I am engaged in the mathematical study of problems arising in fluid mechanics. Fluid motion governed by the Navier-Stokes equations is a nonlinear dynamical system of infinite degrees of freedom, with a hierarchy of modes of various scales coupling with each other, and exhibits a complicated irregular behavior. From a viewpoint vortices and waves as fundamental elements, I make an effort to understand fluid phenomena and then to build mathematical and numerical models for analyzing them. My interest is also in solar magnetic fields and magnetohydrodynamics in space. Recently, I have been working on nonlinear analyses of the three-dimensional vortex motion. By invoking ideas and machinery in integrable systems and Hamiltonian dynamical systems, I derived an accurate formula of translation speed of a vortex ring and discovered its new instability mode.