Studies on Discretizations of Integrable Systems and Their Applications
In this thesis, discretizations of nonlinear integrable systems and their applications to numerical integrators and numerical algorithms are studied. We first discuss an integrable discretization of certain integrable systems which has explicit solutions based on the bilinear formalism. Next we propose a new discretization method which preserves all conserved quantities of the original integrable systems. Moreover, we design a new numerical algorithm for calculating continued fractions by using a discrete integrable system.