- Tytuł:
- Non-Euclidean geometry and differential equations
- Autorzy:
- Popov, A.
- Powiązania:
- https://bibliotekanauki.pl/articles/1359528.pdf
- Data publikacji:
- 1996
- Wydawca:
- Polska Akademia Nauk. Instytut Matematyczny PAN
- Opis:
- In this paper a geometrical link between partial differential equations (PDE) and special coordinate nets on two-dimensional smooth manifolds with the a priori given curvature is suggested. The notion of G-class (the Gauss class) of differential equations admitting such an interpretation is introduced. The perspective of this approach is the possibility of applying the instruments and methods of non-Euclidean geometry to the investigation of differential equations. The equations generated by the coordinate nets on the Lobachevsky plane $Λ^2$ (the hyperbolic plane) take a particular place in this study. These include sine-Gordon, Korteweg-de Vries, Burgers, Liouville and other equations. They form the so-called $Λ^2$-class (the Lobachevsky class). The theorems on the mutual transformation of solutions of $Λ^2$-class equations are formulated. On the base of the developed approach a transformation allowing one to construct global solutions of Liouville type equations from solutions of the Laplace equation is established. Natural generalizations of the well-known nonlinear PDE from the non-Euclidean geometry point of view are proposed. The possibility of the applications of the discussed formalism in the phase spaces theory is stressed.
- Źródło:
-
Banach Center Publications; 1996, 33, 1; 297-308
0137-6934 - Pojawia się w:
- Banach Center Publications
- Dostawca treści:
- Biblioteka Nauki
Artykuł