  The dynamics of initial long-wavelength excitations of the
Fermi-Pasta-Ulam-Tsingou chain has been the subject of intense investigations
since the pioneering work of Fermi and collaborators. We have recently found a
new regime where the spectrum of the Fourier modes decays with a power-law and
we have interpreted this regime as a transient turbulence associated with the
Burgers equation. In this paper we present the full derivation of the latter
equation from the lattice dynamics using a newly developed infinite dimensional
Hamiltonian perturbation theory. This theory allows us to relate the time
evolution of the Fourier spectrum $E_k$ of the Burgers equation to the one of
the Fermi-Pasta-Ulam-Tsingou chain. As a consequence, we derive analytically
both the shock time and the power-law $-8/3$ of the spectrum at this time.
Using the shock time as a unit, we follow numerically the time-evolution of the
spectrum and observe the persistence of the power $-2$ over an extensive time
window. The exponent $-2$ has been widely discussed in the literature on the
Burgers equation. The analysis of the Burgers equation in Fourier space also
gives information on the time evolution of the energy of each single mode
which, at short time, is also a power-law depending on the $k$-th wavenumber
$E_k \sim t^{2k-2}$. This approach to the FPUT dynamics opens the way to a
wider study of scaling regimes arising from more general initial conditions.
