# Questions tagged [non-linear-schroedinger]

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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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### 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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### 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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While going through the derivation of Nonlinear Schrodingers equation from the Helmholtz equation I came across the following two pdes, $\nabla_\perp^2 F + \left[ \epsilon\left(\omega\right) k_0^2 - \... • 11 1 vote 0 answers 40 views ### Solution to periodic potential GP equation Consider the boundary condition$\psi(0)=\psi(2\pi)$and the Hamiltonian (and corresponding nonlinear Schrödinger equation): $$\left[\left(-i\frac{\partial}{\partial\theta}-\Omega\right)^2+2\pi\gamma|\... • 1,047 3 votes 1 answer 282 views ### How to define quantum chaos? I was told that quantum chaos is just a system whose Hamiltonian's classical version shows chaotic behavior. However, I just wondering what happens when one eigenstate of this Hamiltonian evolves? ... • 1,047 1 vote 0 answers 95 views ### Nonlinear Saturated Schrodinger Equation in 1D- Physical Models I'm studying the Nonlinear 1d Schrodinger equation$$i\psi _t + \psi '' + |\psi |^{2p} \psi - \epsilon |\psi | ^{2q} \psi = 0\, , \quad t>0, x\in \mathbb{R}\, ,$$and specifically, its solitary ... • 153 0 votes 1 answer 121 views ### Purpose of the multidimensional NLSE/GNLSE I know the purpose of the NLSE (Evolution of a complex field envelope in a nonlinear dispersive medium). Usually I am solving the 1d-GNLSE when simulating the propagation of a light pulse through a ... • 345 1 vote 0 answers 190 views ### The Nonlinear Schrodinger Equation (NLSE) [closed] I am trying to show that the NLSE:$$\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,\... • 113 1 vote 0 answers 95 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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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}...
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 ...