  The possibility of slow diffusion regions as the origin for extended TeV
emission halos around some pulsars (such as PSR J0633+1746 and PSR B0656+14)
challenges the standard scaling of the electron diffusion coefficient in the
interstellar medium. Self-generated turbulence by electron-positron pairs
streaming out of the pulsar wind nebula was proposed as a possible mechanism to
produce the enhanced turbulence required to explain the morphology and
brightness of these TeV halos. We perform fully kinetic 1D3V particle-in-cell
simulations of this instability, considering the case where streaming electrons
and positrons have the same density. This implies purely resonant instability
as the beam does not carry any current. We compare the linear phase of the
instability with analytical theory and find very reasonable agreement. The
non-linear phase of the instability is also studied, which reveals that the
intensity of saturated waves is consistent with a momentum exchange criterion
between a decelerating beam and growing magnetic waves. With the adopted
parameters, the instability-driven wavemodes cover both the Alfv\'enic (fluid)
and kinetic scales. The spectrum of the produced waves is non-symmetric, with
left-handed circular polarisation waves being strongly damped when entering the
ion-cyclotron branch, while right-handed waves are suppressed at smaller
wavelength when entering the Whistler branch. The low-wavenumber part of the
spectrum remains symmetric when in the Alfv\'enic branch. As a result,
positrons behave dynamically differently compared to electrons. We also
observed a second harmonic plasma emission in the wave spectrum. An MHD-PIC
approach is warranted to probe hotter beams and investigate the Alfv\'en branch
physics. This work confirms that the self-confinement scenario develops
essentially according to analytical expectations [...](abridged)
