What is a soliton?

Question

I have read about solitons and they seem to be a big deal as a phenomenon. The definition I find from wikipedia is that solitons are characterized by the following three properties:

1. They are of permanent form

2. They are localized within a region

3. They can interact with other solitons, and emerge from the collision unchanged, except for a phase shift.

Well, the d'Alembert solution to the wave equation $F(x-ct)$ and $G(x+ct)$ has these properties. We can always choose $F$ and $G$ to be functions with compact support, hence satisfying 2. They are of permanant form as well. I'm not quite sure how to quantify 'interact' in property 3.

So why arent these solutions considered to be 'solitons'?

Answer 1

The main difference with d'Alembert traveling wave solutions is that for solitons,

The speed depends on the size of the wave, and its width on the depth of water.

More precisely, the speed and amplitude of a soliton are constant, and there is a linear dependence of the speed with the amplitude (see e.g. Sec. 13.12 of [1]).

These properties are particularly remarkable. Indeed,

• in the case of d'Alembert's equation (or other non-dissipative linear wave equations), amplitudes of traveling waves are constant, but waves with different amplitudes propagate at the same constant speed $c$: there is no dependence of the speed with the amplitude.

• in the case of Burgers' equation (or other non-dispersive nonlinear wave equations), the amplitude of a smooth traveling wave is not constant. For most of them, a smooth waveform breaks into shock waves.

---

[1] G.B. Whitham, Linear and Nonlinear Waves, Wiley, 1999. doi:10.1002/9781118032954