Topological defects in terms of homotopy classes of maps between spheres

Question

I have some troubles with understanding the concept of topological defects and their treatise via homotopy groups of spheres as discussed in wikipedia article. More precisely, if we encode these in terms of different (pointed) homotopy classes of $[S^n,S^m]_*= \pi_n(S^m,*)$ what from physical point of view plays the role of domain sphre $S^n$ and what of codomain sphere $S^m$?

In wikipedia article the "Overview" paragraph seemingly contradict to "Formal treatment" paragraph on what is actually the domain and what is codomain in this modelling:
The "overview" paragraph tells:

The general characteristic needed for a topological soliton to arise is that there should be some partial differential equation (PDE) - eg the Korteweg-De Vries (KdV) equation - having distinct classes of solutions, with each solution class belonging to a distinct homotopy class. In many cases, this arises because the base space -- 3D space, or 4D spacetime, can be thought of as having the topology of a sphere, obtained by one-point compactification: adding a point at infinity. This is reasonable, as one is generally interested in solutions that vanish at infinity, and so are single-valued at that point. The range (codomain) of the variables in the differential equation can also be viewed as living in some compact topological space. As a result, the mapping from space(time) to the variables in the PDE is describable as a mapping from a sphere to a (different) sphere; the classes of such mappings are given by the homotopy groups of spheres.

How I understood this part so far: We start with a PDE - lets take
eg the Korteweg-De Vries equation - describing the system's dynamics

$$ \partial_t \phi + \partial^3_x \phi - 6\, \phi\, \partial_x \phi =0\, \quad x \in \mathbb{R}, \; t \geq 0 $$

and want to inspect homotopy theoretically the structure of space of its solutions $\phi(x,t)$ where $t \in \Bbb R_{\le 0}$ is the time and $x \in \Bbb R$ resp $\Bbb R^3$ the spatial coordinate.

So let pick such a solution $\phi: \Bbb R_{\le 0} \times \Bbb R^3 \to \Bbb R $.

We assume that it vanishes at infinity as the text assumed, ie $\phi(t, x) \to 0$ for $\vert x \vert, t \to \infty$, so here comes this compactification part into game: We extend the domain $\Bbb R_{\le 0} \times \Bbb R^3$ of $\phi$ by adding "at infinity" the point "$\infty$" such that $\Bbb R_{\le 0} \times \Bbb R^3 \cup \{\infty\} \cong S^4$ and extend $\phi$ to $\infty$ via setting $\phi(\infty):=0$; that seems to be what they meant by "reasonable" assumption that $\phi$ vanish at infinity. Firstly, is that they procisely meant there by that the solution should vanish at infinite?
If yes, so we get $\phi: S^4 \to \Bbb R$.
First intermediate problem: Isn't this already null homolotopical? (Then to analyze it with homotopical methods would be meaningless)

Next they write, that "The range (codomain) of the variables in the differential equation can also be viewed as living in some compact topological space." What do they mean precisely by "range of the variables in the differential equation"? Just the image $\phi(S^4)$? But I doubt that this is what they mean, as $\phi$ constructed that way would be again homotopically to zero map as $\Bbb R$ is contractible.

Question: It seems that either I misunderstand what is written there or the quoted paragraph describes it wrongly as the way how I understood it would lead to conclusion that any such solution $\phi$ considered as map would be null homotopic, but that's not what we want. If the former is the case and I'm missing some issue, could somebody elaborate how correctly to understand the described construction turning certain solutions of the given PDE into homotopy classes of certain maps of spheres?

Even more obscure, what is the connection of this construction to the homotopy groups of type $\pi_i(R)= \pi_i(G/H)=[S^i,G/H]$ where $R$ is the pararameter space and $G$ acts on it with stabilizer $H$ described in "Formal treatment" paragraph? What has the parameter space $R$ to do with the solutions $\phi$ of the equation above?

The strange thing is that in "overview" paragraph once dealt with homotopy classes of maps $\phi:S^n \to S^m$ coming from solutions/configurations of the PDE dscribing the dynamic of the system, so $S^n$ (resp $S^m$) was just the domain (resp. codomain) of a solution function $\phi$. In "Formal treatment" paragraph on the other hand, the codomain is no longer the codomain of the solutions $\phi(x,t)$, but the parameter space $R=G/H$. So its "point" are cerain solutions of the PDE itself.
How to relate the descriptions from "Overview" and "Formal treatment"? Does one consider homotopy classes of solutions of the PDE, or the abstract parameter space?

NB: My problem is not the pure mathematics behind homotopy theory but only the physical part to "merge" things together discussed in wikipedia's "Overview" and "Formal treatment" paragraphs.

Answer 1

I am not overly mathematical myself. I will just give aworking example with the simplest possible topological object: vortex in two spatial dimension.

First of all, there should be a field (solution of PDE) that takes value in $S^{1}$. Think the direction of magnetization in the 2D plane, or the phase of a superfluid order parameter.

And then let us draw a loop in the 2D space, and look at the magnetization direction/order parameter phase as a function of position along the loop. This is your mapping from $S^{1}$ (spatial loop) to $S^{1}$ (solution). You can now talk about the homotopy class, which is just the winding number.

If the winding number is non-zero, you know that inside the loop there must be some singularity; it is thus called a "defect" because you have to remove at least one point in the plane to hide the singularity. Physically this is usually accomplished by the medium not being in the magnetic/superfluid phase in a small region. Or you can literally punch a hole.

Now, why does the field configuration take value in $S^{1}$ in the first place? More often than not it is the result of a spontaneous symmetry breaking. For instance, any quantum theory should be invariant under a global U(1) phase rotation. The ground state is often just invariant as well. But the superfluid order parameter spontaneously breaks this symmetry. U(1) rotation is a symmetry of the entire theory in the sense that, if you U(1)-rotate any superfluid state, you get another distinct but degenerate state. Individually, not even (any one of) the ground state is invariant.

In this simple case, $G = U(1)$ and there is no remaining "unbroken" symmetry, so $H$ is trivial. Then we know that the degenerate solutions can be labelled by $G/H = U(1) = S^{1}$ (abusing the equal sign).

(In fact, if that $S^{1}$ is not somehow symmetry-protected, we can't talk about a definite value in $S^{1}$ in the first place. So I guess symmetry breaking is the mechanism.)

You can now generalize to e.g. magnetization in 3D wrapping around a 2-sphere. A magnetization vector is invariant under axial rotation, so $H = O(2)$, and we assume the base theory is $O(3)$-invariant, fully isotropic in 3D. So the order parameter space is $O(3)/O(2) = S^{2}$. Non-trivial wrapping give a skyrmion. And inside that spatial 2-sphere again you know there must be a defect.

--edit--

Honestly the wiki article is very badly written and I can't decipher half the mumble-jumble. You should read some actual textbooks to get a better idea.

Concrete example of superfluid vortex in 2D:

The simplest form of superfluid can be described by a complex scalar order parameter field $\psi$: $$\psi: \mathbb{R}^2 \rightarrow \mathbb{C}$$

The problem of finding static solution of the full theory corresponds to the minimization of the free energy functional $$ F[\psi] = \int \! d^2 x \; (\nabla\psi^{*})\cdot(\nabla\psi) + \frac{\lambda}{2} (\vert \psi\vert^2 - \mu)^2 $$ $\delta F/\delta\psi = 0$ is a PDE.

The free energy itself is U(1)-invariant under $\psi \rightarrow e^{i\theta} \psi$, but obviously any $\psi \neq 0$ solution is not. At each point where $\psi$ is non-vanishing, it's local value can be characterized by $U(1) = S^1$.

There will be no distinct homotopy class whatsoever if $\psi \neq 0$ everywhere. Set $\psi = 0$ at one point $\vec{r}$, and only now you get non-trivial homotopy when considering $$ \psi/\vert\psi\vert : \mathbb{R}^2/\lbrace\vec{r}\rbrace \rightarrow S^1. $$ And finally you can shrink $\mathbb{R}^2/\lbrace\vec{r}\rbrace$ to just an $S^1$ loop as far as homotopy is concerned.

Alternatively, punch a hole $\mathbb{R}^2$ from the very beginning. Bottomline is, you always need a defect for there to be non-trivial honotopy.