Generic 1-parameter families of binary differential equations of Morse type

J.W. Bruce; F. Tari

doi:10.3934/dcds.1997.3.79

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1997, Volume 3, Issue 1: 79-90. Doi: 10.3934/dcds.1997.3.79

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Generic 1-parameter families of binary differential equations of Morse type

J.W. Bruce^1, and
F. Tari¹

1.
Department of Pure Mathematics, The University of Liverpool, P.O. Box 147, Liverpool L69 3BX

Received: December 31, 1995

Revised: August 31, 1996

Published: December 1996

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Abstract

In a previous paper [2] we made a classification of generic binary differential equations (BDE's)

$a(x,y)dy^2+2b(x,y)dxdy+c(x,y)dx^2=0$

near points at which the discriminant function $b^2-ac$ has a Morse singularity. Such points occur naturally in families of BDE's and here we describe the manner in which the configuration of solution curves change in their natural 1-parameter versal deformations.
The results in this paper can be used to describe, for instance, the changes in the structure of the asymptotic curves on a 1-parameter family of smooth surfaces acquiring a flat umbilic and on integral curves determined by eigenvectors of 1-parameter families of $2\times 2$ matrices. It also sheds light on the structure of the rarefraction curves associated to a $2\times 2$ system of conservation laws in 1 space variable.

Mathematics Subject Classification: 58Fxx, 34Cxx.

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