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

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. en, 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 an -th Bäcklund transformation is also obtained.

In [36,37], a hierarchy called the Frobenius-valued Kakomtsev-Petviashvili hierarchy which takes values in a maximal commutative subalgebra of g푙(푚, C) was constructed. en, 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 -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 푍 2 = C[Γ]/ Γ 2 . We replace the and of (1) with the commutative matrix en, we can get which is called (2 + 1)-dimensional strongly coupled Burgers system.

Residual Symmetries of (2 + 1)-Dimensional Strongly Coupled Burgers System
We first introduce the truncated Painlevé expansion: where , are a set of arbitrary solutions of the equation, and 훼−1 , 훼−2 , …, 0 , 훼−1 , 훼−2 , . . . , 푞 0 to be expressed by derivatives of and . 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: en, plugging (5a and 5b) into (3a and 3b) and vanishing all the coefficients of each power of 휓 + 휙 −푛 It is easy to find that (8) and (9) are just the (2 + 1)-dimensional strongly coupled Burgers system (3) with 1 , 1 , 푟 1 , and 1 as solutions. We then substitute 0 , 0 , 0 , and 0 of (6) into the linearized form of (8) and (9) with (7) that one can find the 0 , 0 , 0 , and 0 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: which is nonlocal for and related to 1 , 1 , 1 , and 1 by (7). en, we introduce auxiliary variables , g, ℎ, and with the relations 푓 = 휓 , g = 휙 , ℎ = 휓 , and 푘 = 휙 to obtain a local symmetry in the following enlarged system: Further, the residual symmetry can be localized into the Lie point symmetry which satisfy erefore, the enlarged system (14) has the Lie point symmetry vector: Next we will give the Bäcklund symmetry theorem, which is obtained by using a finite transformation of the Lie point symmetry (14).
where and 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:

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 subalgebra 2 . en, 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 -th Bäcklund transformation of the strongly coupled Burgers system was obtained by localizing the linear superposition of multiple residual symmetries, and the -th Bäcklund transformation was expressed by determinants in a compact form.

Data Availability
is work does not have any experimental data.

Ethical Approval
is work did not involve any active collection of human data.

Conflicts of Interest
e authors declare that they have no conflicts of interest.