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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)

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)

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