Visualize soliton solutions of a PDE In trying to visualize soliton solutions of a PDE I faced this sentence:
We now think of solitons as self-similar solutions, i.e., solutions which evolve along
symmetries of the flow. 
Question 1: Which solutions are called "self-similar solutions"?
Question 2: What is the meaning of "symmetries of the flow"? and How can one find them in PDEs?
Thanks in advance.
 A: Geometrically self-similarity means some sort of equivariance with respect to a conformal symmetry. 
Let $M,N$ be smooth manifolds. Let $\mathfrak{G}$ be a Lie group, and let $\Phi: \mathfrak{G} \to \mathrm{Diff}(M)$ and $\Psi: \mathfrak{G} \to \mathrm{Diff}(N)$ be homomorphisms. In other words, $\Phi$ and $\Psi$ are actions of the Lie group $\mathfrak{G}$ on $M,N$ respectively. We say that a mapping $f:M\to N$ is $\mathfrak{G}$ equivariant (relative to the actions $\Phi$ and $\Psi$) if
$$ f\circ \Phi_g = \Psi_g \circ f $$
for every $g \in \mathfrak{G}$. 
The interpretation of this is that a Lie group action on $M$ can be construed as a "symmetry". So equivariance says that the mapping "commutes with the symmetry" in an appropriate way. 
Now, given $M,N$ the smooth manifolds, and we consider solutions to a partial differential equation (in the sense which I described here) for $f:M\to N$. That is to say, we consider the product manifold $F = M\times N$ as a fibered manifold over $M$ with the natural projection. A PDE of order $r$ is a positive codimension submanifold $\mathcal{E}$ of the associated $r$-jet bundle $F^{(r)}$. A solution is a holonomic section of the $F^{(r)}$ that sits inside $\mathcal{E}$. The Lie group actions $(\Phi, \Psi^{-1})$ acts on $F$ and induces an action on $F^{(r)}$. We say that the actions $(\Phi, \Psi^{-1})$ is a symmetry of the equation $\mathcal{E}$ if the induced group action preserves $\mathcal{E}$ in $F^{(r)}$. 
The upside to this is that if $f$ is a solution to this PDE, then so is $\Psi^{-1}_g \circ f \circ \Phi_g$. We say that $f$ is an equivariant solution if $\Psi^{-1}_g \circ f \circ \Phi_g = f$ for every $g\in \mathfrak{G}$.

If you don't like the geometrical language above, think of it as the following. Consider a PDE for functions $f:M\to N$. A symmetry of this PDE is a parametrised family of transformations $\Phi_s\times\Psi_s:M\times N \to M\times N$ such that whenever $f$ is a solution to the PDE, so is $\Psi^{-1}_s \circ f \circ \Phi_s$. 
A solution that is equivariant under this symmetry is one that satisfies 
$$ f = \Psi^{-1}_s \circ f \circ \Phi_s $$
Example: Let $P$ be a partial differential operator on $\mathbb{R}^n$. Study the PDE $\partial_t u = Pu$ for $u:\mathbb{R}\times\mathbb{R}^n \to \mathbb{R}$. Since $P$'s coefficients are independent of $t$, we have that time translation is a symmetry. More precisely, our Lie group is $\mathfrak{G} = \mathbb{R}$. The action on the domain is $\Phi_s: (t, \vec{x}) \mapsto (t + s, \vec{x})$. The action on the co-domain is the trivial one: $\Psi_s = \mathrm{Id}$. We see that if $u(t,\vec{x})$ solves the PDE, so does $u(t+s,\vec{x})$. So the action $(\Phi_s, \Psi_s^{-1})$ is a symmetry of the equation. An equivariant solution is one for which $u\circ \Phi_s = u$, or in other words, one that is time-independent. 
Example: Let $P$ be a homogeneous first order partial differential operator on $\mathbb{R}^n$. We consider solutions to $Pu = u^2$ on $u: \mathbb{R}^n \to \mathbb{R}$. We let the Lie group be $(\mathbb{R}^+, \times)$  such that it acts on the domain
$$ \Phi_s(\vec{x}) = s\vec{x} $$
and codomain
$$ \Psi_s(y) = s^{-1} y $$
We verify that
$$ P(\Psi_s^{-1}\circ u \circ \Phi_s) = s^2 (Pu)\circ \Phi_s = s^2 u^2\circ \Phi_s = (\Psi_s^{-1}\circ u\circ \Phi_s)^2 $$
so that this represents a symmetry of the equation. A equivariant solution relative to this symmetry would be a function $u$ such that $s u(s x) = u(x)$ which as we see requires either $u = 0$ everywhere or that $u$ must blow-up as $x \to 0$. 

Now that we have gotten through equivariance, we can talk about self-similarity. To talk about self-similarity we need to have a way of measuring "scale" on the domain, and generally this is implemented by letting $M$ be a (pseudo/semi)Riemannian manifold. 
More precisely, let us be in a situation exactly as above. Now let $M$ be a (pseudo/semi)Riemannian manifold with (pseudo/semi)Riemannian metric $g$. Then if in additional to all of the above, we have that the Lie group action $\Phi$ acting on $M$ gives rise to conformal isometries (which are not isometries, just to rule out the trivial case) of the metric, then we say the symmetry is a scaling symmetry. A solution that is equivariant under a scaling symmetry is said to be self-similar. 
For the case of the Ricci flow, formally the domain manifold is $\mathbb{R}_+\times M$, and on it we put the degenerate metric that equal to the Riemannian metric on $M_t$ for the constant time slices and degenerates in the $t$ direction. Then a self-similar solution can be interpreted largely in the manner described above. (The small technical differences include that parabolic equations are only wellposed one way in time, and so some of the above need to be adapted for semigroups instead of groups.)
