Nakao Hayashi

Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations

We study the asymptotic behavior in time of solutions to the Cauchy problems for the nonlinear Schrödinger equation with a critical power nonlinearity and the Hartree equation. We prove the existence of modified scattering states and the sharp time decay estimate in the uniform norm of solutions to the Cauchy problem with small initial data. This estimate is very important for the proof of the existence of modified scattering states to the nonlinear Schrödinger equations with a critical nonlinearity and the Hartree equation. In order to derive the desired estimates we introduce a certain phase function since the previous methods, based solely on a priori estimates of the operator x + it∇ acting on the solution without specifying any phase function, do not work for the critical case under consideration. The well-known nonexistence of the usual L2 scattering states shows that our result is sharp.

On the derivative nonlinear Schro¨dinger equation

Abstract In this paper we discuss the Cauchy problem for the derivative nonlinear Schrodinger equation: i∂tψ + 2iδ∂x(|;ψ|2ψ) = 0, ψ(0, x) = ф(x) , where δ ≠ 0. Under an explicit smallness condition of the initial data, we prove the unique global existence of solutions to this problem in the usual Sobolev spaces, in the weighted Sobolev spaces, and in the Schwartz class. We describe the smoothing effect in detail. Furthermore, for the data decaying exponentially at infinity we prove that the above equation has unique local solutions which are analytic in the space direction.

Finite energy solutions of nonlinear Schro¨dinger equations of derivative type

This paper is concerned with the initial value problem for nonlinear Schrodinger equations of the form \[ (\dag)\qquad \left\{ \begin{gathered} i\partial _t \psi + \partial \psi = i\lambda \partial \left( {| \psi |^2 \psi } \right) + \lambda _1 | \psi |^{p_1 - 1} \psi + \lambda _2 | \psi |^{p_2 - 1} \psi ,\quad (t,x) \in \mathbb{R} \times \mathbb{R}, \hfill \\ \psi (0,x) = \phi (x),\quad x \in \mathbb{R}, \hfill \\ \end{gathered} \right.\] where

Analyticity and global existence of small solutions to some nonlinear Schrödinger equations

\partial = \partial _x = {\partial /{\partial x}},\lambda ,\lambda _1 ,\lambda _2 \in \mathbb{R}

Smoothing effect for some Schrödinger equations

and

Large-time behaviour of solutions to the dissipative nonlinear Schrödinger equation

2 \leq p_1 < p_2 < 5

Damped wave equation with a critical nonlinearity

. It is shown that if

Gevrey regularizing effect for the (generalized) Korteweg-de Vries equation and nonlinear Schrödinger equations

\phi \in H^1 (\mathbb{R})

Scattering theory for the Hartree equation

and

Analyticity of solutions of the Korteweg-De Vries equation

\| \phi \|_2^2 < {{2\pi } /{| \lambda |}}