I see many mathematicians conflating the definitions of traveling waves and solitons, and I am unable to understand, from a mathematical point of view, the differences between these two types of solutions for a nonlinear dispersive PDE. All I know is the following:


Consider for example a nonlinear dispersive PDE which is [completely integrable](https://en.wikipedia.org/wiki/Integrable_system), i.e. has infinite conservation laws, ( I think the property of complete integrability will not probably add something to the question)


* The **traveling waves** are solutions of the form $u\_0(x+ct)$ where $u\_0$ is the initial data and $c\in\mathbb{R}$.
* The **Solitons** are subset of the traveling waves, that remain with the same shape even after colliding with another soliton. The phenomena of solitons appear after a cancelation between the dispersive effects and the nonlinearity of the equation.


So here are my questions:


1. How do we know if a nonlinear PDE has solitons as solutions, knowing that it has traveling wave solutions ? On other words, how do we prove mathematically that a traveling wave is a soliton (without using simulation).
2. For example, solutions of the form $u(t,x)=e^{i(x-t)}$ or $u(t,x)=\frac{1}{1-\frac12 e^{i(x-ct)}}$, $x\in \mathbb{T}:=\mathbb{R}/(2\pi\mathbb{Z}),$ can be considered as solitons?
3. Does a traveling wave that is almost periodic solution, *i.e. the set $\{u(\cdot+\tau), \tau\in \mathbb{R}\}$ is relatively compact*, can lead to the fact that it is a soliton ?



Mathematical difference between solitons and traveling waves for a non-linear dispersive PDE
A necessary requirement for a traveling wave $u(x,t)=f(x-ct)$ to be a "solitary wave" or "soliton" is that the two limits $\lim\_{s\rightarrow\pm\infty}f(s)=\alpha\_\pm$ exist. This is the condition of *shape invariance* and *localisation*. The stability under collision may or may not be added as extra condition, but in much of the literature any shape-invariant localised wave is called a soliton. One further distinguishes homoclinic and heteroclinic solutions, depending on whether $\alpha\_+$ is equal to $\alpha\_-$ or not.


In response to Q2, the waves $u(t,x)=e^{i(x-t)}$ or $u(t,x)=\frac{1}{1-\frac12 e^{i(x-ct)}}$ are no solitons, because they are not localised.


Concerning Q1, without the "stability upon collision" condition, one way to identify solitonic solutions of a second order wave equation $f''(s)=F[f(s)]$ is to plot the flow lines in the *f-g* plane of the two coupled equations $f'(s)=g(s)$, $g'(s)=F[f(s)]$. [Homoclinic](https://en.wikipedia.org/wiki/Homoclinic_orbit) or [heteroclinic](https://en.wikipedia.org/wiki/Heteroclinic_orbit) orbits then correspond to solitonic solutions.


