Premium
Multi‐valued solutions and branch point singularities for nonlinear hyperbolic or elliptic systems
Author(s) -
Caflisch Russel E.,
Ercolani Nicholas,
Hou Thomas Y.,
Landis Yelena
Publication year - 1993
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160460402
Subject(s) - mathematics , gravitational singularity , singularity , hodograph , singular point of a curve , cube root , mathematical analysis , cusp (singularity) , square root , nonlinear system , point (geometry) , geometry , physics , quantum mechanics
Multi‐valued solutions are constructed for 2 × 2 first‐order systems using 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 point singularities are envelopes for the characteristics and thus move at the characteristic speeds. We perform an analysis of stability of these singularities with respect to perturbations of the initial data. The generic singularity types are folds, cusps, and nondegenerate umbilic points with non‐zero 3‐jet. An isolated singularity is generically a square root branch point corresponding to a fold. Two types of collisions between singularities are generic: At a “tangential” collision between two singularities moving at the same characteristic speed, a cube root branch point is formed, corresponding to a cusp. A “non‐tangential” collision, between two square root branch points moving at different characteristic speeds, remains a square root branch point at the collision and corresponds to a nondegenerate umbilic point. These results are also valid for a diagonalizable n ‐th order system for which there are exactly two speeds. © 1993 John Wiley & Sons, Inc.