Questions tagged [non-linear-schroedinger]
40 questions
25
votes
1 answer
Aether existance in alternate universe made of Bose-Einstein condensate
I came across an interesting question which was shown to me by my professor, it is as follows:
Investigation of an alternative Universe:
This Universe contains three spatial and one time dimensions, and space can be described as a large cuboid. The…
Spoilt Milk
- 1,377
11
votes
4 answers
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 on any other number of real variables). But…
agaminon
- 4,386
10
votes
2 answers
Deducing instability growth rates from the Hamiltonian for the non-linear Schrödinger equation
Consider the following nonlinear Schrödinger equation (NLSE):
$$A_t+iA_{xx}+i|A|^2A = 0, \tag{1}$$
where $A$ is a complex valued function of $(x,t)$.
A solution to this equation is $$A=a_oe^{-ia_o^2t}.\tag{2}$$ We investigate the stability of these…
Nick P
- 1,716
7
votes
2 answers
Difference between real time and imaginary time propagation?
Suppose I want to solve a non-linear Schrödinger equation using imaginary time propagation to get the ground state solution. I choose $t = - i \tau$, and then solve the equation using the split-step Crank-Nicholson method. All the excited states…
user22462
- 71
6
votes
0 answers
Inverse Scattering Transform (IST) for the Linear Schrödinger Equation
I know that the Inverse Scattering Transform (IST) has been employed to solve, for instance, the KdV equation and I believe also other nonlinear PDEs, such as the NLS.
However, if we consider the linear Schrödinger equation with scattering potential…
Jason Born
- 161
5
votes
0 answers
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 the solution by taking the Fourier transform of the…
Niser
- 51
4
votes
2 answers
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 or even linear),
$$S_t |\psi \rangle \rightarrow…
Prem
- 2,356
4
votes
1 answer
Soliton solutions of the Gross-Pitaevskii equation
The Gross-Pitaevskii equation admits soliton solutions such as: $$\psi(x)=\psi_0 sech(x/\xi),$$
where $\xi$ is the healing length defined by: $\xi=\frac{\hbar}{\sqrt{m \mu}}$, with $\mu$ being the chemical potential. To obtain this solution, one of…
T. ssP
- 543
3
votes
1 answer
Solution to two-dimensional PDE (wave/Klein-Gordon type equation)
I'm cross-posting from the Math SE as more people might have relevant knowledge here. I was playing with an optimization problem and ended up reducing it to solving the following PDE:
$$ a^2 xy \frac{\partial^2 f}{\partial x \partial y} + f =…
smalldog
- 229
3
votes
0 answers
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 marginalize a Lagrangian density, but I'm not sure how…
Kantomk
- 31
3
votes
1 answer
correlation function in Fourier space
I'm reading this paper and want to prove eq (8):
The field $\psi(\mathbf{x}) \in \mathbb{C}$ exists in a finite periodic 2D square box (of side length $L$), and has a Fourier series expansion, and corresponding inversion formula for the Fourier…
jms547
- 201
3
votes
1 answer
Energy density in Gross-Pitaevskii equation
I guess this is a straightforward question but I was wondering if I can get an explicit steps toward the answer.
Using the Gross-Pitaevskii equation:
$$ \tag{1} i \hbar\frac{\partial\psi\left(x,t\right)}{\partial t} =\left(-\frac{\hbar^{2}}{2m}…
MA13
- 75
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3
votes
1 answer
How to define quantum chaos?
I was told that quantum chaos is just a system whose Hamiltonian's classical version shows chaotic behavior. However, I just wondering
what happens when one eigenstate of this Hamiltonian evolves?
what the chaotic (nonlinear, of course)…
RoderickLee
- 1,117
2
votes
2 answers
Is there a "measure of nonlinearity" that can be measured when testing quantum mechanics?
For context, I think the comparison to tests of general relativity here is apt. There is the post-Newtonian formalism that has some well-defined parameters that can discriminate between general relativity and other theories.
Is there anything like…
MaximusIdeal
- 8,868
2
votes
1 answer
Gross-Pitaevskii Equation regarding
Sir,
I have been studying the Gross-Pitaevskii equation for weakly interacting Bose gas and I want to find out the Green's function for the equation:
$$i\hbar\frac{\partial}{\partial…
R. Bhattacharya
- 143