"The main aim of this book is to look at the potential of the geometry developed by Lobachevskii in the context of its emergence in various branches of current interest in contemporary mathematics and science, especially in nonlinear problems of mathematical physics. ... the book is well written, very readable, and nicely illustrated throughout with many graphs and figures, especially figures of surfaces. ... This unique book makes this difficult subject interesting and within reach." (Paul F. Bracken, Mathematical Reviews, August, 2015) "The book is original in its form and content. It covers a wide spectrum of geometry and analysis and it displays the Lobachevsky plane as a central object in the study of the classical equations of mathematical physics. The style is expository and clear. This book is a valuable addition to the geometric literature." (Athanase Papadopoulos, zbMATH 1311.51002, 2015)

• Introduction.- 1 Foundations of Lobachevsky geometry: axiomatics, models, images in Euclidean space.- 2 The problem of realizing the Lobachevsky geometry in Euclidean space.- 3 The sine-Gordon equation: its geometry and applications of current interest.- 4 Lobachevsky geometry and nonlinear equations of mathematical physics.- 5 Non-Euclidean phase spaces. Discrete nets on the Lobachevsky plane and numerical integration algorithms for Λ2-equations.- Bibliography.- Index.