Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations
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 th...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2018-01-01
|
| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2018/7590847 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832556025140477952 |
|---|---|
| author | Z. Ok Bayrakdar T. Bayrakdar |
| author_facet | Z. Ok Bayrakdar T. Bayrakdar |
| author_sort | Z. Ok Bayrakdar |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-e268c81dda864889bb71275124d61cb3 |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-e268c81dda864889bb71275124d61cb32025-02-03T05:46:38ZengWileyAdvances in Mathematical Physics1687-91201687-91392018-01-01201810.1155/2018/75908477590847Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential EquationsZ. Ok Bayrakdar0T. Bayrakdar1Department of Physics, Ege University, 35040 İzmir, TurkeyDepartment of Mathematics, Akdeniz University, 07058 Antalya, TurkeyWe 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.http://dx.doi.org/10.1155/2018/7590847 |
| spellingShingle | Z. Ok Bayrakdar T. Bayrakdar Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations Advances in Mathematical Physics |
| title | Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations |
| title_full | Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations |
| title_fullStr | Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations |
| title_full_unstemmed | Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations |
| title_short | Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations |
| title_sort | burgers equations in the riemannian geometry associated with first order differential equations |
| url | http://dx.doi.org/10.1155/2018/7590847 |
| work_keys_str_mv | AT zokbayrakdar burgersequationsintheriemanniangeometryassociatedwithfirstorderdifferentialequations AT tbayrakdar burgersequationsintheriemanniangeometryassociatedwithfirstorderdifferentialequations |