  In this work, we perform a detailed study of heat transport in one
dimensional long-ranged anharmonic oscillator systems, such as the long-ranged
Fermi-Pasta-Ulam-Tsingou model. For these systems, the long-ranged anharmonic
potential decays with distance as a power-law, controlled by an exponent
$\delta \geq 0$. For such a non-integrable model, one of the recent results
that has captured quite some attention is the puzzling ballistic-like transport
observed for $\delta = 2$, reminiscent of integrable systems. Here, we first
employ the reverse nonequilibrium molecular dynamics simulations to look
closely at the $\delta = 2$ transport in three long-ranged models, and point
out a few problematic issues with this simulation method. Next, we examine the
process of energy relaxation, and find that relaxation can be appreciably slow
for $\delta = 2$ in some situations. We invoke the concept of nonlinear
localized modes of excitation, also known as discrete breathers, and
demonstrate that the slow relaxation and the ballistic-like transport
properties can be consistently explained in terms of a novel depinning of the
discrete breathers that makes them highly mobile at $\delta = 2$. Finally, in
the presence of quartic pinning potentials we find that the long-ranged model
exhibits Fourier (diffusive) transport at $\delta = 2$, as one would expect
from short-ranged interacting systems with broken momentum conservation. Such a
diffusive regime is not observed for harmonic pinning.
