Integrability of Systems of Differential Equations

[The Tortoise-Man]

I' m reading wiki article about Solitons and have some some troubles to understand the meaning of the following:

A topological soliton, also called a topological defect, is any solution of a set of partial differential equations that is stable against decay to the "trivial solution". Soliton stability is due to topological constraints, rather than integrability of the field equations. The constraints arise almost always because the differential equations must obey a set of boundary conditions [ ...]

Question: In context of systems of differential equations, what means precisely "integrability of the equations"?
Is there any good intuition how to think about it? Has it some direct connection with naive notion of the concept of an " intergral" in mathematics?

( here I conjecture that the terminology is pure mathematical than physical, so correct me please if that's the wrong subforum to pose this question)
On my search I found following promising concise interpretation here in Mathoverflow for it:

A mechanical system is called integrable if we can reduce its solution to a sequence of quadratures."

But then, what mean for a solution of the given system of DiffEqs to be given as "a sequence of quadratures"? This sounds very "integralic" but I not know what such solutions look like. Are they literally expressible in terms of certain integrals? How such a solution function realized as "sequence of quadratures" looks like? An example?

 

[Haborix]

Integrability is related to underlying conserved quantities (think energy in classical mechanics systems). Sometimes we start with conserved quantities and can derive the differential equations describing the behavior of the variables. Other times we have a set of (possibly coupled) differential equations. Integrability is about being able to write say $y'=(\text{stuff})'$, which we could then read off $y$. That would be solving by quadrature, i.e., direct integration. In some cases, $y=z'$ and so we look to see if we can find $z'=(\text{otherstuff})'$. I think this is what a sequence of quadratures refers to, peeling back the derivatives by integration.

As to your first set of questions. Any particular solution of a differential equation with conserved quantities will have a value of the conserved quantity associated with it. And so will your so-called trivial solution. But the value of the conserved quantity will be different for each solution, and since that quantity is conserved the particular solution cannot "decay" to the trivial solution (this is often put into the language of invariant sub-/manifolds). Solitons on the other hand are about boundary conditions, where the boundary conditions themselves foreclose the "decay" of the soliton into the trivial solution.

 

[The Tortoise-Man]

when you write $y'=(\text{stuff})'$, I assume that you mean that the right hand side not depends on $y$, right? Because, so far I understand the integrable system so 'easy' to solve, is that the goal is then in the next step to integrate independently(!) both sides and obtain $y= \int (\text{stuff})' d$, und then perform some pure algebraic manipulations in order to express the solution in terms of $y$, without trying all this hard business around decoupling the diff equations, that's the advantage of an integrable system, right?

 

[fresh_42]

Question: In context of systems of differential equations, what means precisely "integrability of the equations"?

I interpret this as "finding a solution to an initial value problem" which means ...

Is there any good intuition how to think about it?

... a path (flow) through the vector field described by the differential equations. Examples of stable solutions are attractors, repellers of unstable solutions. A singleton (point) $\{p\}$ can also be a stable solution.

Has it some direct connection with naive notion of the concept of an " intergral" in mathematics?

Only in so far as integration is usually the technique to solve differential equation systems.

Maybe you want to read:
https://www.physicsforums.com/insights/differential-equation-systems-and-nature/
where you can find examples.

On my search I found following promising concise interpretation here in Mathoverflow for it ...

As long as we do not have the book or see what Bühler meant by "quadratures", as long is it meaningless to discuss his quote referenced in MO. "Sequence of integrals" is the closest meaning we can read from the quotation. I was even a bit more cautious when I said "integration as the technique to find flows".

But then, what mean for a solution of the given system of DiffEqs to be given as "a sequence of quadratures"?

A sequence of (possibly numeric) integrations. Or as I would phrase it: "following a flow through the given vector field". Maybe - but this is a guess - the term "quadrature" stems from the Newton-Raphson algorithm to determine numerical solutions (approximations) to non-linear equations. This algorithm can be called quadratic.

This sounds very "integralic" but I not know what such solutions look like. Are they literally expressible in terms of certain integrals? How such a solution function realized as "sequence of quadratures" looks like? An example?

See the article I linked above. Here is another example (the linked section #6)
https://www.physicsforums.com/insig...ion-and-eulers-number/#The-Exponential-Ansatz

 

[Haborix]

I interpret this as "finding a solution to an initial value problem" which means ...

I do think more is meant by integrable in OPs case. Particularly, I think it means something more akin to what is discussed here: integrable systems.

 

[Haborix]

when you write $y'=(\text{stuff})'$, I assume that you mean that the right hand side not depends on $y$, right? Because, so far I understand the integrable system so 'easy' to solve, is that the goal is then in the next step to integrate independently(!) both sides and obtain $y= \int (\text{stuff})' d$, und then perform some pure algebraic manipulations in order to express the solution in terms of $y$, without trying all this hard business around decoupling the diff equations, that's the advantage of an integrable system, right?

Correct, though I'm wondering if that's necessary. Easy is in the eye of the beholder. Even when you can reduce the system to quadrature, the integral rarely can be written in terms of elementary functions (see the whole universe of special functions, e.g., elliptic integrals in classical pendulum problem). Without the decoupling, as you call it, you can't get to the point of writing the "easier" integral.