# Residual Symmetries and Bäcklund Transformations of (2 + 1)-Dimensional Strongly Coupled Burgers System

Authors: Haifeng Wang; Yufeng Zhang
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In this article, we mainly apply the nonlocal residual symmetry analysis to a (2 + 1)-dimensional strongly coupled Burgers system, which is defined by us through taking values in a commutative subalgebra. On the basis of the general theory of Painlevé analysis, we get a residual symmetry of the strongly coupled Burgers system. Then, we introduce a suitable enlarged system to localize the nonlocal residual symmetry. In addition, a Bäcklund transformation is derived by Lie’s first theorem. Further, the linear superposition of the multiple residual symmetries is localized to a Lie point symmetry, and anN-th Bäcklund transformation is also obtained.


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## Body

## 1. Introduction

Nonlinear partial differential equations have wide applications in the field of physical science, engineering, and other applied disciplines, e.g., nonlinear optics [1–4], fluid flows [5–7], plasma physics [8, 9], excitable media, and so on [10–14]. Burgers equation pt=2ppx+pxx is a very important nonlinear partial differential equation occurring in various areas of applied sciences, such as fluid mechanics [15], nonlinear acoustics [15], gas dynamics, and traffic flow [16]. The equation was first introduced by Harry Bateman in 1915 and later studied by Johannes Martinus Burgers in 1948 [17, 18]. The study of symmetries plays an important role in branches of some natural sciences especially in integrable systems [19, 20]. In [21, 22], the authors proposed a residual symmetry in the process of the residue of the truncated Painlevé expansion for the bosonized super symmetric KdV equation which is a nonlocal symmetry [23–33]. In [34, 35], the authors concerned with the application of the nonlocal residual symmetry analysis to (2 + 1)-dimensional Burgers system, which has the form as follows:(1a)pt=ppy+arpx+bpyy+abpxx,(1b)px=ry,wherea and b are arbitrary constants.In [36,37], a hierarchy called the Frobenius-valued Kakomtsev–Petviashvili hierarchy which takes values in a maximal commutative subalgebra of glm,C was constructed. Then, in [38], the authors considered the Hirota quadratic equation of the commutative version of extended multicomponent Toda hierarchy, which should be useful in Frobenius manifold theory [39,40]. Recently, we studied Zn-Painlevé IV equation, Frobenius Painlevé I equation, and Frobenius Painlevé III equation [41]. In this paper, we consider a new (2 + 1)-dimensional strongly coupled Burgers system which is defined by us through taking values in a commutative subalgebra Z2=CΓ/Γ2. We replace the p and r of (1a) and (1b) with the commutative matrix(2)pqqp,rssr.Then, we can get(3a)pt=ppy+qqy+arpx+asqx+bpyy+abpxx,qt=pqy+qpy+arqx+aspx+bqyy+abqxx,(3b)px=ry,qx=sy,which is called (2 + 1)-dimensional strongly coupled Burgers system.The aim of this paper is to promote the (2 + 1)-dimensional Burgers system to a Frobenius integrable systems which is called (2 + 1)-dimensional strongly coupled Burgers system. A suitable enlarged system is given to localize the nonlocal residual symmetry. It follows that a Bäcklund transformation is derived by solving an initial value problem. It means that one can find various solutions of the (2 + 1)-dimensional strongly coupled Burgers system from a seed solution.


## 2. Residual Symmetries of (2 + 1)-Dimensional Strongly Coupled Burgers System

We first introduce the truncated Painlevé expansion:(4)p=∑i=0α0piψ+ϕi−α0+piψ−ϕi−α0−qiψ−ϕi−α0+qiψ+ϕi−α0,q=∑i=0α1qiψ+ϕi−α1+qiψ−ϕi−α1−piψ−ϕi−α1+piψ+ϕi−α1,r=∑i=0α2riψ+ϕi−α2+riψ−ϕi−α2−siψ−ϕi−α2+siψ+ϕi−α2,s=∑i=0α3siψ+ϕi−α3+siψ−ϕi−α3−riψ−ϕi−α3+riψ+ϕi−α3,wherepα, qα are a set of arbitrary solutions of the equation, and pα−1, pα−2, …,p0, qα−1, qα−2, …,q0 to be expressed by derivatives of ψ and ϕ. By balancing the dispersion and nonlinear terms according to the leader order analysis to the system (3a) and (3b), the truncated Painlevé expansion has the following form:(5a)p=p0ψ−q0ϕψ2−ϕ2+p1,q=q0ψ−p0ϕψ2−ϕ2+q1,(5b)r=r0ψ−s0ϕψ2−ϕ2+r1,s=s0ψ−r0ϕψ2−ϕ2+s1.Then, plugging (5a) and (5b) into (3a) and (3b) and vanishing all the coefficients of each power of ψ+ϕ−n+ψ−ϕ−n and ψ−ϕ−n−ψ+ϕ−n, we obtain(6)p0=2bψy,q0=2bϕy,r0=2bψx,s0=2bϕx,(7)aψxr1+aϕxs1+abψxx+p1ψy+q1ϕy+bψyy−ψt=0,aϕxr1+aψxs1+abϕxx+q1ψy+p1ϕy+bϕyy−ϕt=0,(8)p1t−p1p1y−q1q1y−ar1p1x−as1q1x−bp1yy−abp1xx=0,q1t−p1q1y−q1p1y−as1p1x−ar1q1x−bq1yy−abq1xx=0,(9)p1x−r1y=0,q1x−s1y=0.It is easy to find that (8) and (9) are just the (2 + 1)-dimensional strongly coupled Burgers system (3a) and (3b) with p1, q1, r1, and s1 as solutions. We then substitute p0, q0, r0, and s0 of (6) into the linearized form of (8) and (9) with (7) that one can find the p0, q0, r0, and s0 in (6) are the symmetries of the strongly coupled Burgers system.According to the theorem of the residual symmetry [22], the strongly coupled Burgers system has a residual symmetry:(10)σp1=2bψy,σq1=2bϕy,σr1=2bψx,σs1=2bϕx,which is nonlocal forψ and ϕ related to p1, q1, r1, and s1 by (7). Then, we introduce auxiliary variables f, g, h, and k with the relations f=ψx, g=ϕx, h=ψy, and k=ϕy to obtain a local symmetry in the following enlarged system:(11a)pt−ppy−qqy−arpx−asqx−bpyy−abpxx=0,qt−pqy−qpy−aspx−arqx−bqyy−abqxx=0,(11b)px=ry,qx=sy,(11c)aψxr+aϕxs+abψxx+pψy+qϕy+bψyy−ψt=0,aϕxr+aψxs+abϕxx+qψy+pϕy+bϕyy−ϕt=0,(11d)f=ψx,g=ϕx,(11e)h=ψy,k=ϕy.Further, the residual symmetry can be localized into the Lie point symmetry(12a)σp=h,σq=k,(12b)σr=f,σs=g,(12c)σf=−1bfψ+gϕ,σg=−1bgψ+fϕ,(12d)σh=−1bhψ+kϕ,σk=−1bkψ+hϕ,(12e)σψ=−12bψ2+ϕ2,σϕ=−1bψϕ,which satisfy(13a)σtp−abσyxr−aσyrr−aσyss−aryσr−asyσs−bσyyp−pσyp−qσyq−pyσp−qyσq=0,σtq−abσyxs−aσysr−aσyrs−aryσs−asyσr−bσyyq−pσyq−qσyp−pyσq−qyσp=0,(13b)σxp=σyr,σxq=σys,(13c)σtψ−bσyyψ−aσxψr−aσxϕs−aψxσr−aϕxσs−abσxxψ−σpψy−σqϕy−pσyψ−qσyϕ=0,σtϕ−bσyyϕ−aσxψs−aσxϕr−aψxσs−aϕxσr−abσxxϕ−σpϕy−σqψy−pσyϕ−qσyψ=0,(13d)σxψ=σf,σxϕ=σg,(13e)σyψ=σh,σyϕ=σk.Therefore, the enlarged system (11a)–(11e) has the Lie point symmetry vector:(14)V→=h∂p+f∂r−1bfψ+gϕ∂f−1bhψ+kϕ∂h−12bψ2+ϕ2∂ψ,W→=k∂q+g∂s−1bgψ+fϕ∂g−1bkψ+hϕ∂k−1bψϕ∂ϕ.Next we will give the Bäcklund symmetry theorem, which is obtained by using a finite transformation of the Lie point symmetry (14).Theorem 1.
If{p,q,r,s,f,g,h,k,ψ,ϕ} is a solution of the coupled system (11a)–(11e), then so is p¯,q¯,r¯,s¯,f¯,g¯,h¯,k¯,ψ¯,ϕ¯ with(15a)p¯=p+2ϵbϵhψ−ϵkϕ+2bhϵψ+2b2−ϵ2ϕ2,q¯=q+2ϵbϵkψ−ϵhϕ+2bkϵψ+2b2−ϵ2ϕ2,(15b)r¯=r+2ϵbϵfψ−ϵgϕ+2bfϵψ+2b2−ϵ2ϕ2,s¯=s+2ϵbϵgψ−ϵfϕ+2bgϵψ+2b2−ϵ2ϕ2,(15c)ψ¯=2bϵψ2−ϵϕ2+2bψϵψ+2b2−ϵ2ϕ2,ϕ¯=4b2ϕϵψ+2b2−ϵ2ϕ2,(15d)f¯=4b2Cf−DgC2−D2,g¯=4b2Cg−DfC2−D2,(15e)h¯=4b2Ch−DkC2−D2,k¯=4b2Ck−DhC2−D2,
where(16)C=ϵ2ψ2+ϕ2+4bϵψ+4b2,D=2ϵ2ψϕ+4bϵϕ,
andϵ is an arbitrary group parameter.Proof.
According to Lie’s first theorem on vector (14), it is not difficult to find that the key to prove this theorem is to solve the following initial value problem:(17a)dp¯dϵ=h¯,dq¯dϵ=k¯,(17b)dr¯dϵ=f¯,ds¯dϵ=g¯,(17c)df¯dϵ=−1bψ¯f¯+ϕ¯g¯,dg¯dϵ=−1bψ¯g¯+ϕ¯f¯,(17d)dh¯dϵ=−1bψ¯h¯+ϕ¯k¯,dk¯dϵ=−1bψ¯k¯+ϕ¯h¯,(17e)dψ¯dϵ=−12bψ¯2+ϕ¯2,dϕ¯dϵ=−1bψ¯ϕ¯,(17f)p¯0=p,q¯0=q,r¯0=r,s¯0=s,f¯0=f,g¯0=g,h¯0=h,k¯0=k,ψ¯0=ψ,ϕ¯0=ϕ.
And (15a)–(15e) is the solution to the above system. Therefore, we have completed the proof of Theorem 1.


## 3. Bäcklund Transformations of Strongly Coupled Burgers System Related to Multiple Residual Symmetries

For the (2 + 1)-dimensional strongly coupled Burgers system (3a) and (3b), the original residual symmetry(18)σp=2bψy,σq=2bϕy,σr=2bψx,σs=2bϕx,is related to the solution of Equation (11c). Then, according to the linear property of symmetry equations, the multiple residual symmetries are expressed in terms of any linear superposition of symmetry(19)σnp=∑i=1nciψi,y,σnq=∑i=1nciϕi,y,σnr=∑i=1nciψi,x,σns=∑i=1nciϕi,x,n=1,2,3,…,whereψi and ϕi are different solutions of (11c). And the symmetry (19) should be localized to a Lie point symmetry by introducing more variables. In this way, one can find the finite transformation group of the symmetry (19).Theorem 2.
If{p,q,r,s,fi,gi,hi,ki,ψi,ϕi}i=1,2,…,n is a solution of the enlarged system(20a)pt−ppy−qqy−arpx−asqx−bpyy−abpxx=0,qt−pqy−qpy−aspx−arqx−bqyy−abqxx=0,(20b)px=ry,qx=sy,(20c)aψi,xr+aϕi,xs+abψi,xx+pψi,y+qϕi,y+bψi,yy−ψi,t=0,aϕi,xr+aψi,xs+abϕi,xx+qψi,y+pϕi,y+bϕi,yy−ϕi,t=0,(20d)fi=ψi,x,gi=ϕi,x,(20e)hi=ψi,y,ki=ϕi,y,
i
=
1
,
2
,
…
,
n
,
then the symmetry (19) is localized to the Lie point symmetry(21a)σnp=∑i=1ncihi,σnq=∑i=1nciki,(21b)σnr=∑i=1ncifi,σns=∑i=1ncigi,(21c)σψi=−ci2bψi2+ϕi2−∑j≠incj2bψjψi+ϕjϕi,σϕi=−cibψiϕi−∑j≠incj2bψjϕi+ϕjψi,(21d)σfi=−ci2bfiψi+giϕi−∑j≠incj2bfiψj+giϕj+fjψi+gjϕi,σgi=−ci2bfiϕi+giψi−∑j≠incj2bfiϕj+giψj+fjϕi+gjψi,(21e)σhi=−ci2bhiψi+kiϕi−∑j≠incj2bhiψj+kiϕj+hjψi+kjϕi,σki=−ci2bhiϕi+kiψi−∑j≠incj2bhiϕj+kiψj+hjϕi+kjψi.Proof.
The extended system (20a)–(20e) has the following linearized form:(22a)σtp−abσyxr−aσyrr−aσyss−aryσr−asyσs−bσyyp−pσyp−qσyq−pyσp−qyσq=0,σtq−abσyxs−aσysr−aσyrs−aryσs−asyσr−bσyyq−pσyq−qσyp−pyσq−qyσp=0,(22b)σxp=σyr,σxq=σys,(22c)σtψi−bσyyψi−aσxψir−aσxϕis−aψi,xσr−aϕi,xσs−abσxxψi−σpψi,y−σqϕi,y−pσyψi−qσyϕi=0,σtϕi−bσyyϕi−aσxψis−aσxϕir−aψi,xσs−aϕi,xσr−abσxxϕi−σpϕi,y−σqψi,y−pσyϕi−qσyψi=0,(22d)σxψi=σfi,σxϕi=σgi,(22e)σyψi=σhi,σyϕi=σki.
Let us first consider the special case: for any fixedm,cm≠0, while cj=0, j≠m in (19), from (12a)–(12e), the localized symmetry for {p,q,r,s,fm,gm,hm,km,ψm,ϕm} can be obtained as follows:(23a)σp=cmhm,σq=cmkm,(23b)σr=cmfm,σs=cmgm,(23c)σψm=−cm2bψm2+ϕm2,σϕm=−cmbψmϕm,(23d)σfm=−cmbfmψm+gmϕm,σgm=−cmbfmϕm+gmψm,(23e)σhm=−cmbhmψm+kmϕm,σkm=−cmbfmϕm+kmψm.
In (20c), we let i=m and i=j, and then we obtain(24)aψj,xr+aϕj,xs+abψj,xx+pψj,y+qϕj,y+bψj,yy−ψj,t=0,aϕj,xr+aψj,xs+abϕj,xx+qψj,y+pϕj,y+bϕj,yy−ϕj,t=0,(25)aψm,xr+aϕm,xs+abψm,xx+pψm,y+qϕm,y+bψm,yy−ψm,t=0,aϕm,xr+aψm,xs+abϕm,xx+qψm,y+pϕm,y+bϕm,yy−ϕm,t=0.
Then, we substitute (23a) and (23b) into (22c) with i=j yield(26)σtψj−bσyyψj−aσxψjr−aσxϕjs−acmψj,xfm−acmϕj,xgm−abσxxψj−cmhmψj,y−cmkmϕj,y−pσyψj−qσyϕj=0,σtϕj−bσyyϕj−aσxϕjr−aσxψjs−acmψj,xgm−acmϕj,xfm−abσxxϕj−cmhmϕj,y−cmkmψj,y−pσyϕj−qσyψj=0.
Further, one can obtain a solution of Equation (26):(27)σψj=−cm2bψmψj+ϕmϕj,σϕj=−cm2bψmϕj+ϕmψj,
which can be verified by using (20b), (24), and (25). From (22d) and (22e) with i=j, the symmetry for fj, gj, hj, and kj can be given by:(28)σfj=−cm2bfjψm+gjϕm+fmψj+gmϕj,σgj=−cm2bfjϕm+gjψm+fmϕj+gmψj,(29)σhj=−cm2bhjψm+kjϕm+hmψj+kmϕj,σkj=−cm2bhjϕm+kjψm+hmϕj+kmψj.
Further, (21a)–(21e) can be obtained by taking a linear combination of the above results for m=1,2,…,n. Therefore, we have completed the proof of Theorem 2.According to Lie’s first theorem, the initial value problem of the Lie point symmetry (21a)–(21e) has the following form:(30a)dPϵdϵ=∑j=1ncjΨj,yϵ,dQϵdϵ=∑j=1ncjΦj,yϵ,(30b)dRϵdϵ=∑j=1ncjΨj,xϵ,dSϵdϵ=∑j=1ncjΦj,xϵ,(30c)dΨiϵdϵ=−ci2bΨi2ϵ+Φi2ϵ−∑j≠incj2bΨjϵΨiϵ+ΦjϵΦiϵ,dΦiϵdϵ=−cibΨiϵΦiϵ−∑j≠incj2bΨjϵΦiϵ+ΦjϵΨiϵ,(30d)dFi∈d∈=−cibFiϵΨiϵ+GiϵΦiϵ−∑j≠incj2bFiϵΨjϵ+GiϵΦjϵ+FjϵΨiϵ+GjϵΦiϵ,dGi∈d∈=−cibFiϵΦiϵ+GiϵΨiϵ−∑j≠incj2bFiϵΦjϵ+GiϵΨjϵ+FjϵΦiϵ+GjϵΨiϵ,(30e)dHi∈d∈=−cibHiϵΨiϵ+KiϵΦiϵ−∑j≠incj2bHiϵΨjϵ+KiϵΦjϵ+HjϵΨiϵ+KjϵΦiϵ,dKi∈d∈=−cibHiϵΦiϵ+KiϵΨiϵ−∑j≠incj2bHiϵΦjϵ+KiϵΨjϵ+HjϵΦiϵ+KjϵΨiϵ,(30f)P0=p,Q0=q,R0=r,S0=s,Ψi0=ψi,Φi0=ϕi,Fi0=fi,Gi0=gi,Hi0=hi,Ki0=ki,i=1,2,⋯,n.Then, one can get the followingN-th Bäcklund theorem for the extended system (20a)–(20e) by solving (30a)–(30f).Theorem 3.
If{p,q,r,s,fi,gi,hi,ki,ψi,ϕi} is a solution of the coupled system (20a)–(20e), then so is {Pϵ,Qϵ,Rϵ,Sϵ,Fiϵ,Giϵ,Hiϵ,Kiϵ,Ψiϵ,Φiϵ}, i=1,2,…,n, where(31a)Pϵ=p+blnA+B+lnA−By,Qϵ=q+blnA+B−lnA−By,(31b)Rϵ=r+blnA+B+lnA−Bx,Sϵ=s+blnA+B−lnA−Bx,(31c)Ψiϵ=2bA2−B2AiA−BiB,Φiϵ=2bA2−B2BiA−AiB,(31d)Fiϵ=Ψi,xϵ,Giϵ=Φi,xϵ,(31e)Hiϵ=Ψi,xϵ,Kiϵ=Φi,xϵ,
where(32)A=−c12bϵψ1−1c1ϵM1,2⋯c1ϵM1,j⋯c1ϵM1,nc2ϵM1,2−c22bϵψ2−1⋯c2ϵM2,j⋯c2ϵM2,n⋮⋮⋮⋮⋮⋮cjϵM1,jcjϵM2,j⋯−cj2bϵψj−1⋯cjϵMj,n⋮⋮⋮⋮⋮⋮cnϵM1,ncnϵM2,n⋯cnϵMj,n⋯−cn2bϵψn−1,B=−c12bϵϕ1−1c1ϵN1,2⋯c1ϵN1,j⋯c1ϵN1,nc2ϵN1,2−c22bϵϕ2−1⋯c2ϵN2,j⋯c2ϵN2,n⋮⋮⋮⋮⋮⋮cjϵN1,jcjϵN2,j⋯−cj2bϵϕj−1⋯cjϵNj,n⋮⋮⋮⋮⋮⋮cnϵN1,ncnϵN2,n⋯cnϵMj,n⋯−cn2bϵϕn−1,(33)Ai=−c12bϵψ1−1c1ϵM1,2⋯c1ϵM1,i−1c1ϵM1,ic1ϵM1,i+1⋯c1ϵM1,nc2ϵM1,2−c22bϵψ2−1⋯c2ϵM2,i−1c1ϵM2,ic2ϵM2,i+1⋯c2ϵM2,n⋮⋮⋮⋮⋮⋮⋮⋮ci−1ϵM1,i−1ci−1ϵM2,i−1⋯−ci−12bϵψi−1−1ci−1ϵMi−1,ici−1ϵMi−1,i+1⋯ci−1ϵMi−1,nM1,iM2,i⋯Mi−1,i−12bψiMi,i+1⋯Mi,nci+1ϵM1,i+1ci+1ϵM2,i+1⋯ci+1ϵMi−1,i+1−1ci+1ϵMi,i+1−ci+12bϵψi+1−1⋯ci+1ϵMi+1,n⋮⋮⋮⋮⋮⋮⋮cnϵM1,ncnϵM2,n⋯cnϵMi−1,ncnϵMi,ncnϵMi+1,n⋯−cn2bϵψn−1,Bi=−c12bϵϕ1−1c1ϵN1,2⋯c1ϵN1,i−1c1ϵN1,ic1ϵN1,i+1⋯c1ϵN1,nc2ϵN1,2−c22bϵϕ2−1⋯c2ϵN2,i−1c1ϵN2,ic2ϵN2,i+1⋯c2ϵN2,n⋮⋮⋮⋮⋮⋮⋮⋮ci−1ϵN1,i−1ci−1ϵN2,i−1⋯−ci−12bϵϕi−1−1ci−1ϵNi−1,ici−1ϵNi−1,i+1⋯ci−1ϵNi−1,nN1,iN2,i⋯Ni−1,i−12bϕiNi,i+1⋯Ni,nci+1ϵN1,i+1ci+1ϵN2,i+1⋯ci+1ϵNi−1,i+1−1ci+1ϵNi,i+1−ci+12bϵϕi+1−1⋯ci+1ϵNi+1,n⋮⋮⋮⋮⋮⋮⋮cnϵN1,ncnϵN2,n⋯cnϵNi−1,ncnϵNi,ncnϵNi+1,n⋯−cn2bϵϕn−1,
with(34)Mi,j=−14bψiψj+ϕiϕj+ψiϕj+ϕiψj1/2+ψiψj+ϕiϕj−ψiϕj−ϕiψj1/2,Ni,j=−14bψiψj+ϕiϕj+ψiϕj+ϕiψj1/2−ψiψj+ϕiϕj−ψiϕj−ϕiψj1/2.From any seed solution of the (2 + 1)-dimensional strongly coupled Burgers system, one can get an infinite number of new solutions becausen is an arbitrary positive integer. We consider a special case about the solution of the (2 + 1)-dimensional strongly coupled Burgers system. The (2 + 1)-dimensional strongly coupled Burgers system has a soliton solution:(35)p=a0+2b0be4k0−1e2k0+2k1+1e2k0−2k1+1+2b1be2k0+2k1−e2k0−2k1e2k0+2k1+1e2k0−2k1+1,q=a1+2b1be4k0−1e2k0+2k1+1e2k0−2k1+1+2b0be2k0+2k1−e2k0−2k1e2k0+2k1+1e2k0−2k1+1,r=s=0,where(36)k0=b0a0+b1a1t+b0y,k0=b0a0+b1a1t+b0y,anda0, a1, b0, and b1 are arbitrary constants.Then, the solution of (20c) can be obtained(37)ψi=die4k0−1e2k0+2k1+1e2k0−2k1+1+eie2k0+2k1−e2k0−2k1e2k0+2k1+1e2k0−2k1+1,ϕi=die2k0+2k1−e2k0−2k1e2k0+2k1+1e2k0−2k1+1+eie4k0−1e2k0+2k1+1e2k0−2k1+1,wheredi and ei are arbitrary constants.Further, one can obtain theMi,j and Ni,j of Theorem 3(38)Mi,j=−C04be4k0−1e2k0+2k1+1e2k0−2k1+1−D04be2k0+2k1−e2k0−2k1e2k0+2k1+1e2k0−2k1+1,Ni,j=−D04be4k0−1e2k0+2k1+1e2k0−2k1+1−C04be2k0+2k1−e2k0−2k1e2k0+2k1+1e2k0−2k1+1,where(39)C0=didj+eiej+diej+eidj1/2+didj+eiej−diej−eidj1/2,D0=didj+eiej+diej+eidj1/2−didj+eiej−diej−eidj1/2.We writeC1 and D1 as(40)C1=e4k0−1e2k0+2k1+1e2k0−2k1+1,D1=e2k0+2k1−e2k0−2k1e2k0+2k1+1e2k0−2k1+1.Finally, consider the case ofn=2 in Theorem 3, one can get a new soliton solution(41)p=C2c0d1+d2C1+c0e1+e2D1+2b−D2c0e1+e2C1+c0d1+d2D1c0d1+d2+e1+e2C1+D1+2bc0d1+d2−e1−e2C1−D1+2b,q=D2c0d1+d2C1+c0e1+e2D1+2b−C2c0e1+e2C1+c0d1+d2D1c0d1+d2+e1+e2C1+D1+2bc0d1+d2−e1−e2C1−D1+2b,r=s=0,where(42)C2=C1c0a0d1+a1e1+a0d2+a1e2+4b2b0+D1c0a0e1+a1d1+a0e2+a1d2+4b2b1+2bc0b0d1+b1e1+b0d2+b1e2+2ba0,D2=D1c0a0d1+a1e1+a0d2+a1e2+4b2b0+C1c0a0e1+a1d1+a0e2+a1d2+4b2b1+2bc0b1d1+b0e1+b1d2+b0e2+2ba1.


## 4. Conclusion and Discussion

In this paper, we first defined a new (2 + 1)-dimensional strongly coupled Burgers system which takes values in a two component commutative subalgebraZ2. Then, the residual symmetry of the strongly coupled Burgers system was obtained by using the truncated Painlevé expansion, and the corresponding residual system was just the nonlocal symmetry. To localize the residual symmetry, we introduced a suitable enlarged system. According to Lie’s first theorem, the finite Bäcklund transformation was derived. Further, the N-th Bäcklund transformation of the strongly coupled Burgers system was obtained by localizing the linear superposition of multiple residual symmetries, and the N-th Bäcklund transformation was expressed by determinants in a compact form.



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