  We investigate low-energy fluctuations in the real kink crystal phase of
dense quark matter within the Nambu--Jona-Lasinio model. The modulated chiral
condensate breaks both the translational symmetry and chiral symmetry
spontaneously, which leads to the appearance of phonons and pions that are
dominant degrees of freedom in the infrared. Using the Ginzburg-Landau
expansion near the Lifshitz point, we derive elastic free energies for phonons
and pions in dependence on the temperature and chemical potential. We show that
the one-dimensional modulation is destroyed by thermal fluctuations of phonons
at nonzero temperature, and compute the exponent that characterizes the
anisotropic algebraic decay of quasicondensate correlations at long distance.
We also estimate finite-volume effects on the stability of the real kink
crystal and briefly discuss the possibility of its existence in neutron stars.
