Questions tagged [non-linear-schroedinger]

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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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Physical meaning that an energy functional has no minimizer

It is well known that the Hamiltonian of a system might not have a minimizer, even the Hamiltonian is bounded below. For example, let us consider the cubic time independent Schrödinger equation \begin{...
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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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1answer
444 views

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 ...