My research subject is mathematical analysis of partial differential
equations which describe space-time evolution of waves with strong
interactions, arising in various physical contexts, such as plasma,
superfluid, and water waves. The goal is to derive from the equations
universal properties of solutions independent of the physical
backgrounds. Solutions of partial differential equations in general
contain information of infinite dimensions. When the equation has
nonlinearity in addition, it is almost always impossible to write down
the solutions in explicit forms. It has become easier to calculate
approximate solutions thanks to the progress of computers, yet they
can obtain at most finite amount of information. Hence it is a role of
mathematics to capture the whole possibility in infinite dimensions.
The mathematical study of nonlinear wave equations has developed
starting from the most basic problems such as unique existence of
solutions, and mainly by analysis of individual specific solutions
such as blow-up and solitons. Recently, it is gradually exploring new
aspects of infinite dimensions, such as classification of entire set
of solutions, relations between different solutions, and mechanism of
dynamical transitions.