Study of Diophantine Phenomena Appearing in Asymptotic Analysis of Nonlinear Partial Differential Equations

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We have obtained two main results 1) and 2).1) We reveal the role of Diophantine conditions when solving a system of nonlinear partial differential equations or difference equations in the normal form theory. Namely we havea) The simultaneous normal form of a commuting system of maps under Bruno condition is presented. This is also valid for a system of circle maps in a Gevrey class.b) The solvability of nonlinear Fuchsian equations of general variables in a class of finitely smooth functions is proved. This yields a so-called Grobman-Hartmann type theorem.c) The global normal form of a pseudodifferential operator on tori which is a perturbation of a system of constant vector fields is presented. Necessary and sufficient conditions are given in terms of a Diophantine condition.d) To make clear the relations between a simultaneous Diophantine condition of a commuting system of maps or vector fields and the Diophantine conditions for every generator of the system.2) We show that a Riemann-Hilbert problem is closely related to the solvability of equations without Diophantine conditions. Namely we havea) Solvability of nonlinear Fuchsian equations of general variables under a so-called Riemann-Hilbert condition is proved. This is applied to a mixed Monge- Ampere equation.b) Solvability of the same equations as in a) in a class of singular functions under a Poincare condition is presented. This can be regarded as a Frobenius theorem for a partial differential equation. The results are partly reported in ICM conference in Beijing in 2002.