Invariance of the quasilinear equations of hiperbolic type with respect to the three-parametric Lie algebras

We have solved completely the problem of the description of quasi-linear hyperbolic differential equations in two independent variables that are invariant under three-parameter Lie groups. The problem of group classification of differential equations is one of the central problems of modern symmetry analysis of differential equations[1]. One of the important classes of hyperbolic equations. The problem of group classification of such equations has been discussed by many authors (see for instance [2–9]). On this report we consider the problem of the group classification of equations of form: utt = uxx + F (t, x, u, ux), (1) where u = u(t, x)and F is anarbitrary nonlinear differentiable function, with Fux,ux 6= 0 is an arbitrary nonlinear smooth function, which dependet variables u or ux. We use following notation ux = ∂u ∂x , uxx = ∂u ∂x2 , Fux = ∂F ∂ux , ut = ∂u ∂t , utt = ∂u ∂t2 . For the group classification of equation (1) we use the approach proposed in [10]. Here we give three fundamental results (for details, the reader is refered to [11]). Theorem 1 The infinitesimal operator of the symmetry group of the equation (1) has following form: X = (λt+ λ1)∂t + (λx+ λ2)∂x + (h(x)u+ r(t, x))∂u, (2) where λ, λ1, λ2are arbitrary real constants and h(x), r(t, x) are arbitrary functions which satisfy the condition rtt − dh dx2 u− rxx + (h− 2λ)F − (λt+ λ1)Ft − (λx+ λ2)Fx − −(hu + r)Fu − 2ux dh dx − ux(h− λ)Fux − dh dx uFux − rxFux = 0. (3) ∗e-mail: [email protected]