People

Nakao HAYASHI

Email nhayashi(@math.sci.osaka-u.ac.jp)
Research
Partial differential equations
Keywords Schrödinger equation, Klein-Gordon equation, Korteweg-de Fries equation, scattering problem
Office Science Building B-450 (Toyonaka Campus)
URL http://www.math.sci.osaka-u.ac.jp/~nhayashi/

I am interested in asymptotic behavior in time of solutions to nonlinear dispersive equations (1 D nonlinear Schreodinger, Benjamin-Ono, Korteweg-de Vries, modified Korteweg-de Vries, derivative nonlinear Schreodinger equations) and nonlinear dissipative equations Complex Landau-Ginzburg equations, Korteweg-de Vries equation on a half line, Damped wave equations with a critical nonlinearity). These equations have important physical applications. Exact solutions of the cubic nonlinear Schreodinger equations and Korteweg-de Vries can be obtained by using the inverse scattering method. Our aim is to study asymptotic properties of these nonlinear equations with general setting through the functional analysis. We also study nonlinear Schreodinger equations in general space dimensions with a critical nonlinearity of order 1+2/n and the Hartree equation, which is considered as a critical case and the inverse scattering method does not work. On 1995, Pavel I. Naumkin and I started to study the large time behavior of small solutions of the initial value problem for the non-linear dispersive equations and we obtained asymptotic behavior in time of solutions and existence of modified scattering states to nonlinear Schreodinger with critical and subcritical nonlinearities. It is known that the usual scattering states in L2 do not exist in these equations. Recently, E.I.Kaikina and I are studying nonlinear dissipative equations (including Korteweg-de Vries) on a half line and some results concerning asymptotic behavior in time of solutions are obtained.