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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{2.3A.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{2.3A.2} \end{gather} with the second term describing dispersion.
Plugging (\ref{2.3A.1}) into (\ref{2.3A.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}=f+3f^2+A \end{gather*} and integrating we get \begin{gather*} z^2 =f^2+2f^3+Af+B \end{gather*} and \begin{gather*} x= \int \frac{df}{\sqrt{f^2+2f^3+Af+B}}. \end{gather*} Taking $A=0$ we get \begin{gather} u(x,t)=-\frac{1}{2}\cosh^{-2}\Bigl(\frac{\sqrt{c}}{2}(x-ct-a)\Bigr) \label{2.3A.3} \end{gather} describing solitons (solitary waves). Here $a$ is arbitrary.
We took $A=0$ to get solutions decaying as $x\to \pm \infty$; otherway we would get periodic solutions.
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.
Theory of this equation and similary equation was extensively developed in 1970th.