MULTIVALUED SOLUTIONS AND BRANCH POINT SINGULARITIES FOR NONLINEAR HYPERBOLIC OR ELLIPTIC-SYSTEMS

Multi-valued solutions are constructed for 2 x 2 first-order systems u sing a generalization of the hodograph transformation. The solution is found as a complex analytic function on a complex Riemann surface for which the branch points move as part of the solution. The branch poin t singularities are envelopes for the characteristics and thus move at the characteristic speeds. We perform an analysis of stability of the se singularities with respect to perturbations of the initial data. Th e generic singularity types are folds, cusps, and nondegenerate umbili c points with non-zero 3-jet. An isolated singularity is generically a square root branch point corresponding to a fold. Two types of collis ions between singularities are generic: At a ''tangential'' collision between two singularities moving at the same characteristic speed, a c ube root branch point is formed, corresponding to a cusp. A ''non-tang ential'' collision, between two square root branch points moving at di fferent characteristic speeds, remains a square root branch point at t he collision and corresponds to a nondegenerate umbilic point. These r esults are also valid for a diagonalizable n-th order system for which there are exactly two speeds.