Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations

Z Ok Bayrakdar,T Bayrakdar

doi:10.1155/2018/7590847

Abstract

We construct metric connection associated with a first-order differential equation by means of the generator set of a Pfaffian system on a submanifold of an appropriate first-order jet bundle. We firstly show that the inviscid and viscous Burgers’ equations describe surfaces attached to an ODE of the form dx/dt=u(t,x) with certain Gaussian curvatures. In the case of PDEs, we show that the scalar curvature of a three-dimensional manifold encoding a system of first-order PDEs is determined in terms of the integrability condition and the Gaussian curvatures of the surfaces corresponding to the integral curves of the vector fields which are annihilated by the contact form. We see that an integral manifold of any PDE defines intrinsically flat and totally geodesic submanifold.

Highlights

A simple dynamical transport phenomenon such as transport of a pollutant in a fluid, transportation of information, wave propagation, traffic flow, population density, chemical reactions, and gas dynamics is described by a certain first-order partial differential equation (PDE) of space and time variables, the so-called transport equation [1, 2]
As we show in this paper, the identity for the Gaussian curvature of a surface associated with a first-order ordinary differential equation (ODE) in the first-order jet bundle is described by an inhomogeneous Burgers’ equation: x ut + uux = − ∫ K (t, x󸀠) dx󸀠
An ODE or a PDE is identified with an exterior differential system in an appropriate jet bundle and this approach leads to geometrization of differential equations

Summary

Introduction

A simple dynamical transport phenomenon such as transport of a pollutant in a fluid, transportation of information, wave propagation, traffic flow, population density, chemical reactions, and gas dynamics is described by a certain first-order partial differential equation (PDE) of space and time variables, the so-called transport equation [1, 2]. We basically deal with the first-order ODEs and PDEs in the context of Riemannian geometry and Riemannian structure defined on a submanifold corresponding to given equation enables one to give a geometric description of an interesting class of equations in mathematical physics. With this regard, in this paper, we construct o(2, R) and o(3, R)-valued connections from the exterior differential systems encoding first-order ODEs and PDEs on corresponding submanifolds in the appropriate jet bundles. We show that, for all points on an integral submanifold of a flat manifold, sectional curvatures are described by a pair of inhomogeneous Burgers’ equations

Preliminaries

Metric Connections Associated with First-Order Differential Equations