TY - JOUR

T1 - Large Time Behavior of Solutions for Derivative Cubic Nonlinear Schrödinger Equations

AU - Hayashi, Nakao

AU - Naumkin, Pavel I.

AU - Uchida, Hidetake

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1999

Y1 - 1999

N2 - We study the asymptotic behavior in time and scattering problem for the solutions to the Cauchy problem for the derivative cubic nonlinear Schrödinger equations of the following form where are real valued functions. Here the parameters, and λ2, λ3, λ4, λ5 ∈ C are such that λ2-λ3 ∈ R and λ4-λ5 ∈ R. If and λ5=β=±1, equation (A) appears in the classical pseudospin magnet model [9]. We prove that if and the norm ‖u0‖3,0+‖u0‖2,1=∊ is sufficiently small, then the solution of (A) exists globally in time and satisfies the sharp time decay estimate ‖u(t)‖2,0∞≤C∊(1+ |t|)-1/2, where. Furthermore we prove existence of modified scattering states and nonexistence of nontrivial scattering states. Our method is based on a certain gauge transformation and an appropriate phase function.

AB - We study the asymptotic behavior in time and scattering problem for the solutions to the Cauchy problem for the derivative cubic nonlinear Schrödinger equations of the following form where are real valued functions. Here the parameters, and λ2, λ3, λ4, λ5 ∈ C are such that λ2-λ3 ∈ R and λ4-λ5 ∈ R. If and λ5=β=±1, equation (A) appears in the classical pseudospin magnet model [9]. We prove that if and the norm ‖u0‖3,0+‖u0‖2,1=∊ is sufficiently small, then the solution of (A) exists globally in time and satisfies the sharp time decay estimate ‖u(t)‖2,0∞≤C∊(1+ |t|)-1/2, where. Furthermore we prove existence of modified scattering states and nonexistence of nontrivial scattering states. Our method is based on a certain gauge transformation and an appropriate phase function.

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U2 - 10.2977/prims/1195143611

DO - 10.2977/prims/1195143611

M3 - Article

AN - SCOPUS:84979545020

VL - 35

SP - 501

EP - 513

JO - Publications of the Research Institute for Mathematical Sciences

JF - Publications of the Research Institute for Mathematical Sciences

SN - 0034-5318

IS - 3

ER -