The dependence of the Fourier spectrum of oscillations of the amplitude function of the incommensurate superstructure on the magnitude of the anisotropic interaction interacting with Dzialoszynski invariant

Abstract

The Fourier spectra of the incommensurate modulation and the dependence of the wave vector (q) of the incommensurate (IC) superstructure on the magnitude of the anisotropic interaction described by the Dzialoszynski invariant (K) and the long-range interaction (T) are investigated. Fourier spectra in di(cid:27)erent modulation modes of incommensurate superstructure analyzed. The calculation of the spatial changes of the amplitude and phase of the order parameter was performed by the numerical BDF (Backward Di(cid:27)erentiation Formula) method and was performed in the Python environment using the SciPy, JiTCODE libraries according to the method described in the paper. The parameters T and K, which are responsible for the long-range and anisotropic interactions, respectively, in the (cid:28)rst approximation well describe the behavior of the wave vector of the IC modulation, and its modes. The emergence of IC modulation is accompanied by the appearance of an unstable chaotic state in the case of a nonzero value of the anisotropic interaction. The increase in the value of the anisotropic interaction in the IC phase is accompanied by a change in both the frequency of existing oscillations and the magnitude of their contribution to the formation of a wave of the incommensurate modulation, causing a change in the IC modulation regime. The transition from the one mode to another is continuous. The modes themselves are determined by the domination of the contribution of the respective oscillation frequencies. The whole spectrum of possible frequencies arises at the modulation wave origin. With an increase of the anisotropic interaction, their harmonics emerge and their contribution is redistributed.