  Hole-doped high-temperature copper oxide-based superconductors (cuprates)
exhibit complex phase diagrams where electronic orders like a charge density
wave (CDW) and superconductivity (SC) appear at low temperatures. The origins
of these electronic orders are still open questions due to their complex
interplay and correlated nature. These electronic orders can modify the phonons
in the system, which has also been experimentally found in several cuprates as
a softening in the phonon frequency at the CDW vector. Recent experiments have
revealed that the softening in phonons in cuprates due to CDW shows intriguing
behavior with increasing hole doping. Hole doping can also change the
underlying Fermi surface. Therefore, it is an interesting question whether the
doping-induced change in the Fermi surface can affect the softening of phonons,
which in turn can reveal the nature of the electronic orders present in the
system. In this work, we investigate this question by studying the softening of
phonons in the presence of CDW and SC within a perturbative approach developed
in an earlier work. We compare the results obtained within the working model to
some experiments.
