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I have read about the existence of a non-linear scrhödinger equation. What is its utility and application? And how can it be derived? Is it in a relativistic or non-relativistic context?

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Wikipedia article didn't help? –  Marek Jan 29 '11 at 11:38
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Dear Boy, in this case, and many others, Wikipedia (and others) says that the utility and applications don't exist because they don't exist. This is a purely mathematical variation of Schrödinger's equation that doesn't describe any quantum systems because it violates a basic postulate of quantum mechanics, the linearity of operators (including the Hamiltonian that produces evolution). Some people have tried to write down nonlinear Schrödinger equations to "explain" the measurement or the "collapse" of the wave functions but none of these papers makes any sense. –  Luboš Motl Jan 29 '11 at 12:07
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I want to say that Wikipedia clearly answers your question, and, deducing from your comment above, you are even aware of the answer that it gives you - but you seem to dislike the answer, right? The answer is correct, however. Such equations can't describe any physics at the fundamental level. –  Luboš Motl Jan 29 '11 at 12:08
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More misinformation and propaganda in these comments I'm afraid. The NLSE is a widely used tool in theoretical physics, independent of its application to questions of wavefunction collapse. This question is valid and relevant. +1 –  user346 Jan 29 '11 at 13:08
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I would be interested in reading one of these propose about collapsing the wave-function by adding a non-linear term, could someone provide me a reference? –  toot Jun 28 '12 at 10:02
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Broadly speaking nonlinear Schrödinger equations are used to describe wave propagation in nonlinear media. There are many examples, like deep-water waves (where the linear shallow water approximation to the Navier-Stokes equations is not valid), or nonlinear optics.

There is for example an effect called "Kerr effect" in optical fibers where the refractive index depends on the intensity of the optical pulse. You can find a detailed derivation of a nonlinear Schrödinger equation describing electromagnetic waves in such a medium in the book

  • M. J. Ablowitz, B. Prinari, A. D. Trubatch: "Discrete and Continuous Nonlinear Schrödinger Systems", Cambridge University Press, 2004

Edit: Fluids/Hydrodynamics: It is not easy to find a reference for the use of Schrödinger equations in Hydrodynamics, maybe because most authors write for a target audience that is not fluent in quantum mechanics, I don't know. But here is one:

  • Andrew Majda and Andrea Bertozzi: "Vorticity and Incompressible Flow"

chapter 7.1, "Simplified Asymptotic Equations for Slender Vortex Filaments: The Self-Induced Approximation, Hasimoto's Transform and the Nonlinear Schrödinger Equation".

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Thank you! Your reference is what i needed. And do you know a similar reference about detailed derivation in fluids? –  Boy Simone Jan 29 '11 at 12:48
    
@Boy Simone: I tried to answer your question by adding it to my primary answer instead of a comment, see above :-) –  Tim van Beek Jan 29 '11 at 15:38
    
I didn't understand a word of what you said but +1 for a superb answer. –  Trufa Jan 29 '11 at 16:56
    
an important practical application in non-linear fiber optics is soliton-waves. They are derived using nonlinear Schrödinger equations and are often used in telecommunications. –  mirk Aug 15 '12 at 12:46
    
The equation is also quite common in laser plasma interaction problems (self focusing). For a derivation you can read:ijens.org/Vol%2011%20I%2002/118802-3030%20IJBAS-IJENS.pdf –  Shaktyai Aug 22 '12 at 21:23
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You can get some more information about the derivation from the related Wikipedia article about the Gross-Pitaevskii equation which is used in the context of Bose-Einstein condensation. There, one starts with a quantum field theory whose Hamiltonian is nonlinear in the fields, and replaces the field operator by a classical function that describes the condensate. So the reason why the same equation appears in so many different systems is that they are all described by an effective field theory with a nonlinear (fourth-order) interaction term. (The form of this effective theory is dictated by symmetry and the relevant degrees of freedom at low energy.) In the semiclassical approximation, this effective field theory reduces to the nonlinear "Schroedinger" equation.

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thanks for your answer! A thing: can you specify what do you mean with "The form of this effective theory is dictated by symmetry and the relevant degrees of freedom at low energy"? Or can you do an example?In your Gross-Pitaevski equation I have seen that we introduce the delta potential of interaction between bosones. And in Non Linear Schrodingher Equation what is the origin of the "adjunctive terms in the potential"? Or is an effort without specific physical origin to simply see the effects of a term of that type? –  Boy Simone Jan 29 '11 at 12:43
    
Dear Tomáši, that's great but if the function entering the equation describes a classically measurable condensate, then it is not a wave function and the equation is not a Schrödinger equation in the physical sense, is it? Moreover, the form of the field equations for this object won't mimic the non-relativistic Schrödinger equation - it will follow the classical field theories that are typically second-order in time etc. –  Luboš Motl Jan 29 '11 at 12:59
    
@Lubos, if you are going to address people in comments, please append their names with an ampersand so that they are notified and have a chance to respond appropriately. –  user346 Jan 29 '11 at 13:12
    
@space_cadet: you mean prepend the name with an at-sign? :) –  Marek Jan 29 '11 at 16:11
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@Luboš Motl: I have never said that the condensate function is a wave function of an actual quantum system, have I? Please notice that I put the "Schrödinger" in quotation marks. Personally I think that using the term nonlinear Schrödinger equation just leads to confusion as is clear from your comments; the question is not whether the variable in that equation is a quantum wave function, I guess nobody sane claims that. As to the form of the field equations, you can get a nonrelativistic kinetic term (one time derivative) even in a fully relativistic context due to the dense medium - BEC. –  Tomáš Brauner Jan 29 '11 at 21:46
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The nonlinear Schrödinger equation (NLSE) in one dimension is $$ i\frac{\partial\psi}{\partial t}~+~\frac{\partial^2\psi}{\partial x^2}~+~f(|\psi|)\psi~=~0. $$ For $f(|\psi|)~=~|psi|^2$ this is the cubic Schrödinger equation. This is related to a couple of relativistic equations, such as the quartic equation for the Higgs field. The Dirac equation $i\gamma^\mu\partial_\mu\psi~+~({\bar\psi}\gamma^\mu \psi)\gamma_\mu\psi$ is the Thirring model of fermion condensates, related to BCS theory. This is a related nonlinear wave equation. In general the NLSE describes the motion of quantum waves through nonlinear media or in some process where the locality of standard Lagrangians (quadratic etc) is replaced with some nonlocal field.

A separable solution to the NLSE $\psi(x,t)~=~u(x)v(t)$ may be found with $v(t)~=~exp(-i\omega t)$ and the spatial function is given by a DE $u_{xx}~+~\omega u$ $f(|u|)u~=~0$defined as $$ \int\frac{du}{\sqrt{Au^2~-~2F(u)~+~B}}~=~C~\pm~x,~F(u)~=~\int uf(|u|)du $$ which is a traveling wave solution.

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Sorry, but what do you mean with "nonlocal field" ? –  Boy Simone Jan 29 '11 at 14:15
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I should have said theory or Lagrangian. A nonpolynomial potential or $f(\psi)$ or some function which is not defined at a point can be a nonlocal field theory. –  Lawrence B. Crowell Jan 29 '11 at 15:02
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I've seen this used in the context of quantum simulations of bosonic systems. If you have a system of many identical bosons all in the same state, you can treat the squared magnitude of the single particle wave function as a density for all of the particles. Then a single particle interacting with this density becomes a good model for a system of interacting Bosons. See for example Yepez, Phys. Rev. Lett. 103, 084501 (2009).

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