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Long-time asymptotic behavior of the nonlocal nonlinear Schrödinger equation with finite density type initial data

In this work, we employ the $\bar{\partial}$-steepest descent method to investigate the Cauchy problem of the nonlocal nonlinear Schrödinger (NNLS) equation with finite density type initial conditions in weighted Sobolev space $\mathcal{H}(\mathbb{R})$. Based on the Lax spectrum problem, a Riemann-Hilbert problem corresponding to the original problem is constructed to give the solution of the NNLS equation with the finite density type initial boundary value condition. By developing the $\bar{\partial}$-generalization of Deift-Zhou nonlinear steepest descent method, we derive the leading order approximation to the solution $q(x,t)$ in soliton region of space-time, $\left(\frac{x}{2t}\right)=ξ$ for any fixed $ξ=\in (1,K)$($K$ is a sufficiently large real constant), and give bounds for the error decaying as $|t|\rightarrow\infty$. Based on the resulting asymptotic behavior, the asymptotic approximation of the NNLS equation is characterized with the soliton term confirmed by $N(Λ)$-soliton on discrete spectrum and the $t^{-\frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-\frac{3}{4}})$.