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We study the scattering theory for a system of nonlinear Schrödinger equations in space dimension n 3. In the case n 4 , existence of the scattering operator is proved in small data setting in the Sobolev space Hn/2−2. In the case n = 3 , a similar result is proved in the weighted L2 space 〈x〉−1/2L2 = FH−1/2 under the mass resonance condition. Mathematics… (More)

- Nakao Hayashi, Pavel I. Naumkin, Hideaki Sunagawa
- SIAM J. Math. Analysis
- 2008

which suggests dissipativity if Imλ < 0. In fact, it is proved in [17] that the solution decays like O((t log t)−1/2) in Lx as t → +∞ if Imλ < 0 and u0 is small enough. Since the non-trivial free solution (i.e., the solution to (1) for N ≡ 0, u0 = 0) only decays like O(t−1/2), this gain of additional logarithmic time decay reflects a disspative character.… (More)

We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity { ∂2 t u+ ∂tu−∆u+ λu 2 n = 0, x ∈ Rn, t > 0, u(0, x) = εu0 (x) , ∂tu(0, x) = εu1 (x) , x ∈ Rn, where ε > 0, and space dimensions n = 1, 2, 3. Assume that the initial data u0 ∈ H ∩H, u1 ∈ Hδ−1,0 ∩H−1,δ, where δ > n 2 ,… (More)

- Zihua Guo, Nakao Hayashi, Yiquan Lin, Pavel I. Naumkin
- SIAM J. Math. Analysis
- 2013

We study the asymptotic behavior for large time of solutions to the Cauchy problem for the generalized Benjamin-Ono (BO) equation: ut + (|u|ρ−1u)x + Huxx = 0, where H is the Hilbert transform, x, t ∈ R, when the initial data are small enough. If the power ρ of the nonlinearity is greater than 3, then the solution of the Cauchy problem has a quasilinear… (More)

- Nakao Hayashi, Pavel I. Naumkin
- SIAM J. Math. Analysis
- 2001

with φ ∈ L(R), where P is a polynomial vanishing at the origin and ∆ stands for the Laplacian with respect to x. The analyticity in time of the solutions of a semilinear heat equation has been considered by many authors. For example Ōuchi [2] treated the analyticity in time of the solutions of certain initial boundary value problems with bounded continuous… (More)

- Chunhua Li, Nakao Hayashi
- TheScientificWorldJournal
- 2014

The study of nonlinear Schrödinger systems with quadratic interactions has attracted much attention in the recent years. In this paper, we summarize time decay estimates of small solutions to the systems under the mass resonance condition in 2-dimensional space. We show the existence of wave operators and modified wave operators of the systems under some… (More)

We consider the nonlinear Schrödinger systems −i∂tu1 + 1 2 ∆u1 = F (u1, u2), i∂tu2 + 1 2 ∆u2 = F (u1, u2) in n space dimensions, where F is a p-th order local or nonlocal nonlinearity smooth up to order p, with 1 < p ≤ 1 + 2 n for n ≥ 2 and 1 < p ≤ 2 for n = 1. These systems are related to higher order nonlinear dispersive wave equations. We prove the non… (More)

- Nakao Hayashi, Elena I. Kaikina
- Int. J. Math. Mathematical Sciences
- 2010