# Obtaining Solutions of a Vakhnenko Lattice System byN-Fold Darboux Transformation

Authors: Ning Zhang; Xi-Xiang Xu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Using a suitable gauge transformation matrix, we present aN-fold Darboux transformation for a Vakhnenko lattice system. This transformation preserves the form of Lax pair of the Vakhnenko lattice system. Applying the obtained Darboux transformation, we arrive at an exact solution of the Vakhnenko lattice system.


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## Body

## 1. Introduction

Since the beginning of this century, the integrable lattice systems (or lattice soliton systems) have received considerable attention. Many important integrable lattice systems have been studied from the perspective of Mathematics and Physics, for instance, the Ablowitz-Ladik lattice [1], the Toda lattice [2], and the relativistic Toda lattice [3], [4–11].In the soliton theory, the Darboux transformation is a very effective method for solving soliton equations [12, 13]. Later, it also applies to solving integrable lattice equations [7–9, 14, 15]. In reference [10], a Vakhnenko lattice system is introduced:(1)pn,t=pnqnxn−1−pnqn+1xn−rnxn−1+rnxn−qnyn+qn+1yn,qn,t=qnrn−1yn−qnqn+1xn+qn+1zn−pnqn−rn−1+rn,rn,t=rnrn−1yn−qn+1rnxn−pnrn−1+rnzn−qn+qn+1xn,t=qnxnxn−1−rnxnyn+1−xn−1zn+pnxn−yn+yn+1,yn,t=qnxn−1yn−rnynyn+1+pnyn+1−ynzn−xn−1+xn,zn,t=rn−1ynzn−rnyn+1zn−rn−1xn+rnxn−qnyn+qnyn+1,where pn=pn,t,qn=qn,t,rn=rn,t,xn=xn,t, yn=yn,t,zn=zn,t, n is a discrete variable, n∈N, and t is a continuous variable, t∈R. Equation (1) can be rewritten as a discrete zero curvature equation(2)Un,t=EVnUn−UnVn=Vn+1Un−UnVn,of a discrete spatial spectral problem(3)Eφn=Unun,λφn,Unun,λ=λ2+pnλqn+1λrnλxn+1λynzn+1λ2,and a corresponding continuous time evolution equation(4)φn,t=Vnun,λφn,Vnun,λ=λ2−qnxn−1qnλ+rn−11λλqn−1+yn1λ1λ2−rn−1yn.Here, for a lattice functionfn=fn, the shift operator E and the inverse of E are defined as follows.(5)Efn=fn+1,E−1fn=fn−1,n∈Z.In equations (3) and (4), φn1,φn2T is the eigenfunction vector, pn,qn,rn,xn,yn,znT is a potential vector, (1) is a very meaningful lattice system, and many important lattice systems can be reduced from it, such as the nonlinear self-dual network equation, the two coupled discrete nonlinear Schrödinger equation, and the relativistic Volterra lattice [9]. In reference [9], the authors discussed the Darboux transformation of (1), but their results are incorrect [11]. In reference [10], although the author gave some properties of the Darboux transformation of (1), but his approach is different from ours. In reference [11], we derived a 1-fold Darboux transformation of (1). In addition, let pn=zn=wn,qn=un,rn=vn,xn=vn,yn=un, and the equation (1) is reduced to a three-component differential-difference system.(6)un,t=un2vn−1−unun+1vn+un+1wn−unvn−vn−1+vn,vn,t=unvn−1vn−un+1vn2−wnvn−1+vnwn−un+un+1,wn,t=unvn−1wn−un+1vnwn−vn−1vn+vn2−un2+unun+1.In reference [14], its N-fold Darboux transformation is presented. Furthermore, if we set pn=zn=InVn,qn=In,rn=Vn,xn=In,yn=Vn, equation (1) becomes the nonlinear self-dual network equation.(7)In,t=1+In2Vn−1−Vn,Vn,t=1+Vn2In−In+1.In reference [15], the author derived its N-fold Darboux transformation. In this letter, for arbitrary positive integer N, we will present a N-fold Darboux transformation for the Vakhnenko lattice system (1). Finally, an exact solution of (1) is derived.


## 2.N-Fold Darboux Transformation

For any positive integerN, we introduce the following matrix:(8)ΠnNλ=λ2N+∑j=0N−1An2jλ2j+∑j=1NAn−2jλ−2j∑j=1NBn2j−1λ2j−1+∑j=1NCn−2j+1λ−2j+1∑j=1NFn2j−1λ2j−1+∑j=1NGn−2j+1λ−2j+1∑j=1NHn2jλ2j+∑j=0N−1Hn−2jλ−2j+λ−2N.Here,An2j−N≤j≤N−1,Bn2j−1−N≤j≤N, Cn2j−1−N≤j≤N,Fn2j−1−N≤j≤N, Gn2j−N≤j≤N,Hn2j−N+1≤j≤N are 8N undetermined constants.Next, we consider the gauge transformation [7, 8].(9)ϕ^n=ΠnNλϕn.By the transformation (9), the Lax pairs (3) and (4) become(10)Eϕ^n=UnNewϕ^n,(11)ϕ^n,t=VnNewϕ^n,(12)UnNew=Πn+1NλUnΠnNλ−1,(13)VnNew=Πn,tNλ+ΠnNλVnΠnNλ−1.Equations (10) and (11) constitute a new Lax pair. Let us denote(14)ξn1λ,ξn2λT,ηn1λ,ηn2λT,are two real linear independent solutions of equations (3) and (4), and λi1≤i≤4N are 4N distinct eigenvalues of spectral problem (1).Proposition 1.
The matrixUnNew has the same form as Un in equation (3), and the transformation formula is presented by(15)Pn=An+12N−2−An2N−2+pn+xnBn2N−1−Fn2N−1qn−Bn2N−1Hn2N,Qn=qn−Bn2N−1Hn2N,Rn=rnAn+1−2N+Cn+1−2N+1,Xn=xnHn+12N+Fn+12N−1,Yn=yn−Gn−2N+1An−2N,Zn=zn+Hn+1−2N+2−Hn−2N+2+rnGn+1+2N−1−Cn−2N−1yn−Gn−2N+1An−2N.
Namely,(16)UnNew=λ2+PnλQn+1λRnλXn+1λYnZn+1λ2.
In the above matrix,Pn,Qn,Rn,Xn,Yn,Zn are determined by (15), and they are all independent λ. Obviously, (15) transform the old potentials pn,qn,rn,xn,yn,znT of (1) into the new potentials Pn,Qn,Rn,Xn,Yn,ZnT of (9).Proof.
We consider the following linear system:(17)∑j=0N−1An2jλi2j+∑j=1NAn−2jλi−2j+∑j=0N−1Bn2j+1λi2j+1+∑j=0N−1Cn−2j−1λi−2j−1ρin=−λi2N,∑j=0N−1Fn2j+1λi2j+1+∑j=0N−1Gn−2j−1λi−2j−1+∑j=1NHn2jλi2j+∑j=0N−1Hn−2jλi−2jρin=−ρinλi−2N,i=1,…4N.
Here,(18)ρin=ηn2λi+ζiξn2λiηn1λi+ζiξn1λi,i=1,2,…,4N,where λi1≤i≤4N,ζi1≤i≤4N are suitably chosen, such that all the determinants of coefficients for the equation and (17) are nonzero. By solving the linear system (17), we get An2j−N≤j≤N−1,B2j+10≤N−1, Cn−2j−10≤j≤N−1,Fn2j+10≤j≤N−1,Gn−2j−10≤j≤N−1,Hn2j−N+1≤j≤N.
From equations (3) and (18), we have(19)ρin+1=xnλi+1/λiyn+znρin+1/λi2ρinλi2+pn+qnλiρin+1/λirnρin,i=1,2.
For convenience, we set(20)ΠnNλ=ΘnaΘnbΘncΘnd,where(21)Θna=λ2N+∑j=1N−1An2jλ2j+∑j=1NAn−2jλ−2j,Θnb=∑j=0N−1Bn2j+1λ2j+1+∑j=0N−1Cn−2j−1λ−2j−1,Θnc=∑j=0N−1Fn2j+1λ2j+1+∑j=0N−1Gn−2j−1λ−2j−1,Θnd=∑j=1N−1An2jλ2j+∑j=1NAn−2jλ−2j+λ−2N.
Then, we obtain the following equation:(22)Πn+1NUnΠnN∗=Ω11λ,nΩ12λ,nΩ21λ,nΩ22λ,n,where ΠnN∗ is the adjoint matrix of ΠnN and(23)Ω11λ,n=Θn+1aΘndλ2+xnΘn+1bΘnd−qnΘn+1aΘncλ+pnΘn+1aΘnd−znΘn+1bΘnc+ynΘn+1bΘnd−rnΘn+1aΘncλ−1−Θn+1bΘncλ−2,Ω12λ,n=−Θn+1aΘnbλ2+qnΘn+1bΘnb−xnΘn+1aΘnaλ+znΘn+1bΘna−pnΘn+1aΘnb+rnΘnaΘn+1a−ynΘn+1bΘnbλ−1+Θn+1bΘnaλ−2,Ω21λ,n=Θn+1cΘndλ2+xnΘn+1dΘnd−qnΘn+1cΘncλ+pnΘn+1cΘnd−znΘn+1dΘnc+pnΘn+1cΘnd−znΘn+1dΘncλ−1−Θn+1bΘn+1cλ−2,Ω22λ,n=−ΘnaΘndλ2+rnΘn+1cΘna−ynΘn+1dΘnbλ+znΘn+1dΘna−pnΘnbΘn+1c+rnΘn+1cΘna−ynΘn+1dΘnbλ−1+Θn+1dΘnaλ−2,
In the light of (17), we can get(24)DetΠnNλ=Hn2N∏j=14Nλ2−λj2.
Analyzing the coefficients of powers ofλ in (22), we obtain that(25)Πn+1NλUn=αnΠnNλ.
Here,(26)αn=αna,2λ2+αna,0αnb,1λ+αnb,−1λαnc,1λ+αnc,−1λαnd,0+αnd,−2λ2,where αna,2,αna,0,αnb,1,αnb,−1,αnc,1,αnc,−1,αnd,0,αnd,−2 are all independent of λ. By comparing the coefficients of λ in (25), we have(27)αna,2=0,αna,2=An+12N−2−An2N−2+pn+xnBn2N−1−Fn2N−1qn−Bn2N−1Hn2N=Pn,αnb,1=qn−Bn2N−1Hn2N=Qn,αnb,−1=rnAn+1−2N+Cn+1−2N+1=Rn,αnc,1=xnHn+12N+Fn+12N−1=Xn,αnc,−1=yn−Gn−2N+1An−2N=Yn,αnd,−2=1,αnd,0=zn+Hn+1−2N+2−Hn−2N+2+rnGn+1−2N+1−Cn−2N+1yn−Gn−2N+1An−2N=Zn.
The proposition is proved.Proposition 2.
Under the transformation (15), the matrix VnNew has the same form as Vn in equation (2). In other words,(28)VNew=λ2−QnXn−1Qnλ+Rn−11λλQn−1+Yn1λ1λ2−Rn−1Yn.Proof.
We consider(29)Πn,tN+ΠnNVnΠnN∗=Γ11λ,nΓ12λ,nΓ21λ,nΓ22λ,n,where(30)Γ11λ,n=ΘnaΘndλ2+xn−1ΘnbΘndλ+Θn,taΘnd−xn−1qnΘnaΘnd+qnΘnbΘndλ−1,Γ12λ,n=−qnΘnaΘnbλ−ΘnbΘn,tb+rn−1ynΘnb2−rn−1ΘnaΘnbλ−1−Θnb2λ−2,Γ21λ,n=−Θnc2λ2−qnΘnaΘncλ−ΘncΘn,tc+xn−1qnΘnc2−ynΘncΘndλ−1,Γ22λ,n=qnΘnaΘncλ−ΘnaΘn,td−rn−1ynΘnaΘnd+rn−1ΘnaΘncλ−1+ΘnaΘndλ−2.
Owing toDetΠnN=Hn2N∏i=1Nλ2−λi2 and paying attention to the coefficients of powers of λ in (29), we find(31)Πn,tN+ΠnNVn=βnΠnN.
In equation (31),(32)βn=β112λ2+β110β121λ1+β12−1λ−1β211λ+β21−1λ−1β220+β22−2λ−2,where β110,β112,β12−1,β121,β21−1,β121,β220,β22−2 are all independent of spectral parameter λ. Comparing the coefficients of λN+i−2≤i≤2 in equation (31), we arrive at(33)β112=1,β110=−qn−Bn2N−1Hn2Nxn−1Hn2N+Fn2N−1=−QnXn−1,β121=qn−Bn2N−1Hn2N=Qn,β12−1=xn−1Hn2N+Fn2N−1Rn−1,β211=qn−1−Bn−12N−1Hn−12Nβ12−1=yn−Gn−2N+1An−2N=Yn,β22−2=1,β220=−rn−1An−2N+Cn−2N+1yn−Gn−2N+1An−2N=−Rn−1Yn.
The proposition is proved.
In summary, we get the following theorem.Theorem 1.
The equations (9) and (15) constitute a Darboux transformation of (1), that is, from an old solution pn,qn,rn,xn,yn,znT of (1), through transformation (15), a new solution Pn,Qn,Rn,Xn,Yn,ZnT of (1) is derived.


## 3. An Exact Solution

In what follows, we will derive a solution of equation (1) by the Darboux transformation (15). For simplicity, we consider the case of N=1.First, we select the seed solution of the lattice system (1), namely, the simple special solution, pn,qn,rn,xn,yn,znT=1,0,0,0,0,1T. Substituting this solution into the corresponding Lax pair, we get(34)Eϕn=λ2+1001+1λ2ϕn,ϕnt=−λ22−λλλ22+1ϕn.Solving the above two equations, we get two real linear independent solutions:(35)ξnλ=expλ2t1+λ2n−121+1λ2n−1,ηnλ=expλ2t21+λ2n−11+1λ2n−1.Then, we have(36)ρin=ξn2λi+γiηn2λiξn1λi+γiηn1λi=λi2n−21+2γi2+γi,i=1,2,3,4.Here,λi1≤i≤4 are four different arbitrary constants. When N=1,(37)Πn1λ=λ2+An0+An−2λ−2Bn1λ+Cn−1λ−1Fn1λ+Gn−1λ−1Hn2λ2+Hn0+λ−2.According to Proposition1, we can get the following Darboux transformation:(38)ϕ^n=Πn1λϕn,Pn=An+10−An0+pn+xnBn1−Fn1qn−Bn1Hn2,Qn=qn−Bn1Hn2,Rn=rnAn+1−2+Cn+1−1,Xn=xnHn+12+Fn+11,Yn=yn−Gn−1An−2,Zn=zn+Hn+10−Hn0+rnGn+1−1−Cn−1yn−Gn−1An−2.Here,(39)An0=−Detλ12λ1−2λ1ρ1nλ1−1ρ1nλ22λ2−2λ2ρ2nλ2−1ρ2nλ32λ3−2λ3ρ3nλ3−1ρ3nλ42λ4−2λ4ρ4nλ4−1ρ1nDet1λ1−2λ1ρ1nλ1−1ρ1n1λ2−2λ2ρ2nλ2−1ρ2n1λ3−2λ3ρ3nλ3−1ρ3n1λ4−2λ4ρ4nλ4−1ρ1n,An−2=−Det1λ12λ1ρ1nλ1−1ρ1n1λ22λ2ρ2nλ2−1ρ2n1λ32λ3ρ3nλ3−1ρ3n1λ42λ4ρ4nλ4−1ρ1nDet1λ1−2λ1ρ1nλ1−1ρ1n1λ2−2λ2ρ2nλ2−1ρ2n1λ3−2λ3ρ3nλ3−1ρ3n1λ4−2λ4ρ4nλ4−1ρ1n,Bn1=−Det1λ1−2λ12λ1−1ρ1n1λ2−2λ22λ2−1ρ2n1λ3−2λ32λ3−1ρ3n1λ4−2λ42λ4−1ρ1nDet1λ1−2λ1ρ1nλ1−1ρ1n1λ2−2λ2ρ2nλ2−1ρ2n1λ3−2λ3ρ3nλ3−1ρ3n1λ4−2λ4ρ4nλ4−1ρ1n,Cn−1=−Det1λ1−2λ1ρ1nλ121λ2−2λ2ρ2nλ21λ3−2λ3ρ3nλ321λ4−2λ4ρ1nλ42Det1λ1−2λ1ρ1nλ1−1ρ1n1λ2−2λ2ρ2nλ2−1ρ2n1λ3−2λ3ρ3nλ3−1ρ3n1λ4−2λ4ρ4nλ4−1ρ1n,Fn1=Detλ1−1λ1−2ρ1nλ12ρ1nρ1nλ2−1λ2−2ρ2nλ22ρ2nρ2nλ3−1λ3−2ρ3nλ32ρ3nρ3nλ4−1λ4−2ρ4nλ42ρ4nρ4nDetλ1λ1−1λ12ρ1nρ1nλ2λ2−1λ2ρ2nρ2nλ3λ3−1λ32ρ3nρ3nλ4λ4−1λ42ρ4nρ1n,Gn−1=Detλ1−2ρ1nλ1λ12ρ1nρ1nλ2−2ρ2nλ2λ2ρ2nρ2nλ3−2ρ3nλ3λ32ρ3nρ3nλ4−2ρ4nλ4λ42ρ4nρ1nDetλ1λ1−1λ12ρ1nρ1nλ2λ2−1λ2ρ2nρ2nλ3λ3−1λ32ρ3nρ3nλ4λ4−1λ42ρ4nρ1n,Hn2=−Detλ1λ1−1λ1−2ρ1nρ1nλ2λ2−1λ2−2ρ2nρ2nλ3λ3−1λ3−2ρ3nρ3nλ4λ4−1λ4−2ρ4nρ4nDetλ1λ1−1λ12ρ1nρ1nλ2λ2−1λ2ρ2nρ2nλ3λ3−1λ32ρ3nρ3nλ4λ4−1λ42ρ4nρ1n,Hn0=−Detλ1λ1−1λ12ρ1nλ1−2ρ1nλ2λ2−1λ2ρ2nλ2−2ρ2nλ3λ3−1λ32ρ3nλ3−2ρ3nλ4λ4−1λ42ρ4nλ4−2ρ4nDetλ1λ1−1λ12ρ1nρ1nλ2λ2−1λ2ρ2nρ2nλ3λ3−1λ32ρ3nρ3nλ4λ4−1λ42ρ4nρ1n.Thus, we can obtain a solution of equation (1) as follows:(40)Pn=1+An+10−An0+Fn−1Bn1Hn2,Qn=−Bn1Hn2,Rn=Cn+1−1,Xn=Fn+11,Yn=−Gn−1An−2,Zn=1+Hn+10−Hn0+Cn−1Gn−1An−2.In equation (40),(41)An+10=Detλ1−2λ12λ1ρ1n+1λ1−1ρ1n+1λ2−2λ22λ2ρ2n+1λ2−1ρ2n+1λ3−2λ32λ3ρ3n+1λ3−1ρ3n+1λ4−2λ42λ4ρ4n+1λ4−1ρ1n+1Det1λ1−2λ1ρ1n+1λ1−1ρ1n+11λ2−2λ2ρ2n+1λ2−1ρ2n+11λ3−2λ3ρ3n+1λ3−1ρ3n+11λ4−2λ4ρ4n+1λ4−1ρ1n+1,Cn+1−1=Detλ1−21λ1ρ1n+1λ12λ2−21λ2ρ2n+1λ2λ3−21λ3ρ3n+1λ32λ4−21λ4ρ1n+1λ42Det1λ1−2λ1ρ1n+1λ1−1ρ1n+11λ2−2λ2ρ2n+1λ2−1ρ2n+11λ3−2λ3ρ3n+1λ3−1ρ3n+11λ4−2λ4ρ4n+1λ4−1ρ1n+1,Fn+11=Detλ1−1λ1−2ρ1nλ12ρ1n+1ρ1n+1λ2−1λ2−2ρ2nλ22ρ2n+1ρ2n+1λ3−1λ3−2ρ3nλ32ρ3n+1ρ3n+1λ4−1λ4−2ρ4nλ42ρ4n+1ρ4n+1Detλ1λ1−1λ12ρ1n+1ρ1n+1λ2λ2−1λ2ρ2n+1ρ2n+1λ3λ3−1λ32ρ3n+1ρ3n+1λ4λ4−1λ42ρ4n+1ρ1n+1,Hn+10=Detλ1−1λ1λ12ρ1n+1λ1−2ρ1nλ2−1λ2λ2ρ2n+1λ2−2ρ2nλ3−1λ3λ32ρ3n+1λ3−2ρ3nλ4−1λ4λ42ρ4n+1λ4−2ρ4nDetλ1λ1−1λ12ρ1n+1ρ1n+1λ2λ2−1λ2ρ2n+1ρ2n+1λ3λ3−1λ32ρ3n+1ρ3n+1λ4λ4−1λ42ρ4n+1ρ1n+1.Here,(42)ρin+1=ρin+1/λi2ρin1+λi2,i=1,2,3,4.


## 4. Conclusion

In this work, by means of a gauge transformation of Lax pair, we established aN-fold Darboux transformation for a Vakhnenko lattice system. Under this transformation, the structure of the Lax pair remains unchanged. Finally, as an application of this transformation, an exact solution of the Vakhnenko lattice system (1) is given. Starting from the exact solution (40), we apply the Darboux transformation (38) once again; then, another new solution of equation (1) is derived. This process can be performed continually. So, we can get many exact solutions for the lattice system (1).



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