Symmetry and Some Exact Solutions of Non-Linear Polywave Equations

Abstract

We have studied the maximal s m m e t r y group admitted by the non-linear polywave equation O'u = F(u). In particular, we establish that the equation in question admits theSymmetry reduction for the biwave equation U2u = Au -3 is carried out and some exact solutions are obtained.

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

“…(1). Examples of symmetry analysis involving some particular hyperbolic cases of (1) can be found in [5,22,24].…”

“…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 [27,35,6,7,24,43,5,22,25,39]. The interested reader may also consult the book [27] of Ibragimov, where various aspects of symmetry analysis of differential equations on manifolds are presented.…”

Lie symmetry and integrability of ordinary differential equations

“…and what is more this transformation preserves the structure of system (16). The invariance criterion L ;X] = 0, wherẽ…”

“…Lemma 2 Let the system of ODEs (16), where Q 1 ; Q 2 ; Q 3 are constant matrices forming a basis of the Lie algebra so(3), admit a Lie symmetry (17)…”

New conditional symmetries and exact solutions of the nonlinear wave equation

“…; = 0; 1; 2 formulae (13), (16), (17) yield exact solutions of Eq. 14constructed by the symmetry reduction method in 5] and the formulae (18), (19) under f = const, g = const give solutions of Eq. 14obtained by the same method in 26].…”