Συμμετρίες και ολοκληρωσιμότητα διαφορικών και διακριτών εξισώσεων

Abstract

In the present dissertation, we present the study of a family of discrete equations using symmetry-based techniques. Such methods are well established for the study of differential equations. We use the symmetries of discrete equations to establish new connections between discrete and differential equations, as well as to construct new solutions of the former in terms of similarity solutions of the latter. Specifically, we study discrete equations which are affine linear, possess the symmetries of the square and involve four values of an unknown function of two independent discrete variables forming a quadrilateral. The extensive study of this class leads to a conservation law, as well as to linearization conditions. Members of this family are the integrable equations of the Adler, Bobenko, Suris (ABS) classification. The integrability of the ABS equations follows from their multidimensional consistency. The latter implies that, the equation may be extended in a multidimensional lattice. This property allows us to derive directly an auto– Backlund transformation and a Lax pair, using the function defining these equations. These are another evidence of the integrability of the ABS equations. The dependence of these equations on two continuous parameters permits us to study their extended symmetries, i.e. symmetries acting on the parameters as well. These symmetries are our main tool in connecting the ABS equations to integrable systems of differential equations. The integrability of the latter is proved by the construction of an auto–Backlund transformation and a Lax pair. This connection provides us the means to construct solutions of the discrete equations from solutions of the compatible differential system, which are related to solutions of the continuous Painleve equations. On the other hand, we present how these systems lead naturally to generating differential equations, which were presented by Nijhoff, Hone and Joshi starting from another point of view. However, our construction through symmetry reductions is more straightforward. Thus, in the present thesis is presented a novel usage of the symmetries of discrete equations in the construction of solutions and the connection between discrete and differential equations.