  In this paper, we use the $\bar{\partial}$ steepest descent method to study
the initial value problem for focusing nonlinear Schr\"odinger (fNLS) equation
with non-generic weighted Sobolev initial data that allows for the presence of
high-order discrete spectrum. More precisely, we shall characterize the
properties of the eigenfunctions and scattering coefficients in the presence of
high-order poles; further we formulate an appropriate enlarged RH problem;
after a series of deformations, the RH problem is transformed into a solvable
model. Finally, we obtain the asymptotic expansion of the solution of the fNLS
equation in any fixed space-time cone:
  %as $t \to \infty$,
  \begin{equation*}
  \mathcal{S}(x_1,x_2,v_1,v_2):=\left\lbrace (x,t)\in \mathbb{R}^2: x=x_0+vt, \
x_0\in[x_1,x_2]\text{, }v\in[v_1,v_2]\right\rbrace.
  \end{equation*}
  Observing the result indicates that the solution of fNLS equation in this
case satisfies the soliton resolution conjecture. The leading order term of
this solution includes a high-order pole-soliton whose parameters are affected
by soliton-soliton interactions through the cone and soliton-radiation
interactions on continuous spectrum. The error term of this result is up to
$\mathcal{O}(t^{-3/4})$ which comes from the corresponding $\bar{\partial}$
equation.
