Phonon softening: "keyhole" into dynamic stripe-phase
  The highly distinctive phonon self-energy dependences on the wave vector, calculated respectively for the static and dynamic stripe phase models of the underdoped cuprates are presented. The negative values of the real part of the on-stripe holes polarization loop lead to appearance of localized vibration states in the presence of well-separated stripes, relevant for underdoped cuprates. The localized modes split below the bare optical phonon frequencies exhibiting strong ``softening effect''. The calculated gap between localized and propagating phonon frequencies is practically doping independent in accord with the recent neutron measurements. The derived wave vector dependence of the phonon softening nicely coincides with anomalously shallow experimental curve in high-T$_c$ cuprates when stripe meandering is taken into account. Co-centering of the theoretical and experimental curves implicates statistics of on-stripe holes. 
Sergei I. Mukhin