What does it mean for a differential equation "to be integrable"?
Are "integrable" and "solvable" synonyms?
The first thing that comes to my mind is to say: it's integrable if we can find the analytic solution (a function that can be written as y = f(x1, x2, ...)) that when plugged into the differential equation satisfies it.
Sometimes we cannot find such a function in closed form, so we use numerical methods to solve the differential equation. Is the equation still defined "integrable" in this case? If yes, when a differential equation is not integrable?
In the case of the Nonlinear Schrodinger equation, which is a nonlinear partial differential equation:
- we say that the equation is "integrable" because we can solve it using the inverse scattering method, but also this method can provide closed form solutions only in particular cases.
- we can also solve it using finite difference numerical methods. Because of this could we have said that the equation was "integrable" even before discovering the inverse scattering method?
Since apparently the definition of "integrability" is not straigthforward, my main concern is to understand if the inverse scattering technique to solve the equations like the Nonlinear Schrodinger equation or the KdV equation is equivalent to a Finite Difference technique to say that we "solved" the euqation.
Moreover, "solving a differential equation" is equivalent to say "solve the initial value problem"? I guess that when numerical methods are applied (so that we cannot get an analytic solution), we are actually solving the IVP, is that correct?