interesting observations. In this case the dune field cannot break its translational scale invariance to reach a faster colliding of two adjacent dunes. Thus the supposition would be that a smaller dune due to its higher velocity collides with the next bigger one in wind direction without passing over it. That is not the case. As an example see the Figure \[fig:transv\_sol1\]. A small dune climbs up the windward side of the following bigger dune. As it reaches the same height as the following one it seems to hand over the state of being the smaller dune. The dune that was bigger before then wanders towards the following dune of the dune field. This process can be observed several times. The fact that the volume of the smaller dune decreases after every time it passes a dune let us conclude that there will be a final dune which will absorb the small one. Hence, these dunes do not behave exactly like solitons because of their loss of volume. The suggestion of that dunes of different hierarchies (for example dunes and mega dunes) could exist with each other without interactions is not valid in our case. Nevertheless the fact that small 