  We construct a formally time-reversible, one-dimensional forced Burgers
equation by imposing a global constraint of energy conservation, wherein the
constant viscosity is modified to a fluctuating state-dependent dissipation
coefficient. The system exhibits dynamical properties which bear strong
similarities with those observed for the Burgers equation and can be understood
using the dynamics of the poles, shocks, and truncation effects, such as
tygers. A complex interplay of these give rise to interesting statistical
regimes ranging from hydrodynamic behavior to a completely thermalized warm
phase. The end of the hydrodynamic regime is associated with the appearance of
a shock in the solution and a continuous transition leading to a
truncation-dependent state. Beyond this, the truncation effects such as tygers
and the appearance of secondary discontinuity at the resonance point in the
solution strongly influence the statistical properties. These disappear at the
second transition, at which the global quantities exhibit a jump and attain
values that are consistent with the establishment of a quasiequilibrium state
characterized by energy equipartition among the Fourier modes. Our comparative
analysis shows that the macroscopic statistical properties of the formally
time-reversible system and the Burgers equation are equivalent in all the
regimes, irrespective of the truncation effects, and this equivalence is not
just limited to the hydrodynamic regime, thereby further strengthening the
Gallavotti's equivalence conjecture. The properties of the system are further
examined by inspecting the complex space singularities in the velocity field of
the Burgers equation. Furthermore, an effective theory is proposed to describe
the discontinuous transition.
