scholarly journals Instabilities of one-dimensional stationary solutions of the cubic nonlinear Schrödinger equation

2005 ◽  
Vol 39 (1) ◽  
pp. 73-84 ◽  
Author(s):  
Roger J Thelwell ◽  
John D Carter ◽  
Bernard Deconinck
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Stationary Solutions ◽  
Nonlinear Schrodinger Equation ◽  
One Dimensional ◽  
Nonlinear Schrödinger ◽  
Cubic Nonlinear Schrödinger Equation

scholarly journals STOCHASTIC NONLINEAR SCHRÖDINGER EQUATION WITH ALMOST SPACE–TIME WHITE NOISE

2019 ◽  
Vol 109 (1) ◽  
pp. 44-67 ◽  
Author(s):  
JUSTIN FORLANO ◽  
TADAHIRO OH ◽  
YUZHAO WANG
Keyword(s):  
Schrödinger Equation ◽  
White Noise ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Space Time ◽  
Nonlinear Schrodinger Equation ◽  
One Dimensional ◽  
Nonlinear Schrödinger ◽  
The One ◽  
Cubic Nonlinear Schrödinger Equation

We study the stochastic cubic nonlinear Schrödinger equation (SNLS) with an additive noise on the one-dimensional torus. In particular, we prove local well-posedness of the (renormalized) SNLS when the noise is almost space–time white noise. We also discuss a notion of criticality in this stochastic context, comparing the situation with the stochastic cubic heat equation (also known as the stochastic quantization equation).

Stationary solutions to a nonlinear Schrödinger equation with potential in one dimension

2003 ◽  
Vol 133 (2) ◽  
pp. 409-437 ◽  
Author(s):  
Mihai Mariş
Keyword(s):  
Schrödinger Equation ◽  
Sound Velocity ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Stationary Solutions ◽  
Nonlinear Schrodinger Equation ◽  
One Dimension ◽  
One Dimensional ◽  
Nonlinear Schrödinger ◽  
The One

We study the one-dimensional Gross-Pitaevskii-Schrödinger equation with a potential U moving at velocity v. For a fixed v less than the sound velocity, it is proved that there exist two time-independent solutions if the potential is not too big.

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