Email okamoto(
Nonlinear partial differential equations
Keywords Nonlinear Schrödinger equation, nonlinear wave equation

My research interest is in nonlinear partial differential equations. Especially, I am working on the Cauchy problem for nonlinear dispersive equations, such as nonlinear Schrödinger and nonlinear wave equations. The Cauchy problem is a problem to find a solution to a differential equation satisfying what are known as initial conditions (initial data) at a certain (initial) time.

For nonlinear problems, it is usually not possible to obtain explicit solutions, so we need to show an existence result of a solution. It then may happen that the solution may change significantly when the initial data change small. However, this phenomenon causes some difficulties on analysis of physical models and differential equations. Hence, it is often required the well-posedness of the Cauchy problem, which ensures that the solution continuously depends on initial data.

One of the distinguishing features of linear dispersive equations is that the speed of the spread of solutions depends on its frequency. Therefore, some kind of smoothing effect can be obtained by incorporating the oscillation caused by dispersion due to time evolution into the analysis. Furthermore, by grasping what kind of oscillation is generated from each nonlinear term, we can use the smoothing effect in the analysis of nonlinear dispersive equations.

I am studying the well-posedness of the Cauchy problem for dispersive equations by using the smoothing effect due to such a dispersive nature. Recently, I am also interested in asymptotic behavior of solutions to the Cauchy problem and stochastic dispersive equations.