  By treating the centers of solitons as point particles and studying their
discrete dynamics, we demonstrate a new approach to the quantization of the
soliton solutions of the sine-Gordon equation, one of the first model nonlinear
field equations. In particular, we show that a linear superposition of the
non-interacting shapes of two solitons offers a qualitative (and to a good
approximation quantitative) description of the true two-soliton solution,
provided that the trajectories of the centers of the superimposed solitons are
considered unknown. Via variational calculus, we establish that the dynamics of
the quasi-particles obey a pseudo-Newtonian law, which includes cross-mass
terms. The successful identification of the governing equations of the
(discrete) quasi-particles from the (continuous) field equation shows that the
proposed approach provides a basis for the passage from the continuous to a
discrete description of the field.
