Mathematical difference between solitons and traveling waves for a non-linear dispersive PDE 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, i.e. has infinite conservation laws, ( I think the property of complete integrability will not probably add something to the question)

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*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:

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*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).

*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?

*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 ?

 A: 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 or heteroclinic orbits then correspond to solitonic solutions.
A: Terry Tao's expository article Why are solitons stable should clarify some of your questions. See in particular the footnote on page 8.
Consider for example the focusing nonlinear Schrödinger (NLS) equation $i\partial_t u+\Delta u+u|u|^{p-1}=0$ where $u:\mathbb{R}\times \mathbb{R}^d\rightarrow \mathbb{C}$. A stationary state can be defined to be an time-independent solution $u(t,x)=Q(x)$, where $Q$ has to solve $\Delta Q+Q|Q|^{p-1}=0$. A soliton is defined to be a time-periodic solution $u(t,x)=e^{it}Q(x)$, where $Q$ has to solve $\Delta Q-Q+Q|Q|^{p-1}=0$.
Existence of solitons is inferred by variational methods: if you try to minimize the quantity $\| v\|_{H^1(\mathbb{R}^d)}$ over the set $$\{v\in H^1_{\text{radial}}(\mathbb{R}^d),\ \| v\|_{L^{p+1}(\mathbb{R}^d)} =1 \}$$
you can prove (with some work, see for example Terry Tao's book on nonlinear dispersive equations) that the minimizer exists and, up to rescaling and (phase) shift, solves precisely the soliton equation $\Delta Q-Q+Q|Q|^{p-1}=0$.
If your initial data (to focusing NLS) is close to that of the soliton, i.e. $\|u_0-Q\|<\varepsilon$, in a certain range of parameters (for example if $p<1+\frac{4}{d}$) you can show that the solution $u$ stays close (in the $H^1$ norm) to a modulated ((phase) shifted) version of the soliton. To do this, you can use the concentration-compactness lemma which describes the ways in which the embedding $H^1\subset L^p$ can fail to be compact.
