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Preamble: I am a mathematician and not a physicist.

From what little I know about quantum mechanics, Schrödinger's equation is a linear PDE that describes the time-evolution of a system. In general its solution takes values in some infinite-dimensional Hilbert space.

In the case of a system with only finitely many "classical states" (sorry about the probably incorrect terminology), e.g. a particle which is either "spin up" or "spin down", we get a linear ODE taking values in a finite-dimensional Hilbert space. This is from my perspective a bit of a boring equation though...

Question: is there a physical situation whose time evolution is naturally modelled by a non-linear ODE on a finite dimensional Hilbert space? (ex: finitely many atoms which can be either in fundamental or excited state, and for some reason a non-linear term in Schrödinger's equation)? If so, could you please give me a quick description of such a system, its evolution equation, and some references?

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  • $\begingroup$ This post (v2) seems like a list question. $\endgroup$ – Qmechanic Oct 11 '16 at 15:16
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    $\begingroup$ @glougloubarbaki For a brief review of some relevant QM context, complementary to the topics already pointed out in the answers, see lanl.arxiv.org/pdf/quant-ph/0505046v1. $\endgroup$ – udrv Oct 11 '16 at 20:58
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The conventional formalism of QM relies heavily on the theory of linear operators (spectral theorem,...), which would be hard to justify unless the linear structure on the Hilbert space is physically unambiguous, and in particular preserved under the time-evolution (see however udrv's comment below on a non-linear, yet consistent, quantum evolution).

While the Lagrangians used for interacting field theories (eg. the standard model) do lead to non-linear PDEs for the "wave-function", these equations are pathological in the context of QM (in particular, they do not support a healthy probabilistic interpretation, although this is not solely due to their non-linearity), and one has to go to QFT: roughly, quantizing a second time the wave-function allows to recover a linear system that supports the usual quantum interpretation.

This "linearization" occurring during quantization is similar to the one occurring when going from classical mechanics to classical statistical physics. Take some phase space (symplectic manifold) $\mathcal{M}$ with a possibly non-linear time evolution $U(t) : \mathcal{M} \rightarrow \mathcal{M}$. If I look at probability distributions over $\mathcal{M}$, I have the time evolution: $$ \big[ \hat{U}(t) \rho \big](x) := \rho \big( U(t)^{-1} x \big) $$ which is linear (in the space of measurable functions over $\mathcal{M}$ of which probability densities, aka. positive, normalized functions, form a subset). Basically, we are trading a non-linear system for a linear one on a much larger space. I think the resulting linear system can capture a lot of information on the original non-linear one (for example, Gelfand duality is a way to reconstruct an underlying topological space from its algebra of continuous functions).

So, if you are looking for interesting non-linear equations, you will probably find them before quantization, not after (or, depending on your point of view, after semi-classical limit, not before). On finite-dimensional spaces, I can only think of non-linear mechanical systems, like the anharmonic oscillator...

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  • $\begingroup$ thanks for your answer! but I'm especially interested in systems taking place in a complex phase space, with analytic time dependance. as far as I know this doesn't really fit in the hamiltonian or symplectic framework. it does for SE, but then as you said it is linear. I was hoping for a situation where one would have a homogeneous equation that would descend to a projectivized space, but still not be linear. however it very possible that this doesn't really occur in QM $\endgroup$ – Alfred Oct 11 '16 at 17:28
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    $\begingroup$ @glougloubarbaki: Not sure I fully understand what you are looking for... Would a time-evolution on a Kähler manifold (a manifold modeled on a complex Hilbert space = a kind of "non-linear Hilbert space") go in the right direction? Those arise quite naturally in the symplectic context... $\endgroup$ – Luzanne Oct 11 '16 at 17:49
  • $\begingroup$ well I guess that just goes to show I don't know too much about physics... yes, this is exactly the kind of things I am looking for. could you please tell me a bit more about where the complex structure comes in the story from the physics point of view? I knew that classical mechanics can be phrased as an ODE on a symplectic manifold but I didn't know about the additional Kähler structure $\endgroup$ – Alfred Oct 11 '16 at 18:21
  • $\begingroup$ @glougloubarbaki: Unfortunately, I'm far from being a specialist of Kähler manifolds... One way I know them to arise in (mathematical) physics is geometric quantization, specifically holomorphic quantization: this is a systematic way to get a QM theory from a classical one, but it requires 1 additional ingredient on the classical side, namely a complex structure turning the symplectic manifold into a Kähler manifold. $\endgroup$ – Luzanne Oct 11 '16 at 18:44
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    $\begingroup$ "a transformation on a Hilbert space that preserves the norm should at least be real-linear (Mazur–Ulam theorem)": No, isometry ≠ norm conservation, however enticing the Mazur-Ulam theorem may be. There are plenty of well defined nonlinear Schroedinger eqs. that have no problem preserving probability. One of the simplest examples, due to Gisin: $${\dot\psi} = \sigma\left[ A - \frac{\langle\psi|A|\psi\rangle}{\langle\psi|\psi\rangle}\right]|\psi\rangle -\frac{i}{\hbar}H|\psi\rangle$$ where $H$ is linear and hermitian, but $A$ is arbitrary. $\endgroup$ – udrv Oct 11 '16 at 21:55
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If you Google time dependent Hartree Fock, you should get something of interest. In QM the Hartree Fock equations are nonlinear PDE that approx ate many body solutions to the SE. They are nonlinear because the effective potentials also depend upon the wave functions. The TDHF equations satisfy your requirements.

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I don't know of any such equations on a finite-dimensional Hilbert space, but two non-linear variants of the Schrödinger equation that come up naturally in physics are the non-linear Schrödinger equation and the Gross-Pitaevskii equation. Also, the field of quantum chaos studies how the linear Schrödinger equation can lead to non-linear dynamics in the classical or semi-classical limit.

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