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

Author:

Ok Bayrakdar Z.1ORCID,Bayrakdar T.2ORCID

Affiliation:

1. Department of Physics, Ege University, 35040 İzmir, Turkey

2. Department of Mathematics, Akdeniz University, 07058 Antalya, Turkey

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.

Publisher

Hindawi Limited

Subject

Applied Mathematics,General Physics and Astronomy

Cited by 3 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Equivalence problem for first-order and second-order ODEs with a quadratic restriction;International Journal of Geometric Methods in Modern Physics;2024-08-20

2. A geometric description for simple and damped harmonic oscillators;TURKISH JOURNAL OF MATHEMATICS;2019-09-28

3. Minimal Surfaces in Three-Dimensional Riemannian Manifold Associated with a Second-Order ODE;Mediterranean Journal of Mathematics;2018-07-27

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