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11 views

Solution to periodic potential GP equation

Boundary condition: $\psi(0)=\psi(2\pi)$, and Hamiltonian (and corresponding nonlinear Schrödinger equation): $$\left[\left(-i\frac{\partial}{\partial\theta}-\Omega\right)^2+2\pi\gamma|\psi(\theta)|^2\...
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26 views

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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27 views

Nonlinear Saturated Schrodinger Equation in 1D- Physical Models

I'm studying the Nonlinear 1d Schrodinger equation $$i\psi _t + \psi '' + |\psi |^{2p} \psi - \epsilon |\psi | ^{2q} \psi = 0\, , \quad t>0, x\in \mathbb{R}\, q>p \, ,$$ and specifically, its ...
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7 views

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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0answers
54 views

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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35 views

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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1answer
175 views

Deducing instability growth rates from the Hamiltonian for the nonlinear schrodinger equation

Consider the following nonlinear Schrodinger equation (NLSE): $$A_t+iA_{xx}+i|A|^2A = 0$$ where $A$ is a complex valued function of $(x,t)$. A solution to this equation is $A=a_oe^{-ia_o^2t}$. We ...
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933 views

Diifference between real time propagation and imaginary time propagation?

Suppose I want to solve Nonlinear Schrodinger equation using imaginary time propagation to get the ground state solution. I choose $t = - i t$, and then solve the equation using split step Crank ...