  The nonlinear theory of collective plasmon-polariton propagation along the
infinite chain of metallic nanoparticles is developed within RPA quasiclassical
approach to surface plasmons in large nano-spheres (10-50 nm for radius) of Au
or Ag. The wave type self-modes in the chain are determined and analyzed with
respect to nano-sphere size and chain-separation parameters. At some regions
for parameters the undamped modes occur. They were found on the rim of
stability within the linear theory. The nonlinear corrections stabilize,
however, diverging modes of the linear approach and considerably enlarge the
region of undamped propagation. Nonlinearity is associated with relativistic
corrections to the Lorentz friction. According to the nonlinear behavior, the
region of parameters when undamped stable modes occur, covers the instability
region of the linear theory. The remarkable property of fixed amplitude of the
undamped collective propagating wave independently of initial conditions (even
if they are extremely small) has been identified and described. The group
velocity of this wave propagation is calculated.
