  We analyze the interaction of lattice vibrations (phonon wave-packets) with
an asymmetric kink soliton initially at rest. We employ the $\phi^6$ model in
one space and one time dimensions for various lattice spacings and consider two
different discretization prescriptions for the field potential that do not
generate Peierls-Nabarro potentials, i.e. the kink can be placed anywhere along
the lattice beyond discrete translational invariance. Since the $\phi^6$ model
kink is neither symmetric nor anti-symmetric under spatial reflections we
simulate the cases where the wave-packet approaches the kink from negative or
positive spatial infinity. We extract the energy transmission and reflection
coefficients as functions of the central frequency of the phonon wave-packet
for the different lattice spacings. For large lattice spacings the wave-packet
is always fully reflected while for smaller spacing the amount of reflection
and transmission depends on the central frequency. We also identify scenarios
in which the target kink acquires a non-zero velocity from its interaction with
the wave-packet.
