Symmetry and Complete Integrability of Nonlinear Difference Equations

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The purpose of this research is to clarify the mathematical structure of singularity confinement condition for nonlinear difference equations, to extend the condition to ultradiscrete systems, and to study algebraic and geometrical structure of discrete integrable systems. Some of our results are as follows.1. We have studied the relation between continuous and discrete Painleve equations and given a determinant-type solution for them. We also presented a method to describe q-Painleve equations in a unified way. Furthermore, we studied their transformation structure in detail.2. We have presented a few of new integrable nonlinear equations and studied solutions of a coupled KP equation and the higher order nonlinear Schrodinger equations.3.We have proposed an ultradiscrete Sine-Gordon equation, which is considered to be a cellular automaton with soliton-type solutions. We have also given an ultradiscrete modified KdV equation and have shown that the equation directly relates with a box and ball system. Moreover, we proposed a method to ultradiscretize variables without positivity.4.We have studied periodic ultradiscrete systems and determined fundamental period of their solutions exactly and have given an algolithm to constitute the conserved quantities explicitly. We also clarify the relation between conserved quantities of ultradiscrete and continuous systems.