$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$
We know that string equation \begin{gather*} u_{tt}-c^2u_{xx}=0 \end{gather*} has running wave solutions \begin{gather} u(x,t)= f(x-ct) \label{2.1B.1} \end{gather} with an arbitrary function $f(.)$.
In some equations arrising in applications such solutions would be interesting. F.e. consider Korteweg–De Vries equation from the theory of 1D–water waves: \begin{gather} u _{t}+ u_{xxx} -6uu_{x} =0 \label{2.1B.2} \end{gather} with the second term describing dispersion.
Plugging (\ref{2.1B.1}) into (\ref{2.1B.2}) we get
\begin{gather*}
-c f '+ f''' -6ff' =0;
\end{gather*}
integrating it we get
\begin{gather*}
-c f + f'' -3f^2 =A
\end{gather*}
where $A$ is a constant. This is autonomous ODE and to solve it we denote $z=f'$; then
\begin{gather*}
z\frac{dz}{df}=cf+3f^2+A
\end{gather*}
and integrating we get
\begin{gather*}
z^2 =cf^2+2f^3+Af+B
\end{gather*}
and
\begin{gather*}
x= \int \frac{df}{\sqrt{cf^2+2f^3+2Af+B}}.
\end{gather*}
Let
\begin{gather*}
V(f)=-\Bigl(f^3+\frac{1}{2}cf^2+Af\Bigr).
\end{gather*}
If
\begin{gather*}
A=0,\quad c>0
\end{gather*}
then $V(f)$ has local maximum as $f=0$ then there is a solution which decays at $\pm \infty$:
\begin{gather*}
x= \int \frac{df}{f\sqrt{2f +c}}\implies
f(x)=-\frac{1}{2}\cosh^{-2}\Bigl(\frac{\sqrt{c}}{2}(x-a)\Bigr)
\end{gather*}
and
\begin{gather}
u(x,t)=-\frac{1}{2}\cosh^{-2}\Bigl(\frac{\sqrt{c}}{2}(x-ct-a)\Bigr),
\label{2.1B.3}
\end{gather}
describing solitons (solitary waves). Here $a$ is arbitrary.
In contrast to wave equaion shape of solutions (that is function $f$) is not arbitrary and different solitons have different speed of propagation $c$. Further, due to non-linearity sum of solutions is not a solution.
It was proven that an arbitrary solution $u(x,t)$ decaying as $x\to \pm \infty$ asymptotically behaves as superposition of solitary waves (as $|t|\gg 1$ humps of such waves are far away from one another and thus superposition is an asymptotic solution).
Otherway we would get traveling wave solutions, periodic with respect to $x$; they are called cnoidal waves, they are described in terms of non-elementary Jacobi elliptic functions.
Theory of this equation and similary equation was extensively developed in 1970s.
In the next Chapter 3 we consider another class of interesting solutions of PDEs--self-similar solutions.