xvi Introduction was invented. The sine-Gordon equation is the oldest integrable equation and the most important one for geometry. It describes surfaces with con- stant negative Gaussian curvature and goes back at least to Bour (1862) and Bonnet (1867). Many properties of this equation which are nowadays associated with integrability were known in classical surface theory. One can read about the basic structures of the theory of integrable sys- tems in numerous books. We mention just a few of them: Newell (1985), Faddeev-Takhtajan (1986), Hitchin-Segal-Ward (1999), Dubrovin-Kriche- ver-Novikov (2001). The most commonly accepted features of integrable systems include: In the theory of solitons nonlinear integrable equations are usually represented as a compatibility condition of a linear system called the zero curvature representation (also known as Lax or Zakharov- Shabat representations). Various analytic methods of investigation of soliton equations (like the inverse scattering method, algebro- geometric integration, asymptotic analysis, etc.) are based on this representation. Another indispensable feature of integrable systems is that they possess B¨ acklund-Darboux transformations. These special trans- formations are often used to generate new solutions from the known ones. It is a characteristic feature of soliton (integrable) partial differ- ential equations that they appear not separately but are always organized in hierarchies of commuting flows . It should be mentioned that there is no commonly accepted mathematical definition of integrability (as the title of the volume “What is integrabil- ity?”, Zakharov (1991), clearly demonstrates). Different scientists suggest different properties as the defining ones. Usually, one refers to some addi- tional structures, such as those mentioned above. In this book, we propose an algorithmic definition of integrability given in terms of the system itself. In both areas, in differential geometry and in the theory of integrable systems, there were substantial efforts to discretize the fundamental struc- tures. In the theory of solitons the problem is to discretize an integrable dif- ferential equation preserving its integrability. Various approaches to this problem began to be discussed in the soliton literature starting from the mid-1970s. The basic idea is to discretize the zero curvature representation of the smooth system, i.e., to find proper discrete analogues of the corre- sponding linear problems. This idea appeared first in Ablowitz-Ladik (1975).

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