Lobachevsky Geometry and Modern Nonlinear Problems

Description

This monograph presents the basic concepts of hyperbolic Lobachevsky geometry and their possible applications to modern nonlinear applied problems in mathematics and physics, summarizing the findings of roughly the last hundred years. The central sections cover the classical building blocks of hyperbolic Lobachevsky geometry, pseudo spherical surfaces theory, net geometrical investigative techniques of nonlinear differential equations in partial derivatives, and their applications to the analysis of the physical models. As the sine-Gordon equation appears to have profound “geometrical roots” and numerous applications to modern nonlinear problems, it is treated as a universal “object” of investigation, connecting many of the problems discussed. The aim of this book is to form a general geometrical view on the different problems of modern mathematics, physics and natural science in general in the context of non-Euclidean hyperbolic geometry. Front Matter....Pages i-viii Introduction....Pages 1-14 Foundations of Lobachevsky geometry: axiomatics, models, images in Euclidean space....Pages 15-59 The problem of realizing the Lobachevsky geometry in Euclidean space....Pages 61-126 The sine-Gordon equation: its geometry and applications of current interest....Pages 127-223 Lobachevsky geometry and nonlinear equations of mathematical physics....Pages 225-257 Non-Euclidean phase spaces. Discrete nets on the Lobachevsky plane and numerical integration algorithms for $$\Lambda^2$$ -equations....Pages 259-290 Back Matter....Pages 291-310