On the interior scattering of waves, defined by hyperbolic variational principles
type: Item
instance of: scholarly article
title: On the interior scattering of waves, defined by hyperbolic variational principles
zbMATH Open document ID: 0699.35199
DOI: 10.1016/0393-0440(88)90027-7
published in: Journal of Geometry and Physics
publication date: 1988-01-01T00:00:00Z
review text: This paper addresses the problem surrounding the transformation of waves of different kinds at interior points of a nonhomogeneous media. The waves are defined by linear hyperbolic variational systems. An essential review of contact geometry is included together with an important condition namely that the velocity of a moving contact element belongs to the hyperplane of the distribution iff the contact points velocity belongs to the moving contact element. The main result is that the light surface of a typical variational hyperbolic system is reduced in a neighborhood of a typical singular point, to the microlocal formal normal form, \(H=0\), where \(H=p^ 2_ 1\pm q^ 2_ 1-q^ 2_ 2\) for some local Darboux coordinates. That is to say coordinates such that the contact structure is defined by the equation, \(\alpha =0\), where \(\alpha =dz+(p dq-q dp)/2,\quad z\in {\mathbb{R}},\quad p\in {\mathbb{R}}^ D,\quad q\in {\mathbb{R}}^ D,\quad D>1.\) The proof is clearly developed in a series of twelve lemmas.
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 49R50
Mathematics Subject Classification ID: 58E30
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 35L99
zbMATH DE Number: 4146671
zbMATH Keywords: nonstrict hyperbolicity
zbMATH Keywords: wave fronts
zbMATH Keywords: rays
zbMATH Keywords: scattering theory
zbMATH Keywords: linear hyperbolic variational systems
zbMATH Keywords: contact geometry
zbMATH Keywords: hyperbolic system
zbMATH Keywords: singular point
zbMATH Keywords: microlocal formal normal form
author: Vładimir I. Arnold
reviewed by: John Schmeelk
MaRDI profile type: Publication
full work available at URL: 0393-0440(88)90027-7
OpenAlex ID: W2080605248
cites work: Q3805498
cites work: Sur les singularités des formes différentielles
description: scientific article; zbMATH DE number 4146671