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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)
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 [8]. 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)
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)
We discuss the concept of “hydrodynamic” stochastic theory, which is not based on the traditional Markovian concept. A Wigner function developed for friction is used for the study of operators in quantum physics, and for the construction of a quantum equation with friction. We compare this theory with the quantum theory, the Liouville process, and the(More)
We prove the global existence of analytic solutions to the Cauchy problem for a system of nonlinear Schrödinger equations with quadratic interaction in space dimension n 3 under the mass resonance condition. Lagrangian formulation is also described. Global solutions of the derivative Schrödinger equation in a class of functions analytic in a strip,
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)