Discrete Symmetries Analysis and Exact Solutions of the Inviscid Burgers Equation

Abstract

We discuss the Lie point symmetries and discrete symmetries of the inviscid burgers equation. By employing the Lie group method of infinitesimal transformations, symmetry reductions and similarity solutions of the governing equation are given. Based on discrete symmetries analysis, two groups of discrete symmetries are obtained, which lead to new exact solutions of the inviscid Burgers equation.

1. Introduction

Burgers equation is one of the basic partial differential equations of fluid mechanics. It occurs in various fields of applied mathematics, such as modeling of gas dynamics and traffic flow.

For a given velocity 𝑢 and viscosity coefficient 𝜈, the general form of Burgers equation is: 𝑢_𝑡(𝑥,𝑡)+𝑔(𝑢)𝑢_𝑥(𝑥,𝑡)=𝜈𝑢_𝑥𝑥(𝑥,𝑡), (1.1) where 𝑔(𝑢) is a smooth function of 𝑢. If 𝜈=0, Burgers equation reduces to the inviscid Burgers equation: IBE∶𝑢_𝑡(𝑥,𝑡)+𝑔(𝑢)𝑢_𝑥(𝑥,𝑡)=0, (1.2) which is a prototype for equations for which the solution can develop discontinuities (shock waves). There are many methods to solve (1.2). In [1], the authors discussed the matrix exponential representations of solutions to similar equation of (1.2). Here we can use the method of Lie symmetries and discrete symmetries analysis to solve (1.2).

The classical Lie symmetries of the partial differential equations (PDEs) which can be obtained through the Lie group method of infinitesimal transformations were originally developed by Lie [2]. We can use the basic prolongation method and the infinitesimal criterion of invariance to find some particular Lie point symmetries group of the nonlinear partial differential equations. The Lie groups of transformations admitted by a given system of differential equations can be used (1) to lower the order or eventually reduce the equation to quadrature, in the case of ordinary differential equations; (2) to determine particular solutions, called invariant solutions, or generate new solutions, once a special solution is known, in the case of ordinary differential equations or PDEs.

In the past decades, much attention has been paid to the symmetry method and a series of achievements have been obtained [3–9]. Particularly, In [9], a five-dimensional symmetry algebra consisting of Lie point symmetries is firstly computed for the nonlinear Schrödinger equation. But it seems that very few research on discrete symmetries is available up to now. In fact, discrete symmetries also play an important role in solving PDEs. For instance, to understand how a system changes its stability, to simplify the numerical computation of solutions of PDEs and to create new exact solutions from known solutions. Discrete symmetries are usually easy to guess but difficult to get in a systematic way. They can be obtained from the continuous Lie point symmetries. In [10, 11], Hydon studied the application of the method in differential equations.

For the Burgers equation, many researches have been carried on [12–18]. In these papers, Ouhadan and El Kinani used Lie symmetry method for obtaining exact solutions of inviscid Burgers equation in some cases [14]. However, the analysis presented there was not complete. Nadjafikhah extended the study to include other cases of interest [15, 16]. But in all their work, they only used the Lie symmetry method, and the discrete symmetries approach has never been considered. In this work, we obtain the analytical solutions of the inviscid Burgers equation by using Lie group method. Also by applying discrete symmetries, we introduce new groups of analytical solutions of our problem.

2. Lie Symmetries and Lie Algebra

In this section, we recall the general procedure for determining symmetries for any system of PDEs [19–21]. To begin, let us consider the general case of a nonlinear system of PDEs, Δ_𝜈𝑥,𝑢^(𝑛)=0,𝜈=1,…,𝑙, (2.1) that involve 𝑝 independent variables 𝑥=(𝑥¹,…,𝑥^𝑝), 𝑞 dependent variables 𝑢=(𝑢¹,…,𝑢^𝑞), and the derivatives of 𝑢 with respect to 𝑥 up to 𝑛, where 𝑢^(𝑛) represents all the derivatives of 𝑢 of all orders from 0 to 𝑛.

We consider a one-parameter Lie group of infinitesimal transformations acting on the independent and dependent variables of (2.1): 𝑥^𝑖=𝑥^𝑖+𝜖𝜉^𝑖(𝑥,𝑢)+𝑂𝜖²𝑢,𝑖=1,…,𝑝,_𝑗=𝑢^𝑗+𝜖𝜂^𝑗(𝑥,𝑢)+𝑂𝜖²,𝑗=1,…,𝑞, (2.2) where 𝜖 is the parameter of the transformation and 𝜉^𝑖, 𝜂^𝑗 are the infinitesimals of the transformations of the independent and dependent variables, respectively. The infinitesimal generator 𝑋 associated with the above group of transformations can be written as 𝑋=^𝑝_𝑖=1𝜉^𝑖()𝜕𝑥,𝑢𝜕𝑥^𝑖+^𝑞_𝑗=1𝜂^𝑗()𝜕𝑥,𝑢𝜕𝑢^𝑗. (2.3) A symmetry of differential equation is a transformation which maps solutions of the equation to other solutions.

The invariance of system (2.1) under the infinitesimal transformations leads to the invariance condition 𝑝𝑟^(𝑛)𝑋Δ_𝜈𝑥,𝑢^(𝑛)∣_{Δ𝜈(𝑥,𝑢(𝑛))=0}=0,𝜈=1,…,𝑙, (2.4) where 𝑝𝑟^(𝑛) is the 𝑛th-order prolongation of the infinitesimal generator given by 𝑝𝑟^(𝑛)𝑋=𝑋+^𝑞_𝛼=1𝐽𝜓^𝐽_𝛼𝑥,𝑢^(𝑛)𝜕𝜕𝑢^𝛼_𝐽, (2.5) where 𝐽=(𝑗₁,…,𝑗_𝑘), and 1≤𝑗_𝑘≤𝑝,1≤𝑘≤𝑛, 𝜓^𝐽_𝛼𝑥,𝑢^(𝑛)=𝐷_𝐽𝜓_𝛼−^𝑝_𝑖=1𝜉^𝑖𝑢^𝛼_𝑖+^𝑝_𝑖=1𝜉^𝑖𝑢^𝛼_𝐽,𝑖, (2.6) where 𝑢^𝛼_𝑖=𝜕𝑢^𝛼𝜕𝑥^𝑖,𝑢^𝛼_𝐽,𝑖=𝜕𝑢^𝛼_𝐽𝜕𝑥^𝑖. (2.7)

For (1.2), following the general Lie's algorithm [19, 20], we consider the one-parameter Lie group of infinitesimal transformation in (𝑥,𝑡,𝑢) given by 𝑥=𝑥+𝜉(𝑥,𝑡,𝑢)𝜖+𝑂𝜖²,𝑡̂=𝑡+𝜏(𝑥,𝑡,𝑢)𝜖+𝑂𝜖²,𝑢=𝑢+𝜂(𝑥,𝑡,𝑢)𝜖+𝑂𝜖², (2.8) where 𝜖 is the group parameter. The infinitesimal generator of the symmetry algebra takes the form 𝜕𝑋=𝜉𝜕𝜕𝑥+𝜏𝜕𝜕𝑡+𝜂,𝜕𝑢 (2.9) where 𝜉,𝜏,𝜂 are the same as those in (1.2). And the first prolongation is 𝑝𝑟⁽¹⁾𝜕𝑋=𝜉𝜕𝜕𝑥+𝜏𝜕𝜕𝑡+𝜂𝜕𝑢+𝜂_𝑥𝜕𝜕𝑢_𝑥+𝜂_𝑡𝜕𝜕𝑢_𝑡. (2.10) Equation (1.2) can be written as ▵=𝑢_𝑡(𝑥,𝑡)+𝑔(𝑢)𝑢_𝑥(𝑥,𝑡)=0. (2.11) The invariance of (1.2) under the infinitesimal transformations (2.8) needs 𝑝𝑟⁽¹⁾𝑋[▵]∣_▵=0=0. (2.12) From (2.11) and (2.12), we get 𝜂_𝑡+𝑔(𝑢)𝜂_𝑥+𝜂𝑔^′(𝑢)𝑢_𝑥=0, (2.13) where 𝜂_𝑥=𝜕𝜕𝑥𝜂−𝜉𝑢_𝑥−𝜏𝑢_𝑡𝜕+𝜉𝑢𝜕𝑥^𝑥𝜕+𝜏𝑢𝜕𝑡^𝑥,𝜂_𝑡=𝜕𝜕𝑡𝜂−𝜉𝑢_𝑥−𝜏𝑢_𝑡𝜕+𝜉𝑢𝜕𝑥^𝑡𝜕+𝜏𝑢𝜕𝑡^𝑡. (2.14) Conditions on the infinitesimals 𝜉,𝜏, and 𝜂 are determined by equating coefficients of like derivatives of monomials in 𝑢_𝑥 and 𝑢_𝑡 and higher derivatives by zero. This will produce a series of PDEs; by analyzing these equations, we can get that 𝜉,𝜏, and 𝜂 have the following form: 𝜉=𝐶₈𝑥²+𝐶₇𝑥𝑡+𝐶₃𝑥+𝐶₄𝑡+𝐶₁,𝜏=𝐶₈𝑥𝑡+𝐶₇𝑡²−𝐶₅𝑥+𝐶₃−𝐶₆𝑡+𝐶₂,𝐶𝜂=⁷𝑥+𝐶₄+𝐶₈𝑥−𝐶₇𝑡+𝐶₆𝑔(𝑢)+−𝐶₈𝑡+𝐶₅𝑔(𝑢)²𝑔^′(𝑢). (2.15) Here we omit the redundant computational process for simplification. Associated with this Lie group, we have an 8-dimensional Lie algebra that can be represented by the generators 𝑋₁=𝜕𝜕𝑥,𝑋₂=𝜕𝜕𝑡,𝑋₃𝜕=𝑥𝜕𝜕𝑥+𝑡𝜕𝑡,𝑋₄𝜕=𝑡+1𝜕𝑥𝑔^′(𝑢)𝜕,𝑋𝜕𝑢₅𝜕=−𝑥+𝑔𝜕𝑡²(𝑢)𝑔^′(𝑢)𝜕𝜕𝑢,𝑋₆𝜕=−𝑡+𝑔(𝑢)𝜕𝑡𝑔^′(𝑢)𝜕,𝑋𝜕𝑢₇𝜕=𝑥𝑡𝜕𝑥+𝑡²𝜕+(𝑢)𝜕𝑡𝑥−𝑡𝑔𝑔^′(𝑢)𝜕𝜕𝑢,𝑋₈=𝑥²𝜕𝜕𝜕𝑥+𝑥𝑡+𝑔(𝑢)(𝑢)𝜕𝑡𝑥−𝑡𝑔𝑔^′(𝑢)𝜕.𝜕𝑢 (2.16)

3. Discrete Symmetries

In this section, we will derive the discrete symmetries of (1.2), which has a 6-dimensional Lie subalgebra ℒ∶{𝑋₁,𝑋₂,𝑋₃,𝑋₄,𝑋₅,𝑋₆}. we Will calculate the discrete symmetry following the method presented in [11].

By the commutator relation 𝑋_𝑖,𝑋_𝑗=𝑐^𝑘_𝑖𝑗𝑋_𝑘,𝑖<𝑗,𝑖,𝑗=1,2,…,6, (3.1) we can get the nonzero commutators in the following form: 𝑋₁,𝑋₃=𝑋₁,𝑋₁,𝑋₅=−𝑋₂,𝑋₂,𝑋₃=𝑋₂,𝑋₂,𝑋₄=𝑋₁,𝑋₂,𝑋₆=−𝑋₂,𝑋₄,𝑋₅=2𝑋₆+𝑋₃,𝑋₄,𝑋₆=𝑋₄,𝑋₅,𝑋₆=−𝑋₅. (3.2) Then the nonzero structure constants are 𝐶¹₁₃=1,𝐶²₁₅=−1,𝐶²₂₃=1,𝐶¹₂₄=1,𝐶²₂₆=−1,𝐶⁶₄₅=2,𝐶³₄₅𝐶=1,₄₄₆=1,𝐶⁵₅₆=−1,𝐶¹₃₁=−1,𝐶²₅₁=1,𝐶²₃₂=−1,𝐶¹₄₂=−1,𝐶²₆₂𝐶=1,₆₅₄=−2,𝐶³₅₄=−1,𝐶⁴₆₄=−1,𝐶⁵₆₅=1. (3.3) The matrices 𝐶(𝑗) are 𝐶(1)=(2)=,𝐶(3)=(4)=,𝐶(5)=(6)=.000000000000−100000000000010000000000,𝐶0000000000000−10000−100000000000010000100000010000000000000000000000000000,𝐶00000010000000000000000000−100−2000−1000−10000000000000000001002000000000010,𝐶0000000−100000000000001000000−10000000 (3.4) The next step is to calculate the matrices 𝐴(𝜖,𝑗). Exponentiating the matrices 𝜖𝐶(𝑗), we obtain 𝐴()=()=𝐴()=𝑒1,𝜖100000010000−𝜖010000001000𝜖0010000001,𝐴2,𝜖1000000100000−𝜖1000−𝜖001000000100𝜖0001,3,𝜖^𝜖000000𝑒^𝜖()=0000001000000100000010000001,𝐴4,𝜖100000𝜖1000000100000010000−𝜖𝜖²,𝐴()=1−2𝜖000−𝜖015,𝜖1−𝜖000001000000100000𝜖1𝜖²()=2𝜖0000100000𝜖1,𝐴6,𝜖1000000𝑒^−𝜖0000001000000𝑒^𝜖000000𝑒^−𝜖0.000001 (3.5)(3.6) From the nonlinear constants 𝑐^𝑛_𝑙𝑚𝑏^𝑙_𝑖𝑏^𝑚_𝑗=𝑐^𝑘_𝑖𝑗𝑏^𝑛_𝑘,1≤𝑖<𝑗≤6,1≤𝑛≤6, (3.7) and using the adjoint matrices 𝐴(𝑖,𝜖),𝑖=1,…,6, the matrix 𝐵 can be simplified as the following two nonsingular forms: 𝐵₁=𝛼𝛽1000000𝛼0000001000000𝛽000000𝛽0000001,𝐵₂=𝛼0−𝛽0000𝛼000000010000000𝛽⁻¹0000𝛽0000−100−1,𝛼,𝛽∈𝑅. (3.8) The determining equations for the discrete symmetries are given by the system: 𝑋₁𝑥𝑋₁𝑡̂𝑋₁𝑢𝑋₂𝑥𝑋₂𝑡̂𝑋₂𝑢𝑋₃𝑥𝑋₃𝑡̂𝑋₃𝑢𝑋₄𝑥𝑋₄𝑡̂𝑋₄𝑢𝑋₅𝑥𝑋₅𝑡̂𝑋₅𝑢𝑋₆𝑥𝑋₆𝑡̂𝑋₆𝑢𝑋=𝐵¹𝑥𝑋₁𝑡̂𝑋₁𝑢𝑋₂𝑥𝑋₂𝑡̂𝑋₂𝑢𝑋₃𝑥𝑋₃𝑡̂𝑋₃𝑢𝑋₄𝑥𝑋₄𝑡̂𝑋₄𝑢𝑋₅𝑥𝑋₅𝑡̂𝑋₅𝑢𝑋₆𝑥𝑋₆𝑡̂𝑋₆𝑢𝑥̂=𝐵̂1100010𝑡0𝑡0𝑔^′(𝑢)0−𝑥𝑔(𝑢)²𝑔^′(𝑢)𝑡̂𝑔(𝑢)0−𝑔^′(𝑢). (3.9) First we consider 𝐵=𝐵₁. From (3.9), we obtain the following system 𝑋₁𝑥𝑋₁𝑡̂𝑋₁𝑢𝑋₂𝑥𝑋₂𝑡̂𝑋₂𝑢𝑋₃𝑥𝑋₃𝑡̂𝑋₃𝑢𝑋₄𝑥𝑋₄𝑡̂𝑋₄𝑢𝑋₅𝑥𝑋₅𝑡̂𝑋₅𝑢𝑋₆𝑥𝑋₆𝑡̂𝑋₆𝑢=𝛼𝛽𝑥̂𝑡̂000𝛼0𝑡0𝛽01𝛽𝑔^′(𝑢)0−𝛽𝑥𝛽𝑔(𝑢)²𝑔^′(𝑢)𝑡̂𝑔(𝑢)0−𝑔^′(𝑢). (3.10) The solution to system (3.10) is 𝛼𝑥=𝛽𝑥,𝑡̂=𝛼𝑡,𝑢=𝑔⁻¹1𝛽𝑔(𝑢), (3.11) where 𝛼,𝛽 are arbitrary constants. We can prove that (3.11) satisfies the invariant condition 𝑢𝑡̂+𝑔(𝑢)𝑢𝑥=0,when𝑢_𝑡+𝑔(𝑢)𝑢_𝑥=0. (3.12) Therefore, the first group of discrete symmetries is Γ₁∶()↦𝛼𝑥,𝑡,𝑢𝛽𝑥,𝛼𝑡,𝑔⁻¹1𝛽𝑔(𝑢). (3.13) Then we consider 𝐵=𝐵₂. From (3.9), we obtain the following system: 𝑋₁𝑥𝑋₁𝑡̂𝑋₁𝑢𝑋₂𝑥𝑋₂𝑡̂𝑋₂𝑢𝑋₃𝑥𝑋₃𝑡̂𝑋₃𝑢𝑋₄𝑥𝑋₄𝑡̂𝑋₄𝑢𝑋₅𝑥𝑋₅𝑡̂𝑋₅𝑢𝑋₆𝑥𝑋₆𝑡̂𝑋₆𝑢=𝛼0−𝛽0𝑥̂𝑥𝛼00𝑡00−𝛽𝑔(𝑢)²𝛽𝑔^′(𝑢)𝛽̂𝛽𝑡0𝑔^′(𝑢)−𝑔(𝑢)𝑥0−𝑔^′(𝑢). (3.14) The solution to system (3.14) is 𝑥=𝛼𝑡,𝑡̂=−𝛼𝛽𝑥,𝑢=𝑔⁻¹−𝛽𝑔(𝑢), (3.15) where 𝛼,𝛽 are arbitrary constants. We can prove that (3.15) also satisfies the invariant condition 𝑢𝑡̂+𝑢𝑢𝑥=0,when𝑢_𝑡+𝑢𝑢_𝑥=0. (3.16) Therefore, the first group of discrete symmetries is Γ₂∶()↦𝛼𝑥,𝑡,𝑢𝛼𝑡,−𝛽𝑥,𝑔⁻¹−𝛽𝑔(𝑢). (3.17)

Here, we have obtained two groups of discrete symmetries of (1.2). Using them, we can simplify the numerical computation in solving (1.2), create new exact solutions from known solutions and so on. In later chapters, we will introduce how they generate new solutions.

3.1. Lie Symmetries of (1.2)

To obtain the group transformation which is generated by the infinitesimal generators (2.16), we need to solve the system of first-order ordinary differential equations: 𝑑𝑥̂𝑢,𝑥(0)𝑑𝑡̂𝑑𝜖=𝜉𝑥,𝑡,=𝑥,̂𝑢,𝑡̂(0)𝑑𝑢𝑑𝜖=𝜏𝑥,𝑡,=𝑡,̂𝑢,𝑢(0)𝑑𝜖=𝜂𝑥,𝑡,=𝑢. (3.18) Then, we can get the one-parameter groups 𝐺_𝑖 generated by 𝑋_𝑖 for 𝑖=1,…,8: 𝐺₁∶()↦(),𝐺𝑥,𝑡,𝑢𝑥+𝜖,𝑡,𝑢₂∶()↦(),𝐺𝑥,𝑡,𝑢𝑥,𝑡+𝜖,𝑢₃∶(𝑥,𝑡,𝑢)↦𝑥𝑒^𝜖,𝑡𝑒^𝜖,𝐺,𝑢₄∶(𝑥,𝑡,𝑢)↦𝑥+𝜖𝑡,𝑡,𝑔⁻¹(𝑢),𝐺𝜖+𝑔₅∶(𝑥,𝑡,𝑢)↦𝑥,𝑡−𝑥𝜖,𝑔⁻¹𝑔(𝑢)(𝑢),𝐺1−𝜖𝑔₆∶(𝑥,𝑡,𝑢)↦𝑥,𝑡𝑒^−𝜖,𝑔⁻¹𝑔(𝑢)𝑒^𝜖,𝐺₇∶()↦𝑥𝑥,𝑡,𝑢,𝑡1−𝜖𝑡1−𝜖𝑡,𝑔⁻¹𝑔(𝑢)+(𝑢)𝜖,𝐺𝑥−𝑡𝑔₈∶()↦𝑥𝑥,𝑡,𝑢,𝑡1−𝜖𝑥1−𝜖𝑥,𝑔⁻¹𝑔(𝑢)(𝑢)𝜖.1−𝑥−𝑡𝑔 (3.19) In addition, we have two groups of discrete symmetries Γ₁∶()↦𝛼𝑥,𝑡,𝑢𝛽𝑥,𝛼𝑡,𝑔⁻¹1𝛽𝑔(𝑢),Γ₂∶()↦𝛼𝑥,𝑡,𝑢𝛼𝑡,−𝛽𝑥,𝑔⁻¹−𝛽𝑔(𝑢). (3.20) If 𝑢=𝑓(𝑥,𝑡) is a solution of (1.2), so are the functions 𝐺₁()(),𝐺⋅𝑓𝑥,𝑡=𝑓𝑥+𝜖,𝑡₂()(),𝐺⋅𝑓𝑥,𝑡=𝑓𝑥,𝑡+𝜖₃⋅𝑓(𝑥,𝑡)=𝑓𝑥𝑒^𝜖,𝑡𝑒^𝜖,𝐺₄⋅𝑓(𝑥,𝑡)=𝑔⁻¹𝑔𝑓(),𝐺𝑥+𝜖𝑡,𝑡−𝜖₅⋅𝑓(𝑥,𝑡)=𝑔⁻¹𝑔𝑓(𝑥,𝑡−𝑥𝜖)𝑓(),𝐺1+𝜖𝑔𝑥,𝑡−𝑥𝜖₆⋅𝑓(𝑥,𝑡)=𝑔⁻¹𝑒^−𝜖𝑔𝑓𝑥,𝑡𝑒^−𝜖,𝐺₇⋅𝑓(𝑥,𝑡)=𝑔⁻¹𝑔𝑓(𝑥/(1−𝜖𝑡),𝑡/(1−𝜖𝑡))−𝑥𝜖,𝐺1−𝑡𝜖₈⋅𝑓(𝑥,𝑡)=𝑔⁻¹(1−𝜖𝑥)𝑔𝑓(𝑥/(1−𝜖𝑥),𝑡/(1−𝜖𝑥))𝑓(()(,Γ1−𝜖𝑡𝑔𝑥/1−𝜖𝑥,𝑡/1−𝜖𝑥))₁⋅𝑓(𝑥,𝑡)=𝑔⁻¹𝑓𝛼𝛽𝑔𝛽,Γ𝑥,𝛼𝑡₂⋅𝑓(𝑥,𝑡)=𝑔⁻¹−𝛽𝑔𝑓𝑥.𝛼𝑡,−𝛼/𝛽 (3.21)

To illustrate how this technique may be of great interest, we let 𝑢(𝑥,𝑡)=1 is a constant solution of (1.2). We conclude trivial solutions Γ₁⋅1, Γ₂⋅1, 𝐺_𝑖⋅1, 𝑖=1,…,6, and nontrivial solutions for (1.2): 𝐺₇⋅1=𝑔⁻¹𝑔(1)−𝜖𝑥,𝐺1−𝜖𝑡₈⋅1=𝑔⁻¹(1−𝜖𝑥)𝑔(1)(1).1−𝜖𝑡𝑔 (3.22) Now, by applying Γ₂, we get the following solutions: Γ₂⋅𝐺₇⋅1=𝑔⁻¹𝛼𝜖𝑥+𝛽(1),Γ𝛼𝜖𝑡−𝑔₂⋅𝐺₈⋅1=𝑔⁻¹𝛼𝜖𝑔(1)𝑥+𝛽(1)(1).𝛼𝜖𝑔𝑡−𝑔 (3.23)

If we let 𝑔(𝑢)=𝑢, (1.2) becomes 𝑢_𝑡(𝑥,𝑡)+𝑢𝑢_𝑥(𝑥,𝑡)=0. (3.24) From (3.23), we get that 𝑢=𝛼𝜖𝑥+𝛽=𝛼𝜖𝑡−1𝑎𝑥+𝑏𝑎𝑡−1 (3.25) is a solution to (3.24), where 𝑎,𝑏 are arbitrary constants.

If we let 𝑔(𝑢)=(1−𝑢)/(1+𝑢), (1.2) becomes 𝑢_𝑡(𝑥,𝑡)+1−𝑢𝑢1+𝑢^𝑥(𝑥,𝑡)=0. (3.26) From (3.23), we get 𝑎()𝑢=𝑡−𝑥−𝑏𝑎(𝑡+𝑥)+𝑏 (3.27) is a solution to (3.26), where 𝑎,𝑏 are arbitrary constants.

4. Reduction and Invariant Solutions of (1.2)

The first advantage of symmetry group method is to construct new solutions from known solutions. The second is when a nonlinear system of differential equations admits infinite symmetries, so it is possible to transform it to a linear system. In this section, symmetry group method will be applied to the inviscid Burgers equation to be connected directly to some order differential equations. To do this, particular linear combinations of infinitesimals are considered and their corresponding invariants are determined. Using discrete symmetries, a series of new interesting results are obtained.

4.1. Reduction with 𝑋₂+𝑋₄

As a first example, we perform a reduction of (1.2) using the generator 𝑋₂+𝑋₄𝜕=𝑡+𝜕𝜕𝑥+1𝜕𝑡𝑔^′(𝑢)𝜕.𝜕𝑢 (4.1) Having determined the infinitesimals, the symmetry variables can be found by solving the characteristic equation 𝑑𝑥𝑡=𝑑𝑡1=𝑔^′(𝑢)𝑑𝑢. (4.2) The similarity transformation is 1𝜉=2𝑡²−𝑥,𝑢=𝑔⁻¹𝑡−𝑓(𝜉). (4.3) The similarity representation is 1−𝑓(𝜉)𝑓^′(𝜉)𝑔^′(𝑢)=0. (4.4) The similarity solution is 𝑓(𝜉)=±2𝜉+𝐶₁. (4.5) In the end, we obtain that 𝑢=𝑔⁻¹𝑡±𝑡²−2𝑥+𝐶₁ (4.6) is a solution of (1.2). Now, by applying Γ₁, Γ₂ to (4.6), we conclude the other two solutions Γ₁∶𝑢=𝑔⁻¹𝛼𝛽𝑡±𝛽𝛼²𝑡²−2𝛼𝛽𝑥+𝐶₁,Γ₂∶𝑢=𝑔⁻¹𝛽𝛼/𝛽𝑥±((𝛼/𝛽)𝑥)²−2𝛼𝑡+𝐶₁. (4.7)

If we let 𝑔(𝑢)=𝑢 in (1.2), from (4.6), (4.7), we can get the following solutions: 𝑢_1,2=𝑡±𝑡²−2𝑥+𝐶₁,𝑢_3,4=𝛼𝛽𝑡±𝛽𝛼²𝑡²−2𝛼𝛽𝑥+𝐶₁,𝑢_5,6=𝛽𝛼/𝛽𝑥±(𝛼/𝛽𝑥)²−2𝛼𝑡+𝐶₁, (4.8) where 𝛼,𝛽,𝐶₁ are arbitrary constants.

If we let 𝑔(𝑢)=(1−𝑢)/(1+𝑢) in (1.2), from (4.6), (4.7), the following solutions can be obtained: 𝑢_1,2=1−𝑡±𝑡²−2𝑥+𝐶₁1+𝑡±𝑡²−2𝑥+𝐶₁,𝑢_3,4=1−𝛼𝛽𝑡±𝛽𝛼²𝑡²−2𝛼𝑥/𝛽+𝐶₁1+𝛼𝛽𝑡±𝛽𝛼²𝑡²−2𝛼𝑥/𝛽+𝐶₁,𝑢_5,6=𝛼𝑥±𝛽−−𝛼²𝑥²+2𝛼𝛽²𝑡−𝐶₁𝛽²/𝛽²−𝛽²𝛼𝑥±𝛽−−𝛼²𝑥²+2𝛼𝛽²𝑡−𝐶₁𝛽²/𝛽²+𝛽², (4.9) where 𝛼,𝛽,𝐶₁ are arbitrary constants.

4.2. Reduction with 𝑋₂+𝑋₅

Similarly, the generator 𝑋₂+𝑋₅=()𝜕1−𝑥+𝑔𝜕𝑡²(𝑢)𝑔^′(𝑢)𝜕𝜕𝑢 (4.10) with similarity transformation 𝜉=𝑥,𝑢=𝑔⁻¹1−𝑥(1−𝑥)𝑓(𝜉)−𝑡 (4.11) leads to the solution 𝑢=𝑔⁻¹𝑥−1𝑡+𝐶₁. (4.12) By applying Γ₁, Γ₂ to (4.12), we obtain the invariant solution 𝑢=𝑔⁻¹𝛼𝑥−𝛽𝛼𝑡+𝐶₁. (4.13)

If 𝑔(𝑢)=𝑢/(1+𝑢²) in (1.2), then we have the solutions 𝑢_1,2=𝛼𝑡+𝐶₁±𝛼𝑡+𝐶₁²−4𝛼𝑥−𝛽²2.𝛼𝑥−𝛽 (4.14)

4.3. Reduction with 𝑋₁+𝑋₆

In the last, we discuss the generator 𝑋₁+𝑋₆=𝜕𝜕𝜕𝑥−𝑡+𝑔(𝑢)𝜕𝑡𝑔^′(𝑢)𝜕.𝜕𝑢 (4.15) The similarity transformation is 𝜉=𝑥+ln(𝑡),𝑢=𝑔⁻¹1𝑡𝑓(𝜉), (4.16) which leads to the solution 𝑢=𝑔⁻¹1𝑡LW𝑡𝑒^𝑥+𝐶1, (4.17) where LW is the Lambert W-function, which satisfies LW(𝑥)𝑒^LW(𝑥)=𝑥. (4.18) By applying Γ₁, Γ₂ to this solution, we conclude other two solutions: 𝑢₁=𝑔⁻¹𝛽𝛼𝑡LW𝛼𝑡𝑒^{(𝛼/𝛽)𝑥+𝐶1},𝑢₂=𝑔⁻¹𝛼𝑥LW−𝛼/𝛽𝑥𝑒^{𝛼𝑡+𝐶1}. (4.19) Comparing (4.17) and (4.19), we can see (4.17) is a special case of (4.19).

In (1.2), if we let 𝑔(𝑢)=(1−𝑢)/(1+𝑢), (4.19) lead to the following solutions 𝑢₁=𝛼𝑡−𝛽LW𝛼𝑡𝑒^{(𝛼𝑥+𝐶1𝛽)/𝛽}𝛼𝑡+𝛽LW𝛼𝑡𝑒^{(𝛼𝑥+𝐶1𝛽)/𝛽},𝑢₂=LW−𝛼𝑥𝑒^{𝛼𝑡+𝐶1}/𝛽−𝛼𝑥LW−𝛼𝑥𝑒^{𝛼𝑡+𝐶1}./𝛽+𝛼𝑥 (4.20) If we let 𝑔(𝑢)=𝑢/(1+𝑢²), we can get the following solutions: 𝑢_1,2=𝛼𝑡±𝛼²𝑡²−4𝛽²LW𝛼𝑡𝑒^{(𝛼𝑥/𝛽)+𝐶12}2𝛽LW𝛼𝑡𝑒^{(𝛼𝑥/𝛽)+𝐶1},𝑢_3,4=LW−𝛼𝑥𝑒^{𝛼𝑡+𝐶1}/𝛽±LW−𝛼𝑥𝑒^{𝛼𝑡+𝐶1}/𝛽²−4𝛼²𝑥².2𝛼𝑥 (4.21)

5. Summary and Discussion

In this paper, we obtain eight infinitesimal generators for (1.2) by means of the Lie symmetry method. Considering it is 6-dimensional Lie subalgebra, we get two groups of discrete symmetries following the method presented by Hydon. Using the symmetry group, the similarity variable, similarity transformations, and the reduced equations are given. Solving the reduced equations, from the similarity transformations, we get the solutions of the inviscid Burgers equation. In additional, by applying discrete symmetries, we conclude other new solutions. If given 𝑔(𝑢) different forms in (1.2), many different types of solutions can be obtained directly. In forthcoming days, we will further discuss the problem. It is also interesting for us to see how the discrete symmetries will be under the 8-dimensional Lie algebra case.

Acknowledgments

This work was supported by National Natural Science Foundation of China (no. 40976008), Innovation Project of Chinese Academy of Sciences (no. KZCX2-EW-209), National Natural Science Foundation of China (no. 70971079), and Graduate Innovation Foundation from Shandong University of Science and Technology (no. YCA120212).