Nonlinear Mathematical Physics 1996, V.3, N 3–4, 453–457. Application of Differential Forms to Construction of Nonlocal Symmetries

Differential forms are used for construction of nonlocal symmetries of partial differen-tial equations with conservation laws. Every conservation law allows to introduce a nonlocal variable corresponding to a conserved quantity. A prolongation technique is suggested for action of symmetry operators on these nonlocal variables. It is shown how to introduce these variables for the symmetry group to remain the same. A new hidden symmetry and corresponding group-invariant solution are found for gas dynamic equations. 1 Integrable forms and nonlocal symmetries Let us consider a system of differential equations F k(x, t, ux, u t..) = 0, k = 1, 2..m (1) with independent variables x, t and differential variables ui (i = 1, 2..n).We call a differen-tial form ω =α(x, t, ux, u t..)dx+ β(x, t, ux, u t..)dt integrable (on (1)) if Dt(α) = Dx(β) on (1). (It is suggested that ω 6 ≡ 0 on (1).) Here Dt = ∂∂t + uit ∂ui + uit