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1996 | 33 | 1 | 297-308
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Non-Euclidean geometry and differential equations

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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.
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  • Chair of Mathematics, Department of Physics, Moscow State University, 119899 Moscow, Russia
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