Nakao Hayashi Home Page

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I am interested in asymptotic behavior in time of solutions to nonlinear dispersive equations (Nonlinear Schrödinger, nonlinear Klein-Gordon, Benjamin-Ono, Korteweg-de Vries, modified Korteweg-de Vries, derivative nonlinear Schrödinger 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 Schrödinger and the sin Klein-Gordon equation, 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 Schrödinger 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 Schrödinger with critical and subcritical nonlinearities. It is known that the usual scattering states in $L^2$ do not exist in these equations. Elena.I.Kaikina and I studied  nonlinear dissipative equations (including Korteweg-de Vries) on a half line and some results concerning asymptotic behavior in time of solutions were obtained. Recently, Pavel I. Naumkin and I showed existence of the modified wave operator for the cubic nonlinear Klein-Gordon equation and we also constructed the inverse modified wave operator without the compactness assumption on the data.


List of publications

Preprints

Curriculum Vitae


PDE Meetings in Toyonaka Osaka (Japanese)

PDE meetings will be organized in Osaka University several times a year.

(1)    Recent progress on nonlinear waves and dispersive equations , 2002, Oct. 22-24

(2)    On solutions of nonlinear dispersive and dissipative equations, 2002, Nov. 18-20

(3)    Nonlinear waves in Osaka University, 2003, June 3-4

(4)    Nonlinear PDE in Osaka University (recent progress on nonlinear dispersive equations), 2003, July 2-3

(5)    Nonlinear evolution equations in Osaka University, 2003, Oct. 30-31

(6)    Evolution equations in Osaka University, 2003, Dec. 9

(7)    7th  meeting on Evolution equations in Osaka University, 2004, Feb. 2

(8)    1st  meeting  on linear and nonlinear waves, 2004, Feb. 17-18

(9)    8th  meeting on Evolution equations in Osaka University, 2004, June. 1-2

(10)                   2nd  meeting  on linear and nonlinear waves, 2004, Nov. 4-5

(11)                   9th  meeting on Evolution equations in Osaka University, 2004, Nov. 22-23

(12)                   10th meeting on PDEs  in Osaka University, 2005, Jan. 20

(13)                   11th meeting on PDEs  in Osaka University, 2005, Feb. 18

(14)                   12th meeting on PDEs  in Osaka University, 2005, April 15

(15)          3d  meeting  on linear and nonlinear waves, 2005, Nov. 16-18

(16)          13th meeting on PDEs  in Osaka University, 2006, Feb. 16-17

(17)          4th  meeting  on linear and nonlinear waves, 2006, Nov. 6-8

 (18)        14th meeting on PDEs  in Osaka University, 2006, Dec. 6-7

(19)         5th  meeting  on linear and nonlinear waves, 2007, Nov. 1-3