= 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>.
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