Reduction and exact solutions of the eikonal equation
type: Item
instance of: scholarly article
title: Reduction and exact solutions of the eikonal equation
zbMATH Open document ID: 0737.35129
published in: Ukrainian Mathematical Journal
publication date: 1992-06-25T00:00:00Z
review text: The authors find here the wide class of the real exact solutions of the eikonal equation \[ (\partial u/\partial x_ 0)^ 2-(\partial u/\partial x_ 1)^ 2-(\partial u/\partial x_ 2)^ 2-(\partial u/\partial x_ 3)^ 2=1. \leqno (1) \] To solve the equation they find the expressions of \(u\) which reduct the equation (1) to ordinary differential equations. They employ 51 C(1,4)-non equivalent subalgebras of AC(1,4) with rank 3 to give the expressions of \(u\). Here AC(1,4) is the algebra with the bases \(P_ \alpha=\partial_ \alpha\), \(J_{\alpha\beta}=g^{\alpha\nu}x_ \nu\partial_ \beta- g^{\beta\nu}x_ \nu\partial_ \alpha\), \(D=-x^ \alpha\partial_ \alpha\), and \(K_ \alpha=-2(g^{\alpha\beta}x_ \beta)D- (g^{\beta\nu}x_ \beta x_ \nu)\partial_ \alpha (\alpha,\beta,\nu=0,1,\ldots,4)\). Their method is the following: Let \(\omega(x)\), \(\omega(x,u)\) and \(\omega'(x,u)\) be invariants of the subalgebra \(L\). They set (i) \(u=f(x)\varphi(\omega)+g(x)\), \(\omega=\omega(x)\), (ii) \(u^ 2=f(x)\varphi(\omega)+g(x)\), \(\omega=\omega(x)\), or (iii) \(\omega'(x,u)=\varphi(\omega(x,u))\), and have the ordinary differential equations of \(\varphi\) and \(L\) invariant solutions \(u=u(x)\). Finally they show the correspondence between the eikonal equation and the Hamilton-Jacobi equation \(u_ t+(\nabla u)/(2m)=0\) through the bases of AC(1,4).
Mathematics Subject Classification ID: 35Q75
Mathematics Subject Classification ID: 58J70
Mathematics Subject Classification ID: 35A30
Mathematics Subject Classification ID: 35F20
zbMATH DE Number: 12653
zbMATH Keywords: exact solutions
zbMATH Keywords: eikonal equation
zbMATH Keywords: Hamilton-Jacobi equation
zbMATH Keywords: reduction
author: Anatoliy Barannyk
author: Leonid F. Barannyk
author: W. I. Fushchich
reviewed by: Hideo Yamagata
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 12653