  This paper reports the first experimental observation of phonons and their
softening on single crystalline LaPt$_2$Si$_2$ via inelastic neutron
scattering. From the temperature dependence of the phonon frequency in close
proximity to the charge-density wave (CDW) $q$-vector, we obtain a CDW
transition temperature of T$_{CDW}$ = 230 K and a critical exponent $\beta$ =
0.28 $\pm$ 0.03. This value is suggestive of a non-conventional critical
behavior for the CDW phase transition in LaPt$_2$Si$_2$, compatible with a
scenario of CDW discommensuration (DC). The DC would be caused by the existence
of two CDWs in this material, propagating separately in the non equivalent
(Si1-Pt2-Si1) and (Pt1-Si2-Pt1) layers respectively, with transition
temperatures T$_{CDW-1}$ = 230 K and T$_{CDW-2}$ = 110 K. A strong
$q$-dependence of the electron-phonon coupling has been identified as the
driving mechanism for the CDW transition at T$_{CDW-1}$ = 230 K while a CDW
with 3-dimensional character, and Fermi surface quasi-nesting as a driving
mechanism, is suggested for the transition at T$_{CDW-2}$ = 110 K. Our results
clarify some aspects of the CDW transition in LaPt$_2$Si$_2$, which have been
so far misinterpreted by both theoretical predictions and experimental
observations, and give direct insight into its actual temperature dependence.
