scholarly journals Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation

2018 ◽  
Vol 17 (2) ◽  
pp. 487-504
Author(s):  
Yuanyuan Ren ◽  
◽  
Yongsheng Li ◽  
Wei Yan ◽  
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Well Posedness ◽  
Nonlinear Schrödinger ◽  
The Cauchy Problem

GLOBAL WELL-POSEDNESS AND INSTABILITY OF A NONLINEAR SCHRÖDINGER EQUATION WITH HARMONIC POTENTIAL

2014 ◽  
Vol 98 (1) ◽  
pp. 78-103
Author(s):  
T. SAANOUNI
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Harmonic Potential ◽  
Well Posedness ◽  
Potential Well ◽  
Nonlinear Schrödinger ◽  
The Cauchy Problem ◽  
Two Space Dimensions

AbstractThis paper is concerned with the Cauchy problem for a nonlinear Schrödinger equation with a harmonic potential and exponential growth nonlinearity in two space dimensions. In the defocusing case, global well-posedness is obtained. In the focusing case, existence of nonglobal solutions is discussed via potential-well arguments.

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.

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