What is the difference between the notions of travelling wave, solitary wave and soliton? There are three notions I recently heard often:


*

*travelling wave

*solitary wave

*soliton
I am not very familiar with these notions so my question is, what are the differences between these three notions? I tried to google that but only found a lot of different definitions.
For example, travelling wave solutions are of the form
$$
u(x,t)=u(x-ct)
$$
where $c$ is a constant speed of propagation.
 A: A traveling wave is a catch-all for solution to the wave equation in the form as you have expressed, $u(x,t) = f(x\pm c t)$.  It should be understood that the specific solution here is the solution of the wave equation
$$\frac{\partial^2 u}{\partial x^2} - \frac1{c^2} \frac{\partial^2 u}{\partial t^2} = 0$$
where $u(x,0) = f(x)$ and $u_t(x,0)=0$.  Of course, more general solutions are available, but let's not go there right now.
Of course, the above equation describes a wave traveling in a vacuum and thus the wave merely travels and does not lose its shape.  In real life, we are interested in waves traveling in a medium.  In optics, we characterize a medium as having a refractive index $n$; in this case, the wave equation takes the form
$$\frac{\partial^2 u}{\partial x^2} - \frac{n^2}{c^2} \frac{\partial^2 u}{\partial t^2} = 0$$
If $n$ is constant or even piecewise constant, then the solution to this equation takes the form $u(x,t)=f(x \pm \frac{c}{n} t)$ and, again, the shape of the wave is preserved.  
However, lets say that $f$ is a harmonic wave, i.e., 
$$f(x) = \frac1{2 \pi} \int_{-\infty}^{\infty} dk \, F(k) e^{-i k x}$$ 
where $k $ is a frequency of a wave component.  Now, if the refractive index $n$ varies with the frequency, i.e. $n=n(k)$ and $n'(k) \ne 0$, it turns out that this causes a phenomenon known as dispersion in which the shape of the wave is not preserved through propagation. You can see the effects of dispersion by the spreading of the wave as it propagates - a gaussian beam provides ample evidence of this.
However, there are some very special cases in which the wave itself can manage to preserve its shape upon propagation.  For example, if the medium in which the wave propagates also has some nonlinearities, in some cases the nonlinearities can cancel out the dispersive effects and preserve shape.  In such cases, the shape of the wave pulse is something in which the Fourier transform is preserved and independent of width. Such a shape is called a soliton.  One such soliton takes the form of a hyperbolic secant.
(As an aside, for a terrific example of this in physics, I direct you to this paper by my friend B.J. Eggleton, who with his colleagues at the Univ. of Sydney, discovered solitons in Bragg fiber gratings.)
A solitary wave is a long name for a soliton.
A: A solitary wave is a wave which propagates without any temporal evolution in shape or size when viewed in the reference frame moving with the group velocity of the wave. 
On the other hand, a soliton is a nonlinear solitary wave with the additional property that the wave retains its permanent structure, even after interacting with another soliton.
A: A soliton (or, more precisely, a ''one-soliton'' solution) usually refers to a solitary wave solution to an integrable equation. The one-soliton solution together with the multi-soliton solutions of an integrable equation can usually be obtained through the technique of inverse scattering (which requires a Lax pair). So, in a nutshell, we call a solitary wave a soliton if it is a solution of an integrable equation, which also admits $N$-soliton solutions, for any integer $N>0$ (essentially made out of a nonlinear superposition'' of several solitons). If the solitary wave is a solution of a non-integrable equation, then it is not a soliton. However, people do use the word soliton to describe them.
Traveling wave solutions is what you describe although it can be a little bit more general, depending of the symmetries in place. If you think of the nonlinear Schrodinger Equation for example:
$$
i u_t=u_{xx}+u|u|^2,
$$
then what constitutes a traveling wave reduction is $u=e^{i\omega t}f(x-ct)$, where $\omega$ and $c$ are constants. This ansatz leads to an ODEs for the complex function $f$.  
A: According to me, Travelling wave is a main branch and soliton wave or Soliton is its sub branch. The solution that Soliton Wave gives is called Soliton solution. Soliton wave gives Soliton equation i.e, 4th order nonlinear partial differential equation and cannot be solved by First Integral Method.
