Differential invariants for systems of linear hyperbolic equations

Abstract

In this paper we consider a general class of systems of two linear hyperbolic equations. Motivated by the existence of the Laplace invariants for the single linear hyperbolic equation, we adopt the problem of finding differential invariants for the system. We derive the equivalence group of transformations for this class of systems. The infinitesimal method, which makes use of the equivalence group, is employed for determining the desired differential invariants. We show that there exist four differential invariants and five semi-invariants of first order. Applications of systems that can be transformed by local mappings to simple forms are provided.

Symmetry classification and joint invariants for the scalar linear (1+1) elliptic equation

“…Laplace-type and joint invariants for a system of two linear hyperbolic equations are given in the works [5,6]. New invariants and the solution of the equivalence problem for scalar linear (1 + 1) hyperbolic equations are determined in [7][8][9].…”

Cotton‐Type and Joint Invariants for Linear Elliptic Systems

“…A general system of two linear elliptic equations is By means of the complex transformations of the independent variables ( 3 ), this system ( 15 ) is transformed into the system of two linear hyperbolic equations as follows: where This system of linear hyperbolic equations ( 16 ) has five semi-invariants [ 12 ] under the linear change of dependent variables. They are [ 12 ] where The system of linear hyperbolic equations ( 16 ) also has the four joint invariants [ 12 ] We utilize the same approach as in the previous section. Indeed via the transformations ( 3 ), the Laplace-type invariants ( 18 ) transform to the five Cotton-type invariants And the invariant equation is where Note that we have an invariant equation here.…”

“…The solution of the equivalence problem for scalar linear (1 + 1) hyperbolic equations and some new invariants are given in [ 10 , 11 ]. Laplace-type and joint invariants for a system of two linear hyperbolic equations are derived in [ 12 ] and Laplace-type invariants for a subclass of a system of two linear hyperbolic equations obtained from a complex linear hyperbolic equation are presented in [ 13 ]. The approach of complex symmetry analysis (CSA), was utilized in [ 14 ].…”

Laplace type invariants for variable coefficient mKdV equations

“…This method was adopted by various scientists and it was then applied to several linear and nonlinear equations with interesting results. [4,[16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. For example, Ibragimov [16] gave a solution to the Laplace problem which consists of finding all invariants of the hyperbolic equations (1).…”