 Characteristic caustic

A characteristic caustic is a locus where the coordinate map from initial-data and evolution parameters to physical coordinates has zero Jacobian determinant. The inverse function theorem then fails, and different characteristic branches can reach the same point. A classical solution constructed from initial data must be restricted to its appropriate invertible branch before such a fold. In geometrical optics the corresponding ray-envelope phenomenon is a caustic; in nonlinear first-order equations a fold can signal the onset of multiple gradients or shock formation by characteristic intersection.