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 an -th Bäcklund transformation is also obtained.

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 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:

where a 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 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 -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 . We replace the and of (1a) and (1b) with the commutative matrix

Then, we can get

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:

where , are a set of arbitrary solutions of the equation, and , , … , , , 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:

Then, plugging (5a) and (5b) into (3a) and (3b) and vanishing all the coefficients of each power of and , we obtain

It is easy to find that (8) and (9) are just the (2 + 1)-dimensional strongly coupled Burgers system (3a) and (3b) with , , and as solutions. We then substitute , , , and of (6) into the linearized form of (8) and (9) with (7) that one can find the , , , and 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 , , , and by (7). Then, we introduce auxiliary variables , , , and with the relations , , , and to obtain a local symmetry in the following enlarged system:

Further, the residual symmetry can be localized into the Lie point symmetry

which satisfy

Therefore, the enlarged system (11a)–(11e) 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).

Theorem 1. If is a solution of the coupled system (11a)–(11e), then so is withwhereand 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:And (15a)–(15e) is the solution to the above system. Therefore, we have completed the proof of Theorem 1.

For the (2 + 1)-dimensional strongly coupled Burgers system (3a) and (3b), the original residual symmetry

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

where and 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 is a solution of the enlarged system
then the symmetry (19) is localized to the Lie point symmetry

Proof. The extended system (20a)–(20e) has the following linearized form:Let us first consider the special case: for any fixed while , in (19), from (12a)–(12e), the localized symmetry for can be obtained as follows:In (20c), we let and , and then we obtainThen, we substitute (23a) and (23b) into (22c) with yieldFurther, one can obtain a solution of Equation (26):which can be verified by using (20b), (24), and (25). From (22d) and (22e) with , the symmetry for , , , and can be given by:Further, (21a)–(21e) can be obtained by taking a linear combination of the above results for . 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:

Then, one can get the following -th Bäcklund theorem for the extended system (20a)–(20e) by solving (30a)–(30f).

Theorem 3. If is a solution of the coupled system (20a)–(20e), then so is , , wherewherewith

From any seed solution of the (2 + 1)-dimensional strongly coupled Burgers system, one can get an infinite number of new solutions because 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:

where

and , , , and are arbitrary constants.

Then, the solution of (20c) can be obtained

where and are arbitrary constants.

Further, one can obtain the and of Theorem 3

where

We write and as

Finally, consider the case of in Theorem 3, one can get a new soliton solution

where

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 subalgebra . 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 -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

This work does not have any experimental data.

Ethical Approval

This work did not involve any active collection of human data.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Yufeng Zhang contribute the idea and did some calculation. Haifeng Wang get computational results, and wrote the paper. All authors gave final approval for publication.

Acknowledgments

This work is supported by the Fundamental Research Funds for the Central University (No. 2017XKZD11).