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 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.