Lobachevsky geometry and nonlinear equations of mathematical physics

Abstract

In this chapter we present a geometric approach to the interpretation of nonlinear partial differential equations which connects them with special coordinate nets on the Lobachevsky plane $\Lambda^2$.We introduce the class of Lobachevsky differential equations ($\Lambda^2$-class), which admit the aforementioned interpretation. The development of this geometric approach to nonlinear equations of contemporary mathematical physics enables us to apply in their study the rather well developed apparatus and methods of non-Euclidean hyperbolic geometry.