Mathematical difference between solitons and traveling waves for a non-linear dispersive PDE

Question

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)

• 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 ?

Answer 1

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.