scholarly journals The Cauchy problem for the fourth-order nonlinear Schrödinger equation related to the vortex filament

2005 ◽  
Vol 214 (1) ◽  
pp. 1-35 ◽  
Author(s):  
Zhaohui Huo ◽  
Yueling Jia
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Vortex Filament ◽  
Nonlinear Schrödinger ◽  
The Cauchy Problem

Scattering of solutions to the fourth-order nonlinear Schrödinger equation

2016 ◽  
Vol 18 (03) ◽  
pp. 1550035 ◽  
Author(s):  
Nakao Hayashi ◽  
Jesus A. Mendez-Navarro ◽  
Pavel I. Naumkin
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Large Time ◽  
Nonlinear Schrodinger Equation ◽  
Asymptotics Of Solutions ◽  
Nonlinear Schrödinger ◽  
Time Asymptotics ◽  
The Cauchy Problem

We consider the Cauchy problem for the fourth-order nonlinear Schrödinger equation [Formula: see text] where [Formula: see text] [Formula: see text] We introduce the factorization for the free evolution group to prove the large time asymptotics of solutions.

Large-time behaviour of solutions to the dissipative nonlinear Schrödinger equation

2000 ◽  
Vol 130 (5) ◽  
pp. 1029-1043 ◽  
Author(s):  
N. Hayashi ◽  
E. I. Kaikina ◽  
P. I. Naumkin
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Large Time ◽  
Nonlinear Schrodinger Equation ◽  
Time Decay ◽  
Nonlinear Schrödinger ◽  
The Cauchy Problem ◽  
Image Position

We study the Cauchy problem for the nonlinear Schrödinger equation with dissipation where L is a linear pseudodifferential operator with dissipative symbol ReL(ξ) ≥ C1|ξ|2/(1 + ξ2) and |L′(ξ)| ≤ C2(|ξ|+ |ξ|n) for all ξ ∈ R. Here, C1, C2 > 0, n ≥ 1. Moreover, we assume that L(ξ) = αξ2 + O(|ξ|2+γ) for all |ξ| < 1, where γ > 0, Re α > 0, Im α ≥ 0. When L(ξ) = αξ2, equation (A) is the nonlinear Schrödinger equation with dissipation ut − αuxx + i|u|2u = 0. Our purpose is to prove that solutions of (A) satisfy the time decay estimate under the conditions that u0 ∈ Hn,0 ∩ H0,1 have the mean value and the norm ‖u0‖Hn,0 + ‖u0‖H0,1 = ε is sufficiently small, where σ = 1 if Im α > 0 and σ = 2 if Im α = 0, and Therefore, equation (A) is considered as a critical case for the large-time asymptotic behaviour because the solutions of the Cauchy problem for the equation ut − αuxx + i|u|p−1u = 0, with p > 3 have the same time decay estimate ‖u‖L∞ = O(t−½) as that of solutions to the linear equation. On the other hand, note that solutions of the Cauchy problem (A) have an additional logarithmic time decay. Our strategy of the proof of the large-time asymptotics of solutions is to translate (A) to another nonlinear equation in which the mean value of the nonlinearity is zero for all time.

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