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

In this paper, we use the $\bar{\mathrm{\partial }}$ 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 $t\to \mathrm{\infty }$, \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 $\mathcal{O}(t^{-3\mathrm{/}4})$ which comes from the corresponding $\bar{\mathrm{\partial }}$ equation.