Introduction xv The main message of discrete differential geometry, addressed to the inte- grable systems community, becomes: Discretize equations by discretizing the geometry. The profundity and fruitfulness of this principle will be demonstrated throughout the book. Integrability. We will now give a short overview of the historical develop- ment of the integrability aspects of discrete differential geometry. The classi- cal period of surface theory resulted in the beginning of the 20th century in an enormous wealth of knowledge about numerous special classes of surfaces, coordinate systems and their transformations, which is summarized in exten- sive volumes by Darboux (1910, 1914-27), Bianchi (1923), etc. One can say that the local differential geometry of special classes of surfaces and coordi- nate systems has been completed during this period. Mathematicians of that era have found most (if not all) geometries of interest and knew nearly every- thing about their properties. It was observed that special geometries such as minimal surfaces, surfaces with constant curvature, isothermic surfaces, or- thogonal and conjugate coordinate systems, Ribaucour sphere congruences, Weingarten line congruences etc. have many similar features. Among others we mention acklund and Darboux type transformations with remarkable permutability properties investigated mainly by Bianchi, and the existence of special deformations within the class (associated family). Geometers real- ized that there should be a unifying fundamental structure behind all these common properties of quite different geometries and they were definitely searching for this structure see Jonas (1915) and Eisenhart (1923). Much later, after the advent of the theory of integrable systems in the the last quarter of the 20th century, these common features were recognized as being associated with the integrability of the underlying differential equa- tions. The theory of integrable systems (called also the theory of solitons) is a vast field in mathematical physics with a huge literature. It has applica- tions in fields ranging from algebraic and differential geometry, enumerative topology, statistical physics, quantum groups and knot theory to nonlinear optics, hydrodynamics and cosmology. The most famous models of this theory are the Korteweg-de Vries (KdV), the nonlinear Schr¨ odinger and the sine-Gordon equations. The KdV equa- tion played the most prominent role in the early stage of the theory. It was derived by Korteweg-de Vries (1895) to describe the propagation of waves in shallow water. Localized solutions of this equation called solitons gave the whole theory its name. The birth of the theory of solitons is associated with the famous paper by Gardner-Green-Kruskal-Miura (1967), where the inverse scattering method for the analytic treatment of the KdV equation
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