Write down the nonlinear differential equation of Korteveg-de Vries (KdV) in such a form:

here   - is the wave function within a non-linear environment with the dispersion. Authors [1 and 2] were accepting  , hence

Let us rewrite now the periodic boundary conditions of [1,2]as a set:

Thus, the wave lenght is equal to 2, hence the wave function as well as its derivatives are periodic along the coordinate axis. The initial conditions were chosen by the authors [1,2] in the form of harmonic perturbation with unit amplitude:

We can to obtain the numerical solution of above mentioned Cauchy problem as a module, following [2]:

Now we can to present the space-time evolution of solution in two different ways

The animation of Fig. 2  illustrates such effects:

It easy to chek that the other forms of perturbations leads to same results as these mentioned above. Let us exchange the cosine perturbation with the Gauss impulse having the mean value  and the dispersion equal to  

The new solution is:

sol_new:-animate(u(x,t), t=0..2, frames=100,color=blue,thickness=3,gridlines=true,title=`Fig.3 Increasing of the wave front steepness \n and the emergence of solitons-2`,font=[HELVETICA,14],axes=boxed);

This animation shows all the same effects as on the Fig.2.  Therefore, the conclusions, mentioned above, are independent of the form of initial perturbations.

A more direct method of studying the interaction between solitons is the modeling of the solitons doublets, which is one of the cases of the multi-soliton solutions of the Korteveg-de Vries equation  [3].  All multi-soliton solution, the simplest of which is a soliton doublet, that is linked to a pair solitons, are well-known and exact solutions of the equation Korpteveg - de Vrìes. However these multi-solitons are not a simple linear combinations of the individual solitons, though, because the equation of KdS is non-linear, therefore, does not provide a linear combinations of individual solutions as stand-alone solutions. Solìtons in multiplets are bonded in non-linear way.  A soliton doublet, or a pair of solitons, which satisfies the equation KdV as its exact analytical solution is [3]:

here  - are velosities of solitons wich are proportional to its amplitudes, so the faster KdV soliton is also higher and narrower.

Let us show the evolution of doublet into time and space by different ways:

The figure 5 shows that the up to time    the  -coordinates of soliton with smaller amplitude (darker and wider line) are above the -coordinates of solìton with higher amplitude (lighter and narrower line), so the "small" solìton ahead of the "big". After a collision at the point solìtons exchanged places, now the "big" is ahead of the "small". We can notice that the line of evolution of solìtons after the collision is not direct sequels of their lines to clash (there has been some visible shifting).

Animation is probably most popular of these ways:

The animation Fig.6 clearly evident the interaction of solìtons: faster solìton catching up slower. Then they "exchange amplitudes", and front solìton gets the higher speed and amplitude, and the back - lower speed and amplitude. The situation is reminiscent of the collision two elastic balls with various pulses. Note: both solìtons  do not change own shapes after the interaction. While directly at the time of the collision (), both solitons form the wider and lower united complex. It is specially evident at "frame-to-frame" animation.