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In this paper, we use the $\bar{\partial}$ steepest descent method to studythe initial value problem for focusing nonlinear Schr\"odinger (fNLS) equationwith non-generic weighted Sobolev initial data that allows for the presence ofhigh-order discrete spectrum. More precisely, we shall characterize theproperties of the eigenfunctions and scattering coefficients in the presence ofhigh-order poles; further we formulate an appropriate enlarged RH problem;after a series of deformations, the RH problem is transformed into a solvablemodel. Finally, we obtain the asymptotic expansion of the solution of the fNLSequation 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 thiscase satisfies the soliton resolution conjecture. The leading order term ofthis solution includes a high-order pole-soliton whose parameters are affectedby soliton-soliton interactions through the cone and soliton-radiationinteractions 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.

Long time asymptotics for the focusing nonlinear Schr\"odinger equation in the solitonic region with the presence of high-order discrete spectrum