Complete Integrability and Structure of Solutions for Nonlinear Discrete Equations

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For the purpose of clarifying the structure of "singularity confinement", extending the concept to ultra-discrete system, and revealing algebraic and geometrical structure of discrete integrable systems, we have obtained the following results.(1) We have shown the algebraic structure of solution method for nonautonomous discrete integrable systems and constructed solutions expressed by special functions for the systems. We have also proposed new nonlinear relations among the solutions.(2) For some discrete Painleve equations, we have clarified their self-duality and the structure of nonlinear relations among the solutions.(3) We have obtained an ultra-discrete version of the Toda molecule equation and the explicit expression of its solutions. We have also studied its algebraic structure and symmetry, and discussed a possibility of applying the system to the field of mathematical engineering.(4) By considering a nonlinear integro-differential equation as a difference-differential equation, we have studied its integrable structure. We have also studied the structure of solutions for a discrete analogue of coupled nonlinear wave equations.(5) We have presented a geometrical approach to the equation which satisfies the singularity confinement criterion but exhibits chaotic behaviour. We have also given a method to caluculate the algebraic entropy by using the theory of intersection numbers.(6) We have studied geometrical structure of the discrete Painleve equations and shown the relation between the solution space of Toda lattice equation and the solutions of discrete Painleve equations.