Exact solutions and branching of singularities for some hyperbolic equations in two variables

Abstract

Exact global propagators are constructed for the singular hyperbolic operators in two variables x 2 k − 2 ∂ t 2 + λ ( k − 1 ) x k − 2 ∂ t − ∂ x 2 , λ a real parameter, and for the degenerate hyperbolic operators ∂ t 2 − t 2 k − 2 ∂ x 2 − λ ( k − 1 ) t k − 2 ∂ x . Qualitative phenomena such as uniqueness in the Cauchy problem and branching of singularities vary with λ, as shown earlier by Treves and by Taniguchi and Tozaki.