Equivalence Conditions and Invariants for the General Form of Burgers’ Equations

Equivalence of differential equations is one of the most important concepts in the theory of differential equations. In this paper, the moving coframe method is applied to solve the local equivalence problem for the general form of Burgers’ equation, which has two independent variables under action of a pseudo-group of contact transformations. Using this method, we found the structure equations and invariants of these equations, as a result some conditions for equivalence of them will be given.


Introduction
In the begining of twentieth century, Elie Cartan developed a uniform method, called the "Cartan equivalence method", for analyzing the differential invariants of many geometric structures, which is a systematic procedure that allows one to decide whether two sub manifolds can be mapped on each other by a transformation taken in a given pseudo-group [1]. Later, Erhesmann and Chern introduced two important concepts, jets spaces and G-structures to the method of equivalence [2,3]. In recent years, with the help of mathematical software, many authors have successfully applied the method of equivalence to many interesting problems: classifications of differential equations, see [4][5][6] and references therein, holonomy groups [7], inverse variational problems [8,9], general relativity [10,11], and image processing [12]. (We also mention for completeness that in [13,14] the authors classify differential equations using methods from classical theory of surfaces).
In this paper we consider a local equivalence problem for the general form of Burgers' equation: under the contact transformation of a pseudo-group. We use Elie Cartan's method of equivalence [1], in its original form, developed by Fels and Olver [15][16][17], and as stated by morozov [18], to compute the Maurer-Cartan forms, structure equations, basic invariants, and the invariant derivatives for symmetry groups of equations. The symmetry classification problem for classes of differential equations is closely related to the problem of local equivalence. Note that, the symmetry groups of two equations are necessarily isomorphic if these equations are equivalent, while the converse statement is not true in general. For the symmetry analysis of (1) the reader is referred to [19][20][21].

Pseudo-group of Contact Transformations and Symmetries of Differential Equations
In this section we describe the local equivalence problem for differential equations under the action of a pseudo-group of contact transformations. Two equations are said to be equivalent, if there exists a contact transformation that maps the equations to each other. We apply Cartan's structure theory of Lie pseudo-groups to obtain some necessary and sufficient conditions under which equivalence mappings can be found. This theory describes a Lie pseudo-group in terms of a set of invariant differential 1-forms, called Maurer-Cartan forms, which contains all information about the pseudo-group. In particular, they allow us to solve equivalence problem for submanifolds under the action of the pseudo-group. Recall that expansions of exterior differentials of Maurer-Cartan forms in terms of the forms themselves, leads to the Cartan's structure equation for the prescribed pseudo-group. Suppose ∶ ℝ n × ℝ m → ℝ n is a trivial bundle with the local base coordinates (x 1 , … , x n ) and the local fibre coordinates (u 1 , … , u m ) . Then J 1 ( ) is the bundle of the first-order jets of sections of , with local coordinates . For every local section is denoted by j 1 (f ) . A differential 1−form on J 1 ( ) is called a contact form, if it is annihilated by all 1−jets of local sections i.e., j 1 (f ) * = 0 . Note that, in local coordinates every contact 1−form is a linear combination of the forms = du − p i dx i , ∈ {1, … , m} . Recall that a local diffeomorphism is called a contact transformation, if for every contact 1−form , the form Δ * is also a contact. Let R be a first-order differential equation with m dependent and n independent variables. We consider R as a sub-bundle in J 1 ( ) . Suppose Cont(R) is the group of contact symmetries for R , that consists of all contact transformations on J 1 ( ) mapping R to itself. To obtain the collection of invariant 1-forms of the pseudo-group of contact transformations on j 1 ( ) we apply the methods described in [18]. The following differential 1−forms, are the lifted coframe of Cont(J 1 ( )) . They are defined on . They satisfy the following structure equations where the forms Φ , Ψ i j , Π i , Λ i and Ω ij are the modified Maurer-Cartan forms depend on differentials of the coordinates of H.
These differential equations define a submanifold R ⊂ J 1 ( ) , and the Maurer-Cartan forms for its symmetry pseudo-group Cont(R) can be found via the restrictions are defined by our differential equations. In order to compute the Maurer-Cartan forms, for the symmetry pseudo-group, we apply Cartan's equivalence method in the following steps: Step-1: First of all, the forms , i , i are linearly dependent, i.e. there exists a nontrivial set of functions U , Setting these functions equal to some appropriate constants allows us to introduce a part of the coordinates of H as functions of the other coordinates of R × H.
Step-2: We substitute the obtained values in step-1, into the forms = * Φ and i k = * Ψ i k . Then the coefficients of the semi-basic forms at j , j , and the coefficients of the semi-basic forms i j at j are lifted invariants of Cont(R) . We set them equal to an appropriate constants by considering, det(a ). det(b i j ) ≠ 0 , and express them as functions of the other coordinates of R × H.
Step-3: Now, consider the following reduced structure equations If the essential torsion coefficients are dependent to the group parameters, then we may normalize them to some constants to compute some new parts of the group parameters. Substituting these new computed parameters into the reduced modified Maurer-Cartan forms, allows us to repeat the procedure of normalization. This process has two results. First, when the reduced lifted coframe appears to be involutive as explained in [15], this coframe is the set of invariant 1-forms which characterize the pseudo-group Cont(R) . Second, when the coframe is not involutive we should apply the procedure of prolongation as described in [15].

Structure Equations and Invariants of Symmetry Groups for the General Form of Burgers' Equations
Consider the following system equivalent to (1) of first order: We apply the method described in the previous section to the class of equations (5), 2 2 , u x = p 2 1 . We consider this system as a sub-bundle of the bundle J 1 ( ), = ℝ 2 × ℝ 2 ⟶ ℝ 2 , with local coordinates {x 1 , x 2 , u 1 , u 2 , p 1 1 , p 1 2 , p 2 1 , p 2 2 } , where the embedding is defined by the equalities: , are linearly dependent. The group parameters a , b i j must satisfy the conditions det(a ) ≠ 0, det(b i j ) ≠ 0 . linear dependence between the forms i are Computing the linear dependence conditions (7) gives the following group parameters as a functions of other group parameters and the local coordinates {x 1 , x 2 , u 1 , u 2 , p 1 2 , p 2 2 } of R . In particular,

3
Journal of Nonlinear Mathematical Physics (2022) 29:103-114 The expression for f 2 21 , f 2 22 , g 2 22 is too long to be written out in full here. The analysis of the semi-basic modified Maurer-Cartan forms , i k , i at the obtained values of the group parameters gives the following normalizations: The expression for f 1 11 is too long to be written out in full here. The analysis of the structure equations gives the following normalizations: Regarding the appearance of different derivatives of Q(u) in the essential torsion coefficients and with respect to vanishing or non-vanishing of these derivatives and their effects on normalizations process, we have to impose some restrictions on the function Q(u 2 ) . As a result of these restrictions, the following cases arise.
Case-1 After normalization (10), if Q is a constant then we have the following structure equations The structure equations (11) do not contain any torsion coefficient depending on the group parameters. The first reduced character is s � 1 = 5 , and the degree of indeterminancy is 2. The Cartan involutivity test is not satisfied. Therefore we should use the procedure of prolongation, which gives us the following structure equations.
The expression for g 1 22 are too long to be written out in full here. Now, all the group parameters are expressed as functions of the local coordinates {x 1 , x 2 , u 1 , u 2 , , p 1 2 , p 2 2 } . After normalization (13) the structure equations of coframe { 1 , 2 , 1 , 2 , 1 2 , 2 2 } , is where is the only invariant of the symmetry group o equations of the from Case-2. Note that, the exterior differential of I is All derived invariants of the group are expressed as functions of I. Therefore, the rank of the coframe, is 1 and our manifold is 6-dimensional and by theorem 8.22 from [15], we deduce the following theorem.
are invariants of the symmetry group of an equation from Case-3. All derived invariants of the group are expressed as functions of {J 1 , J 2 , J 3 } . Therefore the rank of the coframe, is 3. Again by theorem 8.22 from [15], we have