Equivalence groupoid and group classification of a class of nonlinear wave and elliptic equations

Enhancing and essentially generalizing the results of [Nonlinear Anal. 70 (2009), 3512-3521] on a class on (1+1)-dimensional nonlinear wave and elliptic equations, we apply several new technique to classify admissible point transformations within this class up to the equivalence generated by its equivalence group. This gives an exhaustive description of its equivalence groupoid. After extending the algebraic method of group classification to non-normalized classes of differential equations, we solve the complete group classification problem for the class under study up to both usual and general point equivalences. The solution includes the complete preliminary group classification of the class and the construction of singular Lie-symmetry extensions, which are not related to subalgebras of the equivalence algebra. The complete preliminary group classification is based on classifying appropriate subalgebras of the entire infinite-dimensional equivalence algebra whose projections are qualified as maximal extensions of the kernel invariance algebra.