Xenoliths entrained in alkaline basalts and kimberlites give strong evidence that mantle carbonatitic and carbonated high alkaline mafic silicate melts, which are initially produced at very low degrees of partial melting (≪1%), percolate and accumulate to form impregnations with a melt concentration of up to 10%. At present no compaction model has explained such huge local amplification of melt concentration. Recently, Bercovici et al.  have shown that the commonly used equations of compaction are not sufficiently general to describe all melt percolation processes the mantle. In particular, they show that, when the melt concentration in the mantle is very low, the pressure jump ΔP between the solid and liquid fractions of the mantle mush is very important and plays a driving role during compaction. 1-D compaction waves generated with two different systems of equations are computed. Three types of wave-trains are observed, i.e. (1) sinusoidal waves; (2) periodic waves with flat minima and very acute maxima ('witch hat waves'); (3) periodic solitary waves with flat maxima and extremely narrow minima ('bowler hat waves'). When the initial melt distribution in the mantle is quite homogeneous, the compaction waves have sinusoidal shapes and can locally amplify the melt concentration by a factor less than two. When there is a drastic obstruction at the top of the wetted domain, the pressure jump ΔP between solid and liquid controls the shape of the waves. If the computation assumes the equality of pressure between the two phases (ΔP=0), the compaction wave has a 'bowler hat shape', and locally amplifies the melt concentration by a factor less than 5. Alternatively, simulations taking into account the pressure jump between phases ΔP predict compaction waves with 'witch hat shape'. These waves collect a large quantity of melt promoting the development of magmons with local melt concentration exceeding 100× the background melt concentration. It is inferred that in a mantle with very low concentrations of carbonatitic or high alkaline mafic silicate melt the magmons are about 1 km thick and reach, in less than 1 Ma, a melt concentration of about 10%. The magmons are likely generated below the lithosphere at some distance away from the center of hot spots. This can explain the development of mantle carbonatitic eruptions in the African rift and the carbonatite and high alkaline mafic silicate volcanic activity in oceanic islands.