$\newcommand{\erf}{\operatorname{erf}}$
Like running-wave solutions self-similar solutions is an another class of interesting solutions. For linear equations these solutions can lead to the general solutions. For non-linear equations it is not the case.
Example 1. Self-similar solution to heat equation
\begin{gather}
u_t =u_{xx}
\label{eq-3.B.1}
\end{gather}
We plug
$u_\lambda (x,t)=\lambda^\beta u(\lambda x, \lambda^\alpha t)$ into the same equation and find that it satisfies the same equation as $\alpha=2$. Here $\beta $ can be any and in Section 3.1 we got $\beta=1$ only due to additional conditions $u(x,t)\ge 0$ and
\begin{gather}
\int_{-\infty}^\infty u(x,t)\,dx <\infty.
\label{eq-3.B.2}
\end{gather}
Then we plug $\lambda =t^{-\frac{1}{2}}$ and get
\begin{gather}
u(x,t)=t^{-\frac{1}{2}} u(xt^{-\frac{1}{2}},1) =
t^{-\frac{1}{2}}\phi (xt^{-\frac{1}{2}})
\label{eq-3.B.3}
\end{gather} and this expression we plugged into (\ref{eq-3.B.1}) to get ODE for $\phi$ and solve it.
Example 2. Self-similar solution to linearized Korteweg–De Vries equation \begin{gather} u_t =u_{xxx}. \label{eq-3.B.4} \end{gather} We do the same thing albeit now $\alpha=3$ and to get $\beta=1$ we need to appeal to (\ref{eq-3.B.2}) or something similar. Then \begin{gather*} -\frac{1}{3}(\xi \phi)'= \phi'''\implies -\frac{1}{3}\xi \phi = \phi'' \end{gather*} where again we neglect a constant. This is an Airy equation (which appears in the theory of diffraction) and its solutions are Airy functions.
Example 3. Let us try \begin{gather*} u_t =u_x + u_{xxx}. \end{gather*} There are 3 terms and they scale differently and we cannot select $\alpha$ (in the linear equations $\beta$ is arbitrary).
Example 4. Self-similar solution to viscous Burgers equation \begin{gather} u_t + uu_{x}=u_{xx}. \label{eq-3.B.5} \end{gather} Again three terms scale differently but now non-linearity helps: \begin{align*} & u_{\lambda\,t} (x,t)= \lambda^\beta u (\lambda x, \lambda^\alpha t),\\ & u_{\lambda\,xx} (x,t)= \lambda^{\beta+2\alpha} u (\lambda x, \lambda^\alpha t),\\ &u_\lambda u_{\lambda\,x} =\lambda^{2\beta+1} (uu_x)(\lambda x, \lambda^\alpha t) \end{align*} and we need to have all three powers equal, which leads to $\alpha=2$ and $\beta=1$ without reference to (\ref{eq-3.B.2}). It leads to (\ref{eq-3.B.3}) and plugging into (\ref{eq-3.B.5}) we get \begin{gather} -\frac{1}{2} (\xi \phi)' + \phi\phi' = \phi''. \label{eq-3.B.6} \end{gather} Integrating we get \begin{gather} -\frac{1}{2} \xi \phi + \frac{1}{2} \phi^2 = \phi' \label{eq-3.B.7} \end{gather} where we took a constant $0$ because we still want solutions to decay at infinity.
This is Bernoulli's equation. Rewriting it as \begin{multline*} (\phi e^{\xi^2/4} )' = \frac{1}{2} \phi ^2 e^{\xi^2/4} \implies (\phi e^{\xi^2/4} )^{-2}(\phi e^{\xi^2/4} )'= e^{-\xi^2/4}\implies\\ (\phi e^{\xi^2/4} )^{-1} = \sqrt{\pi}\Bigl( \erf (\xi /2) +D\Bigr)\implies \end{multline*} and finally \begin{gather*} \phi(\xi) = \frac{e^{-\xi^2/4}}{\sqrt{\pi} \Bigl( \erf (\xi /2) +D\Bigr)} \end{gather*} with $|D|>1$ to prevent vanishing determinator. And we get expression for $u(x,t)$.
Example 5. Self-similar solution to Korteweg-De Wries equation \begin{gather} u_t +uu_x =u_{xxx}. \label{eq-3.B.8} \end{gather} Repeating arguments of Example 4 we arrive to \begin{gather} u(x,t)= t^{-\frac{1}{3}}\phi (xt^{-\frac{1}{3}}). \label{eq-3.B.9} \end{gather} Then \begin{gather*} -\frac{1}{3} (\xi \phi )'+\phi '\phi = \phi'''\implies -\frac{1}{3}\xi \phi +\frac{1}{2}\phi ^2 =\phi'' \end{gather*} where we again negect a constant. This is a non-linear second order equation and we cannot solve it explicitly (but can investigate its properties).