Spatially extended particles hidden in line-soliton dynamics in more   than one space dimension
  A dynamics of spatially extended particles, hidden in the dynamics of line solitons in more than one space dimension, is revealed through conservation laws obeyed by the single-soliton solution. These are functions of the solution of a nonlinear evolution equation and its derivatives, which vanish for a single-soliton solution. They map multi-line-soliton solutions into systems of vertices - spatially extended structures, localized around the soliton-collision regions. In more than one space dimension, vertices move in space, emulating the dynamics of spatially extended particles. Examples are provided through the analysis of several soliton solutions of the Kadomtsev-Petviashvili (KP) equation in (1+2)-dimensions. The solution with one collision region is mapped onto a single vertex, which moves at a constant velocity, preserving its spatial structure, thereby emulating a free particle. In solutions with several collision regions, each region is mapped onto a vertex. When vertices are well separated, they also emulate free particles. They move in space, coalesce upon collision, and then split up. The total particle linear momentum is conserved through the collision in all solutions studied. However, depending on the soliton solution, particle masses and the total kinetic energy may change. 
Yair Zarmi
nlin.SI