Questions tagged [non-linear-schroedinger]
The non-linear-schroedinger tag has no usage guidance.
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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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Derivation of non-linear Schrödinger equation from many-body QM
I hope this (and not MathOverflow) is the right place to post this question. I am a math student taking a methods of mathematical physics course, in which we cover the solution theory the non-linear ...
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Soliton solution of the NLS equation
My understanding of soliton - it is a moving pulse in a medium which does not change its structure with time. It has other properties like no interaction with other solitons (this could certainly be ...
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Lax pairs, Nonlinear Schrodinger Equation
Briefly: I have two Lax pairs in matrix form and using compatibility condition I have found a nonlinear partial differential equation system. I have searched this system very much seems the nonlinear ...
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Intuition behind focusing vs defocusing in integrable systems like NLS, KdV, mKdV
The following are examples of integrable systems arising from the AKNS system (check out AKNS paper here and a short Wikipedia description)
Non-Linear Schrodinger equation
Korteweg-de Vries equation
...
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Conversion of the nonlinear schrodinger equation from $\partial_zE$ to $\partial_tE$
While reading some papers about the nonlinear schrodinger equation (NLS) I noticed that the authors sometimes use (for the linear case)
$$\partial_zE=\frac{i}{2k_0}\nabla^2E$$
and sometimes
$$\...
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Linearizing the Lugiato-Lefever Partial Differential Equation
Problem Statement:
Given the Lugiato-Lefever equation, linearize the equation and determine the dynamics near a stationary solution by looking for a stationary solution with a small perturbation.
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Does the non-linear Schrodinger equation satisfy quantum mechanics rules?
Thinking about the 0+1 dimensional (time-only) non-linear Schrodinger equation:
$$i\frac{\partial}{\partial t} \psi(t) =\kappa |\psi(t)|^2 \psi(t).$$
Treating $\psi$ as a wave function instead of a ...
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correlation function in Fourier space
I'm reading this paper and want to prove eq (8):
The field $\psi(\mathbf{x}) \in \mathbb{C}$ exists in a finite periodic 2D square box (of side length $L$), and has a Fourier series expansion, and ...
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Physical significance of orbital stability
I saw the orbital stability in Wiki, I just understand it from mathematics angle. But in physical, what is its mean? Since I saw many paper talk about the stability of Schrödinger equation, I think ...
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Is the Schrödinger-Newton equation time-reversal symmetric? What about PT-symmetry or similar symmetries?
I tried to figure it out myself. If you take the integro-differential form of the equation, a complex square of the time-dependent wavefunction appears.. It seems to me that this means the equation ...
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Is there anywhere online where I could simulate the schrodinger equation with different Hamiltonian's? [duplicate]
I have a code which simulates the Schrodinger equation and it works good for the harmonic potential, I have checked that. Now, i have changed the Hamiltonian, I have the results but I want to compare ...
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Inverse Scattering Transform (IST) for the Linear Schrödinger Equation
I know that the Inverse Scattering Transform (IST) has been employed to solve, for instance, the KdV equation and I believe also other nonlinear PDEs, such as the NLS.
However, if we consider the ...
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Split step method for nonlinear Schrodinger equation does not result in self focusing
I'm trying to simulate self focusing in the case of anomalous dispersion and positive Kerr nonlinearity in the nonlinear Schrödinger equation $$\frac{\partial a}{\partial t} - i\frac{\partial^2 a}{\...
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Applications of optical rogue waves
It has been recently (2014) discovered that rogue waves arise not only in the context of deep sea waves, but also in that of fiber optics. To be precise, consider a single-mode fiber, which its slowly ...
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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 ...
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Solving Higher-Order Kinetic Energy Term (Gross-Pitaevskii equation) [closed]
Consider now propagation of non-linear waves in one-dimensional chain of dimers governed by the non-linear Schrödinger equation for the normalized wave envelope $\Psi(x,t)$,
$$
i \frac{\partial \Psi}{...
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Energy density in Gross-Pitaevskii equation
I guess this is a straightforward question but I was wondering if I can get an explicit steps toward the answer.
Using the Gross-Pitaevskii equation:
$$ \tag{1} i \hbar\frac{\partial\psi\left(x,t\...
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Mode distribution and Phase constant for nonlinear optical fiber
While going through the derivation of Nonlinear Schrodingers equation from the Helmholtz equation I came across the following two pdes,
$\nabla_\perp^2 F + \left[ \epsilon\left(\omega\right) k_0^2 -
\...
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Solution to periodic potential GP equation
Consider the boundary condition $\psi(0)=\psi(2\pi)$ and the Hamiltonian (and corresponding nonlinear Schrödinger equation):
$$\left[\left(-i\frac{\partial}{\partial\theta}-\Omega\right)^2+2\pi\gamma|\...
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How to define quantum chaos?
I was told that quantum chaos is just a system whose Hamiltonian's classical version shows chaotic behavior. However, I just wondering
what happens when one eigenstate of this Hamiltonian evolves?
...
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Nonlinear Saturated Schrodinger Equation in 1D- Physical Models
I'm studying the Nonlinear 1d Schrodinger equation
$$i\psi _t + \psi '' + |\psi |^{2p} \psi - \epsilon |\psi | ^{2q} \psi = 0\, , \quad t>0, x\in \mathbb{R}\, ,$$
and specifically, its solitary ...
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Purpose of the multidimensional NLSE/GNLSE
I know the purpose of the NLSE (Evolution of a complex field envelope in a nonlinear dispersive medium). Usually I am solving the 1d-GNLSE when simulating the propagation of a light pulse through a ...
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The Nonlinear Schrodinger Equation (NLSE) [closed]
I am trying to show that the NLSE:
$$\frac{\partial A(z,T)}{\partial z} = -i \frac{\beta_2}{2} \frac{\partial^2A}{\partial T^2} + i \gamma |A|^2 A$$
may be cast in the form:
$$\frac{\partial U(z,\...
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Nonlinear Schrodinger Equation perturbation stability
Consider the nonlinear Schrödinger equation
$i\frac{\partial A}{\partial z} -\frac{\beta_2}{2}\frac{\partial^2 A}{\partial T^2}+\gamma|A|^2A=0$
This has steady state solution $A(z,T)=\sqrt{P_0}\exp\...
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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}...
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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 ...
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