  Moved by the pressing need for rigorous and reliable numerical tools for the
analysis of peridynamic materials, the authors propose a model able to capture
the dispersive features of nonlocal soliton like solutions obtained by a
peridynamic formulation of the Sine-Gordon equation. The analysis of the Cauchy
problem associated to the peridynamic Sine Gordon equation with local Neumann
boundary condition is performed in this work through a spectral method on
Chebyshev polynomials nodes joined with the Stormer Verlet scheme for the time
evolution. The choice for using the spectral method resides in the resulting
reachable numerical accuracy, while, indeed, Chebyshev polynomials allow
straightforward implementation of local boundary conditions. Several numerical
experiments are proposed for thoroughly describe the ability of such scheme.
Specifically, dispersive effects as well as the ability of preserving the
internal energy of the specific peridynamic kernel are demonstrated.
