# Characters of Explicit Solutions for a Semidiscrete Integrable Coupled Equation

Authors: Qiulan Zhao; Qianqian Yang; Xiangwen Qu
Journal: Advances in Mathematical Physics
 (2021)

---

## Abstract

A semidiscrete integrable coupled system is obtained by embedding a free function into the discrete zero-curvature equation. Then, explicit solutions of the first two nontrivial equations in this system are derived directly by the Darboux transformation method. Finally, in order to compare the solutions before and after coupling intuitively, their structure figures are presented and analyzed.


---

## Body

## 1. Introduction

Integrable coupled equations have attracted more attention in soliton theory in recent years. For a given integrable system, we can construct a nontrivial system of differential equations which is still integrable and includes the original integrable system as a subsystem [1]. It is interesting to find integrable coupled systems for a given integrable equation. Hirota and Satsuma in 1981 introduced a coupled KdV system [2]. Fuchssteiner in 1982 proposed the important question: how should completely systems interact without losing complete integrability [3]? Then, the method for constructing integrable coupling systems by perturbation was first proposed by Ma and Fuchssteiner [4]. Later, the method has been developed. So far, it mainly includes perturbations, enlarging spectral problems [5, 6], creating new loop algebras [7], and multi-integrable couplings. With the development of integrable coupling theory, it is verified that integrable coupled equations are usually used for describing phenomena related to dark and antidark solitons. Such as the integrable coupled generalized nonlinear Schrödinger equations can exhibit N-bright-bright and N-dark-dark soliton solutions [8].Obtaining explicit solutions for integrable equations is the main mission in nonlinear science research. Many methods have been developed to solve integrable equations, such as Darboux transformation [9–15], inverse scattering transformation [16], algebra-geometric approach [17], Lie symmetry method [18], and Hirota bilinear method [19]. Darboux transformation is a useful tool for solving integrable equations. It can obtain its nontrivial solutions in accordance with an arbitrary seed solution of the integrable equations. Solving integrable coupled equations by the Darboux transformation method is a meaningful investigation. Explicit solutions of an integrable coupled system of Merola-Ragnisco-Tu lattice equation (24) are investigated, and explicit solutions of a new discrete integrable soliton hierarchy with 4×4 Lax pair [20] are discussed by the Darboux transformation.In this paper, our main consideration is the following semidiscrete integrable coupled equations:(1)qn,t=−qnrnqn+1+1rn+qn,rn,t=rnrn−1qn−1qn−rn,un,t=un+1qnqn+12rn−1−unrnqn+1+1rn−snrn2+un,sn,t=unrn−1−rnqn2rn−1+snrn−1−rnqnrn−12+rnrn−1qn−1qn−sn,where qn,rn,un, and sn are the potentials. The spectral problems deriving the semidiscrete integrable coupled system are an extension of a spectral problem introduced by Sun et al. [21]. Another extension of the spectral problems in [21] was investigated by Xue et al. [22]. They also constructed infinitely many conversation laws and Darboux transformations for the first nonlinear integrable equations. In this paper, we will concentrate on investigating explicit solutions of Equation (1) by means of the Darboux transformation method based on its Lax pair.The outline of this paper is as follows. In Section2, we will derive Equation (1) by means of the discrete zero-curvature equations and construct the Darboux transformation for Equation (1) based on its spectral problems. In Section 3, we will obtain the explicit solutions of Equation (1) and discuss the properties of solutions by means of different figures. In Section 4, some conclusions will be given.


## 2. A New Integrable Coupled Equation and Its Darboux Transformation

### 2.1. Constructing for a New Integrable Coupled Equation

In order to obtain Equation (1), we consider the following discrete spectral problem:
(2)Eφn=Unu,λφn,Un=0λqn0λun−λrnλ2qnrn+1−λsnλ2qnsn+unrn000λqn00−λrnλ2qnrn+1,and its auxiliary problem
(3)φn,tm=Γnφn,Γn=∑i≥0aniλ−2i∑i≥0bniλ−2i+1∑i≥0eniλ−2i∑i≥0fniλ−2i+1∑i≥0cniλ−2i+1−∑i≥0aniλ−2i∑i≥0gniλ−2i+1−∑i≥0eniλ−2i00∑i≥0aniλ−2i∑i≥0bniλ−2i+100∑i≥0cniλ−2i+1−∑i≥0aniλ−2i,where λ is independent of t and E is a shift operator defined by Efn=Efn+1.Then, we embed a free functionδnm into the following modification term Δnm:
(4)Δnm=δnm0000anm0enm00δnm0000anm.LetVnm=λ2mΓnm+Δnm. Solving the stationary discrete zero-curvature equations EΓnUn−UnΓn=0 and the discrete zero-curvature equations Un,t−EVnmUn+UnVnm=0, we can obtain the following integrable coupled hierarchy:
(5)qn,t=an+1mqn+bn+1m+qnδn+1m,rn,t=−anmrn+cnm−rnδnm,un,t=en+1mqn+an+1mun+fn+1m+unδn+1m,sn,t=−enmrn−anmsn+gnm−snδnm.Then, the semidiscrete integrable coupled Equation (1) can be obtained by setting m=1 and δn1=1. The detailed symbolic computation steps are similar to [22].


### 2.2. Darboux Transformation

In this section, we will investigate the Darboux transformation of Equation (1).The spectral problems of Equation (1) are presented by
(6)Eφn=Unu,λφn,Un=0λqn0λun−λrnλ2qnrn+1−λsnλ2qnsn+unrn000λqn00−λrnλ2qnrn+1,(7)φn,t1=Vn1φn,Vn1=V11λrn−1V13λrn−1−sn−1rn−12−λqn12λ2λun−qnqn212λ200V11λrn−100−λqn12λ2,where
(8)V11=−12λ2−1qnrn−1+1,V13=−12λ2+unqn2rn−1−1rn−1qn+snqnrn−12.Firstly, we choose a proper Darboux matrix:(9)Tn=T11T12T13T14T21T22T23T24T31T32T33T34T41T42T43T44=a2,nλ2+a0,nb1,nλe2,nλ2+e0,nf1,nλc1,nλa0,nλ2+1g1,nλe0,nλ2+100a2,nλ2+a0,nb1,nλ00c1,nλa0,nλ2+1.The Darboux transformation can transformUn,Vn into U~n,V~n, i.e.,
(10)Eφ~n=U~nφ~n,φ~t=V~nφ~n,where(11)φ~n=Tnφn,U~n=Tn+1UnTn−1,V~n=Tn,t+TnVnTn−1.And the potential functionsU~n and V~n have the same form as Un and Vn, respectively. Then, we need to define a solution matrix Φn; it can be represented as
(12)Φn=φ1nψ1nφ2nψ2nφ3nψ3nφ4nψ4n.If we assume that the terms bothφn=φ1n,φ2n,φ3n,φ4nT and ψn=ψ1n,ψ2n,ψ3n,ψ4nT are two linear independent solutions of the 4×4 Lax pair of Equation (1), we can obtain
(13)δi,1n=φ2,n−κiψ2,nφ1,n−κiψ1,n,δi,2n=φ3,n−κiψ3,nφ1,n−κiψ1,n,δi,3n=φ4,n−κiψ4,nφ1,n−κiψ1,n,by means of the transformation
(14)Φ~n=TnΦn.So we have an algebraic system:(15)λi2a2,n+a0,n+b1,nλiδi,1n+e2,nλi2+e0,nδi,2n+f1,nλi=0,c1,nλi+λi2+a0,n+1δi,1n+g1,nλiδi,2n+λi2e0,n+1δi,3n=0,λi2a2,n+a0,nδi,1n+b1,nλiδi,3n=0,c1,nλiδi,2n+a0,nλi2+1δi,3n=0.Solving the linear algebraic system, we can obtain the expressions fora0,n,a2,n,b1,n,c1,n,e0,n,e2,n, f1,n,g1,n in Appendix A. Furthermore, the following equations can be obtained by Equation (2):
(16)δi,1n+1=−λi+1+λi2δi,1n−λiδi,2n+2λ2δi,3nλiδi,1n+λiδi,3n,δi,2n+1=δi,2nδi,1n+δi,3n,δi,3n+1=−λiδi,2n+1+λi2δi,3nλiδi,1n+λiδi,3n.From Equation (9) to Equation (16), it can be seen that ±λ1 and ±λ2 are the roots of DetTn=0. So, we have
(17)DetTn=αntλ−λ12λ+λ12λ−λ22λ+λ22,where αnt is a function with respect to t. From the above conclusion, we can prove the following proposition.Proposition 1.
The form of the matrixU~n is
(18)U~n=0λq~n0λu~n−λr~nλ2q~nr~n+1−λs~nλ2q~ns~n+u~nr~n000λq~n00−λr~nλ2q~nr~n+1,and the transformation from the old potentials qn,rn,un,sn to new ones q~n,r~n,u~n,s~n is given by
(19)q~n=a0,n+1qn+b1,n+1,r~n=rn+c1,na0,n,u~n=a0,n+1un−qn+e0,n+1qn+f1,n+1−b1,n+1,s~n=sna0,n+rna0,n+g1,na0,n−rne0,na0,n2−c1,ne0,na0,n2.Proof.
LetTn−1=Tn∗/detTn and
(20)Tn+1UnTn∗=Θ11,nλΘ12,nλΘ13,nλΘ14,nλΘ21,nλΘ22,nλΘ23,nλΘ24,nλΘ31,nλΘ32,nλΘ33,nλΘ34,nλΘ41,nλΘ42,nλΘ43,nλΘ44,nλ,
where all the expressionsΘijλ,i,j=1,2,3,4 are the functions with respect to n and t. It is easy to verify that the terms ±λ1 and ±λ2 are the roots of Θijλ,i,j=1,2,3,4 except Θ31λ,Θ32λ,Θ41λ,Θ42λ,Θ11λ,Θ13λ, and Θ33λ. We can also prove that Θ31λ,Θ32λ,Θ41λ,Θ42λ,Θ11λ,Θ13λ,Θ33λ≡0, the terms Θ12,nλ,Θ14,nλ,Θ21,nλ,Θ23,nλ,Θ34,nλ,andΘ43,nλ are ninth-order polynomials with respect to λ, and the terms Θ22,nλ,Θ24,nλ, and Θ44,nλ are tenth-order polynomials with respect to λ. The equation Tn+1UnTn∗=detTnPn can be written as Tn+1Un=PnTn, with
(21)Pn=0P12λ0P14λP21λP222λ2+P220P23λP24λ2000P12λ00P21λP222λ2+P220.
By comparing the coefficients inTn+1Un=PnTn, we find
(22)P12=a0,n+1qn+b1,n+1=q~n,P21=−rn−c1,na0,n=−r~n,P220=1,P222=1a0,na0,n+1qn+b1,n+1rn+c1,n=q~nr~n,P14=a0,n+1un−qn+e0,n+1qn+f1,n+1−b1,n+1=u~n,P23=−sna0,n−rna0,n−g1,na0,n+rne0,na0,n2+c1,ne0,na0,n2=−s~n,P24=a0,n+1qn+b1,n+1sna0,n+rna0,n+g1,na0,n−rne0,na0,n2−c1,ne0,na0,n2+a0,n+1un−qn+a0,n+1qn+f1,n+1−b1,n+1rn+c1,na0,n=q~ns~n+u~nr~n.
From the above equations, we can seePn=U~n. The proof is completed.Proposition 2.
Under the transformations Equations (10) and (19), the term V~n has the form
(23)V~n=−12λ2−1q~nr~n−1+1λr~n−1−12λ2+u~nq~n2r~n−1−1r~n−1q~n+s~nq~nr~n−12λr~n−1−s~n−1r~n−12−λq~n12λ2λu~n−q~nq~n212λ200−12λ2−1q~nr~n−1+1λr~n−100−λq~n12λ2.Proof.
(24)Tn,t+TnVn1Tn∗=Ω11,nλΩ12,nλΩ13,nλΩ14,nλΩ21,nλΩ22,nλΩ23,nλΩ24,nλΩ31,nλΩ32,nλΩ33,nλΩ34,nλΩ41,nλΩ42,nλΩ43,nλΩ44,nλ,
where all the expressionsΩiji,j=1,2 are the functions with respect to n and t. It is obvious that the terms Ω11,nλ,Ω13,nλ,Ω22,nλ,Ω24,nλ,Ω33,nλ, and Ω34,nλ are tenth-order polynomials with respect to λ, the terms Ω14,nλ,Ω21,nλ,Ω23,nλ,Ω34,nλ, and Ω43,nλ are ninth-order polynomials with respect to λ, the term Ω12,nλ is eighth-order polynomials with respect to λ, and the terms Ω31,nλ,Ω32,nλ,Ω41,nλ, and Ω42,nλ are all zero. In addition, the terms ±λ1 and ±λ2 are the roots of the Ωijλ. So, we have
(25)Tn,t+TnVn1Tn∗=detTnQn,with
(26)Qn=Q112λ2+Q110Q121Q132λ2+Q130Q141λQ211λQ222λ2Q131λQ242λ200Q112λ2+Q110Q121λ00Q211λQ222λ2.
Equation (25) can be rewritten as
(27)Tn,t+TnVn1=QnTn.
By comparing the coefficients in Equation (27), we obtain
(28)Q112=−12,Q110=−1q~nr~n−1+1,Q12=1rn−1,Q132=−12,Q130=u~nq~n2~r~n−1−1q~nr~n−1+s~nq~nr~n−12,Q14=r~n−1−s~n−1r~n−12,Q21=−1q~n,Q22=12,Q23=u~n−q~nq~n2,Q24=12.
It is obvious thatQn=V~n. The proof is completed.Equations (14) and (19) constitute the Darboux transformation for the semidiscrete integrable coupled systems based on Equations (6) and (7).


## 2.1. Constructing for a New Integrable Coupled Equation

In order to obtain Equation (1), we consider the following discrete spectral problem:
(2)Eφn=Unu,λφn,Un=0λqn0λun−λrnλ2qnrn+1−λsnλ2qnsn+unrn000λqn00−λrnλ2qnrn+1,and its auxiliary problem
(3)φn,tm=Γnφn,Γn=∑i≥0aniλ−2i∑i≥0bniλ−2i+1∑i≥0eniλ−2i∑i≥0fniλ−2i+1∑i≥0cniλ−2i+1−∑i≥0aniλ−2i∑i≥0gniλ−2i+1−∑i≥0eniλ−2i00∑i≥0aniλ−2i∑i≥0bniλ−2i+100∑i≥0cniλ−2i+1−∑i≥0aniλ−2i,where λ is independent of t and E is a shift operator defined by Efn=Efn+1.Then, we embed a free functionδnm into the following modification term Δnm:
(4)Δnm=δnm0000anm0enm00δnm0000anm.LetVnm=λ2mΓnm+Δnm. Solving the stationary discrete zero-curvature equations EΓnUn−UnΓn=0 and the discrete zero-curvature equations Un,t−EVnmUn+UnVnm=0, we can obtain the following integrable coupled hierarchy:
(5)qn,t=an+1mqn+bn+1m+qnδn+1m,rn,t=−anmrn+cnm−rnδnm,un,t=en+1mqn+an+1mun+fn+1m+unδn+1m,sn,t=−enmrn−anmsn+gnm−snδnm.Then, the semidiscrete integrable coupled Equation (1) can be obtained by setting m=1 and δn1=1. The detailed symbolic computation steps are similar to [22].


## 2.2. Darboux Transformation

In this section, we will investigate the Darboux transformation of Equation (1).The spectral problems of Equation (1) are presented by
(6)Eφn=Unu,λφn,Un=0λqn0λun−λrnλ2qnrn+1−λsnλ2qnsn+unrn000λqn00−λrnλ2qnrn+1,(7)φn,t1=Vn1φn,Vn1=V11λrn−1V13λrn−1−sn−1rn−12−λqn12λ2λun−qnqn212λ200V11λrn−100−λqn12λ2,where
(8)V11=−12λ2−1qnrn−1+1,V13=−12λ2+unqn2rn−1−1rn−1qn+snqnrn−12.Firstly, we choose a proper Darboux matrix:(9)Tn=T11T12T13T14T21T22T23T24T31T32T33T34T41T42T43T44=a2,nλ2+a0,nb1,nλe2,nλ2+e0,nf1,nλc1,nλa0,nλ2+1g1,nλe0,nλ2+100a2,nλ2+a0,nb1,nλ00c1,nλa0,nλ2+1.The Darboux transformation can transformUn,Vn into U~n,V~n, i.e.,
(10)Eφ~n=U~nφ~n,φ~t=V~nφ~n,where(11)φ~n=Tnφn,U~n=Tn+1UnTn−1,V~n=Tn,t+TnVnTn−1.And the potential functionsU~n and V~n have the same form as Un and Vn, respectively. Then, we need to define a solution matrix Φn; it can be represented as
(12)Φn=φ1nψ1nφ2nψ2nφ3nψ3nφ4nψ4n.If we assume that the terms bothφn=φ1n,φ2n,φ3n,φ4nT and ψn=ψ1n,ψ2n,ψ3n,ψ4nT are two linear independent solutions of the 4×4 Lax pair of Equation (1), we can obtain
(13)δi,1n=φ2,n−κiψ2,nφ1,n−κiψ1,n,δi,2n=φ3,n−κiψ3,nφ1,n−κiψ1,n,δi,3n=φ4,n−κiψ4,nφ1,n−κiψ1,n,by means of the transformation
(14)Φ~n=TnΦn.So we have an algebraic system:(15)λi2a2,n+a0,n+b1,nλiδi,1n+e2,nλi2+e0,nδi,2n+f1,nλi=0,c1,nλi+λi2+a0,n+1δi,1n+g1,nλiδi,2n+λi2e0,n+1δi,3n=0,λi2a2,n+a0,nδi,1n+b1,nλiδi,3n=0,c1,nλiδi,2n+a0,nλi2+1δi,3n=0.Solving the linear algebraic system, we can obtain the expressions fora0,n,a2,n,b1,n,c1,n,e0,n,e2,n, f1,n,g1,n in Appendix A. Furthermore, the following equations can be obtained by Equation (2):
(16)δi,1n+1=−λi+1+λi2δi,1n−λiδi,2n+2λ2δi,3nλiδi,1n+λiδi,3n,δi,2n+1=δi,2nδi,1n+δi,3n,δi,3n+1=−λiδi,2n+1+λi2δi,3nλiδi,1n+λiδi,3n.From Equation (9) to Equation (16), it can be seen that ±λ1 and ±λ2 are the roots of DetTn=0. So, we have
(17)DetTn=αntλ−λ12λ+λ12λ−λ22λ+λ22,where αnt is a function with respect to t. From the above conclusion, we can prove the following proposition.Proposition 1.
The form of the matrixU~n is
(18)U~n=0λq~n0λu~n−λr~nλ2q~nr~n+1−λs~nλ2q~ns~n+u~nr~n000λq~n00−λr~nλ2q~nr~n+1,and the transformation from the old potentials qn,rn,un,sn to new ones q~n,r~n,u~n,s~n is given by
(19)q~n=a0,n+1qn+b1,n+1,r~n=rn+c1,na0,n,u~n=a0,n+1un−qn+e0,n+1qn+f1,n+1−b1,n+1,s~n=sna0,n+rna0,n+g1,na0,n−rne0,na0,n2−c1,ne0,na0,n2.Proof.
LetTn−1=Tn∗/detTn and
(20)Tn+1UnTn∗=Θ11,nλΘ12,nλΘ13,nλΘ14,nλΘ21,nλΘ22,nλΘ23,nλΘ24,nλΘ31,nλΘ32,nλΘ33,nλΘ34,nλΘ41,nλΘ42,nλΘ43,nλΘ44,nλ,
where all the expressionsΘijλ,i,j=1,2,3,4 are the functions with respect to n and t. It is easy to verify that the terms ±λ1 and ±λ2 are the roots of Θijλ,i,j=1,2,3,4 except Θ31λ,Θ32λ,Θ41λ,Θ42λ,Θ11λ,Θ13λ, and Θ33λ. We can also prove that Θ31λ,Θ32λ,Θ41λ,Θ42λ,Θ11λ,Θ13λ,Θ33λ≡0, the terms Θ12,nλ,Θ14,nλ,Θ21,nλ,Θ23,nλ,Θ34,nλ,andΘ43,nλ are ninth-order polynomials with respect to λ, and the terms Θ22,nλ,Θ24,nλ, and Θ44,nλ are tenth-order polynomials with respect to λ. The equation Tn+1UnTn∗=detTnPn can be written as Tn+1Un=PnTn, with
(21)Pn=0P12λ0P14λP21λP222λ2+P220P23λP24λ2000P12λ00P21λP222λ2+P220.
By comparing the coefficients inTn+1Un=PnTn, we find
(22)P12=a0,n+1qn+b1,n+1=q~n,P21=−rn−c1,na0,n=−r~n,P220=1,P222=1a0,na0,n+1qn+b1,n+1rn+c1,n=q~nr~n,P14=a0,n+1un−qn+e0,n+1qn+f1,n+1−b1,n+1=u~n,P23=−sna0,n−rna0,n−g1,na0,n+rne0,na0,n2+c1,ne0,na0,n2=−s~n,P24=a0,n+1qn+b1,n+1sna0,n+rna0,n+g1,na0,n−rne0,na0,n2−c1,ne0,na0,n2+a0,n+1un−qn+a0,n+1qn+f1,n+1−b1,n+1rn+c1,na0,n=q~ns~n+u~nr~n.
From the above equations, we can seePn=U~n. The proof is completed.Proposition 2.
Under the transformations Equations (10) and (19), the term V~n has the form
(23)V~n=−12λ2−1q~nr~n−1+1λr~n−1−12λ2+u~nq~n2r~n−1−1r~n−1q~n+s~nq~nr~n−12λr~n−1−s~n−1r~n−12−λq~n12λ2λu~n−q~nq~n212λ200−12λ2−1q~nr~n−1+1λr~n−100−λq~n12λ2.Proof.
(24)Tn,t+TnVn1Tn∗=Ω11,nλΩ12,nλΩ13,nλΩ14,nλΩ21,nλΩ22,nλΩ23,nλΩ24,nλΩ31,nλΩ32,nλΩ33,nλΩ34,nλΩ41,nλΩ42,nλΩ43,nλΩ44,nλ,
where all the expressionsΩiji,j=1,2 are the functions with respect to n and t. It is obvious that the terms Ω11,nλ,Ω13,nλ,Ω22,nλ,Ω24,nλ,Ω33,nλ, and Ω34,nλ are tenth-order polynomials with respect to λ, the terms Ω14,nλ,Ω21,nλ,Ω23,nλ,Ω34,nλ, and Ω43,nλ are ninth-order polynomials with respect to λ, the term Ω12,nλ is eighth-order polynomials with respect to λ, and the terms Ω31,nλ,Ω32,nλ,Ω41,nλ, and Ω42,nλ are all zero. In addition, the terms ±λ1 and ±λ2 are the roots of the Ωijλ. So, we have
(25)Tn,t+TnVn1Tn∗=detTnQn,with
(26)Qn=Q112λ2+Q110Q121Q132λ2+Q130Q141λQ211λQ222λ2Q131λQ242λ200Q112λ2+Q110Q121λ00Q211λQ222λ2.
Equation (25) can be rewritten as
(27)Tn,t+TnVn1=QnTn.
By comparing the coefficients in Equation (27), we obtain
(28)Q112=−12,Q110=−1q~nr~n−1+1,Q12=1rn−1,Q132=−12,Q130=u~nq~n2~r~n−1−1q~nr~n−1+s~nq~nr~n−12,Q14=r~n−1−s~n−1r~n−12,Q21=−1q~n,Q22=12,Q23=u~n−q~nq~n2,Q24=12.
It is obvious thatQn=V~n. The proof is completed.Equations (14) and (19) constitute the Darboux transformation for the semidiscrete integrable coupled systems based on Equations (6) and (7).


## 3. Explicit Solutions

### 3.1. Explicit Solutions for Equation (1)

In order to obtain the explicit solutions of the semidiscrete integrable coupled equations (1), we choose the trivial solution qn=un=et,rn=sn=e−t. Then, the spectral problems Equations (2) and (7) become
(29)Eφn=0λet0λet−λe−tλ2−λe−t2λ2000λet00−λe−tλ2φn,φn,t=−12λ2λet−12λ2+10−λe−t12λ2012λ200−12λ2λet00−λe−t12λ2φn.By solving Equation (29), we achieve the two basic solutions of Equation (1), φn and ψn,
(30)φn=φ1,nφ2,nφ3,nφ4,n=−λ3λ2t−t+12λ2−2e−1/2λ2−2t2−λ4+t−3λ22λ2−2e−1/2λ2tλe−1/2λ2−2te1/2λ2t,ψn=ψ1,nψ2,nψ3,nψ4,n=λ2n−1λ2+2λ2t−t−12λ2−2e1/2λ2tλ2n−2λ2+2λ4t−λ2t−1+λ2−2λ2−12λ2−2e12λ2−1tλ2n−1e1/2λ2tλ2ne1/2λ2−2t.Then, based on Equation (13), we can obtain
(31)δi,1n=−−2t−4λi2n+2+t+1λi2n+4+tλi2n+6κietλi2−1+−λi6+λi4t−3λi4+2λi22κi−1/2t+1/2λi2n+2−1/2λi2n+4t+λi2nt+1etλi2−1/2λi3etλi+1tλi2−t+1λi,δi,2n=−2λi2−1λi2ne1/2tλi2κi−λi2−1/2tλi2+t−λi3λi+1tλi2−t+1e−1/2tλi2+t+2e1/2tλi2−1/2t+1/2λi2n+2−1/2λi2n+4t+λi2nt+1κi,δ1,3n=−2κiλi2+2κiλi2n+1etλi2−1+2λi3−2λi2κi−1/2t+1/2λi2n+2−1/2λi2n+4t+λi2nt+1etλi2−λi3etλi+1tλi2−t+1.According to the transformations Equations (19), (16), and (31), we can obtain the explicit solutions of Equation (1) by symbolic computing software maple. It is presented by δi,1n,δi,2n,δi,3n in Appendix B.


### 3.2. Discussions

The integrable equations investigated in [22] are presented as follows:
(32)qn,t=−qnqn+1rn+1rn+qn,rn,t=rnqnrn−1−1qn−rn.The explicit solutions of Equation (32) by Darboux transformation method are given in [22]
(33)q~n=−ρ1κ1λ12nλ22λ12−1e2t+ρ2κ2λ22nρ1ρ3et−λ12e2t+λ12λ22e2tρ1κ1λ12n+2λ22λ12−1et−ρ2κ2λ12λ22n+2ρ1ρ3−et+λ22et,r~n=λ12+λ22−ρ1κ1λ12n+2λ22−1e−t+ρ2κ2λ22n+2λ12−1e−tλ12et−λ22et+ρ1κ1λ12nλ22−1−ρ2κ2λ22nλ12−1,where ρ1=eλ12t,ρ2=eλ22t, and ρ3=κ1λ12nλ12−λ22.In order to compare the differences of the explicit solutions between integrable coupled Equation (1) and integrable Equation (32), we first investigate the three-dimensional structure and the evolution properties of explicit solutions Equation (33) with two sets of parameters, λ1=0.4,λ2=25,κ1=32, and κ2=0.8 and λ1=0.3, λ2=0.6I, κ1=3I, and κ2=2 in Figures 1 and 2. Then, we plot the three-dimensional structures, density figures, the evolution properties of q~n,r~n, and s~n with parameters λ1=0.3, λ2=0.6I, and κ1=3I, and u~n with parameters λ1=0.4,λ2=25,κ1=32, and κ2=0.8 in Figures 3–5 (The expression for u~n is so complicated that Maple cannot calculate its figure when the parameters contain a complex number. The specific expression of u~n is in the appendix.).Figure 1
Three-dimensional structure figures of explicit solutions from Equation (32) and the evolutions of these solutions with parameters λ1=0.3, λ2=0.6I, κ1=3I, and κ2=2.
(a)
qn(b)
rn(c)
qn in Equation (24)(d)
rn in Equation (29)Figure 2
Three-dimensional structure figures of explicit solutions from Equation (32) and the evolutions of these solutions with parameters λ1=0.4, λ2=25, κ1=32, and κ2=0.8.
(a)
qn(b)
rn(c)
qn in Equation (29)(d)
rn in Equation (29)Figure 3
Three-dimensional structure figures ofq~n, r~n, and s~n with parameters λ1=0.3, λ2=0.6I, κ1=3I, and κ2=2 and density figures of u~n with parameters λ1=0.4, λ2=25, κ1=32, and κ2=0.8.
(a)
qn(b)
rn(c)
sn(d)
unFigure 4
Density figures ofq~n, r~n, and s~n with parameters λ1=0.3, λ2=0.6I, κ1=3I, and κ2=2 and density figures of u~n with parameters λ1=0.4, λ2=25, κ1=32, and κ2=0.8.
(a)
Density plot ofqn(b)
Density plot ofrn(c)
Density plot ofsn(d)
Density plot ofunFigure 5
The evolution of solutionq~n, r~n, and s~n with parameters λ1=0.3, λ2=0.6I, κ1=3I, and κ2=2 and evolution of solution u~n with parameters λ1=0.4, λ2=25, κ1=32, and κ2=0.8.
(a)
The evolution ofqn(b)
The evolution ofrn(c)
The evolution ofsn(d)
The evolution ofunFrom Figures1 and 2, it can be observed that the solitary waves move from right to left whether the parameter contains complex numbers or not. From the evolutions of solutions in Figure 1, we see that these one-soliton solutions are not stable. From Figure 2, we observed that q~n and r~n have the kink-shaped structures when we take the appropriate parameters. From Figure 5, it is easy to see that the solitary waves of q~n, r~n, and s~n also move from right to left, and they are also not stable. From Figures 1–5, we find that the shapes, amplitudes, and wavelengths of solutions have a big difference before and after coupling, but the directions of solitary wave solution propagation have not changed.


## 3.1. Explicit Solutions for Equation (1)

In order to obtain the explicit solutions of the semidiscrete integrable coupled equations (1), we choose the trivial solution qn=un=et,rn=sn=e−t. Then, the spectral problems Equations (2) and (7) become
(29)Eφn=0λet0λet−λe−tλ2−λe−t2λ2000λet00−λe−tλ2φn,φn,t=−12λ2λet−12λ2+10−λe−t12λ2012λ200−12λ2λet00−λe−t12λ2φn.By solving Equation (29), we achieve the two basic solutions of Equation (1), φn and ψn,
(30)φn=φ1,nφ2,nφ3,nφ4,n=−λ3λ2t−t+12λ2−2e−1/2λ2−2t2−λ4+t−3λ22λ2−2e−1/2λ2tλe−1/2λ2−2te1/2λ2t,ψn=ψ1,nψ2,nψ3,nψ4,n=λ2n−1λ2+2λ2t−t−12λ2−2e1/2λ2tλ2n−2λ2+2λ4t−λ2t−1+λ2−2λ2−12λ2−2e12λ2−1tλ2n−1e1/2λ2tλ2ne1/2λ2−2t.Then, based on Equation (13), we can obtain
(31)δi,1n=−−2t−4λi2n+2+t+1λi2n+4+tλi2n+6κietλi2−1+−λi6+λi4t−3λi4+2λi22κi−1/2t+1/2λi2n+2−1/2λi2n+4t+λi2nt+1etλi2−1/2λi3etλi+1tλi2−t+1λi,δi,2n=−2λi2−1λi2ne1/2tλi2κi−λi2−1/2tλi2+t−λi3λi+1tλi2−t+1e−1/2tλi2+t+2e1/2tλi2−1/2t+1/2λi2n+2−1/2λi2n+4t+λi2nt+1κi,δ1,3n=−2κiλi2+2κiλi2n+1etλi2−1+2λi3−2λi2κi−1/2t+1/2λi2n+2−1/2λi2n+4t+λi2nt+1etλi2−λi3etλi+1tλi2−t+1.According to the transformations Equations (19), (16), and (31), we can obtain the explicit solutions of Equation (1) by symbolic computing software maple. It is presented by δi,1n,δi,2n,δi,3n in Appendix B.


## 3.2. Discussions

The integrable equations investigated in [22] are presented as follows:
(32)qn,t=−qnqn+1rn+1rn+qn,rn,t=rnqnrn−1−1qn−rn.The explicit solutions of Equation (32) by Darboux transformation method are given in [22]
(33)q~n=−ρ1κ1λ12nλ22λ12−1e2t+ρ2κ2λ22nρ1ρ3et−λ12e2t+λ12λ22e2tρ1κ1λ12n+2λ22λ12−1et−ρ2κ2λ12λ22n+2ρ1ρ3−et+λ22et,r~n=λ12+λ22−ρ1κ1λ12n+2λ22−1e−t+ρ2κ2λ22n+2λ12−1e−tλ12et−λ22et+ρ1κ1λ12nλ22−1−ρ2κ2λ22nλ12−1,where ρ1=eλ12t,ρ2=eλ22t, and ρ3=κ1λ12nλ12−λ22.In order to compare the differences of the explicit solutions between integrable coupled Equation (1) and integrable Equation (32), we first investigate the three-dimensional structure and the evolution properties of explicit solutions Equation (33) with two sets of parameters, λ1=0.4,λ2=25,κ1=32, and κ2=0.8 and λ1=0.3, λ2=0.6I, κ1=3I, and κ2=2 in Figures 1 and 2. Then, we plot the three-dimensional structures, density figures, the evolution properties of q~n,r~n, and s~n with parameters λ1=0.3, λ2=0.6I, and κ1=3I, and u~n with parameters λ1=0.4,λ2=25,κ1=32, and κ2=0.8 in Figures 3–5 (The expression for u~n is so complicated that Maple cannot calculate its figure when the parameters contain a complex number. The specific expression of u~n is in the appendix.).Figure 1
Three-dimensional structure figures of explicit solutions from Equation (32) and the evolutions of these solutions with parameters λ1=0.3, λ2=0.6I, κ1=3I, and κ2=2.
(a)
qn(b)
rn(c)
qn in Equation (24)(d)
rn in Equation (29)Figure 2
Three-dimensional structure figures of explicit solutions from Equation (32) and the evolutions of these solutions with parameters λ1=0.4, λ2=25, κ1=32, and κ2=0.8.
(a)
qn(b)
rn(c)
qn in Equation (29)(d)
rn in Equation (29)Figure 3
Three-dimensional structure figures ofq~n, r~n, and s~n with parameters λ1=0.3, λ2=0.6I, κ1=3I, and κ2=2 and density figures of u~n with parameters λ1=0.4, λ2=25, κ1=32, and κ2=0.8.
(a)
qn(b)
rn(c)
sn(d)
unFigure 4
Density figures ofq~n, r~n, and s~n with parameters λ1=0.3, λ2=0.6I, κ1=3I, and κ2=2 and density figures of u~n with parameters λ1=0.4, λ2=25, κ1=32, and κ2=0.8.
(a)
Density plot ofqn(b)
Density plot ofrn(c)
Density plot ofsn(d)
Density plot ofunFigure 5
The evolution of solutionq~n, r~n, and s~n with parameters λ1=0.3, λ2=0.6I, κ1=3I, and κ2=2 and evolution of solution u~n with parameters λ1=0.4, λ2=25, κ1=32, and κ2=0.8.
(a)
The evolution ofqn(b)
The evolution ofrn(c)
The evolution ofsn(d)
The evolution ofunFrom Figures1 and 2, it can be observed that the solitary waves move from right to left whether the parameter contains complex numbers or not. From the evolutions of solutions in Figure 1, we see that these one-soliton solutions are not stable. From Figure 2, we observed that q~n and r~n have the kink-shaped structures when we take the appropriate parameters. From Figure 5, it is easy to see that the solitary waves of q~n, r~n, and s~n also move from right to left, and they are also not stable. From Figures 1–5, we find that the shapes, amplitudes, and wavelengths of solutions have a big difference before and after coupling, but the directions of solitary wave solution propagation have not changed.


## 4. Conclusions

In this paper, we have constructed a coupled4×4 Lax pair by enlarging the 2×2 Lax pair and derived new semidiscrete integrable coupled equations which include the original integrable equations as subequations. Next, we have found a suitable Darboux matrix and obtained the explicit solutions of Equation (1) according to the seed solution qn=un=et,rn=sn=e−t by means of the Darboux transformation method. Then, we plot three-dimensional structure figures, the evolution properties of solutions before and after coupling, and the density plot of solutions after coupling. Finally, we analyze the properties of solutions before and after coupling. All the results in this paper may be helpful in understanding some physical phenomena.



---

*Source: 9964540-2021-12-16.xml*