  The equilibration of sinusoidally modulated distribution of the kinetic
temperature is analyzed in the $\beta$-Fermi-Pasta-Ulam-Tsingou chain with
different degrees of nonlinearity and for different wavelengths of temperature
modulation. Two different types of initial conditions are used to show that
either one gives the same result as the number of realizations increases and
that the initial conditions that are closer to the state of thermal equilibrium
give faster convergence. The kinetics of temperature equilibration is monitored
and compared to the analytical solution available for the linear chain in the
continuum limit. The transition from ballistic to diffusive thermal
conductivity with an increase in the degree of anharmonicity is shown. In the
ballistic case, the energy equilibration has an oscillatory character with an
amplitude decreasing in time, and in the diffusive case, it is monotonous in
time. For smaller wavelength of temperature modulation, the oscillatory
character of temperature equilibration remains for a larger degree of
anharmonicity. For a given wavelength of temperature modulation, there is such
a value of the anharmonicity parameter at which the temperature equilibration
occurs most rapidly.
