  We propose an analytical Landau-Ginzburg theory of the charge density waves
coupled with lattice and electronic long-range order parameters. Examples of
long-range order include electronic wave function of superconducting Cooper
pairs, structural distortions, electric polarization, and magnetization. We
formulate the Landau-Ginzburg free energy density as power expansion with
respect to the charge density and other long-range order parameters, as well as
their spatial gradients, and biquadratic coupling terms. We introduced a
biquadratic coupling between the charge density gradient and long-range order
parameters, as well as nonlinear higher gradients of the long-range order
parameters. The biquadratic gradient coupling is critical to the appearance of
different spatially-modulated phases in charge-ordered ferroics and
high-temperature superconductors. We derived the thermodynamic conditions for
the stability of the spatially-modulated phases, which are the intertwined
spatial waves of charge density and lattice/electronic long-range order. The
analytical expressions for the energies of different phases, corresponding
order parameters, charge density waves amplitudes and modulation periods,
obtained in this work, can be employed to guide the comprehensive physical
explanation, deconvolution and Bayesian analysis of experimental data on
quantum materials ranging from charge-ordered ferroics to high-temperature
superconductors.
