Mathematics on discrete and ultradiscrete integrable systems and its application

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Among the systems of discrete equations, quantum mechanical lattice models and Cellular Automata (discrete dynamical systems which take finite number of states), there exist a class of systems named integrable systems in which exact solutions and/or statistical quantities can be expressed by analytic functions. In this research, we investigated mathematical aspects of these discrete integrable systems, in particular we clarified the mathematical structure of the Box-Ball system (BBS) which is a typical ultradiscrete system. The BBS is a dynamical system of balls moving in an array of boxes and shows solitonic behavior; it has soliton solutions, sufficient number of conserved quantities, and its initial value problem is solvable. Furthermore it has combinatorial features related to quantum integrable models; soliton scattering satisfies the Yang-Bater relation, its phase space and dynamics are related to some quantum algebra of deformation parameter q→0. For this BBS, we studied the ini … More tial value problem from various points of view such as elementary combinatorial methods, KKR bijection, soliton solutions of ultradiscrete KdV equation, and linealisation on the Jacobian variety. We found interesting relations between BBS and other mathematical objects such as Weyl groups, representation theory of quantum algebra, distribution of primes and so on. For example, conserved quantities of a generalized BBS are expressed by paths on a graph, the operation to determine the paths is equivalent to action of Weyl groups. We have also established the relation between fundamental cycles of the periodic BBS and eigenvalues of the transfer matrices of integrable lattice modes with quantum algebraic symmetry and showed that the string hypothesis of the Bethe ansatz equation is characterized by the conserved quantities of the periodic BBS. Furthermore we have made significant progress in the correlation functions of quantum integrable models, fundamental problems in ultradiscrete systems and application to traffic flows. Less