On linearization of hyperbolic equations using differential invariants

Abstract

In this paper we consider the general class of hyperbolic equations u xt = F (x, t, u, u x , u t ). We use equivalence transformations to derive differential invariants for this class and for two subclasses. Then we employ these invariants to construct equations that can be linearized via local mappings. Further applications of the differential invariants are given.

Differential invariants for systems of linear hyperbolic equations

“…Equivalently, we may say that all five semi-invariants (22) vanish. However, for sufficiency we require one further condition.…”

“…In fact, he has proposed a simple method for constructing differential invariants of families of linear and non-linear differential equations admitting infinite equivalence transformation groups [14][15][16]. There is a continuing interest in applying this method on various families of linear and non-linear differential equations [17][18][19][20][21][22][23][24][25][26]. An alternative approach for deriving differential invariants is Cartan's method [27,28].…”

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).…”

Lie group analysis of two-dimensional variable-coefficient Burgers equation

“…The method was employed first for understanding the group theoretic nature of the well-known Laplace invariants for the linear hyperbolic pdes and then to derive the Laplace type invariants for the parabolic equations. Since then, the method was applied to families of linear and nonlinear odes and pdes [2,17,19,[37][38][39]. Here, we employ this method to derive differential invariants for the class (1.3).…”