponent coupled dispersionless integrable system Theor. Math. Phys. 197 1714 DOI:10.1134/S0040577918120036
RajanM S MMahalingamA2013 Multi-soliton propagation in a generalized inhomogeneous nonlinear Schrödinger-Maxwell-Bloch system with loss/gain driven by an external potential J. Math. Phys. 54 043514 DOI:10.1063/1.4798477
Mani RajanM SMahalingamAUthayakumarAPorseziancK2013 Observation of two soliton propagation in an erbium doped inhomogeneous lossy fiber with phase modulation Commun. Nonlinear Sci. Numer. Simulat. 18 1410 DOI:10.1016/j.cnsns.2012.10.008 [Cited within: 1]
TakasakiK1995 Dispersionless Toda hierarchy and two-dimensional string theory Commun. Math. Phys. 170 743 DOI:10.1007/BF02099441 [Cited within: 1]
CarrollRKodamaY1995 Solution of the dispersionless Hirota equations J. Phys. A: Math. Gen. 28 6373 DOI:10.1088/0305-4470/28/22/013
DunajskiM2008 Interpolating dispersionless integrable system J. Phys. A: Math. Theor. 41 315202 DOI:10.1088/1751-8113/41/31/315202
RajanM S MMahalingamA2015 Nonautonomous solitons in modified inhomogeneous Hirota equation: soliton control and soliton interaction Nonlinear Dyn. 79 2469 DOI:10.1007/s11071-014-1826-y
RajanM S M2016 Dynamics of optical soliton in a tapered erbium-doped fiber under periodic distributed amplification system Nonlinear Dyn. 85 599 DOI:10.1007/s11071-016-2709-1 [Cited within: 1]
HirotaRTsujimotoS1994 Note on new coupled integrable dispersionless equations J. Phys. Soc. Japan 63 3533 DOI:10.1143/JPSJ.63.3533
KakuhataHKonnoK1996 A generalization of coupled integrable, dispersionless system J. Phys. Soc. Japan 65 340 DOI:10.1143/JPSJ.65.340 [Cited within: 1]
HassanM2009 Darboux transformation of the generalized coupled dispersionless integrable system J. Phys. A: Math. Theor. 42 65203 DOI:10.1088/1751-8113/42/6/065203
LouS YYuG F2016 A generalization of the coupled integrable dispersionless equations Math. Meth. Appl. Sci. 39 4025 DOI:10.1002/mma.3844
,Department of Mathematics, 
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