  A functional of the solution of the Kadomtsev-Petviashvili II equation maps
multi-soliton solutions onto systems of vertices - structures that are
localized around soliton junctions. A solution with one junction is mapped onto
a single vertex, which emulates a free, spatially extended, particle. In
solutions with several junctions, each junction is mapped onto a vertex. Moving
in the x-y plane, the vertices collide, coalesce upon collision, and then split
up. When well separated, they emulate free particles. Multi-soliton solutions,
whose structure does not change under space-time inversion as for infinite
times, are mapped onto vertex systems that undergo elastic collisions.
Solutions, whose structure does change, are mapped onto systems that undergo
inelastic collisions. The inelastic vertex collisions generated from the
infinite family of (M,1) solutions (M external solitons, (M-2) Y-shaped soliton
junctions, M greater than 3) play a unique role: The only definition of vertex
mass consistent with momentum conservation in these collisions is the spatial
integral of the vertex profile. This definition ensures, in addition, that, in
these collisions, the total mass and kinetic energy due to the motion in the
y-direction are conserved. In general, the kinetic energy due to the motion in
the x-direction is not conserved in these collisions.
