Lie Symmetry Exact Explicit Solutions for Nonlinear Burgers’ Equation

In light of Liu at el. ’s original works, this paper revisits the solution of Burgers’s nonlinear equation u t = a ( u x ) 2 + bu xx . The study found two exact and explicit solutions for groups G 4 and G 6 , as well as a general solution. A numerical simulation is carried out. In the appendix a Maple code is provided.


I. INTRODUCTION
It is well known that many, if not all, of the fundamental equations of physics are nonlinear and that linearity is achieved as an approximation. Slving the nonlinear equations is a difficulty task. Sophus Lie [1][2][3] studied mathematics systems from the perspective of those transformation groups which left the system invariant and developed his notion of continuous transformation groups and their role in the theory of differential equations. The continuous transformation groups can simplify the original differential equations to obtain an exact solution. The Lie group method is a systematic and powerful analytical tool, therefore Cantwell [4] even asserts that students in science and engineering should learn both dimensional analysis and the Lie group (symmetric analysis) as the former one establishes a physical model (equation), and the latter solves the solutions.
In mathematics the Lie group is a group, as well as a differential manifold with the property of maintaining a smooth structure under group operation. Lie began to create a single-parameter continuous group theory (later known as the Lie group) around 1875, and completed the famous monographs about the Lie group [1][2][3]. After publication of Lie's three volume monographs, the first textbook on the Lie group was completed in 1897 by American scholar, J.M. Page [5], who was one of the first students to attend Lie's lecture on the Lie group. American scholar, A. Cohen, published the second textbook in 1911 [6]. Ince, a scholar at the American University of Egypt in 1926, introduced the Lie group symmetry method in his ordinary differential equation book [7]. After a long period of silence, in 1960 American scholar Birkhoff, published a monograph on the study of fluid mechanics with the Lie group in 1960. Due to the high-level study of Birkhoff, the Lie group symmetry method has started to attract more attention [8]. In respect of the application development of the Lie group symmetry method, it must be mentioned that the Lie group analysis monograph of former Soviet scholar, Ovsiannikov [9] who led the Soviet school, promoted the development of the Lie group application and computer computing. Bluman translated Ovsiannikov's monograph in 1967. Hansen's 1964 book discusses fluid mechanics and heat transfer problems. In 1965, Ames' monograph studied nonlinear problems in engineering [10], while Ibragimov and Bluman made an influential contribution to the education and application of the Lie group later [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].
Liu et al. [23] applied the Lie group to study the following nonlinear Burgers' equation [27]: where u = u(x, t) is a real function, and parameters a, b ∈ R with ab = 0, and u t = ∂ ∂t , u x = ∂ ∂x and u xx = ∂ 2 ∂x 2 . For the general nonlinear Burgers' equation, Liu et al. [23] obtained six one-parameter Lie groups G i , i = 1...6, and four exact explicit solutions corresponding to the group G 1 , G 2 , G 3 and G 5 . However, Liu et al. [23] could not find the exact and explicit solutions for both groups G 4 and G 6 . Liu et al. [23] did not finishing the integration below for group G 4 as well and just keeping the solution in following unfinished integration: Liu et al. [23] derived a nonlinear ordinary differential equation for group G 6 : and stated that "we cannot obtain the exact and explicit solutions for Eq.(3) by using the elementary functions.". To solve the Eq.(3), they cleverly proposed a generalized power series solution: where β and η are arbitrary numbers. General speaking, Liu et al.'s results were very good, however, it would be pity to leave some solutions that are not completed. To make up the shortcoming, namely did not find the solutions corresponding to groups G 4 and G 6 that can be expressed in elementary functions, this paper will following the footstep of Liu et al. [23] and to solve the problems.
The research article is organized as follows: following an introduction, Section 2 reviews Lie infinitesimal generator, Lie algebra obtained by Liu et al.; Section 3 gives exact solution in form for each group; Section 4 propose two exact and explicit solutions. Section 5 propose a general solution for u t = a(u x ) 2 + bu xx for a combined group; Section 6 provides a numerical calculation. Finally, concludes with perspectives. In the appendix A, a Maple code is provided.
For the sake of clarity, it is our honor to summarize some results from Liu et al. [23] as follows: The infinitesimal transformation of x, t, u The corresponding infinitesimal generator X can be expressed as: and the second order prolongation X (2) is where and D x and D t are the total differential derivative respect to x and t, respectively The Lie group symmetry of the equation (1) is The relevant expression can be substituted into the symmetry determining equation (14), paying attention to replacing u t by au 2 x + bu xx , and comparing the coefficient functions to get infinitesimal generators (one may also use Maple to obtain, see Maple code in Appendix): where c i (i = 1, 2, · · · , 6) is an arbitrary constant, α = α(x, t) is a function that satisfies the following conduction equation: Hence the problem of infinitesimal generators are where the base vector are and X α is an infinite dimensional subalgebra. This verifies that X i satisfies the closure of Lie algebra, namely X i , and X j ∈ L, then [X i , X j ] ∈ L. The vector field commutator-[X i , X j ] structure coefficient list, is as follows: The Lie group, G i , generated by X i , can take advantage of the previously obtained transformation relationship e Xi (x, t, u) = (x,t,û), such asx for G 1 : and Other situations can be calculated in the same manner. The Lie groups are listed below: (24) G 3 : (x, t, u) → (x, t, u + ); (25) G 4 : (x, t, u) → (e x, e 2 t, u); (26) The above shows that the G 1 group is spatial translation, G 2 is time translation, G 3 is the translation of the dependent variable, G 4 is the extension transformation, G 5 is the Galileo (speed) transformation, and G 6 is a true local transformation group, whose physical meaning is ambiguous but very important, which can be used to find similarity solutions.

III. EXACT SOLUTION OF ut = a(ux) 2 + buxx
Since G i is a symmetric group, if u = f (x, t) is one solution of the equation (1), then the solution u (i) corresponding to G i can be obtained as well. Liu et al. [23] obtained following solutions u i , i = 1, ..., 6 as follows: With an infinitesimal generator X i , one can determine the corresponding invariant,whilst using it to simplify the equation (1) and to find its solution.
It should be noted that the combination of X 1 and X 2 produces a moving wave solution. Hence, the new variable µ = x − ct, therefore, u = u(x, t) = f (µ), where c is the wave speed. Thus, the equation (1) becomes bf + af 2 + cf = 0. (37) where f = df dµ . The solution of this equation can be integrated as follows In the above formula, c 1 , c 2 are integral constants. The solution in Eq. 38 is different from the solution obtained by Liu et al. [23], whose solution is u( For the stretching group G 4 , one can derive an invariant and similarity variable, namely The Lie group invariant solution is ω = f (θ) = f (xt −1/2 ), namely Substituting this formula into (1), which can be simplified into ordinary differential equation as follows where f = df dθ and Liu at el. [23] did not complete the above integration, and left it as is.
If we check the integration (42) carefully, it is not difficulty to recognise that the integration exp( −θ 2 4b )dθ is actually an error function. After simple calculation, the solution of the equation (41) can be expressed by the error function as: where erf(θ) = 2 π θ 0 e −θ 2 dθ is error function.
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 29 November 2021 doi:10.20944/preprints201908.0150.v3 There are similarity variable for the Galileo group G 5 , namely This group invariant solution is ω = f (θ), namely Substituting (45) into equation (1), which is reduced to an ordinary differential equation in the following where f = df dθ . By integration, we obtain its solution as follows The solution in Eq. 47 was obtained by Liu et al. [23]. For group G 6 , results in the first set of similarity variables as follows: This Lie group invariant solution is Substituting (49) into equation (1) produces an ordinary differential equation Hence, in this case, corresponding to group G 6 , the solution is as follows The solution in Eq. 52 was obtained by Liu et al. [23]. Corresponding to group G 6 , another similarity variable, the ln t in the expression (48), is changed to ln x, and the second set of similarity variables is Hence, the Lie group invariant solution is: Substituting the expression (54) into equation (1) yields the equation for the function f (θ) where f = df dθ . Liu at el. [23] pointed out that the exact and explicit solutions of equation (55) cannot be obtained, therefore they proposed a series solution.
In fact, this equation can be solved with the following exact and explicit elementary solution: Hence, we have second exact solution, namely The exact solution for the equations (56) and (57)  Besides those solutions corresponding to each group G i given in the section 3, it is necessary to combine them to form a general solution based on the general symmetry group X = c 1 X 1 + ... + c 6 X 6 + X α . An exponential map g: g → G can be constructed as: If u = f (x, t) is known as a solution for the equation (1) This solution has not been reported by Liu et al. [23].

VI. SUMMARY AND REMARKS
We have fixed the problems left by Liu at el.. Although we have found the above two similarity solutions, it would be honor to say that the similarity solutions of the nonlinear Burgers' equation in Eq. (1) can be easily obtained by Maple code in the appendix A. Regarding the further investigation, this study may be useful to deal with high order nonlinear PDE such as nonlinear Schrödinger equations in [28,29].