Non-Euclidean geometry and differential equations

A. Popov

Banach Center Publications (1996)

  • Volume: 33, Issue: 1, page 297-308
  • ISSN: 0137-6934

Abstract

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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.

How to cite

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Popov, A.. "Non-Euclidean geometry and differential equations." Banach Center Publications 33.1 (1996): 297-308. <http://eudml.org/doc/262603>.

@article{Popov1996,
abstract = {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.},
author = {Popov, A.},
journal = {Banach Center Publications},
keywords = {Lobachevsky plane; sine-Gordon; Korteweg-de Vries; Burgers; Liouville; Lobachevsky class; transformation of solutions},
language = {eng},
number = {1},
pages = {297-308},
title = {Non-Euclidean geometry and differential equations},
url = {http://eudml.org/doc/262603},
volume = {33},
year = {1996},
}

TY - JOUR
AU - Popov, A.
TI - Non-Euclidean geometry and differential equations
JO - Banach Center Publications
PY - 1996
VL - 33
IS - 1
SP - 297
EP - 308
AB - 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.
LA - eng
KW - Lobachevsky plane; sine-Gordon; Korteweg-de Vries; Burgers; Liouville; Lobachevsky class; transformation of solutions
UR - http://eudml.org/doc/262603
ER -

References

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