  Digital computing currently uses irreversible logic gates whose energy
dissipation is fundamentally limited. Reversible logic gates can provide an
energy-efficient alternative since they can operate with reversible processes
that have no dissipation, such as with scattering processes involving elastic
particles. The presented logic uses fluxons, topological solitons in Long
Josephson Junctions (LJJs), as inputs into and outputs from logic gates. An
advantage of using LJJs for connections is that they restrict scattering to 1D
paths, in contrast to previous ballistic logic which is based on 2D scattering.
Furthermore, we find through simulation that there is almost no energy loss in
the scattering of fluxons between LJJs of designed unpowered circuit gates. To
switch bit states, the fluxons are made to change polarity during operations --
fluxons in an input LJJ freely propagate into the gate circuit, excite a
nonlinear oscillatory interface mode, and quickly scatter deterministically to
an output LJJ as a fluxon or antifluxon. The numerically simulated soliton
dynamics shows that over $97\%$ of the total energy is preserved as fluxon
energy after gate operations. These phenomena are further analyzed with a
collective coordinate ansatz, reducing the dynamics of many degrees of freedom
to two coordinates which characterize fluxon and antifluxon type excitations in
the input and the output LJJs. The solutions of the reduced model accurately
describe the four possible energy-conserving scattering processes found in
simulations of one-bit gate circuits (with two scattering directions and two
output polarities). Calculated parameter tolerances indicate that the gates can
be manufactured and tested. Results are shown for 1-bit gates as well as a
fundamental 2-bit gate.
