Nanometer-scale modulations can spontaneously emerge in complex materials when multiple degrees of freedom interact. Here we demonstrate that ferroelectric ${\mathrm{Sr}}_{1-x}{\mathrm{Ca}}_x{\mathrm{TiO}}_3$ lies in close proximity to a finite-$q$ lattice instability associated with a tendency toward structural modulation. Using inelastic neutron and x-ray scattering, we show that upon cooling, dipolar fluctuations strongly couple to and soften the $c_{44}$ transverse acoustic mode. We identify the wave vector at which this softening is maximal, thereby defining the characteristic length scale of the instability. Calcium substitution enhances both the amplitude and the wavevector of the softening by strengthening the ferroelectric and antiferrodistortive instabilities. Our results demonstrate that the flexoelectric phonon coupling promotes a tendency toward a dynamically modulated state that cooperates with, rather than competes against, the other lattice instabilities in ${\mathrm{SrTiO}}_3$.

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• Anharmonic lattice dynamics
• Dynamical phase transitions
• Ferroelasticity
• Phononics
• Ferroelectrics
• Paraelectrics
• Polarization
• Inelastic X-ray scattering
• Landau-Ginzburg-Devonshire theory
• Neutron scattering