Context.The stability of parallel propagating circularly polarized Alfvén waves (pump waves) has been studied for more than four decades with the use of normal mode analysis. It is well known that the normal mode analysis does not answer the question if a pump wave looks stable or unstable in a particular reference frame. To answer this question it is necessary to find out if the instability is absolute or convective in this reference frame.\ud \ud Aims.We extend our previous study of absolute and convective instabilities of pump waves with small amplitude to pump waves with arbitrary amplitude.\ud \ud Methods.To study the absolute and convective instabilities of pump waves with arbitrary amplitude we numerically implement Brigg's method.\ud \ud Results.We show that the wave is absolutely unstable in a reference frame moving with the velocity U with respect to the rest plasma if U satisfies the inequality Ul < U < Ur and convectively unstable otherwise, and calculate the dependences of Ul and Ur on the dimensionless wave amplitude a . We also calculate the dependence of the increment of the absolute instability on U for different values of a . When the instability is convective (U < Ul or U > Ur) we study the signalling problem. We show that spatially amplifying waves exist only when the signalling frequency is in two symmetric frequency bands, and calculate the dependences of the boundaries of these bands on U for different values of a . We also obtain the dependences of the maximum spatial amplification rate on U for different values of a . The implication of these results on the interpretation of observational data from space missions is discussed. In particular, it is shown that circularly polarized Alfvén waves propagating in the solar wind are convectively unstable in a reference frame of any realistic spacecraft
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