Lie Symmetries of the Wave Equation on the Sphere Using Geometry

A semilinear quadratic equation of the form Aij(x)u(ij)=Bi(x,u)u(i)+F(x,u)defines a metric Aij; therefore, it is possible to relate the Lie point symmetries of the equation with the symmetries ofthis metric. The Lie symmetry conditions break into two sets: one set containing the Lie derivative ofthe metric wrt the Lie symmetry generator, and the other set containing the quantities Bi(x,u),F(x,u).From the first set, it follows that the generators of Lie point symmetries are elements of the conformal algebra of the metric Aij, while the second set serves as constraint equations, which select elementsfrom the conformal algebra of Aij. Therefore, it is possible to determine the Lie point symmetriesusing a geometric approach based on the computation of the conformal Killing vectors of the metric Aij. In the present article, the nonlinear Poisson equation triangle(g)u-f(u) =0 is studied. The metric defined by this equation is 1 + 2 decomposable along the gradient Killing vector partial derivative t. It is a conformally flat metric, which admits 10 conformal Killing vectors. We determine the conformal Killing vectors of this metric using a general geometric method, which computes the conformal Killing vectors of a general1+ (n-1) decomposable metric in a systematic way. It is found that the nonlinear Poisson equation triangle(g)u-f(u) =0 admits Lie point symmetries only when f(u) =ku, and in this case, only the Killing vectors are admitted. It is shown that the Noether point symmetries coincide with the Liepoint symmetries. This approach/method can be used to study the Lie point symmetries of morecomplex equations and with more degrees of freedom