length of two kilometers. The shear velocity is $u_*=0.45$ m s$^{-1}$ after $31.7$ years, the boundary conditions are periodic. The structure is quite irregular.[]{data-label="fig:transv_1dp1"}](Figure13.eps){width="80.00000%"}

![Surface of a dune field with a length of two kilometers. The shear velocity is $u_*=0.45 $m s$^{-1}$ after $95.1$ years, the boundary conditions are periodic. There are three dunes left with similar heights.[]{data-label="fig:transv_1dp2"}](Figure14.eps){width="80.00000%"}

![Surface of a dune field with a length of two kilometers. The shear velocity is $u_*=0.45$ m s$^{-1}$ after $190.2$ years, the boundary conditions are periodic. The number of dunes will decrease by one after coalescence of two dunes.[]{data-label="fig:transv_1dp3"}](Figure15.eps){width="80.00000%"}

![The number of dunes decreases quite regularly in time.[]{data-label="fig:transv_nu_d"}](Figure16.eps){width="80.00000%"}

Do transversal dunes behave like solitons? {#sec:trans_soliton}
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proposed that barchan dunes behave like solitons. Solitons are self-stabilizing wave packs which do not change their shape during their propagation not even after collision with other waves. They are found in non-linear systems like for example in shallow water waves. Observations of barchan dunes seem to give evidence that small dunes can migrate over bigger ones without being absorbed completely. A closer examination of the merging of two or more dunes from calculations of a two dimensional dune field with periodic boundary let to some 