Lie Symmetries of the Wave Equation on the Sphere Using Geometry.

A semilinear quadratic equation of the form A i j (x) u i j = B i (x , u) u i + F (x , u) defines a metric A i j ; therefore, it is possible to relate the Lie point symmetries of the equation with the symmetries of this metric. The Lie symmetry conditions break into two sets: one set containing the Lie derivative of the metric wrt the Lie symmetry generator, and the other set containing the quantities B i (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 A i j , while the second set serves as constraint equations, which select elements from the conformal algebra of A i j. Therefore, it is possible to determine the Lie point symmetries using a geometric approach based on the computation of the conformal Killing vectors of the metric A i j . In the present article, the nonlinear Poisson equation Δ g u − f (u) = 0 is studied. The metric defined by this equation is 1 + 2 decomposable along the gradient Killing vector ∂ 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 general 1 + (n − 1) decomposable metric in a systematic way. It is found that the nonlinear Poisson equation Δ g u − f (u) = 0 admits Lie point symmetries only when f (u) = k u , and in this case, only the Killing vectors are admitted. It is shown that the Noether point symmetries coincide with the Lie point symmetries. This approach/method can be used to study the Lie point symmetries of more complex equations and with more degrees of freedom. [ABSTRACT FROM AUTHOR]