What is meant by integrable equation/system?

Question

In mathematical physics one frequently encounters the idea of an integrable equation/system. E.g., people talk about integrability of Schrodinger's Equation and KdV equation. To be frank, I can't understand the mumbo-jumbo written in Wikipedia article on Integrable Systems. Can anyone explain in a simpler manner? And what is exactly a scattering transform?

Answer 1

We refer to a system to be integrable if the number of symmetries $N_s$ and the number of degrees of freedoms $N_\text{dof}$ obey the relation

$N_s \ge N_\text{dof}$

For example the two bodies in three dimensional space have 2 times 3 degrees of freedom. So $N_\text{dof} = 6$ And we have the following symmetries

• three translations
• three rotations
• energy conservation

so we have $N_s = 7$ and therefore an integrable system.

While the famous three body problem has $9$ degrees of freedom but also only $7$ symmetries it is not integrable (and that is also part of the difficulty of the three body problem)

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Edit example how to integrate a system.

the harmonic Oscilator $\ddot x = - k x$ Has one variable $x$ so one degree of freedom, it preserves the energy $E = \dot x^2 + k x^2 $ this what we would call a symmetry (it is time translation symmetry by the way (if you are interested read about Noether's theorem)).

Now if we want to integrate that system we can start by computing the Energy of that initial state. Since we know that this energy does not change we can rearange the equation for the energy $\sqrt{E - k x^2} = \dot x $

and then

$\sqrt{E - k x^2}dt = dx $

$dt = \frac{dx}{\sqrt{E - k x^2}} $

and now all we need to do is integrate that relation to obtain $t(x)$ inverting that function gives us $x(t)$ and we have solved for the motion.