scholarly journals The Nonlinear Schrödinger Equation on Z and R with Bounded Initial Data: Examples and Conjectures

2020 ◽  
Vol 180 (1-6) ◽  
pp. 910-934
Author(s):  
Benjamin Dodson ◽  
Avraham Soffer ◽  
Thomas Spencer
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger

scholarly journals Finite speed of disturbance for the nonlinear Schrödinger equation

2018 ◽  
Vol 149 (6) ◽  
pp. 1405-1419
Author(s):  
Simão Correia
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Finite Speed Of Propagation ◽  
Finite Speed ◽  
Nonlinear Schrödinger ◽  
The Cauchy Problem ◽  
Whole Space

AbstractWe consider the Cauchy problem for the nonlinear Schrödinger equation on the whole space. After introducing a weaker concept of finite speed of propagation, we show that the concatenation of initial data gives rise to solutions whose time of existence increases as one translates one of the initial data. Moreover, we show that, given global decaying solutions with initial data u0, v0, if |y| is large, then the concatenated initial data u0 + v0(· − y) gives rise to globally decaying solutions.

scholarly journals Energy scattering for a class of inhomogeneous nonlinear Schrödinger equation in two dimensions

2021 ◽  
Vol 18 (01) ◽  
pp. 1-28
Author(s):  
Van Duong Dinh
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Two Dimensions ◽  
Nonlinear Schrödinger ◽  
Energy Scattering ◽  
The One ◽  
Inhomogeneous Nonlinear Schrödinger Equation

We consider a class of [Formula: see text]-supercritical inhomogeneous nonlinear Schrödinger equations in two dimensions [Formula: see text] where [Formula: see text] and [Formula: see text]. Using a new approach of Arora et al. [Scattering below the ground state for the 2D radial nonlinear Schrödinger equation, Proc. Amer. Math. Soc. 148 (2020) 1653–1663], we show the energy scattering for the equation with radially symmetric initial data. In the focusing case, our result extends the one of Farah and Guzmán [Scattering for the radial focusing INLS equation in higher dimensions, Bull. Braz. Math. Soc. (N.S.) 51 (2020) 449–512] to the whole range of [Formula: see text] where the local well-posedness is available. In the defocusing case, our result extends the one in [V. D. Dinh, Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation, J. Evol. Equ. 19(2) (2019) 411–434], where the energy scattering for non-radial initial data was established in dimensions [Formula: see text].

scholarly journals The Effect of Gain and Strong Dissipative Structures on Nonlinear Schrödinger Equations in Optical Fiber

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Xintong Yuan ◽  
Chunhua Li
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Gain Coefficient ◽  
Dissipative Structures ◽  
Nonlinear Schrodinger Equation ◽  
Large Initial Data ◽  
Nonlinear Schrödinger ◽  
Global Existence Of Solutions

We consider the initial value problem for the nonlinear Schrödinger equation satisfying the strong dissipative condition Iλ<0 and Iλ>p-1/2pRλ in one space dimension. Our purpose in this paper is to study how the gain coefficient μ(t) and strong dissipative nonlinearity λvp-1v affect solutions to the nonlinear Schrödinger equation for large initial data. We prove global existence of solutions and present some time decay estimates of solutions for large initial data.

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