  A number of possible pulsar radio emission mechanisms is based on
instabilities of the relative streaming and beams in their relativistic
electron-positron pair plasma. At saturation the unstable waves can form, in
principle, stable solitary waves which could emit the observed intense radio
signals. We searched for the proper plasma parameters which would lead to the
formation of solitons, investigated their properties and dynamics as well as
the resulting oscillations of electrons and positrons possibly leading to radio
wave emission. We utilized a one-dimensional version of the relativistic
Particle-in-Cell code Acronym, initialized an appropriately parameterized
one-dimensional Maxwell-J\"uttner velocity space particle distribution and
studied the evolution of the resulting streaming instability in the strong
pulsar magnetic fields. We found that for plasmas with inverse temperatures
$\rho \geq 1.66$ or relative electron-positron drift speeds with Lorentz
factors $\gamma > 40$, strong electrostatic solitons form associated with L-
and A-mode plasma waves. The parameters of the solitons fulfill the wave
emission conditions. For appropriate pulsar parameters the resulting energy
densities of L-mode solitons can reach up to $1.1 \times 10^5$
erg$\cdot$cm$^{-3}$ while those of A-mode solitons reach only up to $1.2 \times
10^4$ erg$\cdot$cm$^{-3}$. Estimated energy densities of up to $7 \times
10^{12}$ erg$\cdot$cm$^{-3}$ suffice to explain pulsar nanoshots.
