Conditional Symmetry and Exact Solutions of the Multidimensional Nonlinear d'Alembert Equation

Abstract

In the present paper new classes of exact solutions of the nonlinear d'Alembert equation in the space R 1,n , n ≥ 2, 2u + λu k = 0 (1) are constructed. Here

Special conformal groups of a Riemannian manifold and Lie point symmetries of the nonlinear Poisson equation

“…g μ = 0 ( 6 ) and ξ = ξ i (x) ∂ ∂x i is a conformal Killing vector field such that L ξ g ij = μg ij .…”

“…We merely mention the Poisson equations in R n , in particular those involving critical exponents, taking in (1) f (u) = u n+2 n−2 and the Euclidean metric; the Klein-Gordon equation, taking in (1) the metric ds 2 = −dt 2 + dx 2 + dy 2 + dz 2 and f (u) = u; the semilinear wave equations in R 1+n , with f ′′ (u) = 0 and the metric ds 2 = −dt 2 + δ ij dx i dx j in (1); and the Klein-Gordon equation on the S 2 sphere. These particular equations have been studied in [26,34,6,7,23,41,5,22]. The interested reader may also consult the book [26] of Ibragimov, where various aspects of symmetry analysis of differential equations on manifolds are presented.…”

Q-Conditional Symmetries and Exact Solutions of Nonlinear Reaction–Diffusion Systems

“…Substituting q 2 = 0, obtained from (12), into the third equation of system (11), we obtain λ 1 b(bv+c) = 0, that is λ 1 b = 0, but that contradicts the above restrictions. Thus, in the case λ 2 = −λ 1 we do not obtain any Q-conditional operator of the form (5).…”

“…All possible Lie symmetries of system (2) have been found, in [8][9][10][11]. In particular, Q-conditional symmetries of (2) were found in [12]. Reference [13] also contains some results related with system (2).…”

Infinite-dimensional symmetry for wave equation with additional condition

“…the classical texts [11,12,13]) of the same equation with a certain additional condition. Conditional symmetries of wave equation are specifically discussed in [14,15,16].…”