I want to preface this question by saying that I am not a mathematician, I am a chemist. (My mathematical knowledge is pretty good on DEs and basic analysis, but I have almost no algebra or topology skills) I am taking a class in advanced methods in differential equations, and we have been working with the non-linear Schrodinger and Korteweg-De Vries equations. Some of the solutions to these equations form solitons, which I thought I had a pretty good handle on.
However, I have read some of the literature that uses the NLS equation in discussions of N-body nucleus problems and describes the soliton solutions as non-topological solitons [1]. However, when reading about topological solitons, they are described on Wiki as the solutions to some weakly non-linear PDEs (e.g., the NLS equation) [2].
Are the solutions to the NLS equation topological or non-topological? More fundamentally, what is the difference between topological and non-topological singularities. Thanks in advance.
References [1] https://iopscience.iop.org/article/10.1088/1367-2630/ab0e58/pdf [2] https://en.wikipedia.org/wiki/Topological_defect
An example of a topological soliton is the Sine-Gordon Equation. Notice that the solutions of this equation looks like a type of coil that wraps around an axis some number of times. Solutions are actually classified into equivalence classes based on the number of times (the winding number) they wrap around an axis. This solution can be described using an angle $\theta\in(0,2\pi] = S^1$ about the axis and a time variable $t\in\mathbb{R}$. The phase space (set of all configurations of solutions) is geometrically a cylinder: $S^1\times\mathbb{R}$. This phase space contains 1 hole.
Topology is the general study of smooth deformations of shapes. This requires functions to smoothly deform one shape into another. If such a function exists (homotopy functions), the two shapes are said to be equivalent. This is how equivalence classes of solutions are created.
In contrast, solutions of NLS and KDV are non-topological solutions. Why? Because the domain they're defined on (x-y-t plane or just x-t) doesn't have the same freedom in geometry (there are no holes in the domain or phase space). Phase space is described by position and velocity of number of solitons described. Geometrically equivalent to $\mathbb{R}^{2N}$ for $N$ solitons. Even though this is a space, it has $0$ holes.
Solitons (or regular solitons) can be one or two dimensional and are local perturbations of the domain that travel in time (Galilean invariant). Singular solitons blow up locally in space. Consider the example of a black hole (classified as a singular soliton solution - of the density profile blows up locally in space)(see this physics discussion on black holes)
To take it one step further, this raises the question of how to tell whether a nonlinear pde (specifically, an integrable system) has a topological soliton or not. To my knowledge, the key is to look at the dispersion relation, ($w(k)$), from these equations. If the dispersion relation (determined simply by the linear derivative terms) is polynomial (NLS: $w \propto k^2$, KDV: $w\propto k^3$), it's soliton solutions are non-topological. If the dispersion relation is rational (SG: $w\propto \frac{1}{k}$), the solitons will be topological. I even bet that the poles of the rational dispersion relation (forming a meromorphic function, or even Riemann surfaces) allow us to design crazy complicated topological solitons.