  The $J_1-J_2$ spin chain model with nearest neighbor $J_1$ and next nearest
neighbor anti-ferromagnetic $J_2$ interaction is one of the most popular
frustrated magnetic models. This model system has been extensively studied
theoretically and applied to explain the magnetic properties of the real
low-dimensional materials. However, existence of different phases for the
$J_1-J_2$ model in an axial magnetic field $h$ is either not understood or has
been controversial. In this paper we show the existence of higher order $p>4$
multipolar phase near the critical point $(J_2/J_1)_c=-0.25$. The criterion to
detect the quadrupolar or spin nematic (SN)/spin density wave of type two
(SDW$_2$) phase using the inelastic neutron scattering (INS) experiment data is
also discussed, and INS data of LiCuVO$_4$ compound is modelled. We discuss the
dimerized and degenerate ground state in the quadrupolar phase. The major
contribution of binding energy in the spin-1/2 system comes from the
longitudinal component of the nearest neighbor bonds. We also study spin
nematic/SDW$_2$ phase in spin-1 system in large $J_2/J_1$ limit.
