Multi‐valued solutions and branch point singularities for nonlinear hyperbolic or elliptic systems

Russel E. Caflisch, Nicholas M Ercolani, Thomas Y. Hou, Yelena Landis

Research output: Contribution to journalArticle

23 Citations (Scopus)

Abstract

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.

Original languageEnglish (US)
Pages (from-to)453-499
Number of pages47
JournalCommunications on Pure and Applied Mathematics
Volume46
Issue number4
DOIs
StatePublished - 1993

Fingerprint

Nonlinear Hyperbolic Systems
Nonlinear Elliptic Systems
Branch Point
Multivalued
Singularity
Collision
Square root
Cusp
Fold
Cube root
Isolated Singularity
Complex Functions
Riemann Surface
Envelope
Analytic function
Valid
Perturbation

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Multi‐valued solutions and branch point singularities for nonlinear hyperbolic or elliptic systems. / Caflisch, Russel E.; Ercolani, Nicholas M; Hou, Thomas Y.; Landis, Yelena.

In: Communications on Pure and Applied Mathematics, Vol. 46, No. 4, 1993, p. 453-499.

Research output: Contribution to journalArticle

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