Non-local Symmetry of the 3-Dimensional Burgers-Type Equation

Non-local transformation, which connects the 3-dimensional Burgers-type equation with a linear heat equation, is constructed. Via this transformation, nonlinear superposition formulae for solutions are obtained and the conditional non-local symmetry of this equation is studied. The multidimensional generalization of the Burgers equation L1(u) = u0 − u|∇u| − u = 0, (1) is called further the Burgers-type equation. This equation was suggested by W. Fushchych in [1]. We use here such notations: ∂μu = ∂u ∂xμ , {xμ} = (x0,x1, ..., xn−1), ∇u = ‖u1, u2, ..., un‖ , (μ = 0, n− 1), |∇u| = √ (∇u)2, = (∇)2 = ∂ 1 + ∂ 2 + · · ·+ ∂ n−1. In the present paper, we construct the non-local transformation, which connects the 3dimensional equation (1) with a linear heat equation. Via this transformation, we obtain nonlinear superposition formulae for solutions of equation (1). Also we investigate the conditional non-local symmetry of equation (1) and obtain formulae generating solutions of this equation. 1. Conditional non-local superposition Let us consider the 3-dimensional scalar heat equation L2(w) = w0 − w = 0. (2) For the vector-function H, such equations are fulfiled: H = 2∇ lnw, H = ‖h1, h, h3‖T , (3) 1 2 (H) + (∇ ·H) = ∂0 lnw, ∇×H = 0. (4) Non-local Symmetry of the 3-Dimensional Burgers-Type Equation 173 From the integrability condition for equations (3), (4), it follows that equation (2) is connected with the vector equations H0 − 12(H) 2 −∇(∇ ·H) = 0, ∇×H = 0. (5) Let |H|2 = (h1)2 + (h2)2 + (h3)2 = u2 for H = θ · u, where |θ| = 1, θ = |θ1, θ2, θ3|T . Then we obtain from (5) that ∇× θ = 0 and the equality θ [u0 − u|∇u| − u] = u [θ0 − 2∇ lnu(∇ · θ)−∇(∇ · θ)] . Let relations L1(u) = u0 − u|∇u| − u = 0, (6) θ0 − 2∇ lnu(∇ · θ)−∇(∇ · θ) = 0, ∇× θ = 0. (7) be fulfiled on the some subset of solutions of equation (5). System (3), (4) in new variables, which connect equations (2) and (6), has the form