Semirational solutions to the coupled Fokas–Lenells equations

Abstract

The Darboux transformation for the coupled Fokas–Lenells equations and the special vector solution of the corresponding Lax pair are constructed. Utilizing a limiting process, some novel high-order semirational solutions of the coupled system are given. They include high-order rogue waves interacting with multi-bright or dark solitons, and high-order rogue waves interacting with multi-breathers. Also, the dynamic structures of the first- and second-order semirational solutions are discussed. Furthermore, it is shown that the free parameter $\gamma$ in the special vector solution can influence the interactional processes (fusion or separation) among different nonlinear waves. Compared to the uncoupled systems, there may exist more abundant and interesting solutions in the coupled ones.

1 Introduction

It is well known that the nonlinear localized waves have been widely researched in a lot of documents, which usually include rogue waves (RWs) [1, 2], breathers [3,4,5], solitons [6, 7] and lump solutions [8, 9]. RWs are commonly defined as the gigantic waves owning extreme amplitudes and seem to appear from nowhere and disappear without a trace [10]. Moreover, breathers are periodic in space (Kuznetsov-Ma breathers) [3] or time (Akhmediev breathers) [4, 5] or both. Owing to the balance between nonlinearity and dispersion in the nonlinear models, solitons can be generated [7]. Additionally, they always keep their amplitudes and speeds unchanged during propagation. Furthermore, lump solutions are special rational localization solutions, which propagate in all directions both in time and in space [8]. In reality, actual wave dynamics is a superposition of various types of nonlinear waves [11,12,13].

In recent years, there have been a variety of hybrid solutions among different nonlinear waves researched in many physical models [14,15,16,17]. The higher-order semirational solutions including higher-order RWs and higher-order breathers were constructed in the derivative NLS equation [17]. The RW was constructed by the interaction between lump soliton and a pair of resonance kink strip solitons in ($2+1$)-dimensional Korteweg–de Vries (KdV) equation [18]. In [13] and [19], some novel semirational solutions were constructed in local and nonlocal Davey–Stewartson (DS) equations, respectively. Using the Darboux transformation (DT) technique, the hybrid solutions simultaneously including RWs, dark and anti-dark rational traveling waves are exhibited in the nonlocal DS equations [20]. From the mathematical expressions, semirational solution can be defined as a combination of rational and exponential functions [17, 19]. In this paper, we focus on semirational solutions of the following coupled Fokas–Lenells (FL) equations [21,22,23,24]

$$\begin{aligned} \begin{aligned}&u_{xt}+u+i\left( |u|^2+\frac{1}{2}\sigma |v|^2\right) u_x+\frac{i}{2}\sigma uv^{*}v_x=0,\\&v_{xt}+v+i\left( \sigma |v|^2+\frac{1}{2}|u|^2\right) v_x+\frac{i}{2}vu^{*}u_x=0. \end{aligned} \end{aligned}$$

(1)

Here, $\sigma =\pm \,1$ and the symbol $*$ denotes complex conjugation, and u and v are all complex functions of x and t. Besides, $u^{*}$ and $v^{*}$ represent the complex conjugations of u and v, respectively, and “i” is the imaginary unit. The subscripted variables x and t in Eq. (1) denote the corresponding partial differentiation. The symbol “||” is the modulus of a complex function. As the relationship exists between the Camassa–Holm equation and the Korteweg–de Vries (KdV) equation, the FL equation has similar relationship with the nonlinear Schrödinger (NLS) equation [25, 26]. Actually, the FL equation is the first negative flow of the hierarchy for the derivative NLS equation [24, 27].

In this paragraph, some research history on the FL system will be introduced. The single-component FL equation was first derived in [25] by Fokas. Utilizing the bi-Hamiltonian structure, the Lax pair and conservation laws of the FL system were constructed by Lenells and Fokas [26]. In addition, the high-order RW solutions of the uncoupled FL equation were constructed through the DT technique [27]. In [28], multi-soliton solutions of the single-component FL equation were obtained through DT method. There are many other papers on single-component FL equation, such as dark soliton [29], algebraic geometry solutions [30] and long-time asymptotic behavior of the solution [31]. Similar to the multi-component NLS equations [32], the coupled FL systems are supposed to own more novel and abundant solutions than the ones in the single-component FL equation. With the aid of the spectral gradient method, the coupled FL system was rediscovered. Besides, its Lax pair and conservation laws were also obtained [21]. The coupled FL equations (1) were derived in [23] with $\alpha =-\,3,\beta =\tfrac{1}{4}$ in the operator L. The general soliton solutions of the coupled system (1) were constructed by DT method [24], such as bright–dark soliton, dark–anti-dark soliton, breather-like soliton and multi-bright (or dark) soliton. In [22], the authors obtained soliton, breathers and RWs for the coupled FL equations. As pointed out in [23], the coupled FL system (1) is equivalent to the coupled FL system in [22] by a gauge transformation. Higher-order RWs of the coupled FL equations were constructed by DT method [33]. Using two integration schemes, optical solitons of the coupled FL equations with differential group delay were given in [34]. The initial boundary value problem for the coupled Fokas–Lenells equations on the half-line was considered by the Riemann–Hilbert approach in [35].

Baronio et al. [11] reported that there existed some semirational solutions in the coupled NLS system, which include the first-order RW interacting with one bright or dark soliton and one breather interacting with the first-order RW. Meanwhile, the experimental conditions to observe these kinds of semirational solutions were given in [11]. RWs on a multi-soliton were obtained in the vector NLS equations by Darboux dressing method [32]. As far as we know, there have not been any reports on semirational solutions of the coupled FL equations (1). It is very necessary to investigate some novel semirational solution of Eq. (1), which includes high-order RWs interacting with multi-bright (or dark) solitons and multi-breathers interacting with high-order RWs.

From the special vector solutions of the Lax pair of the coupled FL equations (1), the concrete expressions of the high-order semirational solutions are given in determinant forms by DT technique [36]. When $\gamma =0$ (the free parameter in the vector solutions of the Lax pair), the semirational solutions degenerate to the rational ones and they are all RWs. When $\gamma \ne 0$, the semirational solutions can be mainly classified as two types: (1) One component is high-order RWs interacting with multi-bright solitons, and the other one is high-order RWs interacting with multi-dark solitons; (2) two components are all high-order RWs interacting with multi-breathers. By increasing the absolute values of $\gamma$, different nonlinear waves can merge with each other.

The paper is organized as follows. In Sect. 2, the Lax pair and DT of the coupled FL equations are constructed. In Sect. 3, the special vector solutions of the Lax pair are skillfully given; then, some semirational solutions are obtained by DT method. Besides, some dynamics of the first- and second-order semirational solutions are discussed in detail. The last section includes several conclusions and discussions.

2 Lax pair and Darboux transformation for the coupled FL equations

The Lax pair of the coupled FL equations (1) can be given as [21,22,23,24]

$$\begin{aligned} \varPhi _x= & {} (i\lambda ^{-2}J+\lambda ^{-1}P_x)\varPhi =U\varPhi , \end{aligned}$$

(2)

$$\begin{aligned} \varPhi _t= & {} i\left( \frac{1}{4}\lambda ^2J+\frac{1}{2}J\left( P^2-\lambda P\right) \right) \varPhi =V\varPhi , \end{aligned}$$

(3)

with

$$\begin{aligned} J= & {} \begin{pmatrix}1&{}\quad 0&{}\quad 0\\ 0&{}\quad -1&{}\quad 0\\ 0&{}\quad 0&{}\quad -1\end{pmatrix},\quad P=\begin{pmatrix}0&{}\quad u^{*}&{}\quad \sigma v^{*}\\ u&{}\quad 0&{}\quad 0\\ v&{}\quad 0&{}\quad 0 \end{pmatrix},\\ U= & {} i\lambda ^{-2}J+\lambda ^{-1}P_x,\quad V=i(\frac{1}{4}\lambda ^2J\\&+\frac{1}{2}J(P^2-\lambda P)). \end{aligned}$$

Here, “i” is the imaginary unit and the symbol “$^*$” indicates the conjugation of a vector or matrix. Besides, $\varPhi (x,t)=(\psi (x,t),\chi (x,t),\phi (x,t))^T$ (“T” denotes the transposition of a vector or matrix) and $\lambda$ is a spectral parameter. The above-mentioned U, V, P and J are all $3\times 3$ matrices. Additionally, we can derive the coupled FL system (1) through the following compatibility relationship $U_t{-}V_x{+}UV{-}VU=0$.

Based on the DT constructed in [23, 24], the first-step fundamental DT of the coupled FL equations (1) can be expressed as follows

$$\begin{aligned} T[1]= & {} I+\frac{B_1}{\lambda -\lambda _1^{*}}-\frac{JB_1J}{\lambda +\lambda _1^{*}}, \end{aligned}$$

(4)

$$\begin{aligned} u[1]= & {} u[0]+\frac{2\chi _1\psi _1^{*}}{\varGamma _1}, \end{aligned}$$

(5)

$$\begin{aligned} v[1]= & {} v[0]+\frac{2\phi _1\psi _1^{*}}{\varGamma _1}, \end{aligned}$$

(6)

with

$$\begin{aligned} \varGamma _1= & {} \dfrac{2[\lambda _1|\psi _1|^2-\lambda _1^{*} (|\chi _1|^2+|\phi _1|^2)]}{\lambda _1^{*2}-\lambda _1^{2}},\quad \\ B_1= & {} L_1|y_1{>}{<}y_1|K , \\ L_1= & {} diag(-\varGamma _1^{*-1},\varGamma _1^{-1},\varGamma _1^{-1}),~\\&K=diag(1,-1,-\sigma ), \end{aligned}$$

where $\sigma =\pm \,1$ and the vector function $\varPhi _1(x,t)=(\psi _1(x,t),\chi _1(x,t),\phi _1(x,t))^{T}$ is the special solution of Lax pair (2)–(3) with $\lambda =\lambda _1$. Here, u[0] and v[0] are the seed solutions of Eq. (1); thus, u[1] and v[1] denote the first-step solutions of Eq. (1) through the above first-step fundamental DT. In the above expressions, I is the $3\times 3$ identity matrix, and $B_1$ is a $3\times 3$ matrix which is written as $B_1=L_1|y_1{>}{<}y_1|K$ and $|y_1{>}=\varPhi _1(x,t)=(\psi _1(x,t),\chi _1(x,t),\phi _1(x,t))^T$. Additionally, $\psi _1^{*}(x,t)$ and $\varGamma _1^{*}$ are the complex conjugations of $\psi (x,t)$ and $\varGamma$, respectively. Here, the symbol “(x, t)” is omitted in the expression $\phi$. In the whole contents, the symbol “diag” denotes $3\times 3$ diagonal matrix and “$|>$” represents a column vector; then, “$< |$” indicates the Hermite conjugation of the corresponding column vector. For example, $|y_1{>}=\varPhi _1=(\psi _1,\chi _1,\phi _1)^{T}$ and ${<}y_1|=|y_1{>}^{\dag }=(\psi _1^{*},\chi _1^{*},\phi _1^{*})$; the symbol $^{\dag }$ denotes Hermite conjugation.

Based on the related results received in [23, 24], the inverse of Darboux matrix T[1] can be written as

$$\begin{aligned} T[1]^{-1}=I+\frac{KB_1^{\dag }K}{\lambda -\lambda _1}-\frac{JKB_1^{\dag }KJ}{\lambda +\lambda _1}, \end{aligned}$$

(7)

where $B_1^{\dag }=(L_1|y_1{>}{<}y_1|K)^{\dag }$ and $^{\dag }$ represents Hermite conjugation in the whole contents. Enlightened by the method to construct the N-step DT for the derivative NLS equation PROPOSITION 2. and THEOREM 2. in [36], we give the following proposition and theorem.

Proposition 1

The N-step DT for the coupled FL equations (1) can be written as follows

$$\begin{aligned} T_N= & {} T[N]T[N-1]\ldots T[1]=I\nonumber \\&+\sum _{i=1}^{N}\left( \frac{C_i}{\lambda -\lambda _i^{*}}- \frac{JC_iJ}{\lambda +\lambda _i^{*}}\right) , \end{aligned}$$

(8)

and the corresponding N-step inverse of DT for the coupled FL equations (1) can also be written as

$$\begin{aligned} T_N^{-1}= & {} T[1]^{-1}T[2]^{-1}\ldots T[N]^{-1}=I\nonumber \\&+\sum _{i=1}^{N}\left( \frac{KD_i^{\dag }K}{ \lambda -\lambda _i}-\frac{JKD_i^{\dag }KJ}{\lambda +\lambda _i}\right) . \end{aligned}$$

(9)

Here, $C_i$ and $D_i$ are all $3\times 3$ undermined matrices,

$$\begin{aligned}&T[i]=I+\frac{B_i}{\lambda -\lambda _i^{*}}-\frac{JB_iJ}{\lambda +\lambda _i^{*}},\\&T[i]^{-1}=I+\frac{KB_i^{\dag }K}{\lambda -\lambda _i}-\frac{JKB_i^{\dag }KJ}{\lambda + \lambda _i},\\&B_i=L_i|y_i{>}{<}y_i|K,\\&\varGamma _i=\dfrac{2[\lambda _i|\psi _i|^2- \lambda _i^{*}(|\chi _i|^2+|\phi _i|^2)]}{\lambda _i^{*2}-\lambda _i^{2}}, \\&L_i=diag(-\varGamma _i^{*-1},\varGamma _i^{-1},\varGamma _i^{-1})~(1\le i\le N); \end{aligned}$$

the column vector $|y_i{>}=\varPhi _i=(\psi _i,\chi _i,\phi _i)^{T}$ is the special solution of Lax pair (2)–(3) with $\lambda =\lambda _i$ and ${<}y_i|=|y_i{>}^{\dag }=(\psi _i^{*},\chi _i^{*},\phi _i^{*})$. In Proposition 1, the N-step DT and the corresponding N-step inverse of DT are only written in compact formats using the undermined matrices $C_i$ and $D_i$. Utilizing the formulas (8) and (9), these two matrices $C_i$ and $D_i$ can be determined in the following contents.

Proof

From the N-step iterative formula, the expression of $T_N$ can be directly written as

$$\begin{aligned} T_N= & {} T[N]T[N-1]\ldots T[1]\nonumber \\= & {} \left( I{+}\frac{B_N}{\lambda {-}\lambda _N^{*}}{-}\frac{JB_NJ}{\lambda {+}\lambda _N^{*}}\right) \left( I{+}\frac{B_{N-1}}{\lambda {-}\lambda _{N{-}1}^{*}} \right. \nonumber \\&-\left. \frac{JB_{N-1}J}{\lambda {+} \lambda _{N{-}1}^{*}}\right) \ldots \left( I{+}\frac{B_1}{\lambda {-}\lambda _1^{*}}{-} \frac{JB_1J}{\lambda {+}\lambda _1^{*}}\right) \nonumber \\= & {} I+\sum _{i=1}^{N}\left( \frac{C_i}{\lambda - \lambda _i^{*}}+\frac{F_i}{\lambda +\lambda _i^{*}}\right) , \end{aligned}$$

(10)

Taking the residues for both sides of Eq. (10) with $\lambda =\lambda _i^{*}$ and $\lambda =-\lambda _i^{*}$, respectively, we can get the two expressions of residues as

$$\begin{aligned}&\mathrm{Res}|_{\lambda =\lambda _i^{*}}T_N=\left( I{+}\frac{B_N}{\lambda _i^{*}{-}\lambda _N^{*}}{-} \frac{JB_NJ}{\lambda _i^{*}{+}\lambda _N^{*}}\right) \ldots \nonumber \\&\quad B_i \ldots \left( I{+}\frac{B_1}{\lambda _i^{*}{-}\lambda _1^{*}}{-}\frac{JB_1J}{\lambda _i^{*}{+} \lambda _1^{*}}\right) =C_i, \end{aligned}$$

(11)

and

$$\begin{aligned}&\mathrm{Res}|_{\lambda =-\lambda _i^{*}}T_N=-\left( I{+}\frac{B_N}{-\lambda _i^{*}{-}\lambda _N^{*}}{-} \frac{JB_NJ}{-\lambda _i^{*}{+}\lambda _N^{*}}\right) \ldots \nonumber \\&\quad JB_iJ \ldots \left( I{+}\frac{B_1}{-\lambda _i^{*}{-}\lambda _1^{*}}{-}\frac{JB_1J}{-\lambda _i^{*}{+} \lambda _1^{*}}\right) =F_i. \end{aligned}$$

(12)

Here, the symbol “Res” indicates residue of a matrix. From Eq. (12), we can directly calculate that $-J\mathrm{Res}|_{\lambda =-\lambda _i^{*}}T_NJ=\mathrm{Res}|_{\lambda =\lambda _i^{*}}T_N$; namely, $F_i=-JC_iJ$, and Eq. (8) is proved. Similarly, Eq. (9) can be also proved.

This completes the proof. $\square$

Setting $\varPhi _i=(\psi _i,\chi _i,\phi _i)^T$ is the special solutions of Lax pair (2)–(3) with $u=u[0]$, $v=v[0]$ and $\lambda =\lambda _i$ $(1\le i \le N)$, besides, $\varPhi _i$, $\psi _i$,$\chi _i$ and $\phi _i$ are all complex functions of x and t. Furthermore, the transformations between the new potential functions u[N], v[N] and the seed solutions u[0], v[0] can be constructed by the N-step DT $T_N$, which is shown in the following theorem.

Theorem 1

The transformations for the potential functions in the N-step DT can be expressed in the following determinant forms

$$\begin{aligned} u[N]= & {} u[0]-2\frac{\begin{vmatrix}M&Y_1^{\dag }\\ Y_2&0 \end{vmatrix}}{|M|},\nonumber \\ v[N]= & {} v[0]-2\frac{\begin{vmatrix}M&Y_1^{\dag }\\ Y_3&0 \end{vmatrix}}{|M|}, \end{aligned}$$

(13)

where $M=(m_{ij})_{N\times N}$,

$$\begin{aligned} Y_1= & {} (\psi _1,\psi _2,\ldots ,\psi _N), \quad Y_1^{\dag }=(\psi _1^{*},\psi _2^{*},\ldots ,\psi _N^{*})^{T}, \\ Y_2= & {} (\chi _1,\chi _2,\ldots ,\chi _N), Y_3=(\phi _1,\phi _2,\ldots ,\phi _N),\\ m_{ij}= & {} \frac{\varPhi _i^{\dag }K\varPhi _j}{\lambda _i^{*}-\lambda _j}{-} \frac{\varPhi _i^{\dag }KJ\varPhi _j}{\lambda _i^{*}+\lambda _j}~(1\le i,j \le N), \end{aligned}$$

where $\varPhi _i=(\psi _i,\chi _i,\phi _i)^{T}$ and $\varPhi _j=(\psi _j,\chi _j,\phi _j)^{T}$ are the corresponding column vector solutions of the Lax pair (2)–(3) with $\lambda =\lambda _i$ and $\lambda =\lambda _j$, respectively.

Proof

Proposition 1 indicates that $C_i=Res|_{\lambda =-\lambda _i^{*}}T_N=(I{+}\frac{B_N}{\lambda _i^{*}{-}\lambda _N^{*}}{-} \frac{JB_NJ}{\lambda _i^{*}{+}\lambda _N^{*}}) \ldots B_i$ $\ldots (I{+}\frac{B_1}{\lambda _i^{*}{-}\lambda _1^{*}}{-}\frac{JB_1J}{\lambda _i^{*}{+} \lambda _1^{*}})$, and the ranks of $C_i$ admit the following inequality $1\le r(C_i)\le$ $min(r(B_i),r(I{+}\frac{B_j}{\lambda _i^{*}{-}\lambda _j^{*}}{-}\frac{JB_jJ}{\lambda _i^{*}{+}\lambda _j^{*}}))~(j\ne i,1\le i,j\le N)$; here the symbol “r” represents the rank of a matrix. Since $B_i=L_i|y_i{>}{<}y_i|K$, we can directly calculate that $r(B_i)=1$ and then $r(C_i)=1$.

From $r(C_i)=1$ and Eq. (11), we can directly calculate that $C_i=|x_i{>}{<}y_i|$, where $|x_i{>}$ is an undetermined 3-tuple column vector and ${<}y_i|=|y_i{>}^{\dag }=(\psi _i^{*},\chi _i^{*},\phi _i^{*})$. We consider the conjugate form of the linear system (2)–(3), which can be written as

$$\begin{aligned} \varPsi _x=-\varPsi U, \quad \varPsi _t=-\varPsi V. \end{aligned}$$

(14)

Here, $\varPsi$ is a 3-tuple row vector and U, V own the same forms of the ones in the Lax pair (2)–(3).

It can be directly calculated that U and V in the Lax pair (2)–(3) have the following symmetry:

$$\begin{aligned} U(\lambda )=-KU(\lambda ^{*})^{\dag }K,\quad V(\lambda )=-KV(\lambda ^{*})^{\dag }K. \end{aligned}$$

(15)

From Eqs. (14) and (15), we can find that if $\varPhi _i$ is a solution for the Lax pair(2)–(3) with $\lambda =\lambda _i$, then $\varPhi _i^{\dag }K$ is a solution for the conjugated system (14) with $\lambda =\lambda _i^{*}$.

Additionally, the equality holds $T_NT_N^{-1}=I$, then we can have the following residue of $T_NT_N^{-1}$ that $Res|_{\lambda =\lambda _i^{*}}T_NT_N^{-1}=0$. It can be rewritten as follows

$$\begin{aligned} {<}y_i|T_N^{-1}|_{\lambda =\lambda _i^{*}}=0. \end{aligned}$$

(16)

Besides, $\varPhi _i^{\dag }K$ is the special solution of (14) with $\lambda =\lambda _i^{*}$ and admits the following equality

$$\begin{aligned} \varPhi _i^{\dag }KT_N^{-1}|_{\lambda =\lambda _i^{*}}=0. \end{aligned}$$

(17)

Comparing Eqs. (16) and (17), we can choose that ${<}y_i|=\varPhi _i^{\dag }K$.

In order to calculate the concrete expressions of $C_i=|x_i{>}{<}y_i|=|x_i{>}\varPhi _i^{\dag }K$, we should first derive out the undetermined 3-tuple column vector $|x_i{>}$. Since $T_N|_{\lambda =\lambda _j}\varPhi _j=0~(1\le j\le N)$, one can get the following equality by N-step DT formula (8)

$$\begin{aligned}&\varPhi _j+\sum _{i=1}^{N}\left( \frac{|x_i{>}\varPhi _i^{\dag }K\varPhi _j}{\lambda _j-\lambda _i^{*}}- \frac{J|x_i{>}\varPhi _i^{\dag }KJ\varPhi _j}{\lambda _j+\lambda _i^{*}}\right) =0 \nonumber \\&\quad (j=1,2,\ldots ,N). \end{aligned}$$

(18)

Solving (18), we can get

$$\begin{aligned}&(|x_1{>},|x_2{>},\ldots ,|x_N{>})_1=(\psi _1,\psi _2,\ldots ,\psi _N)H^{-1},\\&(|x_1{>},|x_2{>},\ldots ,|x_N{>})_2=(\chi _1,\chi _2,\ldots ,\chi _N)M^{-1},\\&(|x_1{>},|x_2{>},\ldots ,|x_N{>})_3=(\phi _1,\phi _2,\ldots ,\phi _N)M^{-1}, \end{aligned}$$

with

$$\begin{aligned} H= & {} (h_{ij})_{N\times N},\quad h_{ij}=\frac{\varPhi _i^{\dag }K\varPhi _j}{\lambda _i^{*}- \lambda _j}{+}\frac{\varPhi _i^{\dag }KJ\varPhi _j}{\lambda _i^{*}+\lambda _j},\\ M= & {} (m_{ij})_{N\times N},\quad m_{ij}=\frac{\varPhi _i^{\dag }K\varPhi _j}{\lambda _i^{*}- \lambda _j}{-}\frac{\varPhi _i^{\dag }KJ\varPhi _j}{\lambda _i^{*}+\lambda _j}, \end{aligned}$$

where these subscripts “$_1$,” “$_2$” and “$_3$” stand for the first, second and third rows of the 3-tuple column vector $|x_i{>}~(1\le i \le N)$, respectively.

Since $T_N$ is the N-step DT for the coupled FL equations (1), we can get

$$\begin{aligned} T_{N,x}+T_NU=U[N]T_N. \end{aligned}$$

(19)

Multiplying both sides of Eq. (19) by $\lambda$ and letting $\lambda \rightarrow 0$, we arrive at

$$\begin{aligned} P[N]=P[0]+\sum _{i=1}^{N}(C_i-JC_iJ), \end{aligned}$$

(20)

and here $C_i=|x_i{>}{<}y_i|=|x_i{>}\varPhi _i^{\dag }K$, the capital letters U[N] and V[N] denote the N-step transformed matrices of U and V in the Lax pair (2)–(3) through the N-step DT, respectively. Additionally, the small letters u[N] and v[N] indicate the N-step transformed solutions of Eq. (1) with the seed solutions u[0] and v[0] through the N-step DT separately. Hence, the P[N] and P[0] are the results that the elements u,v in the matrix P are replaced by u[N],v[N] and u[0],v[0], respectively.

Substituting the above concrete expressions of $|x_i{>}$ and ${<}y_i|$ into Eq. (20), it follows that

$$\begin{aligned} u[N]= & {} u[0]+\sum _{i=1}^{N}(C_i-JC_iJ)_{21}=u[0]\nonumber \\&+\,2(\chi _1,\chi _2,\ldots ,\chi _N)M^{-1} \begin{pmatrix} \psi _1^{*}\\ \psi _2^{*}\\ \vdots \\ \psi _N^{*} \end{pmatrix}\nonumber \\= & {} u[0]-2\frac{\begin{vmatrix}M&Y_1^{\dag }\\ Y_2&0 \end{vmatrix}}{|M|}, \end{aligned}$$

(21)

$$\begin{aligned} v[N]= & {} v[0]+\sum _{i=1}^{N}(C_i-JC_iJ)_{31}=v[0]\nonumber \\&+\,2(\phi _1,\phi _2,\ldots ,\phi _N)M^{-1} \begin{pmatrix} \psi _1^{*}\\ \psi _2^{*}\\ \vdots \\ \psi _N^{*} \end{pmatrix}\nonumber \\= & {} v[0]-2\frac{\begin{vmatrix}M&Y_1^{\dag }\\ Y_3&0 \end{vmatrix}}{|M|}, \end{aligned}$$

(22)

where

$$\begin{aligned} Y_1= & {} (\psi _1,\psi _2,\ldots ,\psi _N),\quad Y_2=(\chi _1,\chi _2,\ldots ,\chi _N),\\ Y_3= & {} (\phi _1,\phi _2,\ldots ,\phi _N); \end{aligned}$$

besides, the subscripts “$_{21}$” and “ $_{31}$” denote the second row of the first column of a matrix and the third row of the first column of a matrix, respectively.

To construct the uniform formulae (21) and (22), the following identities have been used. Suppose M is a $N \times N$ matrix and  $\phi$ and $\psi$ are N-tuple row vectors, then the following identities hold

$$\begin{aligned} \phi M^{-1}\psi ^{\dag }=\frac{\begin{vmatrix}M&\psi ^{\dag }\\ -\phi&0 \end{vmatrix}}{|M|}. \end{aligned}$$

This completes the proof. $\square$

3 High-order semirational solutions for the coupled FL equations

In order to utilize a limiting process for constructing semirational solutions of Eq. (1), the appropriate solution of the Lax pair (2)–(3) should be derived first. The seed solution of Eq. (1) can be directly chosen as

$$\begin{aligned} u[0]=a_1e^{i\eta },\quad v[0]=a_2e^{i\eta }, \end{aligned}$$

(23)

where $\eta =\frac{1}{a_1^2+\sigma a_2^2}x$, $a_1$ and $a_2$ are all real constants ($a_1\ne a_2$). Considering the above seed solution Eq. (23) and the spectrum parameter $\lambda$, a special vector solution of the Lax pair (2)–(3) can be constructed as follows

$$\begin{aligned} \varPhi =\begin{pmatrix}\psi \\ \chi \\ \phi \end{pmatrix} =\begin{pmatrix} (K_1e^{H_1}-K_2e^{-H_1})e^{-\tfrac{i}{2}\eta }\\ l_1(K_2e^{H_1}-K_1e^{-H_1})e^{\tfrac{i}{2}\eta }-\gamma \sigma a_2e^{H_2}\\ l_2(K_2e^{H_1}-K_1e^{-H_1})e^{\tfrac{i}{2}\eta }+\gamma a_1e^{H_2} \end{pmatrix}, \nonumber \\ \end{aligned}$$

(24)

where

$$\begin{aligned}&H_1=\dfrac{i\sqrt{\lambda ^4+4(a_1^2+\sigma a_2^2)^2}}{4\lambda ^2 (a_1^2+\sigma a_2^2)}\left[ 2x+(a_1^2+\sigma a_2^2)\lambda ^2t\right. \\&\left. \quad \quad \quad +\sum _{i=1}^{N}S_i\epsilon ^{2i}\right] ,\quad H_2=-\dfrac{i(4x+\lambda ^4t)}{4\lambda ^2},\\&l_1=\frac{a_1}{\sqrt{a_1^2+\sigma a_2^2}},\quad l_2=\frac{a_2}{\sqrt{a_1^2+\sigma a_2^2}},\quad S_i=m_i+in_i,\\&K_1=\dfrac{\left[ \lambda ^2+2(a_1^2+\sigma a_2^2)+\sqrt{\lambda ^4+4(a_1^2+\sigma a_2^2)^2}\right] ^{\tfrac{1}{2}}}{\sqrt{\lambda ^4+4(a_1^2+\sigma a_2^2)^2}},\\&K_2=\dfrac{\left[ \lambda ^2+2(a_1^2+\sigma a_2^2)-\sqrt{\lambda ^4+4(a_1^2+\sigma a_2^2)^2}\right] ^{\tfrac{1}{2}}}{\sqrt{\lambda ^4+4(a_1^2+\sigma a_2^2)^2}}. \end{aligned}$$

Here, $m_i$, $n_i$ and $\gamma$ are three arbitrary real constants. For convenience, the parameter $\sigma$ in the coupled FL equations (1) is chosen as $\sigma =1$ in the following contents.

For deriving the solution (24), the variable coefficient differential expressions in the Lax pair (2)–(3) with the seed solution Eq. (23) should be transformed to constant coefficient ones by the gauge transformation $\varPhi =N\varPsi$. The transformed Lax pair reads as

$$\begin{aligned} \varPsi _x= & {} U_0\varPsi =(N^{-1}UN-N^{-1}N_x),\\ \varPsi _t= & {} V_0\varPsi =(N^{-1}VN-N^{-1}N_t), \end{aligned}$$

and $N=diag(e^{-\tfrac{2i}{3}\eta },e^{\tfrac{i}{3}\eta },e^{\tfrac{i}{3}\eta })$. In this paper, we consider the case that the characteristic equation of $U_0$ owns a double root. Under this condition, the spectrum parameter $\lambda$ should be chosen as $\lambda _0=(1+i)\sqrt{a_1^2+a_2^2}$. Substituting $\lambda =\lambda _1=(1+i+\epsilon ^2)\sqrt{a_1^2+a_2^2}$ (the parameter $\epsilon$ is a complex infinitesimal) into the vector solution (24), the Taylor expansion of $\varPhi _1=\varPhi |_{\lambda =\lambda _1}$ can be expanded as

$$\begin{aligned} \varPhi _1= & {} (\psi _1,\chi _1,\phi _1)^T=\varPhi |_{\lambda =\lambda _1}= \sum _{j=0}^{N-1}\varPhi _1^{[j]}\epsilon ^{2j}\nonumber \\&+O(\epsilon ^{2N}), \end{aligned}$$

(25)

where $\varPhi _1^{[j]}{=}(\psi _1^{[j]},\chi _1^{[j]},\phi _1^{[j]})^T=\dfrac{1}{(2j)!}\dfrac{\partial ^j \varPhi _1}{\partial \epsilon ^j}|_{\epsilon =0}$.

Here, we only give the concrete expressions of the first two terms $\varPhi _1^{[0]}$ and $\varPhi _1^{[1]}$ in Eq. (25) as follows

$$\begin{aligned} \psi _1^{[0]}= & {} \dfrac{x+ix-\rho ^2t+i\rho ^2t+\rho }{\rho \sqrt{(2+2i)\rho }}e^{- \tfrac{ix}{2\rho }}, \end{aligned}$$

(26)

$$\begin{aligned} \chi _1^{[0]}= & {} \dfrac{a_1(x+ix-\rho ^2t+i\rho ^2t-\rho )}{\rho ^2\sqrt{2+2i}}e^{\tfrac{ix}{2\rho }} \nonumber \\&-\,\gamma a_2e^{-\tfrac{x}{2\rho }+\tfrac{\rho t}{2}}, \end{aligned}$$

(27)

$$\begin{aligned} \phi _1^{[0]}= & {} \dfrac{a_2(x+ix-\rho ^2t+i\rho ^2t-\rho )}{\rho ^2\sqrt{2(1+i)}}e^{\tfrac{ix}{2\rho }} \nonumber \\&+\,\gamma a_1e^{-\tfrac{x}{2\rho }+\tfrac{\rho t}{2}}, \end{aligned}$$

(28)

$$\begin{aligned} \psi _1^{[1]}= & {} \frac{\sqrt{2(1+i)}}{48\rho ^{\tfrac{7}{2}}}(2ix^3-2x^3-6i\rho ^2x^2t -6\rho ^2x^2t\nonumber \\&-\,6i\rho ^4xt^2+6\rho ^4xt^2+2i\rho ^6t^3+2\rho ^6t^3+6i\rho x^2\nonumber \\&-\,12\rho ^3xt{-}6i\rho ^5t^2{-}15\rho ^2x{+}21i\rho ^2x{+}3\rho ^4t\nonumber \\&{+}\, 9i\rho ^4t{-}3\rho ^3{+}6i\rho ^3{+}12\rho ^2m_1\nonumber \\&{+}\,12i\rho ^2n_1) e^{{-}\tfrac{ix}{2\rho }}, \end{aligned}$$

(29)

$$\begin{aligned} \chi _1^{[1]}= & {} \frac{ia_1}{24\sqrt{2(1+i)}\rho ^4} \Omega e^{\tfrac{ix}{2\rho }} \nonumber \\&+\frac{a_2\gamma (ix-x+i\rho ^2t-\rho ^2t)}{2\rho }e^{-\tfrac{x}{2\rho }+\tfrac{\rho t}{2}}, \end{aligned}$$

(30)

$$\begin{aligned} \phi _1^{[1]}= & {} \frac{ia_2}{24\sqrt{2(1+i)}\rho ^4}\Omega e^{\tfrac{ix}{2\rho }} \nonumber \\&-\frac{a_1\gamma (ix-x+i\rho ^2t-\rho ^2t)}{2\rho }e^{-\tfrac{x}{2\rho }+\tfrac{\rho t}{2}}, \end{aligned}$$

(31)

with

$$\begin{aligned} \Omega= & {} 4ix^3-12\rho ^2x^2t-12i\rho ^4xt^2+4\rho ^6t^3-6\rho x^2\\&-\,6i\rho x^2+12\rho ^3xt-12i\rho ^3xt+6\rho ^5t^2\\&+\,6i\rho ^5t^2+6\rho ^2x+36i\rho ^2x+6i\rho ^4t+12\rho ^4t\\&-\,9i \rho ^3-12i\rho ^2m_1+12i\rho ^2n_1+12\rho ^2m_1\\&+\,12\rho ^2n_1-3\rho ^3,~~\rho =\sqrt{a_1^2+a_2^2}. \end{aligned}$$

Moreover, it is easy to find that $m_{ij}$ in Theorem 1 can be expressed as

$$\begin{aligned}&m_{ij}|_(\lambda _i=\lambda _1,\lambda _j=\lambda _1)\nonumber \\&\quad = \left( \frac{\varPhi _i^{\dag }K\varPhi _j}{\lambda _i^{*}-\lambda _j} \frac{\varPhi _i^{\dag }KJ\varPhi _j}{\lambda _i^{*} +\lambda _j}\right) |_{(\lambda _i=\lambda _1,\lambda _j=\lambda _1)}\nonumber \\&\quad =\dfrac{{-}2[(1{+}i{+}\epsilon ^2)|\psi _1|^2{-}(1{-}i{+} \epsilon ^{*2})(|\chi _1|^2{+}|\phi _1|^2)]}{\sqrt{a_1^2{+}a_2^2}(2{+}\epsilon ^2{+} \epsilon ^{*2})(2i{+}\epsilon ^2{-}\epsilon ^{*2})}\nonumber \\&\quad = \sum _{j,l{=}1}^{N}m^{[j,l]}\epsilon ^{2(l-1)}\epsilon ^{*2(j-1)}+O(|\epsilon |^{4N}), \end{aligned}$$

(32)

where

$$\begin{aligned} m^{[j,l]}= & {} \lim _{\epsilon ,\epsilon ^{*}{\rightarrow } 0}\dfrac{1}{(2(j{-}1))!(2(l{-}1)!)}\\&\dfrac{\partial ^{2(j+l-2)}m_{ij}|_{(\lambda _i=\lambda _1,\lambda _j=\lambda _1)}}{\partial \epsilon ^{2(l-1)}\partial \epsilon ^{*2(j-1)}}. \end{aligned}$$

In what follows, taking the limit approach of the N-step DT in Theorem 1, one can construct the Nth-order semirational solutions for the coupled FL equation (1) as

$$\begin{aligned} u[N]= & {} u[0]-2\frac{\begin{vmatrix}M&Y_1^{\dag }\\ Y_2&0 \end{vmatrix}}{|M|}, \end{aligned}$$

(33)

$$\begin{aligned} v[N]= & {} v[0]-2\frac{\begin{vmatrix}M&Y_1^{\dag }\\ Y_3&0 \end{vmatrix}}{|M|}, \end{aligned}$$

(34)

with

$$\begin{aligned} M= & {} (m^{[j,l]})_{1\le j,l\le N}, Y_1=(\psi _1^{[0]},\psi _1^{[1]},\ldots ,\psi _1^{[N-1]}),\\ Y_2= & {} (\chi _1^{[0]},\chi _1^{[1]},\ldots ,\chi _1^{[N-1]}),\\ Y_3= & {} (\phi _1^{[0]},\phi _1^{[1]},\ldots ,\phi _1^{[N-1]}). \end{aligned}$$

Choosing $N=1$ in the general expressions of N-order semirational solutions (33) and (34), we can straightway derive the explicit expressions for the first-order semirational solutions of the coupled FL equations (1) as

$$\begin{aligned}&u[1]=a_1e^{\tfrac{ix}{\rho }}\nonumber \\&\quad -\dfrac{16a_1L_1e^{\tfrac{ix}{\rho }}+16a_2\gamma L_2e^{\tfrac{ix}{2\rho }-\tfrac{x}{\rho }+\tfrac{\rho t}{2}}}{4(1-i)(a_1+ia_2)(ia_2-a_1)\rho \gamma ^2e^{\rho t-\tfrac{x}{\rho }}+L_3}, \nonumber \\\end{aligned}$$

(35)

$$\begin{aligned}&v[1]=a_2e^{\tfrac{ix}{\rho }}\nonumber \\&\quad +\dfrac{-16a_2L_1e^{\tfrac{ix}{\rho }}+16a_1\gamma L_2e^{\tfrac{ix}{2\rho }-\tfrac{x}{\rho }+\tfrac{\rho t}{2}}}{4(1-i)(a_1+ia_2)(ia_2-a_1)\rho \gamma ^2e^{\rho t-\tfrac{x}{\rho }}+L_3},\nonumber \\ \end{aligned}$$

(36)

where

$$\begin{aligned} L_1= & {} \dfrac{2ix^2+2i\rho ^4t^2-2\rho x-2\rho ^3t-i\rho ^2}{\sqrt{2-2i}\sqrt{2+2i}\rho ^{2}},\\ L_2= & {} \dfrac{-x-ix-\rho ^{2}t+i\rho ^2t-i\rho }{\sqrt{2-2i}},\\ L_3= & {} \frac{\sqrt{2}(1+i)}{\rho ^3}(2\rho x^2+2ia_1^2x^2+2ia_2^2x^2\\&+\,2i\rho ^4a_1^2t^2+2\rho ^5t^2+ 2i\rho ^4a_2^2t^2-2i\rho a_1^2x\\&-\,2i\rho a_2^2x+2\rho ^2x+2i\rho ^3a_2^2t+2i\rho ^3a_2^2t-2 \rho ^4t\\&+\,i\rho ^2a_2^2+\rho ^3+i\rho ^2a_1^2). \end{aligned}$$

From the above formulae (35) and (36), we can find that these solutions are combinations of rational and exponential functions, which is called semirational solutions in many documents [11, 13, 19]. When the free parameter is chosen as $\gamma =0$, the above semirational solutions are reduced to rational ones and they are all first-order RWs. When $\gamma \ne 0$, various interactional solutions can be generated in the coupled FL equations (1), which include RWs $+$ bright solitons (‘+’ stands for interaction among different nonlinear waves), RWs $+$ dark solitons and RWs $+$ breathers.

Fig. 1

a, b First-order semirational solutions of case (1) with parameters chosen by $\gamma =\tfrac{1}{200}$, $a_1=1$, $a_2=0$

Fig. 2

One dark and bright soliton merges with the first-order RW in case (1) with parameters chosen by $\gamma =1$, $a_1=1$, $a_2=0$: a, b the three-dimensional plots; c, d the corresponding plane evolution plots

(1) If $\gamma \ne 0$, one of the two parameters $a_1$ and $a_2$ is zero, and the semirational solutions including soliton and RW can be constructed in Eq. (1). It is shown that one component is the first-order RW $+$ one bright soliton, and the other one is the first-order RW $+$ one dark soliton in Figs. 1 and 2. One first-order RW exists at $t=0$ on one dark soliton background, see Fig. 1a; one first-order RW appears on one bright soliton background, see Fig. 1b. It is shown that the first-order RW on the left of one bright soliton in Fig. 1b is only a small bump, because the RW is generated on a plane with almost zero amplitude. Besides, the first-order RW and one dark (bright) soliton separate in u and v components with a small value of $|\gamma |$, respectively.

Fig. 3

First-order semirational solution of case (2) with parameters chosen by $a_1=1$, $a_2=\tfrac{1}{2}$: a, b a first-order RW and a breather separate in two components with $\gamma =\tfrac{1}{100000}$: c, d a first-order RW merge with a breather in two components with $\gamma =\tfrac{1}{10}$

It demonstrates that the first-order RW merges with one dark or one bright soliton by increasing the value of $|\gamma |$ in Fig. 2a, b. At this point, we can easily find the first-order RW on the top of one bright soliton in Fig. 2b, because the RW is generated on a plane with almost 1 amplitude. The evolutionary processes of these hybrid solutions at different moments are shown in Fig. 2c, d. In Fig. 2c, only one dark soliton propagates if $t<0$, and a first-order RW exists at $t=0$. Then, the RW disappears and one dark soliton is still preserved if $t>0$. Additionally, it demonstrates that the collision process in Fig. 2c is not elastic, because the amplitude of the right dark soliton is bigger than the amplitude of the left one. However, the collision between the first-order RW and one bright soliton is elastic, as shown in Fig. 2d.

(2) When $\gamma \ne 0$ and $a_1a_2\ne 0$, we can obtain the second kind of interactional solutions where the first-order RW interacts with a breather for the coupled FL equations (1). As shown in Figs. 3a, b, the first-order RW and one breather separate in both u and v components. Moreover, the breathers in these two figures are very different. In Fig. 3a, we can find that the amplitudes of the breather above the background plane are smaller than ones under the background plane. Nevertheless, the corresponding amplitudes of the breather are inverse in Fig. 3b. By increasing the values $|\gamma |$, the first-order RW merges with one breather obviously, see Figs. 3c, d.

Fig. 4

a, b Second-order semirational solutions including the fundamental second-order RW interacting with two dark or bright solitons for case (1) with parameters chosen by $a_1=\tfrac{5}{4}$, $a_2=0$, $\gamma =\tfrac{1}{100000}$

Fig. 5

Second-order semirational solutions including the second-order RW of triangular pattern interacting with two solitons for case (1) with parameters chosen by $a_1=\tfrac{5}{4}$, $a_2=0$, $m_1=100$, $n_1=-100$: a, b three first-order RWs and two solitons are separated with $\gamma =\tfrac{1}{100000}$; c, d three first-order RWs and two solitons are fused with $\gamma =\tfrac{1}{100}$

Analogously, fixing $N=2$ in the universal formulae (33) and (34), the second-order semirational solutions for the coupled FL equations (1) can be directly given as

$$\begin{aligned} u[2]= & {} a_1e^{\dfrac{ix}{a_1^2+a_2^2}}-2\frac{\begin{vmatrix}m^{[1,1]}&m^{[1,2]}&\psi _1^{[0]*}\\m^{[2,1]}&m^{[2,2]}&\psi _1^{[1]*}\\ \chi _1^{[0]}&\chi _1^{[1]}&0 \end{vmatrix}}{\begin{vmatrix} m^{[1,1]}&m^{[1,2]}\\m^{[2,1]}&m^{[2,2]}\end{vmatrix}}, \end{aligned}$$

(37)

$$\begin{aligned} v[2]= & {} a_2e^{\dfrac{ix}{a_1^2+a_2^2}}-2\frac{\begin{vmatrix}m^{[1,1]}&m^{[1,2]}&\psi _1^{[0]*}\\m^{[2,1]}&m^{[2,2]}&\psi _1^{[1]*}\\ \phi _1^{[0]}&\phi _1^{[1]}&0 \end{vmatrix}}{\begin{vmatrix} m^{[1,1]}&m^{[1,2]}\\m^{[2,1]}&m^{[2,2]}\end{vmatrix}}. \end{aligned}$$

(38)

Here, $\psi _1^{[0]*}$ and $\psi _1^{[1]*}$ stand for the complex conjugations of $\psi _1^{[0]}$ and $\psi _1^{[1]}$, respectively. The concrete expressions of $\psi _1^{[0]},\chi _1^{[0]},\phi _1^{[0]}$ and $\psi _1^{[1]},\chi _1^{[1]},\phi _1^{[1]}$ are given in Eqs. (26)–(31). Additionally, the expressions of $m^{[j,l]}~(1\le j,l\le 2)$ are calculated out by Eq. (32). If $\gamma =0$, the second-order semirational solutions become rational ones and they are the second-order RWs. Similar to the first-order case, the second-order semirational solutions are also classified as two types with $\gamma \ne 0$.

Fig. 6

Second-order semirational solution consisted of the second-order RW of triangular pattern and two breathers for case (2) with parameters chosen by $a_1=1$, $a_2=-1$: a, b these two kinds of nonlinear waves are separated with $\gamma =\tfrac{1}{1000000}$; c, d they merge with each other with $\gamma =\tfrac{1}{100}$

(1) If $\gamma \ne 0$, $a_1\ne 0$ and $a_2=0$, we can get the first kind of the second-order semirational solutions consisted of the second-order RW and two dark (bright) solitons. It is shown that a second-order fundamental RW $+$ two dark solitons exists in u component, and a second-order fundamental RW $+$ two bright solitons appears in v component, see Fig. 4. Similar to Fig. 1b, the second-order RW is not easily seen in Fig. 4b for the reason that the amplitude of background plane on which RW appears are almost zero.

The patterns of the high-order RWs in these semirational solutions are determined by the parameters $S_i$ ($S_i=m_i+in_i$) in Eq. (24). Choosing $S_1\ne 0$, the second-order fundamental RW in Fig. 4 is decomposed into three first-order ones in Fig. 5. Compared to Fig. 5a, b, these three first-order RWs merge with two dark or bright solitons in Fig. 5c, d through increasing the value of $|\gamma |$. These first-order RWs are easily observed in Fig. 5d, because they are generated on significant nonzero plane.

(2) If $\gamma \ne 0$ and $a_1a_2\ne 0$, the second kind of the second-order semirational solutions is exhibited, which includes the second-order RW and two breathers. As shown in Fig. 6a, b, a second-order RW of triangular pattern and two breathers coexist in both u and v components. Furthermore, the breathers in these two figures are same. Similar to the first-order case, it is shown that these three first-order RWs merge with two breathers through increasing the value of $|\gamma |$ in Fig. 6c, d.

From Eqs. (33)–(34), some much higher-order semirational solutions to the coupled FL equations (1) can be derived. Similar to the first- and second-order cases, these much higher-order semirational solutions can also be classified as two types with $\gamma \ne 0$: (1) One component is high-order RW $+$ multi-dark solitons, and the other one is high-order RW $+$ multi-bright solitons; (2) two components are all high-order RW $+$ multi-breathers. Moreover, various patterns of high-order RWs in these semirational solutions can be generated by choosing different combinations of $m_i$ and $n_i$. Here, we only discuss the first- and second-order semirational solutions in detail. We find that these semirational solutions can not be generated in single-component FL equation. It is shown that the solutions in coupled systems are more abundant and interesting than ones in uncoupled systems. Besides, we expect that these semirational solutions can be observed in the physical experiments in the future.

4 Conclusion

A special vector solution of the Lax pair (2)–(3) and the DT for the coupled FL equations (1) are constructed, respectively. Using the limiting technique, some novel interactional solutions to Eq. (1) are exhibited. During the computational processes of the vector solution (24), all solutions in the fundamental solution of the transformed $U_0$ are reserved and it is very important to generate various semirational solutions. Additionally, the parameter $\gamma$ in Eq. (24) plays an important role in generating these semirational solutions. If $\gamma =0$, these semirational solutions are reduced to rational ones RWs; these hybrid solutions exist only if $\gamma \ne 0$.

Moreover, these semirational solutions are mainly classified as two types: (1) One component is RWs $+$ dark solitons, and the other one is RWs $+$ bright solitons; (2) two components are all RWs $+$ breathers. Here, the dynamics of the first- and second-order semirational solutions are discussed in detail. It is shown that these different nonlinear waves can merge with each other significantly by increasing the value of $|\gamma |$. Actually, wave dynamics are superposition of various kinds of nonlinear waves [11, 13, 32]. It is necessary to investigate the interactional solutions of nonlinear models because we believe the experimental conditions for observing hybrid solutions among first-order RW, one bright (dark) soliton and one breather in [11] in the future could depend on high-order semirational solutions of coupled FL equations.