Within the framework of Landau-Ginzburg-Devonshire (LGD) theory we studied the role of the flexocoupling between the order parameter and elastic strain gradients in the stability of a spatially-modulated phase (SMP) in ferroics with commensurate and incommensurate long-range ordered phases under the presence of squired elastic strain gradient. The squired elastic strain gradient is required for the free energy stability to arbitrary strain gradients.Obtained analytical expressions showed that the fundamental upper limit for the magnitude of the static bulk flexoelectric effect strength, established by Yudin and Tagantsev under the absence of squired elastic strain gradient and higher order gradients terms, should be substituted by the temperature-dependent condition on the flexoelectric coupling strength under the presence of the gradient terms. Moreover, we established that the SMP appears and becomes stable in commensurate ferroics if the flexocoupling constant exceeds the critical value, defined by the reduced temperature, strain and order parameter gradients constants, striction and expansion coefficients in the LGD functional.We calculated the soft phonon dispersion in ferroics with commensurate and incommensurate longrange ordered phases allowing for the squired elastic strain gradient, as well as static and dynamic flexocoupling.Appeared that the dispersion for the optic mode is slightly sensitive to the flexocoupling, while the dispersion of acoustic mode strongly depends on the coupling strength. Obtained results demonstrate that the non-trivial differences in the dispersion of optic and acoustic modes appear under the change of flexocoupling constant.Hence the experimental determination of soft phonon dispersion can give the important information about the influence of the strain gradient and flexocoupling on the SMP in ferroics with commensurate and incommensurate long-range order. These theoretical predictions require experimental verification.

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Analytical expressions describing the frequency dispersion of the soft transverse acoustic (TA)and optic (TO) phonon modes in uniaxial ferroelectrics, as well as their dependence on theflexoelectric coupling constant f, have been analyzed in the framework of the Landau–Ginzburg–Devonshire theory. A critical behavior of the TA mode with respect to the f magnitude isrevealed.

Within the Landau–Ginzburg–Devonshire (LGD) theory framework, the impact of the flexocoupling and electrostriction on the soft phonon dispersion and neutron scattering spectra is analyzed, and analytical expressions are derived. The impact of the higher gradient term in the LGD functional is studied analytically. The existence of incommensurate modulations in a temperature interval higher than the Curie temperature TC, but lower than the temperature of incommensurate phase transition TIC, TC < T < TIC, and under the condition of the flexocoefficient magnitude ranging over the critical value, |f| > fcr(T), is established. The influence of the dynamic flexocoefficient M, suggested in [1], on phonon spectra is studied. We consider various parameters of the free energy functional; especially, we have discovered a significant contribution of the electrostriction to the appearance of a commensurate phase. In the recent years, the various methods based on the neutron scattering acquire more applications to study phonons in solid crystals. We have analyzed neutron scattering spectra and shown that theoretical predictions are in a very good agreement with observed data.