This paper presents an investigation of the filamentation process of a uniform plane Alfvén wave for a compressible magnetohydrodynamics case, when a transverse perturbation (perturbation having non-uniform intensity distribution in a plane perpendicular to the direction of wave propagation) is present on it. The physical mechanism of the filament formation is discussed in detail with the applications to the Solar wind turbulence and the coronal heating. The dependence of the critical field of the perturbation, its critical transverse size and the formation of filaments, on the plasma $\beta $, main Alfvén field and other parameters has been studied in detail.