The absolute and convective instabilities of circularly polarized Alfven waves propagating along an ambient magnetic field are studied. The approximation of ideal magnetohydrodynamics is used. The analysis is restricted to the decay instability that occurs when the sound speed is smaller than the Alfven speed. In addition, it is assumed that the amplitude a of an unstable Alfven wave (pump wave) is small. This assumption allows us to study the problem analytically using expansions in power series with respect to a. It is shown that there are quantities, U-l<0 and U-r>0, such that the pump wave is absolutely unstable in a reference frame moving with velocity U with respect to the rest plasma if U-l<U<U-r. If either U<U-l or U>U-r, then the pump wave is convectively unstable. The expressions for U-l and U-r are found. The signaling problem is studied in a reference frame where the pump wave is convectively unstable. It is shown that spatially amplifying waves exist only when the signaling frequency is in two narrow symmetric frequency bands with the widths of the order of a. The implication of the obtained results on the interpretation of observational data obtained in space missions is discussed. It is shown that circularly polarized Alfven waves propagating in the solar wind are convectively unstable in a reference frame of any spacecraft moving with the velocity not exceeding a few tens of km/s in the solar reference frame. The spatial amplification scale of these waves is very large, of the order of 1/6 a.u. In view of these results it is not surprising at all that evidence of the decay instability of Alfven waves in the solar wind is sparse
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