Questions tagged [non-linear-schroedinger]
The non-linear-schroedinger tag has no usage guidance.
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Asking advice for numerical implementation of Conservative Finite Difference Method for solving Gross-Pitaveskii equation
I am trying to numerically solve the Gross-Pitaevskii equation for an impurity coupled with a one-dimensional weakly-interacting bosonic bath, given by (in dimensionless units):
\begin{align}
i \frac{\...
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Is it physically relevant to restrict the solution of a nonlinear PDE to positive frequencies in the Fourier transfrom?
I would like to mention that I am a mathematician and not a physicist, so I apologize in advance if my question seems obvious.
Considering any linear PDE, it is common to understand the behavior of ...
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Solitons and traveling waves for a Schrödinger type equation
I am a mathematician and not a physicist.
I came across a non--linear PDE whose linear part is a Schrödinger equation (i.e. a dispersive equation) and we know that this equation has a solution for $x\...
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What is the proof of Davies theorem: if a map on pure quantum states transforms equivalent ensembles to equivalent ones, then the map is unitary?
In the following paper (Dynamical Reduction Models by Bassi and Ghirardi), at the end of section 5.3, the following claim is made.
Consider a bijective(*) map on pure states (not necessarily unitary ...
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What Class of Interactions, excluding spontaneous pair creation, Entangle two particles [duplicate]
Un-Entangled particles have state vector: $|\psi\rangle = |p_1, p_2\rangle$, while entangled particles have a composite state vector $|\psi\rangle = 1√2(|p_1,p_2\rangle\pm |p^\prime_1,p^\prime_2\...
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Are there any nonlinear Schrödinger equations?
The 1D Schrödinger equation reads:
$$\frac{\partial \Psi}{\partial t}=\frac{i\hbar}{2m}\frac{\partial^2 \Psi}{\partial x^2}-\frac{i}{\hbar}V\Psi.$$
Now, generally we have $V=V(x)$ (or it dependending ...
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How to evaluate a non-banal derivate?
I need to evaluate the following derivate:
$$\frac{dF}{d\Psi} = \frac{d}{d\Psi}\left[\beta\Delta\Psi+\alpha\left|\Psi\right|^2\Psi+\mu\Psi-i\vec{v}\cdot\bar{\nabla}\Psi\right]$$
where $\Psi$ is a ...
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Is there a "measure of nonlinearity" that can be measured when testing quantum mechanics?
For context, I think the comparison to tests of general relativity here is apt. There is the post-Newtonian formalism that has some well-defined parameters that can discriminate between general ...
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Solution to two-dimensional PDE (wave/Klein-Gordon type equation)
I'm cross-posting from the Math SE as more people might have relevant knowledge here. I was playing with an optimization problem and ended up reducing it to solving the following PDE:
$$ a^2 xy \frac{\...
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How to marginalize a Lagrangian density?
I'm trying to replicate a result from this paper: Physical Review A 76, 063614 (2007). It's for a class in classical mechanics, so we're only concerned with Lagrangian densities and such. I must ...
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Gross-Pitaevskii Equation regarding
Sir,
I have been studying the Gross-Pitaevskii equation for weakly interacting Bose gas and I want to find out the Green's function for the equation:
$$i\hbar\frac{\partial}{\partial t}\psi(r,t)=\Big[-...
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Soliton solutions of the Gross-Pitaevskii equation
The Gross-Pitaevskii equation admits soliton solutions such as: $$\psi(x)=\psi_0 sech(x/\xi),$$
where $\xi$ is the healing length defined by: $\xi=\frac{\hbar}{\sqrt{m \mu}}$, with $\mu$ being the ...
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Derivation of the Zakharov-Shabat System and Lax Pair for the Gross–Pitaevskii Equation
Question: In addition to showing that the nonlinear Schrodinger equation $i \Psi_t + \Psi_{xx} - 2|\Psi|^2 \Psi = 0$ (without a potential) is integrable and isospectral, the existence of a Lax pair ...
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Nonlinear Schrödinger equation in a potential
I've recently become interested in the integrability of nonlinear PDEs while reading these lecture notes.
Question 1: Would the equation $i\Psi_t + \Psi_{xx} - (2|\Psi|^2 + V) \Psi = 0$ for a ...
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Fourier Decomposition of Schrödinger's Equation with a Potential ${V}{\left({x}\right)}=e^x$
Question: Can the equation ${\psi}_{{{t}}}-{i}{\psi}_{{{x}{x}}}={e}^{{{x}}}{\psi}$ be solved with a canonical Fourier transform? If it requires a Fokas transform or inverse scattering transform, how ...
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