We study the asymptotic behavior for large time of solutions to the Cauchy problem for the generalized Benjamin-Ono (BO) equation: ut + (|u| ρ−1 u)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)
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 H n/2−2. In the case n = 3 , a similar result is proved in the weighted L 2 space x −1/2 L 2 = F H −1/2 under the mass resonance condition.
This talk is based on a joint work with Nakao Hayashi and Pavel Naumkin . We consider the initial value problem for the nonlinear Shrödinger equation of the derivative type: i∂ t u + 1 2 ∂ 2 x u = N(u, ∂ x u), t > 0, x ∈ R, u(0, x) = u 0 (x), x∈ R. (1) where i = √ −1, ∂ t = ∂/∂t, ∂ x = ∂/∂x and u is a complex-valued unknown function. We will occasionally… (More)
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 study asymptotic behavior in time of global small solutions to the quadratic nonlinear Schrödinger equation in two-dimensional spaces i∂ t u + (1/2)∆u = ᏺ(u), (t, x) ∈ R × R 2 ; u(0,x) = ϕ(x), x ∈ R 2 , where ᏺ(u) = 2 j,k=1 (λ jk (∂ x j u)(∂ x k u)+µ jk (∂ x j u)(∂ x k u)), where λ jk ,µ jk ∈ C. We prove that if the initial data ϕ satisfy some… (More)
We study large-time asymptotic behavior of solutions to the Cauchy problem for a model of nonlinear dissipative evolution equation. The linear part is a pseudodifferential operator and the nonlinearity is a cubic pseudodifferential operator defined by means of the inverse Fourier transformation and represented by bilinear and trilinear forms with respect to… (More)