  Following the idea of three wave resonant interactions of lower hybrid waves
it is shown that quantum -modified lower hybrid (QLH) wave in electron positron
ion plasma with spatial dispersion can decay into another QLH wave ( where
electron and positrons are activated whereas ions remain in the background) and
another ultra low frequency QULH (where ions are mobile). Quantum effects like
Bohm potential, exchange correlation and Fermi pressure on the lower hybrid
wave significantly reshaped the dispersion properties of lower hybrid waves.
Later a set of nonlinear Zakharov equations have been derived to consider the
formation of QLH wave solitons with the nonlinear contribution coming from the
QLH waves. Further, modulational instability of the lower hybrid wave solitons
is investigated and consequently it's growth rates are examined for different
limiting cases. Since the growth rate associated with the three-wave resonant
interaction are generally smaller than the growth associated with the
modulational instability, therefore only latter have been investigated. Soliton
solutions from the set of coupled Zakharov and NLS equations in the
quasi-stationary regime have been studied. Ordinary solitons are attribute of
nonlinearity whereas a cusp soliton solution featured by nonlocal nonlinearity
have also studied. Such an approach to lower hybrid waves and cusp solitons
study in Fermi gas comprising electron positron and ions is new and important.
The general results obtained in this quantum plasma theory will have widespread
applicability, particularly for processes in high energy plasma-laser
interactions set for laboratory astrophysics and solid state plasmas.
