  A new relativistic integrable nonlinear model for real, Majorana type spinor
fields in 1+1 dimensions, gauge equivalent to Papanicolau spin model, defined
on the one sheet hyperboloid is introduced. By using the double numbers, the
model is represented as hyperbolic complex valued relativistic massive Thirring
type model. By Hirota's bilinear method, an exact one and two dissipative
soliton solutions of this model are constructed. Calculation of first three
integrals of motion for one dissipaton solution shows that the last one
represents a particle-like nonlinear excitation, with relativistic dispersion
and highly nonlinear mass. A nontrivial solution of the system of algebraic
equations, showing fusion and fission of relativistic dissipatons is found.
Asymptotic analysis of exact two dissipaton solution confirms resonant
character of our dissipaton interactions.
