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.

Asymptotics for the fourth-order nonlinear Schrödinger equation in 2D

2020 ◽  
pp. 2050090
Author(s):  
Pavel I. Naumkin
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Large Time ◽  
Pseudodifferential Operators ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Two Space Dimensions ◽  
Time Decay Rate

Our aim is to study the large time asymptotics of solutions to the fourth-order nonlinear Schrödinger equation in two space dimensions [Formula: see text] where [Formula: see text] We show that the nonlinearity has a dissipative character, so the solutions obtain more rapid time decay rate comparing with the corresponding linear case, if we assume the nonzero total mass condition [Formula: see text] We continue to develop the factorization techniques. The crucial points of our approach presented here are the [Formula: see text] — estimates of the pseudodifferential operators and the application of the Kato–Ponce commutator estimates.

SUBCRITICAL QUADRATIC NONLINEAR SCHRÖDINGER EQUATION

2011 ◽  
Vol 13 (06) ◽  
pp. 969-1007
Author(s):  
NAKAO HAYASHI ◽  
PAVEL I. NAUMKIN
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Large Time ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Self Similar Solution ◽  
The Cauchy Problem ◽  
Self Similar ◽  
The Fourier Transform

We study the initial value problem for the quadratic nonlinear Schrödinger equation [Formula: see text] where γ > 0. Suppose that the Fourier transform [Formula: see text] of the initial data u1satisfies estimates [Formula: see text], where ε > 0 is sufficiently small. Also suppose that [Formula: see text] for |ξ| ≤ 1. Assume that γ > 0 is small: [Formula: see text]. Then we prove that there exists a unique solution u ∈ C([1, ∞);L2) of the Cauchy problem (*). Moreover, the solution u approaches for large time t → +∞ a self-similar solution of the quadratic nonlinear Schrödinger equation (*).

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