  Over the past two decades high energy-resolution inelastic proton scattering
studies were used to gain an understanding of the origin of fine structure
observed in the isoscalar giant quadrupole resonance (ISGQR) and the isovector
giant dipole resonance (IVGDR). Recently, the isoscalar giant monopole
resonance (ISGMR) in $^{58}$Ni, $^{90}$Zr, $^{120}$Sn and $^{208}$Pb was
studied at the iThemba Laboratory for Accelerator Based Sciences (iThemba LABS)
by means of inelastic $\alpha$-particle scattering at very forward scattering
angles (including $0\circ$). The good energy resolution of the measurement
revealed significant fine structure of the ISGMR.~To extract scales by means of
wavelet analysis characterizing the observed fine structure of the ISGMR in
order to investigate the role of different mechanisms contributing to its decay
width. Characteristic energy scales are extracted from the fine structure using
continuous wavelet transforms. The experimental energy scales are compared to
different theoretical approaches performed in the framework of quasiparticle
random phase approximation (QRPA) and beyond-QRPA including complex
configurations using both non-relativistic and relativistic density functional
theory. All models highlight the role of Landau fragmentation for the damping
of the ISGMR especially in the medium-mass region. Models which include the
coupling between one particle-one hole (1p-1h) and two particle-two hole
(2p-2h) configurations modify the strength distributions and wavelet scales
indicating the importance of the spreading width. The effect becomes more
pronounced with increasing mass number. Wavelet scales remain a sensitive
measure of the interplay between Landau fragmentation and the spreading width
in the description of the fine structure of giant resonances.
