N-Fold Darboux Transformation and Various Solutions for the Coupled mKdV Equations

In this paper, we get the N-fold Darboux transformation with multi-parameters for the coupled mKdV equations with the help of a guage transformation of the spectral problem. As an application, some new multi-soliton solutions and complexiton solutions are obtained from choosing the appropriate seed solution. All obtained solutions and N-fold Darboux transformations are expressed using the Vandermonde-like determinants.

• The N-fold Darboux Transformation
• The Coupled mKdV Equations
• Multi-Soliton Solutions
• Complexiton Solutions

1. Introduction

Soliton equations play an important role in the field of nonlinear science, whose have specific solutions to describe and explain the nonlinear phenomena, for example, super conductivity, plasma and elastic media, etc. [1] - [3] . In the past decades, a considerable number of methods [4] - [10] have been developed for obtaining explicit solutions of nonlinear evolution equations. Among them, the Darboux transformation is one of effective algorithmic procedures to generate explicit solution of some nonlinear evolution equations from the trivial seeds [11] [12] . The key step for constructing the Darboux transformation is to keep the corresponding spectral problems. The N-fold Darboux transformation can be regarded as the superposition of single Darboux transformation, thus allowing for the further acquisition of various solutions through symbolic computation [13] - [15] .

In this paper, we construct the N-fold Darboux transformation for the coupled mKdV equations

$\begin{array}{l}{u}_t+\frac{1}{2}{u}_{xxx}-3u{u}_{x}v=0,\\ v_t+\frac{1}{2}{v}_{xxx}-3v{v}_{x}u=0,\end{array}$ (1)

which are still unknown to our knowledge. Equation (1) is an important member of the AKNS hierarchy and has various applications in mathematical and physical fields.

This paper is organized as follows: In Section 2, based on Neugebauer’s idea, the N-fold DT of system (1) is constructed. In Section 3, the 2N-soliton solutions and N-complexiton solutions of system (1) are obtained, which are also expressed as vandermonde-like determinants. In Section 4, some conclusion is given.

2. N-Fold Darboux Transformation

In order to construct a N-fold DT of the system (1), let us consider the following spectral problem

${\varphi }_x=M\varphi ,\varphi ={\left({\varphi }_1,{\varphi }_2\right)}^{\text{T}},M=\left(\begin{array}{cc}-i\lambda & u\\ v& i\lambda \end{array}\right),$ (2)

and its auxiliary problem

${\varphi }_t=N\varphi ,N=\left(\begin{array}{cc}-2i{\lambda }^3-iuv\lambda +\frac{1}{2}\left(u_{x}v-u{v}_x\right)& 2u{\lambda }^2+i{u}_x\lambda -\frac{1}{2}\left(u_{xx}-2u^{2}v\right)\\ 2v{\lambda }^2-i{v}_x\lambda -\frac{1}{2}\left(v_{xx}-2v^{2}u\right)& 2i{\lambda }^3+iuv\lambda -\frac{1}{2}\left(u_{x}v-u{v}_x\right)\end{array}\right),$ (3)

the compatibility condition ${\varphi }_{xt}={\varphi }_{tx}$ yields a zero curvature equation

$M_t-N_x+\left[M,N\right]=0,$

which yields coupled mKdV system (1) by a direct computation.

Now, we consider a gauge transformation of the spectral problems (2) and (3)

$\bar{\varphi }=T\varphi ,$ (4)

where $T$ is defined by

$\left(T_x+TM\right)=\bar{M}T,$ (5)

$\left(T_t+TN\right)=\bar{N}T.$ (6)

By cross differentiating (5) and (6), we get

${\bar{M}}_t-{\bar{N}}_x+\bar{M}\bar{N}-\bar{N}\bar{M}=T\left(M_t-N_x+MN-NM\right)T^{-1}.$ (7)

The lax pair (2) and (3) are transformed to

${\bar{\varphi }}_x=\bar{M}\bar{\varphi },{\bar{\varphi }}_t=\bar{N}\bar{\varphi }.$ (8)

(7) and (8) implie that $\bar{M},\bar{N}$ has the same form as $M,N$ expect replacing $u$ and $v$ with $\bar{u}$ and $\bar{v}$.

Suppose the Darboux matrix $T$ in the forms of

$T=T\left(\lambda \right)=\left(\begin{array}{cc}A\left(\lambda \right)& B\left(\lambda \right)\\ C\left(\lambda \right)& D\left(\lambda \right)\end{array}\right),$ (9)

where

$A\left(\lambda \right)=A_N\left({\lambda }^N+\underset{k=0}{\overset{N-1}{{\displaystyle \sum }}}{A}_k{\lambda }^k\right),B\left(\lambda \right)=A_N\left(\underset{k=0}{\overset{N-1}{{\displaystyle \sum }}}{B}_k{\lambda }^k\right),$

$C\left(\lambda \right)=\frac{1}{A_N}\left(\underset{k=0}{\overset{N-1}{{\displaystyle \sum }}}{C}_k{\lambda }^k\right),D\left(\lambda \right)=\frac{1}{A_N}\left({\lambda }^N+\underset{k=0}{\overset{N-1}{{\displaystyle \sum }}}{D}_k{\lambda }^k\right),$

$A_N,A_k,B_k,C_k,D_k\left(0\ge k\le N-1\right)$ are functions of $x$ and $t$.

Let $\varphi \left({\lambda }_j\right)={\left({\varphi }_1\left({\lambda }_j\right),{\varphi }_2\left({\lambda }_j\right)\right)}^{\text{T}}$, $\psi \left({\lambda }_j\right)={\left({\psi }_1\left({\lambda }_j\right),{\psi }_2\left({\lambda }_j\right)\right)}^{\text{T}}$ be two basic solutions of the spectral problems (2) and (3). From (5) and (6), there exist constants $r_j\left(1\ge j\le 2N\right)$, which satisfies

$A\left({\lambda }_j\right){\varphi }_1\left({\lambda }_j\right)+B\left({\lambda }_j\right){\varphi }_2\left({\lambda }_j\right)-r_j\left(A\left({\lambda }_j\right){\psi }_1\left({\lambda }_j\right)+B\left({\lambda }_j\right){\psi }_2\left({\lambda }_j\right)\right)=0,$

$C\left({\lambda }_j\right){\varphi }_1\left({\lambda }_j\right)+D\left({\lambda }_j\right){\varphi }_2\left({\lambda }_j\right)-r_j\left(C\left({\lambda }_j\right){\psi }_1\left({\lambda }_j\right)+D\left({\lambda }_j\right){\psi }_2\left({\lambda }_j\right)\right)=0.$

That is $A_k,B_k,C_k$ and $D_k$ are given by a linear algebraic system

$\underset{k=0}{\overset{N-1}{{\displaystyle \sum }}}\left(A_k+{\alpha }_j{B}_k\right){\lambda }_j^k=-{\lambda }_j^N,\underset{k=0}{\overset{N-1}{{\displaystyle \sum }}}\left(C_k+{\alpha }_j{D}_k\right){\lambda }_j^k=-{\alpha }_j{\lambda }_j^N$ (10)

with

${\delta }_j=\frac{{\varphi }_2\left({\lambda }_j\right)-r_j{\psi }_2\left({\lambda }_j\right)}{{\varphi }_1\left({\lambda }_j\right)-r_j{\psi }_1\left({\lambda }_j\right)},0\ge j\le 2N,$ (11)

where constants ${\lambda }_j$ and $r_j$ ( ${\lambda }_k\ne {\lambda }_j$, $r_k\ne r_j$, as $k\ne j$) are chosen properly such that the determinant of coefficients for (10) are nonzero. So, $A_k,B_k,C_k$ and $D_k\left(0\ge k\le N-1\right)$ are uniquely determined by (10).

From Equation (9), we have

$\det T\left({\lambda }_j\right)=A\left({\lambda }_j\right)D\left({\lambda }_j\right)-B\left({\lambda }_j\right)C\left({\lambda }_j\right).$ (12)

On the other side, from system (10), we know that

$A\left({\lambda }_j\right)=-{\alpha }_{j}B\left({\lambda }_j\right),C\left({\lambda }_j\right)=-{\alpha }_{j}D\left({\lambda }_j\right).$ (13)

Hence, it implies that

$\det T\left({\lambda }_j\right)=0,$

which shows that ${\lambda }_j\left(1\ge j\le 2N\right)$ are $2N$ roots of $\det T\left(\lambda \right)$, in other words,

$\det T\left(\lambda \right)=\underset{j=1}{\overset{2N}{{\displaystyle \prod }}}\left(\lambda -{\lambda }_j\right).$ (14)

Based on the above facts, we will prove the following propositions.

Proposition 1 Assume that $A_N$ satisfies

${\partial }_x\ln \left(A_N\right)=0,A_N^2=1,$ (15)

then the matrix $\bar{M}$ has the same form as $M$, respectively, where the transformation between the old potential $u,v$ into new ones are defined as

$\bar{u}=u+2i{B}_{N-1},$ (16)

$\bar{v}=v-2i{C}_{N-1},$ (17)

and

$\begin{array}{l}{A}_{N-m,x}=2i{B}_{N-1}{C}_{N-m}+C_{N-m}u-B_{N-m}v,\\ B_{N-m,x}=-2i\left(B_{N-M-1}-B_{N-1}{D}_{N-m}\right)-A_{N-m}u+D_{N-m}u,\\ C_{N-m,x}=-2i\left(A_{N-M}{C}_{N-1}-C_{N-M-1}\right)+\left(A_{N-m}-D_{N-m}\right)v,\\ D_{N-m,x}=-2i{B}_{N-m}{C}_{N-1}-C_{N-m}u-B_{N-m}v.\end{array}$ (18)

Proof Let $T^{-1}=T^{\text{*}}/\text{det}T$ and

$\left(T_x+TM\right)T^{\text{*}}=\left(\begin{array}{cc}{g}_{11}\left(\lambda \right)& g_{12}\left(\lambda \right)\\ g_{21}\left(\lambda \right)& g_{22}\left(\lambda \right)\end{array}\right),$ (19)

where $g_{11}\left(\lambda \right)$ and $g_{22}\left(\lambda \right)$ are $\left(2N+3\right)$th-order polynomials in $\lambda$, $g_{12}\left(\lambda \right),g_{21}\left(\lambda \right)$ are $\left(2N+2\right)$th-order polynomials in $\lambda$, from Equation (2) and (11), we find a Riccati equation

${\delta }_{jx}=v+2i\lambda {\delta }_j-u{\delta }_j^2.$ (20)

From (14) and (20), we can get that ${\lambda }_j\left(1\ge j\le 2N\right)$ are roots of $g_{ki}\left(\lambda \right)\left(k,i=1,2\right)$, in this way, together with (14) and (19) gives

$\left(T_x+TM\right)T^{*}=\left(\det T\right)p\left(\lambda \right),$ (21)

with

$p\left(\lambda \right)=\left(\begin{array}{cc}{p}_{11}^{\left(1\right)}\lambda +p_{11}^{\left(0\right)}& p_{12}^{\left(0\right)}\\ p_{21}^{\left(0\right)}& p_{22}^{\left(1\right)}\lambda +p_{22}^{\left(0\right)}\end{array}\right),$ (22)

where $p_{kj}^{\left(l\right)}\left(k,j=1,2;l=0,1\right)$ are undetermined functions independent of $\lambda$. And Equation (21) can be written as

$\left(T_x+TM\right)=p\left(\lambda \right)T.$ (23)

By compared with the coefficients of ${\lambda }^{N+1},{\lambda }^N$ in Equation (23), we can find that

$p_{11}^{\left(1\right)}=p_{22}^{\left(1\right)}=-i,$ (24)

$p_{21}^{\left(0\right)}=\frac{v-2i{C}_{N-1}}{A_N^2},p_{11}^{\left(0\right)}=-p_{22}^{\left(0\right)}={\partial }_x\ln \left(A_N\right)$ (25)

$p_{12}^{\left(0\right)}=A_N^2\left(u+2i{B}_{N-1}\right).$ (26)

Substituting (15) and (16) into (24)-(26), we can obtain that

$p_{11}^{\left(1\right)}=p_{22}^{\left(1\right)}=-i,p_{11}^{\left(0\right)}=-p_{22}^{\left(0\right)}=0,$

$p_{12}^{\left(0\right)}=u+2i{B}_{N-1},p_{21}^{\left(0\right)}=v-2i{C}_{N-1}.$

From (5) and (23), it is easy to see that $\bar{M}=p\left(\lambda \right)$. Therefore, the proof of Proposition 1 is completed.

Next, let the solutions $\varphi \left({\lambda }_j\right)$ and $\psi \left({\lambda }_j\right)$ also satisfy (3), we try to prove $\bar{N}$ in (6) has the same form as $N$ under the transformation (4) and (17).

Proposition 2 Assume that $A_N$ submits the differential equation with respect to the variable $t$

${\partial }_{t}ln\left(A_N\right)=4u{C}_{N-1}\left(A_{N-1}-D_{N-1}\right).$ (27)

Then, the matrix $\bar{N}$ has the same form as $N$, namely,

$\bar{N}=\left(\begin{array}{cc}-2i{\lambda }^3-i\bar{u}\bar{v}\lambda +\frac{1}{2}\left({\bar{u}}_x\bar{v}-\bar{u}{\bar{v}}_x\right)& 2\bar{u}{\lambda }^2+i{\bar{u}}_x\lambda -\frac{1}{2}\left({\bar{u}}_{xx}-2{\bar{u}}^2\bar{v}\right)\\ 2\bar{v}{\lambda }^2-i{\bar{v}}_x\lambda -\frac{1}{2}\left({\bar{v}}_{xx}-2{\bar{v}}^2\bar{u}\right)& 2i{\lambda }^3+i\bar{u}\bar{v}\lambda -\frac{1}{2}\left({\bar{u}}_x\bar{v}-\bar{u}{\bar{v}}_x\right)\end{array}\right),$ (28)

the old potentials $u,v$ are mapped into new ones $\bar{u},\bar{v}$ according to the same Darboux transformation (4), (16) and (17).

Proof Let $T^{-1}=T^{*}/\det T$ and

$\left(T_t+TN\right)T^{\text{*}}=\left(\begin{array}{cc}{f}_{11}\left(\lambda \right)& f_{12}\left(\lambda \right)\\ f_{21}\left(\lambda \right)& f_{22}\left(\lambda \right)\end{array}\right),$ (29)

where $f_{11}\left(\lambda \right)$ and $f_{22}\left(\lambda \right)$ are $\left(2N+3\right)$th-order polynomials in $\lambda$, $f_{12}\left(\lambda \right),f_{21}\left(\lambda \right)$ are $\left(2N+2\right)$th-order polynomials in $\lambda$. On the basic of (3), (11) and (13), we can get a Riccati equation

$\begin{array}{c}{\delta }_{jt}=2v{\lambda }_j^2-i{v}_x{\lambda }_j-\frac{1}{2}\left(v_{xx}-2v^{2}u\right)+\left(4i{\lambda }_j^3+2iuv\lambda -j-\left(u_{x}v-u{v}_x\right)\right){\delta }_j\\ -\left(2u{\lambda }_j^2+i{u}_x{\lambda }_j\frac{1}{2}\left(u_{xx}-2u^{2}v\right)\right){\delta }_j^2.\end{array}$ (30)

Through a series of calculation, we can get that ${\lambda }_j\left(1\ge j\le 2N\right)$ are roots of $f_{ki}\left(\lambda \right)\left(k,i=1,2\right)$. From (14) and (29), we arrive at

$\left(T_t+TN\right)T^{*}=\left(\det T\right)Q\left(\lambda \right),$ (31)

with

$Q\left(\lambda \right)=\left(\begin{array}{cc}{q}_{11}^{\left(3\right)}{\lambda }^3+q_{11}^{\left(2\right)}{\lambda }^2+q_{11}^{\left(1\right)}\lambda +q_{11}^{\left(0\right)}& q_{12}^{\left(2\right)}{\lambda }^2+q_{12}^{\left(1\right)}\lambda +q_{12}^{\left(0\right)}\\ q_{21}^{\left(2\right)}{\lambda }^2+q_{21}^{\left(1\right)}\lambda +q_{21}^{\left(0\right)}& q_{22}^{\left(3\right)}{\lambda }^3+q_{22}^{\left(2\right)}{\lambda }^2+q_{22}^{\left(1\right)}\lambda +q_{22}^{\left(0\right)}\end{array}\right),$

where $q_{n,s}^{\left(l\right)}\left(n,s=1,2;l=0,1,2\right)$ are undetermined functions independent of $\lambda$. Equation (31) can be written as

$\left(T_t+TN\right)=Q\left(\lambda \right)T.$ (32)

According to compare the coefficients of ${\lambda }^{N+3},{\lambda }^{N+2},{\lambda }^{N+1}$ in Equation (32), we have

$q_{11}^{\left(2\right)}=q_{22}^{\left(2\right)}=0,q_{11}^{\left(3\right)}=-q_{22}^{\left(3\right)}=-2i,$

$q_{21}^{\left(2\right)}=2\left(v-2i{C}_{N-1}\right),q_{12}^{\left(2\right)}=2\left(u+2i{B}_{N-1}\right),$

$q_{11}^{\left(1\right)}=-iuv+2v{B}_{N-1}-4i{B}_{N-1}{C}_{N-1}-2u{C}_{N-1},$

$q_{22}^{\left(1\right)}=2u{C}_{N-1}+iuv-2v{B}_{N-1}+4i{C}_{N-1}{B}_{N-1},$

$q_{12}^{\left(1\right)}=i{u}_x+2u{A}_{N-1}+4i{B}_{N-2}-4i{B}_{N-1}{D}_{N-1}-2u{D}_{N-1},$

$q_{21}^{\left(1\right)}=-i{v}_x+2v{D}_{N-1}-4i{C}_{N-2}-2v{A}_{N-1}+4i{C}_{N-1}{A}_{N-1},$

$\begin{array}{c}{q}_{12}^{\left(0\right)}=-\frac{1}{2}\left(u_{xx}-2u^{2}v\right)+2u{A}_{N-2}+i{u}_x{A}_{N-1}+4i{B}_{N-3}+2iuv{B}_{N-1}\\ -\left(2v{B}_{N-1}-4i{B}_{N-1}{C}_{N-1}-2u{C}_{N-1}\right)B_{N-1}-D_{N-2}\left(4i{B}_{N-1}+2u\right)\\ -D_{N-1}\left(i{u}_x+2u{A}_{N-1}+4i{B}_{N-2}-4i{B}_{N-1}{D}_{N-1}-2u{D}_{N-1}\right),\end{array}$

$\begin{array}{c}{q}_{21}^{\left(0\right)}=-4i{C}_{N-3}-2iuv{C}_{N-1}-\frac{1}{2}\left(v_{xx}-2v^{2}u\right)+2v{D}_{N-2}\\ -i{v}_x{D}_{N-1}-\left(2v-4i{C}_{N-1}\right)A_{N-2}\\ -\left(-i{v}_x+2v{D}_{N-1}-4i{C}_{N-2}-2v{A}_{N-1}+4i{C}_{N-1}\right)A_{N-1})A_{N-1}\\ -\left(2u{C}_{N-1}-2v{B}_{N-1}+4i{C}_{N-1}{B}_{N-1}\right)C_{N-1},\end{array}$

$\begin{array}{c}{q}_{11}^{\left(0\right)}={\partial }_t\ln \left(A_N\right)+\frac{1}{2}\left(u_{x}v-u{v}_x\right)+2v{B}_{N-2}-i{v}_x{B}_{N-1}\\ -\left(2v{B}_{N-1}-4i{B}_{N-1}\right)C_{N-1})A_{N-1}-\left(-4i{B}_{N-1}+2u\right)C_{N-2}\\ -C_{N-1}\left(i{u}_x+4i{B}_{N-2}-4i{B}_{N-1}{D}_{N-1}-2u{D}_{N-1}\right),\end{array}$

$\begin{array}{c}{q}_{22}^{\left(0\right)}=-{\partial }_t\ln \left(A_N\right)+2u{C}_{N-2}+i{u}_x{C}_{N-1}-\frac{1}{2}\left(u_{x}v-u{v}_x\right)\\ -\left(2v-4i{C}_{N-1}\right)B_{N-2}-\left(-i{v}_x-4i{C}_{N-2}-2v{A}_{N-1}+4i{C}_{N-1}\right)A_{N-1})B_{N-1}\\ -\left(2u{C}_{N-1}+4i{C}_{N-1}\right)B_{N-1})D_{N-1}.\end{array}$

Substituting (15)-(17) into above expressions, we can obtain

$\begin{array}{l}{q}_{11}^{\left(2\right)}=q_{22}^{\left(2\right)}=0,q_{11}^{\left(3\right)}=-q_{22}^{\left(3\right)}=-2i,\\ q_{21}^{\left(2\right)}=2\left(v-2i{C}_{N-1}\right)=2\bar{v},q_{12}^{\left(2\right)}=2\left(u+2i{B}_{N-1}\right)=2\bar{u},\\ q_{11}^{\left(1\right)}=-i\bar{u}\bar{v},q_{22}^{\left(1\right)}=i\bar{u}\bar{v},q_{12}^{\left(1\right)}=i{\bar{u}}_x,q_{21}^{\left(1\right)}=i{\bar{v}}_x,\\ q_{12}^{\left(0\right)}=-\frac{1}{2}\left({\bar{u}}_{xx}-2{\bar{u}}^2\bar{v}\right),q_{21}^{\left(0\right)}=-\frac{1}{2}\left({\bar{v}}_{xx}-2{\bar{v}}^2\bar{u}\right),\\ q_{11}^{\left(0\right)}=\frac{1}{2}\left({\bar{u}}_x\bar{v}-\bar{u}{\bar{v}}_x\right),q_{22}^{\left(0\right)}=-\frac{1}{2}\left({\bar{u}}_x\bar{v}-\bar{u}{\bar{v}}_x\right).\end{array}$ (33)

Therefore, we have $Q\left(\lambda \right)=\bar{N}$, that is, the proof is completed.

The two propositions indicate that both of the lax pairs (2), (3) and (8) lead to the same system (1). The transformation (4), (16) and (17), $\left(\varphi ,u,v\right)\to \left(\bar{\varphi },\bar{u},\bar{v}\right)$ are called a DT of system (1).

Theorem The N-fold DT (16), (17) in terms of Vandermonde-like determinants can be written as

$\begin{array}{l}\bar{u}\left[N\right]=u+2i\frac{\text{Δ}{B}_{N-1}}{{\text{Δ}}_0},\hfill \\ \bar{v}\left[N\right]=v-2i\frac{\text{Δ}{C}_{N-1}}{{\text{Δ}}_0},\hfill \end{array}$ (34)

where $\text{Δ}$ is determinant of the coefficients for the linear algebraic system (10), that is

${\text{Δ}}_0=\left|\begin{array}{ccccccc}1& {\delta }_1& {\lambda }_1& {\delta }_1{\lambda }_1& \cdots & {\lambda }_1^{N-1}& {\delta }_1{\lambda }_1^{N-1}\\ 1& {\delta }_2& {\lambda }_2& {\delta }_2{\lambda }_2& \cdots & {\lambda }_2^{N-1}& {\delta }_2{\lambda }_2^{N-1}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 1& {\delta }_{2N-1}& {\lambda }_{2N-1}& {\delta }_{2N-1}{\lambda }_{2N-1}& \cdots & {\lambda }_{2N-1}^{N-1}& {\delta }_{2N-1}{\lambda }_{2N-1}^{N-1}\\ 1& {\delta }_{2N}& {\lambda }_{2N}& {\delta }_{2N}{\lambda }_{2N}& \cdots & {\lambda }_{2N}^{N-1}& {\delta }_{2N}{\lambda }_{2N}^{N-1}\end{array}\right|$

with $\text{Δ}{B}_{N-1}$ is produced from $\text{Δ}$ by replacing its $2N$-th column with ${\left(-{\lambda }_1^N,-{\lambda }_2^N,\cdots ,-{\lambda }_{2N}^N\right)}^{\text{T}}$, $\text{Δ}{C}_{N-1}$ is produced from $\text{Δ}$ by replacing its $\left(2N-1\right)$-th column with ${\left(-{\delta }_1{\lambda }_1^N,-{\delta }_2{\lambda }_2^N,\cdots ,-{\delta }_{2N}{\lambda }_{2N}^N\right)}^{\text{T}}$, and ${\delta }_j\left(j=1,2,\cdots ,2N\right)$ are given by (11).

3. The (2N)-Soliton Solutions

In this section, we shall apply the N-fold DT to obtain multi-soliton solutions of system (1). We start from the trivial solution $u=\alpha +\beta i$, $v=-\alpha +\beta i$ (as $\alpha$ and $\beta$ are constants ) as our seed solutions, then we choose two basic solutions of (2) and (3)

$\varphi \left({\lambda }_j\right)=\left(\begin{array}{c}\cosh {\mu }_j\\ \frac{i{\lambda }_j}{u}\cosh {\mu }_j+\frac{c_j}{u}\sinh {\mu }_j\end{array}\right),\psi \left({\lambda }_j\right)=\left(\begin{array}{c}\sinh {\mu }_j\\ \frac{i{\lambda }_j}{u}\sinh {\mu }_j+\frac{c_j}{u}\cosh {\mu }_j\end{array}\right),$

with

${\mu }_j=c_j\left(x+\left(uv+2{\lambda }_j^2\right)t\right),c_j=\sqrt{uv+{\lambda }_j^2}\left(1\ge j\le 2N\right).$

thus, ${\delta }_j$ can be written as

${\delta }_j=\frac{i{\lambda }_j}{u}+\frac{c_j}{u}\frac{{\text{tanh}}_{{\mu }_j}-r_j}{1-r_j{\text{tanh}}_{{\mu }_j}},\left(1\ge j\le 2N\right).$ (35)

In what follows, we discuss soliton solutions of system (1) for the case of $N=1$, and $N=2$.

Let $N=1,\lambda ={\lambda }_j\left(j=1,2;j\ne k\right)$, solving the linear algebraic system (10), we have

$A_0=\frac{\text{Δ}{A}_0}{{\text{Δ}}_0},B_0=\frac{\text{Δ}{B}_0}{{\text{Δ}}_0},C_0=\frac{\text{Δ}{C}_0}{{\text{Δ}}_0},D_0=\frac{\text{Δ}{D}_0}{{\text{Δ}}_0},$ (36)

where

${\text{Δ}}_0=\left|\begin{array}{cc}1& {\delta }_1\\ 1& {\delta }_2\end{array}\right|,\text{Δ}{B}_0=\left|\begin{array}{cc}1& -{\lambda }_1\\ 1& -{\lambda }_2\end{array}\right|,\text{Δ}{C}_0=\left|\begin{array}{cc}-{\delta }_1{\lambda }_1& {\delta }_1\\ -{\delta }_2{\lambda }_2& {\delta }_2\end{array}\right|,$

Resorting to expressions (16), (17), a multi-soliton solution of system (1) can be obtained as follows,

$u\left[1\right]=\alpha +i\beta +\frac{2i\left({\lambda }_1-{\lambda }_2\right)}{{\delta }_2-{\delta }_1},v\left[1\right]=-\alpha +i\beta -\frac{2i{\delta }_1{\delta }_2\left({\lambda }_2-{\lambda }_1\right)}{{\delta }_2-{\delta }_1},$ (37)

where

${\delta }_j=\frac{i{\lambda }_j}{u}+\frac{c_j}{u}\frac{{\text{tanh}}_{{\mu }_j}-r_j}{1-r_j{\text{tanh}}_{{\mu }_j}},{\mu }_j=c_j\left(x+\left(uv+2{\lambda }_j^2\right)t\right),c_j=\sqrt{uv+{\lambda }_j^2},$

and $r_j,{\lambda }_j$ are arbitrary parameters.

For the case of $N=2,\lambda ={\lambda }_j\left(j=1,2,3,4\right)$, solving the linear algebraic system (10), we have

$A_1=\frac{\text{Δ}{A}_1}{{\text{Δ}}_1},B_1=\frac{\text{Δ}{B}_1}{{\text{Δ}}_1},C_1=\frac{\text{Δ}{C}_1}{{\text{Δ}}_1},D_1=\frac{\text{Δ}{D}_1}{{\text{Δ}}_1},$ (38)

where

${\text{Δ}}_1=\left|\begin{array}{cccc}1& {\delta }_1& {\lambda }_1& {\lambda }_1{\delta }_1\\ 1& {\delta }_2& {\lambda }_2& {\lambda }_2{\delta }_2\\ 1& {\delta }_3& {\lambda }_3& {\lambda }_3{\delta }_3\\ 1& {\delta }_4& {\lambda }_4& {\lambda }_4{\delta }_4\end{array}\right|,\text{Δ}{B}_1=\left|\begin{array}{cccc}1& {\delta }_1& {\lambda }_1& -{\lambda }_1^2\\ 1& {\delta }_2& {\lambda }_2& -{\lambda }_2^2\\ 1& {\delta }_3& {\lambda }_3& -{\lambda }_3^2\\ 1& {\delta }_4& {\lambda }_4& -{\lambda }_4^2\end{array}\right|,$

$\text{Δ}{C}_1=\left|\begin{array}{cccc}1& {\delta }_1& -{\delta }_1{\lambda }_1^2& {\lambda }_1{\delta }_1\\ 1& {\delta }_2& -{\delta }_2{\lambda }_2^2& {\lambda }_2{\delta }_2\\ 1& {\delta }_3& -{\delta }_3{\lambda }_3^2& {\lambda }_3{\delta }_3\\ 1& {\delta }_4& -{\delta }_4{\lambda }_4^2& {\lambda }_4{\delta }_4\end{array}\right|.$

Resorting to expressions (16) and (17), a multi-soliton solution of system (1) can be obtained as follows,

$u\left[2\right]=\alpha +i\beta +\frac{2i\text{Δ}{B}_1}{{\text{Δ}}_1},v\left[2\right]=-\alpha +i\beta -\frac{2i\text{Δ}{C}_1}{{\text{Δ}}_1}.$ (39)

4. The Complexiton Solutions

If we take the trivial solution $u=v=0$, and choose conjugated spectral parameters as follows,

$\begin{array}{l}{\lambda }_{4j-3}={\alpha }_j+i{\beta }_j={\lambda }_j^{\left(1\right)},{\lambda }_{4j-2}=-{\alpha }_j-i{\beta }_j={\lambda }_j^{\left(2\right)},\\ {\lambda }_{4j-1}={\alpha }_j-i{\beta }_j={\bar{\lambda }}_j^{\left(1\right)},{\lambda }_{4j}=-{\alpha }_j+i{\beta }_j={\bar{\lambda }}_j^{\left(2\right)}.\end{array}$ (40)

where $j=1,2,\cdots$. then the corresponding compatible solution of the lax pairs (2) and (3) can be choosed as

$\begin{array}{l}\varphi \left({\lambda }_{4j-3}\right)=\left(\begin{array}{l}{\varphi }_1\left({\lambda }_{4j-3}\right)\hfill \\ {\varphi }_2\left({\lambda }_{4j-3}\right)\hfill \end{array}\right)=\left(\begin{array}{c}\exp \left({\eta }_j^{-}\right)\left(\cos \left({\xi }_j^{-}\right)+i\sin \left({\xi }_j^{-}\right)\right)\\ \exp \left(-{\eta }_j^{-}\right)\left(\cos \left({\xi }_j^{-}\right)-i\sin \left({\xi }_j^{-}\right)\right)\end{array}\right)\\ \varphi \left({\lambda }_{4j-2}\right)=\left(\begin{array}{l}{\varphi }_1\left({\lambda }_{4j-2}\right)\hfill \\ {\varphi }_2\left({\lambda }_{4j-2}\right)\hfill \end{array}\right)=\left(\begin{array}{c}\exp \left({\eta }_j^{+}\right)\left(\cos \left({\xi }_j^{+}\right)-i\sin \left({\xi }_j^{+}\right)\right)\\ \exp \left(-{\eta }_j^{+}\right)\left(\cos \left({\xi }_j^{+}\right)+i\sin \left({\xi }_j^{+}\right)\right)\end{array}\right)\\ \varphi \left({\lambda }_{4j-1}\right)=\left(\begin{array}{l}{\varphi }_1\left({\lambda }_{4j-1}\right)\hfill \\ {\varphi }_2\left({\lambda }_{4j-1}\right)\hfill \end{array}\right)=\left(\begin{array}{c}\exp \left({\eta }_j^{-}\right)\left(\cos \left({\xi }_j^{-}\right)-i\sin \left({\xi }_j^{-}\right)\right)\\ \exp \left(-{\eta }_j^{-}\right)\left(\cos \left({\xi }_j^{-}\right)+i\sin \left({\xi }_j^{-}\right)\right)\end{array}\right)\\ \varphi \left({\lambda }_{4j}\right)=\left(\begin{array}{l}{\varphi }_1\left({\lambda }_{4j}\right)\hfill \\ {\varphi }_2\left({\lambda }_{4j}\right)\hfill \end{array}\right)=\left(\begin{array}{c}\exp \left({\eta }_j^{+}\right)\left(\cos \left({\xi }_j^{+}\right)+i\sin \left({\xi }_j^{+}\right)\right)\\ \exp \left(-{\eta }_j^{+}\right)\left(\cos \left({\xi }_j^{+}\right)-i\sin \left({\xi }_j^{+}\right)\right)\end{array}\right)\end{array}$ (41)

where

${\eta }_j^{-}=-i{\alpha }_{j}x+\left(6i{\alpha }_j{\beta }_j^2-2i{\alpha }_j^3\right)t,{\eta }_j^{+}=i{\alpha }_{j}x-\left(6i{\alpha }_j{\beta }_j^2-2i{\alpha }_j^3\right)t,$

${\xi }_j^{-}=-i{\beta }_{j}x+\left(6i{\alpha }_j^2{\beta }_j+2i{\beta }_j^3\right)t,{\xi }_j^{+}=-i{\beta }_{j}x+\left(6i{\alpha }_j^2{\beta }_j+2i{\beta }_j^3\right)t,$

and ${\alpha }_j,{\beta }_j$ are arbitrary real constants.

For the case of $r_j=0,1,\left(1\le j\le 4\right)$, according to (11), we have

$\begin{array}{l}{\delta }_{4j-3}=\frac{{\varphi }_2\left({\lambda }_{4j-3}\right)}{{\varphi }_1\left({\lambda }_{4j-3}\right)}=\exp \left(-2{\eta }_j^{-}\right)\left(\cos 2\left({\xi }_j^{-}\right)-i\sin 2\left({\xi }_j^{-}\right)\right)={\delta }_j^{\left(1\right)},\\ {\delta }_{4j-2}=\frac{{\varphi }_2\left({\lambda }_{4j-2}\right)}{{\varphi }_1\left({\lambda }_{4j-2}\right)}=\exp \left(-2{\eta }_j^{+}\right)\left(\cos 2\left({\xi }_j^{+}\right)+i\sin 2\left({\xi }_j^{+}\right)\right)={\delta }_j^{\left(2\right)},\\ {\delta }_{4j-1}=\frac{{\varphi }_2\left({\lambda }_{4j-1}\right)}{{\varphi }_1\left({\lambda }_{4j-1}\right)}=\exp \left(-2{\eta }_j^{-}\right)\left(\cos 2\left({\xi }_j^{-}\right)+i\sin 2\left({\xi }_j^{-}\right)\right)={\bar{\delta }}_j^{\left(1\right)},\\ {\delta }_{4j}=\frac{{\varphi }_2\left({\lambda }_{4j}\right)}{{\varphi }_1\left({\lambda }_{4j}\right)}=\exp \left(-2{\eta }_j^{+}\right)\left(\cos 2\left({\xi }_j^{+}\right)-i\sin 2\left({\xi }_j^{+}\right)\right)={\bar{\delta }}_j^{\left(2\right)}.\end{array}$ (42)

Resorting to expressions (40), (42) and DT (16) and (17), we can obtain the complexiton solutions of system (1),

$\begin{array}{l}\bar{u}\left[2N\right]=u+2i\frac{\text{Δ}{B}_{2N-1}}{{\text{Δ}}_{2N-1}},\\ \bar{v}\left[2N\right]=v-2i\frac{\text{Δ}{C}_{2N-1}}{{\text{Δ}}_{2N-1}},\end{array}$ (43)

where

${\Delta }_{2N-1}=\det {\left[{\sigma }_1^{\left(1\right)},{\sigma }_1^{\left(2\right)},\overline{{\sigma }_1^{\left(1\right)}},\overline{{\sigma }_1^{\left(2\right)}},\cdots ,{\sigma }_N^{\left(1\right)},{\sigma }_N^{\left(2\right)},\overline{{\sigma }_N^{\left(1\right)}},\overline{{\sigma }_N^{\left(2\right)}}\right]}^{\text{T}},$

$\Delta B_{2N-1}=\det {\left[b_1^{\left(1\right)},b_1^{\left(2\right)},\overline{b_1^{\left(1\right)}},\overline{b_1^{\left(2\right)}},\cdots ,b_N^{\left(1\right)},b_N^{\left(2\right)},\overline{b_N^{\left(1\right)}},\overline{b_N^{\left(2\right)}}\right]}^{\text{T}},$

$\Delta C_{2N-1}=\det {\left[c_1^{\left(1\right)},c_1^{\left(2\right)},\overline{c_1^{\left(1\right)}},\overline{c_1^{\left(2\right)}},\cdots ,c_N^{\left(1\right)},c_N^{\left(2\right)},\overline{c_N^{\left(1\right)}},\overline{c_N^{\left(2\right)}}\right]}^{\text{T}},$

with

$\begin{array}{l}{\sigma }_j^{\left(l\right)}=\left(1,{\delta }_j^{\left(l\right)},{\lambda }_j^l,{\delta }_j^{\left(l\right)}{\lambda }_j^{\left(l\right)},\cdots ,{\lambda }_j^{{\left(l\right)}^{2N-2}},{\delta }_j^{\left(l\right)}{\lambda }_j^{{\left(l\right)}^{2N-2}},{\lambda }_j^{{\left(l\right)}^{2N-1}},{\delta }_j^{\left(l\right)}{\lambda }_j^{{\left(l\right)}^{2N-1}}\right),\\ b_j^{\left(l\right)}=\left(1,{\delta }_1^{\left(l\right)},{\lambda }_1^{\left(l\right)},-{\lambda }_1^{{\left(l\right)}^2},\cdots ,{\lambda }_j^{{\left(l\right)}^{2N-2}},{\delta }_j^{\left(l\right)}{\lambda }_j^{{\left(l\right)}^{2N-2}},{\lambda }_j^{{\left(l\right)}^{2N-1}},-{\lambda }_j^{{\left(l\right)}^{2N}}\right),\\ c_j^{\left(l\right)}=\left(1,{\delta }_1^{\left(l\right)},-{\delta }_1^{\left(l\right)}{\lambda }_1^{{\left(l\right)}^2},{\delta }_1^{\left(l\right)}{\lambda }_1^{\left(l\right)},\cdots ,{\lambda }_j^{{\left(l\right)}^{2N-2}},{\delta }_j^{\left(l\right)}{\lambda }_j^{{\left(l\right)}^{2N-2}},-{\delta }_j^{\left(l\right)}{\lambda }_j^{{\left(l\right)}^{2N}},{\delta }_j^{\left(l\right)}{\lambda }_j^{{\left(l\right)}^{2N-1}}\right),\end{array}$ (44)

and $\overline{{\sigma }_j^{\left(1\right)}},\overline{b_j^{\left(1\right)}},\overline{c_j^{\left(1\right)}}$ are conjugated functions of ${\sigma }_j^{\left(l\right)},b_j^{\left(l\right)},c_j^{\left(l\right)},\left(l=1,2\right)$.

According to properties of determinant, N-complexiton solutions of system (1) can be obtained as follows.

$\begin{array}{l}\bar{u}=u+2i\frac{\text{Λ}{B}_{2N-1}}{{\text{Λ}}_{2N-1}},\\ \bar{v}=v-2i\frac{\text{Λ}{C}_{2N-1}}{{\text{Λ}}_{2N-1}},\end{array}$ (45)

where

${\Lambda }_{2N-1}=\det {\left[\mathrm{Re}{\sigma }_1^{\left(1\right)},\mathrm{Re}{\sigma }_1^{\left(2\right)},\mathrm{Im}{\sigma }_1^{\left(1\right)},\mathrm{Im}{\sigma }_1^{\left(2\right)},\cdots ,\mathrm{Re}{\sigma }_N^{\left(1\right)},\mathrm{Re}{\sigma }_N^{\left(2\right)},\mathrm{Im}{\sigma }_N^{\left(1\right)},\mathrm{Im}{\sigma }_N^{\left(2\right)}\right]}^{\text{T}},$

${\Lambda }_{B_{2N-1}}=\det {\left[\mathrm{Re}{b}_1^{\left(1\right)},\mathrm{Re}{b}_1^{\left(2\right)},\mathrm{Im}{b}_1^{\left(1\right)},\mathrm{Im}{b}_1^{\left(2\right)},\cdots ,\mathrm{Re}{b}_N^{\left(1\right)},\mathrm{Re}{b}_N^{\left(2\right)},\mathrm{Im}{b}_N^{\left(1\right)},\mathrm{Im}{b}_N^{\left(2\right)}\right]}^{\text{T}},$

${\Lambda }_{C_{2N-1}}=\det {\left[\mathrm{Re}{c}_1^{\left(1\right)},\mathrm{Re}{c}_1^{\left(2\right)},\mathrm{Im}{c}_1^{\left(1\right)},\mathrm{Im}{c}_1^{\left(2\right)},\cdots ,\mathrm{Re}{c}_N^{\left(1\right)},\mathrm{Re}{c}_N^{\left(2\right)},\mathrm{Im}{c}_N^{\left(1\right)},\mathrm{Im}{c}_N^{\left(2\right)}\right]}^{\text{T}},$

with ${\sigma }_j^{\left(l\right)},b_j^{\left(l\right)},c_j^{\left(l\right)},\left(l=1,2\right)$ are given by (44).

For simplicity, we shall discuss complexiton solutions of system (1) with a special case $j=1$, which we called 1-complexiton solutions of system (1).

$\begin{array}{l}\bar{u}=u+2i\frac{{\text{Λ}}_{B_1}}{{\text{Λ}}_1},\\ \bar{v}=v-2i\frac{{\text{Λ}}_{C_1}}{{\text{Λ}}_1},\end{array}$ (46)

where

${\Lambda }_1=\det {\left[\mathrm{Re}{\sigma }_1^{\left(1\right)},\mathrm{Re}{\sigma }_1^{\left(2\right)},\mathrm{Im}{\sigma }_1^{\left(1\right)},\mathrm{Im}{\sigma }_1^{\left(2\right)}\right]}^{\text{T}},$

${\Lambda }_{B_1}=\det {\left[\mathrm{Re}{b}_1^{\left(1\right)},\mathrm{Re}{b}_1^{\left(2\right)},\mathrm{Im}{b}_1^{\left(1\right)},\mathrm{Im}{b}_1^{\left(2\right)}\right]}^{\text{T}},$

${\Lambda }_{C_1}=\det {\left[\mathrm{Re}{c}_1^{\left(1\right)},\mathrm{Re}{c}_1^{\left(2\right)},\mathrm{Im}{c}_1^{\left(1\right)},\mathrm{Im}{c}_1^{\left(2\right)}\right]}^{\text{T}},$

with

${\sigma }_1^{\left(l\right)}=\left(1,{\delta }_1^{\left(l\right)},{\lambda }_1^{\left(l\right)},{\delta }_1^{\left(l\right)}{\lambda }_1^{\left(l\right)}\right),$

$b_1^{\left(l\right)}=\left(1,{\delta }_1^{\left(l\right)},{\lambda }_1^{\left(l\right)},-{\lambda }_1^{{\left(l\right)}^2}\right),$

$c_1^{\left(l\right)}=\left(1,{\delta }_1^{\left(l\right)},-{\delta }_1^{\left(l\right)}{\lambda }_1^{{\left(l\right)}^2},{\delta }_1^{\left(l\right)}{\lambda }_1^{\left(l\right)}\right),$

$\left(l=1,2\right),{\delta }_1^{\left(l\right)}$ are given by (42), and ${\lambda }_1^{\left(l\right)}$ are given by (40).

5. Conclusion and Suggestions

In this paper, an explicit N-fold Darboux transformation is constructed for the coupled mKdV equation, and through these transformations, the determinant forms of the multi-soliton and complexiton solutions of system (1) are obtained.

Acknowledgements

This work is supported by the NSFC (11975306), the Higher Educational Scientific Research Projects of Inner Mongolia Autonomous Region (NJZY20248) and the Scientific Research Project of Jining Normal University (Grant No. jsky2024008).

References

1. Liu, H., Tian, B., Chen, Y., Cheng, C. and Gao, X. (2024) N-Soliton, Hth-Order Breather, Hybrid and Multi-Pole Solutions for a Generalized Variable-Coefficient Gardner Equation with an External Force in a Plasma or Fluid. Nonlinear Dynamics, 113, 3655-3672. >https://doi.org/10.1007/s11071-024-10397-1

2. Zhao, Y., Wang, Y., Yang, S., Zhang, X. and Chen, Y. (2024) Soliton, Breather and Rogue Wave Solutions of the Higher-Order Modified Gerdjikov-Ivanov Equation. Chaos, Solitons&Fractals, 185, Article ID: 115147. >https://doi.org/10.1016/j.chaos.2024.115147

3. Zhai, Y., Ji, T. and Geng, X. (2021) Coupled Derivative Nonlinear Schrödinger III Equation: Darboux Transformation and Higher-Order Rogue Waves in a Two-Mode Nonlinear Fiber. Applied Mathematics and Computation, 411, Article ID: 126551. >https://doi.org/10.1016/j.amc.2021.126551

4. Wu, X., Gao, Y., Yu, X., Ding, C., Hu, L. and Li, L. (2022) Binary Darboux Transformation, Solitons, Periodic Waves and Modulation Instability for a Nonlocal Lakshmanan-Porsezian-Daniel Equation. Wave Motion, 114, Article ID: 103036. >https://doi.org/10.1016/j.wavemoti.2022.103036

5. Zhang, H., Zhang, Y., Feng, B. and Afzal, F. (2024) The Exact Solutions for the Non-Isospectral Kaup-Newell Hierarchy via the Inverse Scattering Transform. Applied Mathematics Letters, 152, Article ID: 109011. >https://doi.org/10.1016/j.aml.2024.109011

6. Li, Q., Chen, D., Zhang, J. and Chen, S. (2012) Solving the Non-Isospectral Ablowitz-Ladik Hierarchy via the Inverse Scattering Transform and Reductions. Chaos, Solitons&Fractals, 45, 1479-1485. >https://doi.org/10.1016/j.chaos.2012.08.010

7. Zhang, X. and Chen, Y. (2019) Inverse Scattering Transformation for Generalized Nonlinear Schrödinger Equation. Applied Mathematics Letters, 98, 306-313. >https://doi.org/10.1016/j.aml.2019.06.014

8. Yang, X., Xue, L. and Liu, Q.P. (2024) Supersymmetric KdV Equation and Its Darboux-Bäcklund Transformations. Communications in Theoretical Physics, 76, Article ID: 115002. >https://doi.org/10.1088/1572-9494/ad6a04

9. Peng, W., Tian, S., Wang, X., Zhang, T. and Fang, Y. (2019) Riemann-Hilbert Method and Multi-Soliton Solutions for Three-Component Coupled Nonlinear Schrödinger Equations. Journal of Geometry and Physics, 146, Article ID: 103508. >https://doi.org/10.1016/j.geomphys.2019.103508

10. Zhang, H., Pei, Z. and Ma, W. (2019) Riemann-Hilbert Approach for a Coherently-Coupled Nonlinear Schrödinger System Associated with a 4 × 4 Matrix Spectral Problem. Chaos, Solitons&Fractals, 123, 429-434. >https://doi.org/10.1016/j.chaos.2019.04.017

11. Zhang, H., Zhang, M. and Hu, R. (2018) Darboux Transformation and Soliton Solutions in the Parity-Time-Symmetric Nonlocal Vector Nonlinear Schrödinger Equation. Applied Mathematics Letters, 76, 170-174. >https://doi.org/10.1016/j.aml.2017.09.002

12. Lei, W., Yi-Tian, G., Xiao-Ling, G., De-Xin, M., Xing, L. and Xin, Y. (2010) N-Fold Darboux Transformation and Bidirectional Solitons for Whitham-Broer-Kaup Model in Shallow Water. Communications in Theoretical Physics, 53, 413-422. >https://doi.org/10.1088/0253-6102/53/3/03

13. Raslan, K.R. (2004) The Decomposition Method for a Hirota-Satsuma Coupled KdV Equation and a Coupled MKdV Equation. International Journal of Computer Mathematics, 81, 1497-1505. >https://doi.org/10.1080/0020716042000261405

14. Fan, E. (2001) Soliton Solutions for a Generalized Hirota-Satsuma Coupled KdV Equation and a Coupled MKdV Equation. Physics Letters A, 282, 18-22. >https://doi.org/10.1016/s0375-9601(01)00161-x

15. Yong, C., Zhen-Ya, Y., Biao, L. and Hong-Qing, Z. (2003) New Explicit Exact Solutions for a Generalized Hirota-Satsuma Coupled KdV System and a Coupled MKdV Equation. Chinese Physics, 12, Article 1. >https://doi.org/10.1088/1009-1963/12/1/301