Analytic Solutions and Conserved Quantities of Wave Equation on Torus

This paper discusses the wave equation on torus in terms of classical Lie theory. The symmetry algebra is computed and found solvable. A general element of one-dimensional Lie algebra is used to compute similarity variables. Further invariant solutions are obtained for the considered equation by using similarity variables, while two- and three-dimensional algebras are reported for which the considered equation is investigated by quadrature. Conserved quantities for the considered equation are obtained. The components of conserved vectors are then associated with the translational symmetry generators using the symmetry conservation laws relation given by Kara and Mahomed (2000) [12].