\section{Introduction}
Topological quantum phase is currently a mainstream of research in condensed matter physics~\cite{klitzing1980new,tsui1982two,hasan2010colloquium,wen2017colloquium,kosterlitz2017nobel}. At equilibrium, the topological phases can be characterized by nonlocal topological invariants~\cite{wen1990topological,qi2011topological} defined in ground states. This classifies the gapped band structures into distinct topological states, with great success having been achieved in study of topological insulators~\cite{fu2007topological,fu2007topologicalPRB,fu2011topological,chang2013experimental}, topological semimetals~\cite{burkov2011weyl,young2012dirac,xu2015discovery,lv2015experimental}, and topological superconductors~\cite{qi2009time,bernevig2013topological,ando2015topological,sato2017topological,zhang2018observation}. In general, a noninteracting topological phase can be greatly affected after considering many-body interaction~\cite{wang2010topological,pesin2010mott,lee2011effects,castro2011topological,araujo2013change,yao2013interaction,messer2015exploring,spanton2018observation,rachel2018interacting,viyuela2018chiral,andrews2020fractional,mook2021interaction}. For instance, the repulsive Hubbard interaction can drive a trivial insulator into a topological Mott insulator~\cite{raghu2008topological,rachel2010topological,vanhala2016topological}, while the attractive Hubbard interaction may drive a trivial phase of two-dimensional (2D) quantum anomalous Hall (QAH) system into a topological superconductor/superfluid~\cite{qi2006topological,qi2010chiral,liu2014realization,poon2018semimetal}. Nevertheless, how to accurately identify the topological phases driven by the interactions is still a fundamental issue and usually hard in experiments.

In recent years, the rapid development of quantum simulations~\cite{hofstetter2018quantum,fauseweh2021digital,monroe2021programmable}
provides new realistic platforms to explore exotic interacting physics, such as ultracold atoms in optical lattices~\cite{bloch2008many,langen2015ultracold,gross2017quantum,schreiber2015observation,smith2016many,bordia2016coupling,choi2016exploring} and superconducting qubits~\cite{ramasesh2017direct,jiang2018quantum,yan2019strongly}. A number of topological models have been realized in experiments, such as the 1D Su-Schriffer-Heeger model~\cite{su1980soliton,atala2013direct}, 1D AIII class topological insulator~\cite{liu2013manipulating,song2018observation}, 1D bosonic symmetry-protected phase~\cite{haldane1983continuum,de2019observation}, 2D Haldane model~\cite{jotzu2014experimental}, the spin-orbit coupled QAH model~\cite{wu2016realization,sun2018highly,liang2021realization}, and the 3D Weyl semimetal band~\cite{he2016realization,wang2018dirac,lu2020ideal,wang2021realization,li2021weyl}. Accordingly, the various detection schemes for the exotic topological physics are also developed, ranging from the measurements of equilibrium topological physics~\cite{liu2013detecting,hafezi2014measuring,wu2014topological,price2016measurement} to non-equilibrium quantum dynamics~\cite{vajna2015topological,hu2016dynamical,budich2016dynamical,wilson2016remnant,heyl2018dynamical,hu2020dynamical,hu2020topological,kuo2021decoherent,cai2022synthetic}. In particular, the dynamical characterization~\cite{zhang2018dynamical,zhang2019characterizing,ye2020emergent,jia2021dynamically,li2021direct,zhang2021universal} shows the correspondence between broad classes of equilibrium topological phases and the emergent dynamical topology in far-from-equilibrium quantum dynamics induced by quenching such topological systems, which brings about the systematic and high-precision schemes to detect the topological phases based on quantum dynamics and has advanced broad studies in experiment~\cite{sun2018uncover,tarnowski2019measuring,yi2019observing,wang2019experimental,xin2020quantum,ji2020quantum,niu2020simulation,chen2021digital,yu2021topological,zhang2022topological}. Nevertheless, in the current these studies have been mainly focused on the noninteracting topological systems, while the particle-particle interactions are expected to have crucial effects on the topological phases, whose detection is typically hard to achieve. For example, when quenching an interacting topological system~\cite{manmana2007strongly,polkovnikov2011colloquium,kiendl2017many,zhang2021nonequilibrium}, both the interactions and many-body states of the system evolve simultaneously after quenching, leading to complex nonlinear quantum dynamics~\cite{moeckel2008interaction,han2012evolution,foster2013quantum,dong2015dynamical} and exotic nonequilibrium phenomena~\cite{gornyi2005interacting,eisert2015quantum,yao2017discrete,peotta2021determination}.

In this paper, we propose a novel scheme based on quench dynamics to detect the mean-field topological phase diagram of an interacting Chern insulator, with nontrivial dynamical quantum physics being uncovered. Specifically, we consider a 2D QAH system in presence of a weak to intermediate Hubbard interaction which mainly induces a ferromagnetic order under the mean-field level. The interaction corrects the Zeeman coupling to an effective form and the equilibrium topological properties are fully determined by the self-consistent Zeeman potential. By quenching the system from an initial near fully polarized trivial state to a parameter regime in which the equilibrium phase is topologically nontrivial, we uncover two dynamical phenomena rendering the dynamical signals of the equilibrium mean-field phase. (i) There are two characteristic times $t_s$ and $t_c$ capturing the emergence of dynamical self-consistent particle number density and dynamical topological phase transition for the time-dependent Hamiltonian, respectively, with $t_s>t_c$ ($t_s<t_c$) occurring in the repulsive (attractive) interaction and $t_s=t_c$ on topological phase boundaries. (ii) The topological number of the equilibrium mean-field phase after quench is determined by the spin polarization of four Dirac points at the time $t_s$ in examining spin dynamics. Under the two fundamental properties, we employ the characteristic times to establish a dynamical detection scheme of topological phase in the interacting system. With feasibility of measuring the time scales in dynamical evolution, this scheme provides a simplified way to detect the mean-field phase diagram in an interacting Chern matter and may be applied to the quantum simulation experiments in the near future.

The remaining part of this paper is organized as follows.
In Sec.~\ref{sec:QAH-Hubbard model}, we introduce the QAH-Hubbard model.
In Sec.~\ref{sec:Quench dynamics}, we study the quench dynamics of the system.
In Sec.~\ref{sec:Nontrivial dynamical properties}, we reveal the nontrivial dynamical properties in quench dynamics.
In Sec.~\ref{sec:Determination of mean-field topological phases}, we determine the equilibrium mean-field topological phases via the dynamical properties.
In Sec.~\ref{sec:Experimental detection}, we establish the nonequilibrium detection scheme for the real measurements of mean-field topological phase diagram.
Finally, we summarize the main results and provide the brief discussion in Sec.~\ref{sec:Conclusion}.

\section{QAH-Hubbard model}\label{sec:QAH-Hubbard model}
Our starting point is a minimal 2D QAH model~\cite{qi2006topological,liu2014realization}, which has been recently realized in cold atoms~\cite{wu2016realization,sun2018uncover,yi2019observing,liang2021realization}, together with an attractive or repulsive on-site Hubbard interaction of strength $U$. The QAH-Hubbard Hamiltonian reads
\begin{equation}\label{eq:1}
\begin{aligned}
H & =\sum_{\mathbf{k}}C^{\dag}_{\mathbf{k}}\mathcal{H}^{(0)}_{\mathbf{k}}C_{\mathbf{k}}+U\sum_{\mathbf{j}}n_{\mathbf{j}\uparrow}n_{\mathbf{j}\downarrow},\\
\mathcal{H}^{(0)}_{\mathbf{k}} & = \mathbf{h}_{\mathbf{k}}\cdot\boldsymbol{\sigma} =\left[m_z-2t_0(\cos k_x+\cos k_y)\right]\sigma_z\\
&\quad+2t_{\text{so}}\sin k_y\sigma_x+2t_{\text{so}}\sin k_x\sigma_y,
\end{aligned}
\end{equation}
where $C_{\mathbf{k}}=(c_{\mathbf{k}\uparrow},c_{\mathbf{k}\downarrow})^{T}$ is the spinor operator of momentum $\mathbf{k}$, $n_{\mathbf{j}s}=c^{\dagger}_{\mathbf{j}s}c_{\mathbf{j}s}$ with $s=\uparrow$ or $\downarrow$ is the particle number operator at site $\mathbf{j}$, $\sigma_{x,y,z}$ are the Pauli matrices, and $m_z$ is the Zeeman coupling. Here $t_0$ and $t_{\text{so}}$ are the spin-conserved and spin-flipped hopping coefficients, respectively. In the noninteracting case, the Bloch Hamiltonian $\mathcal{H}^{(0)}_{\mathbf{k}}$ produces two energy bands $\pm e_{\mathbf{k}}=\pm\sqrt{h_{x,\mathbf{k}}^2+h_{y,\mathbf{k}}^2+h_{z,\mathbf{k}}^2}$, for which the gap can be closed at Dirac points $\mathbf{D}_i\in\{\mathbf{X}_1,\mathbf{X}_2,\boldsymbol{\Gamma},\mathbf{M}\}$ with $\mathbf{X}_1=(0,\pi)$, $\mathbf{X}_2=(\pi,0)$, $\boldsymbol{\Gamma}=(0,0)$, and $\mathbf{M}=(\pi,\pi)$ for certain Zeeman coupling. When the system is fully gapped, the corresponding band topology can be characterized by the first Chern number $\text{Ch}_1$, determining the QAH topological region $0<|m_z|<4t_0$ with $\text{Ch}_1=\text{sgn}(m_z)$ and the trivial region $|m_z|>4t_0$. This noninteracting topological property can also be captured by more intuitive and physical quantities, such as the spin textures on band inversion surfaces~\cite{zhang2018dynamical,yi2019observing} and spin polarizations on four Dirac points~\cite{liu2013detecting,wu2016realization}.

The presence of nonzero interactions can greatly affect the physics of Hamiltonian \eqref{eq:1} and may induce various interesting quantum phases at suitable interaction strength, such as the superfluid phase in attractive interaction~\cite{jia2019topological,powell2022superfluid} and the antiferromagnetic phase in repulsive interaction~\cite{esslinger2010fermi,tarruell2018quantum}. Especially for a general strong interaction, the system may host rich magnetic phases~\cite{ziegler2020correlated,ziegler2022large,tirrito2022large}. Yet, here we focus on the system at half filling~\cite{guo2011topological} within a weak to intermediate interacting regime, where the interaction mainly induces ferromagnetic order. In this case, under the mean-field level the Hubbard interaction can be rewritten as
\begin{equation}
\sum_{\mathbf{j}}n_{\mathbf{j}\uparrow}n_{\mathbf{j}\downarrow}=n_{\uparrow}\sum_{\mathbf{k}}c_{\mathbf{k} \downarrow}^{\dagger} c_{\mathbf{k} \downarrow}+n_{\downarrow}\sum_{\mathbf{k}}c_{\mathbf{k} \uparrow}^{\dagger} c_{\mathbf{k} \uparrow}-Nn_{\uparrow}n_{\downarrow}
\end{equation}
with $n_{s}=({1}/{N})\sum_{\mathbf{k}} \langle c_{\mathbf{k}s}^{\dagger} c_{\mathbf{k}s}\rangle$. Here $N$ is the total number of sites. It is clear that the nonzero interaction corrects the Zeeman coupling to an effective form
\begin{equation}\label{m_eff}
m^{\text{eff}}_z=m_z-U\frac{(n_{\uparrow}-n_{\downarrow})}{2}=m_z-Un_d,
\end{equation}
where $n_d\equiv (n_\uparrow-n_\downarrow)/2$ is the difference of density for spin-up and spin-down particles. The effective Zeeman coupling shifts the topological region to $0<|m^{\text{eff}}_z|<4t_0$ with $\text{Ch}_1=\text{sgn}(m^{\text{eff}}_z)$ in the interacting regime.

\begin{figure}[t]
\includegraphics[width=\columnwidth]{figure1.pdf}
\caption{(a)-(b) Self-consistent results of $n_{\uparrow}$ and $n_{\downarrow}$.
(c)~Mean-field phase diagram for the nonzero $U$ and $m_z$. (d)~Contour lines of $n_d$, where the topological phase boundaries (two dashed lines) have $n^{*}_d=\pm 0.4068$. Here we set $t_{\text{so}}=t_0$ and $N=60\times 60$.}
\label{Fig1}
\end{figure}

By self-consistently calculating the particle number density $n_{\uparrow (\downarrow)}$ [Figs.~{\ref{Fig1}}(a) and {\ref{Fig1}}(b)], we show the mean-field phase diagram of QAH-Hubbard model \eqref{eq:1} in Fig.~{\ref{Fig1}}(c). An important feature is that the topological phase boundaries depend on the strength of nonzero interaction and Zeeman coupling linearly~[see Appendix \ref{appendix-1}]. Indeed, this linear behavior of topological boundaries is essentially a natural consequence of the linear form of the contour lines of $n_d$ in terms of the interaction strength and Zeeman coupling, as we show in Fig.~{\ref{Fig1}}(d). On these lines, $m^{\mathrm{eff}}_{z}$ is also unchanged, since $n_{d}$ is fully determined by the effective Zeeman coupling. Specifically, we have $n^{*}_d=\pm 0.4068$ on the topological phase boundaries $|m^{\mathrm{eff}}_{z}|=4t_{0}$ for $t_{\mathrm{so}}=t_{0}$. This linear scaling form of $n_{d}$ also greatly affects the dynamical properties of the system and leads to novel phenomena in the quench dynamics.

\section{Quench dynamics}\label{sec:Quench dynamics}

It is known that the quench dynamics has been widely used in cold atoms~\cite{sun2018highly,sun2018uncover,song2019observation,wang2021realization}. The above mean-field topological phase diagram can be dynamically detected by employing the quantum quench scheme. Unlike the interaction quench in Refs.~\cite{moeckel2008interaction,foster2013quantum}, here we choose the Zeeman coupling as quench parameter, which has the following advantages: (i) The effective Zeeman coupling directly determines the topology of mean-field ground states at equilibrium. (ii) The Zeeman field in spin-orbit coupled quantum gases is controlled by the laser intensity and/or detuning and can be changed in a very short time scale, fulfilling the criterion for a sudden quench. (iii) The real experiments~\cite{lin2011spin,wang2012spin,williams2013raman} have demonstrated that both the magnitude and the sign of Zeeman field can be tuned, which is more convenient to operate.

The quench protocol is as follows. We first initialize the system into a near fully polarized state for time $t<0$ by taking a very large constant magnetization $m^{(c)}_z$ along the $\sigma_{z}$ axis but a very small $m^{(c)}_{x(y)}$ along $\sigma_{x(y)}$ axis. In this case, the effect of interaction $U$ can be ignored, and the spin of initial state is almost along the $z$ axis with only a very small component in the $x$-$y$ plane. At $t=0$, we suddenly change the Zeeman coupling $m^{(c)}_z$ to the postquench value $m_z$ and remove $m^{(c)}_{x(y)}$, then the near fully polarized state begins to evolve under the equation of motion $i \dot{\Psi}_{\mathbf{k}}(t)=\mathcal{H}_{\mathbf{k}}(t)\Psi_{\mathbf{k}}(t)$ with
\begin{equation}\label{eq:2}
\mathcal{H}_{\mathbf{k}}(t)={\left[ \begin{array}{cc}
h_{z,\mathbf{k}}+Un_\downarrow(t) & h_{x,\mathbf{k}}-ih_{y,\mathbf{k}} \\
h_{x,\mathbf{k}}+ih_{y,\mathbf{k}} & -h_{z,\mathbf{k}}+Un_{\uparrow}(t)
\end{array}
\right]},
\end{equation}
where $\Psi_{\mathbf{k}}(t)=[\chi_{\mathbf{k}}(t),\eta_{\mathbf{k}}(t)]^T$ is the instantaneous many-body state and $\mathcal{H}_{\mathbf{k}}(t)$ is the postquench Hamiltonian. To facilitate the capture of nontrivial properties in quantum dynamics, here $m^{(c)}_z$ should have the same sign as the postquench $m_z$.

Note that both the state $\Psi_{\mathbf{k}}(t)$ and Hamiltonian $\mathcal{H}_{\mathbf{k}}(t)$ are time-evolved, where the instantaneous particle number density is determined as
\begin{equation}\label{eq:3}
n_{\uparrow}(t)=\frac{1}{N} \sum_{\mathbf{k}}|\chi_\mathbf{k}(t)|^2,~
n_{\downarrow}(t)=\frac{1}{N} \sum_{\mathbf{k}}|\eta_\mathbf{k}(t)|^2.
\end{equation}
This is completely different from the noninteracting quantum quenches, where the postquench Hamiltonian keeps unchanged~\cite{zhang2018dynamical,zhang2019characterizing,ye2020emergent,jia2021dynamically,li2021direct,zhang2021universal}. We note that even if the Hamiltonian $\mathcal{H}_{\mathbf{k}}(t)$ may become steady after the long time evolution, it is still different from the equilibrium mean-field Hamiltonian with the postquench parameter $m_{z}$ and $U$, i.e.,
\begin{equation}\label{ap-1}
\mathcal{H}^{\text{MF}}_\mathbf{k}={\left[ \begin{array}{cc}
h_{z,\mathbf{k}}-Un_d+\frac{U}{2} & h_{x,\mathbf{k}}-ih_{y,\mathbf{k}} \\
h_{x,\mathbf{k}}+ih_{y,\mathbf{k}} & -h_{z,\mathbf{k}}+Un_d+\frac{U}{2}
\end{array}
\right]}.
\end{equation}
Here $n_{d}$ is the equilibrium value. Hence the detection scheme for noninteracting topological phases is inapplicable in this case.

\begin{figure}[t]
\includegraphics[width=\columnwidth]{figure2.pdf}
\caption{Numerical results for $n_{\uparrow}(t)$ (red solid lines), $n_{\downarrow}(t)$ (blue solid lines), $m^{\text{eff}}_z(t)$ (green solid lines), $\tilde{n}_{\uparrow}(t)$ (light-blue dash-dotted lines), and $\tilde{n}_{\downarrow}(t)$ (black dash-dotted lines) for different interaction and equilibrium mean-field topological phases. Two characteristic time $t_s$ and $t_c$ emerge in the dynamical time evolution. Here the postquench parameters are $\left(m_z,U\right)=(1.4,5.7)$, $(3.0,3.5)$, $(4.9,-2.7)$, and $(5.8,-4.0)$ for (a)-(d), respectively. We set $t_{\text{so}}=t_0=1$, $m^{(c)}_{x}=m_z$, $m^{(c)}_y=0$, and $m^{(c)}_z=100$.
}
\label{Fig2}
\end{figure}

Next we shall propose a feasible dynamical scheme to detect the mean-field topological phases of $\mathcal{H}^{\text{MF}}_{\mathbf{k}}$ based on the above novel dynamical time evolution. By defining the time-evolved quantity $n_{d}(t)\equiv[n_{\uparrow}(t)-n_{\downarrow}(t)]/2$, we notice that the time-dependent effective Zeeman coupling
\begin{equation}
m^{\text{eff}}_z(t)=m_z-Un_d(t)
\end{equation}
determines the topology $W(t)$ of the instantaneous postquench Hamiltonian $\mathcal{H}_{\mathbf{k}}(t)$ through the relation $W(t)=\text{sgn}[m^{\text{eff}}_z(t)]$ for $0<|m^{\text{eff}}_z(t)|<4t_0$ and $W(t)=0$ otherwise. Since the equilibrium mean-field Hamiltonian $\mathcal{H}^{\text{MF}}_{\mathbf{k}}$ is actually determined by the self-consistent particle number density $n_{\uparrow(\downarrow)}$ (i.e., the equilibrium effective Zeeman coupling $m^{\text{eff}}_z$; see the discussions in Sec.~\ref{sec:QAH-Hubbard model}), a key idea of identifying the equilibrium mean-field topological phases is to find a characteristic time $t_{s}$ such that
\begin{equation}
        n_{\uparrow(\downarrow)}(t_{s})=n_{\uparrow(\downarrow)}.
\end{equation}
Namely, the evolved particle number density $n_{\uparrow(\downarrow)}(t)$ at the time $t_s$ captures the equilibrium self-consistent $n_{\uparrow(\downarrow)}$, giving the dynamical self-consistent particle number density $n_{\uparrow(\downarrow)}(t_{s})$. Then the instantaneous Hamiltonian $\mathcal{H}_{\mathbf{k}}(t_s)$ is the same as the equilibrium mean-field Hamiltonian $\mathcal{H}^{\text{MF}}_{\mathbf{k}}$, and the time-dependent effective Zeeman coupling $m^{\mathrm{eff}}_{z}(t_{s})$ determines the corresponding topology. We also introduce another characteristic time $t_{c}$ such that
\begin{equation}
|m^{\text{eff}}_z(t_c)|=4t_0,
\end{equation}
characterizing the critical time of the dynamical topological phase transition of $\mathcal{H}_{\mathbf{k}}(t)$. Since Hamiltonian (\ref{ap-1}) behaves as a single particle model under mean-field level, $\mathcal{H}_{\mathbf{k}}(t_s)$ can essentially describe the ground state properties of the mean-field Hamiltonian $\mathcal{H}^{\text{MF}}_{\mathbf{k}}$ and its topological phase transition is obtained by $\mathcal{H}_{\mathbf{k}}(t_c)$. With this, identifying the topology of $\mathcal{H}_{\mathbf{k}}(t_s)$ as well as that of $\mathcal{H}^{\text{MF}}_{\mathbf{k}}$ is equivalent to comparing the characteristic time $t_{s}$ and $t_{c}$ in the quench dynamics [i.e., comparing $m^{\mathrm{eff}}_{z}(t_{s})$ and $m^{\mathrm{eff}}_{z}(t_{c})$].

In Fig.~\ref{Fig2}, we show the time evolution of $n_{\uparrow(\downarrow)}(t)$ and $m^{\text{eff}}_z(t)$ and confirm the existence of characteristic time $t_{s}$ and $t_{c}$. Since the system is initially prepared in the near fully polarized state, the short-time evolution of $n_{d}(t)$ increases (decays) from $n_{d}(t=0)\approx -0.5$ ($+0.5$), when the postquench Zeeman coupling $m_z$ is positive (negative). We also observe that the two characteristic time both emerge in the short-time evolution, and their relative relation ($t_{s}<t_{c}$ or $t_{s}>t_{c}$) behaves differently for different interaction and equilibrium mean-field phases. In the following, we show how to identify the equilibrium mean-field topological phases via $t_{s}$ and $t_{c}$ in detail.

\section{Nontrivial dynamical properties}\label{sec:Nontrivial dynamical properties}

In this section, we determine the characteristic time $t_{s}$ and $t_{c}$ entirely from the quantum dynamics without knowing the postquench equilibrium mean-field value of $n_{\uparrow(\downarrow)}$, and provide the analytical results under certain conditions. The nontrivial dynamical properties in this interacting system are uncovered.

\subsection{Dynamical self-consistent particle number density and characteristic time $t_s$}

Since the equilibrium mean-field value of $n_{\uparrow(\downarrow)}$ is in general unknown, we need to first figure out how to determine the characteristic time $t_{s}$ in the quantum dynamics. A notable fact is that $n_{\uparrow(\downarrow)}$ is self-consistently obtained for the mean-field Hamiltonian $\mathcal{H}^{\text{MF}}_{\mathbf{k}}$, and the time-evolved particle number density $n_{\uparrow(\downarrow)}(t)$ can be considered as a specific path for updating the mean-field parameters. For this, we introduce another set of time-dependent particle number density
\begin{equation}\label{eq:4}
\tilde{n}_{\uparrow}(t)=\frac{1}{N} \sum_{\mathbf{k}}|\tilde{\chi}_\mathbf{k}(t)|^2,\quad
\tilde{n}_{\downarrow}(t)=\frac{1}{N} \sum_{\mathbf{k}}|\tilde{\eta}_\mathbf{k}(t)|^2
\end{equation}
for the eigenvector $\tilde{\Psi}_{\mathbf{k}}(t) =[\tilde{\chi}_{\mathbf{k}}(t),\tilde{\eta}_{\mathbf{k}}(t)]^T$ of $\mathcal{H}_{\mathbf{k}}(t)$ with negative energy, which can be obtained once we know the instantaneous particle number density $n_{\uparrow(\downarrow)}(t)$. We see that $\mathcal{H}_{\mathbf{k}}(t)$ is self-consistent when the two sets of time-dependent particle number density coincide, giving the characteristic time
\begin{equation}\label{eq:5}
t_s\equiv \text{min}\{t|n_{\uparrow(\downarrow)}(t)=\tilde{n}_{\uparrow(\downarrow)}(t)\},
\end{equation}
at which we have $n_{\uparrow(\downarrow)}=n_{\uparrow(\downarrow)}(t_s)=\tilde{n}_{\uparrow(\downarrow)}(t_s)$, as shown in Fig.~\ref{Fig2}. This gives the dynamical self-consistent particle number density $n_{\uparrow(\downarrow)}(t_{s})$, capturing the equilibrium self-consistent $n_{\uparrow(\downarrow)}$.

Note that here only the time-dependent particle number density becomes self-consistent, while the instantaneous wavefunction is in general not, which has subtle difference from the equilibrium mean-field Hamiltonian. This also implies that different instantaneous states may lead to the same $n_{\uparrow(\downarrow)}(t_s)$, and there may be multiple time points fulfilling the self-consistent condition. We hereby choose the smallest time point as the characteristic time $t_{s}$ due to the consideration of short-time evolution [see Fig.~\ref{Fig2}(a)]. Also, the characteristic time $t_s$ shall definitely appear in the short-time evolution, since the $n_{\uparrow(\downarrow)}(t)$ increases or decays in this regime and the state gradually reaches a steady state after $t>t_s$.

Although the explicit form of the characteristic time $t_{s}$ is in general very complex, we can obtain a analytical result in the short-time evolution for a weak interaction strength $U$ or $|Un_{d}|$, which is given by [see Appendix~\ref{appendix-2}]
\begin{equation}\label{a-tn}
t_s\approx \sqrt{\frac{N_1}{N_0+\sqrt{N_2+N_3 (m_z-n_dU)^2}}}
\end{equation}
with $N_0=12t_{\text{so}}^2$, $N_1=3(1\pm 2n_d)$, $N_2=-28t_{\text{so}}^4(\pm 10n_d-1)-72t_0^2t_{\text{so}}^2(\pm 2n_d+1)$, and $N_3=-28t_{\text{so}}^2(\pm 2n_d+ 1)$, where the symbol $\pm$ corresponds to the sign of the postquench $m_z$. From this result, we observe that the characteristic time $t_{s}$ is fixed by $n_{d}$. Since the contour lines of $n_{d}$ have the linear form of interaction strength $U$ and Zeeman coupling $m_{z}$, then $t_{s}$ has a similar linear scaling, which fully matches with the numerical results [see Fig.~\ref{Fig3}(a)].

\begin{figure}[t]
\includegraphics[width=\columnwidth]{figure3.pdf}
\caption{(a)-(b) Analytical results of $t_s$ and $t_c$, where $t^*=t_s=t_c=0.176$ is on the topological boundaries (black solid lines). The dark-brown regions present $t_s>0.35$ in (a) and $t_c=+\infty$ in (b) respectively, while the dark-purple regions in (b) give $t_c=0$. These regions actually correspond to the complex values of $t_s$ and $t_c$ since a large truncation $\mathcal{O}(U^3n_d^3)$ is taken. After reducing the truncation error, $t_s$ gives the real value but $t_c$ still gives complex value due to no solution. (c) Sign of $t_s-t_c$, where the shaded area is the equilibrium topological phase region. (d) Analytical results of $t_s$, $n_d$ and $\text{Re}[t_c]$ at $n_d\approx -0.09$, showing the linear scaling of $t_s$ and $n_d$. Here we set $t_{\text{so}}=t_0=1$.}
\label{Fig3}
\end{figure}

\subsection{Dynamical topological phase transition and characteristic time $t_c$}

We now determine the characteristic time $t_{c}$ from the dynamical topological phase transition of $\mathcal{H}_{\mathbf{k}}(t)$. Since the initial state is prepared into a near fully polarized state for $t<0$ and has $n_d(t=0)\approx\pm 0.5$ for negative or positive postquench $m_{z}$, the absolute value of $n_{d}(t)$ in general decays in the time evolution and approaches a steady value. We find that when the interaction strength satisfies $U>-8t_0$, the time-dependent effective Zeeman coupling $m^{\text{eff}}_z(t)$ passes through only one phase transition point $4t_{0}$ for $m_{z}>0$ or $-4t_{0}$ for $m_{z}<0$, although the corresponding crossing may occurs multiple times. Therefore, $\mathcal{H}_{\mathbf{k}}(t)$ can only change from the trivial (topological) regime at $t=0$ to the topological (trivial) regime for $t>0$, when the interaction is repulsive (attractive). The characteristic time $t_{c}$, characterizing the critical time of the dynamical topological phase transition of $\mathcal{H}_{\mathbf{k}}(t)$, can be naturally defined as
\begin{equation}\label{eq:6}
t_{c}\equiv\text{min}\{t|m^{\text{eff}}_z(t)=\pm 4t_0\}
\end{equation}
Correspondingly, we have $t_c=+\infty$ and $0$ ($t_c=0$ and $+\infty$) in the repulsive (attractive) interaction regime for $|m^{\text{eff}}_z(t)|>4t_0$ and $|m^{\text{eff}}_z(t)|<4t_0$, respectively. The numerical results for $m_z>0$ are shown in Fig.~\ref{Fig2}, where $m^{\text{eff}}_z(t)$ only passes though the phase transition point $m^{\text{eff}}_z(t_c)=4t_0$ and decays (increases) for the repulsive (attractive) interaction in the short time region. This feature actually can be described by the analytical result for $n_d(t)$ in Appendix \ref{appendix-2}, where we have $\dot{m}_z^{\text{eff}}(t)\gtrless 0$ for $Um_z\lessgtr 0$ due to $\dot{m}_z^{\text{eff}}(t)=-U\dot{n}_d(t)$.

Like Eq.~\eqref{a-tn}, we can also obtain a analytical result for $t_c$ in the short-time evolution [see Appendix \ref{appendix-2}]:
\begin{equation}\label{a-tp}
t_c\approx \sqrt{\frac{P_0-\sqrt{{P_2-3P_1{{(\pm m_z-4t_0)}/{U}}}}}{P_1}}
\end{equation}
with $P_0=6t_{\text{so}}^2$, $P_1=4t_{\text{so}}^2(5t_{\text{so}}^2+19t_0^2)$, and $P_2=6t_{\text{so}}^2(t_{\text{so}}^2-19t_0^2)$. Compared with $t_s$, the characteristic time $t_c$ presents a completely different scaling form in terms of the Zeeman coupling and interaction strength, expect for the topological phase boundaries~[see Fig.~\ref{Fig3}(b)]. Particularly, the finite $t_c$ only appears around the topological phase boundaries, since $\mathcal{H}_{\mathbf{k}}(t)$ always remains topological or trivial when it is far away from the phase boundaries; see Fig.~\ref{Fig5}(b).

\subsection{Linear scaling of characteristic time on phase boundaries}

We now give a special time $t^{*}$ on the topological phase boundaries, which shows the linear scaling for the Zeeman coupling and interaction. By taking $m_z-Un_d=\pm 4 t_0$ for the above two analytical characteristic time, we obtain
\begin{equation}\label{tst}
t^{*}=t_s=t_c\approx\sqrt{\frac{B_1}{B_0+\sqrt{B_2}}},
\end{equation}
where $B_0=6t_\text{so}^2$, $B_1=3(1\pm 2n^{*}_d)/2$, and $B_2=-114(1\pm 2n^*_d)t_0^2t_\text{so}^2-6(\pm 10n^{*}_d-1)t_\text{so}^4$; see Appendix~\ref{appendix-2}. This is a necessary condition for the topological transition of $\mathcal{H}^{\text{MF}}_{\mathbf{k}}$. Especially, we have $t^*\propto 1/t_0$ for a strong coupling with $t_{\text{so}}=t_0$, which gives $t^*= 0.176$ when $t_{\text{so}}=t_0=1$, matching the numerical result $t^*=0.180$ [see Fig.~\ref{Fig5}(b)]. With the time $t_{s}$, $t_{c}$, and $t^{*}$, the equilibrium mean-field topological phases is next characterized.

\section{Determination of mean-field topological phases}\label{sec:Determination of mean-field topological phases}

Based on the properties of $n_d(t)$ and $m^{\text{eff}}_z(t)$, the system shall have $t_s>t_c$ ($t_s<t_c$) in the quench dynamics for topological phases with the repulsive (attractive) interaction, if $\mathcal{H}_\mathbf{k}(t)$ is trivial (topological) at $t=0$. The correspondence between the characteristic time and the equilibrium topological properties is given by
\begin{equation}\label{eq:7}
|\text{Ch}_1|=\left\{
\begin{aligned}
1 & , & \text{for}
\begin{cases}
        t_s>t^*>t_c,~U>0\\
        t_s<t^*<t_c,~U<0
\end{cases}\\
0 & , & \text{for}
\begin{cases}
        t^*<t_s<t_c,~U>0\\
        t^*>t_s>t_c,~U<0
\end{cases}\\
\end{aligned}
\right.
\end{equation}
where $t^*$ is for the gap closing at Dirac points $\boldsymbol{\Gamma}$ or $\mathbf{M}$. The above formula provides a convenient method to determine the mean-field topological phase diagram by observing the short-time evolution of the system after quantum quench. Moreover, both the numerical results [Fig.~\ref{Fig2}] and the analytical results [Fig.~\ref{Fig3}(c)] confirm this scheme, especially for weak interaction $U$ or small $|Un_d|$, as we show in Fig.~\ref{Fig3}(d).

\begin{figure}[t]
\includegraphics[width=\columnwidth]{figure4.pdf}
\caption{(a) Schematic diagram for the motion of the near fully polarized spin at Dirac points $\mathbf{D}_i$. The clockwise (counterclockwise) motion of the spin implies the upward (downward) polarization of Hamiltonian at these points. (b) The polarization directions at Dirac points $\mathbf{D}_i$ determine the topology of postquench Hamiltonian. (c) Topological number $W(t)$ for $\mathcal{H}_{\mathbf{k}}(t)$. We have $\text{Ch}_1=1$ for the phase with $(m_z,U)=(2,2)$ due to $t_{c}=0<t_{s}=0.165$, while $\text{Ch}_1=0$ for the phase with $(m_z,U)=(3,4)$ due to $t_{c}=9.485>t_{s}=0.230$. The insets show the changes of polarization directions at Dirac points $\mathbf{D}_i$ for $\mathcal{H}_{\mathbf{k}}(t)$. Here we set $t_{\text{so}}=t_0=1$, $m^{(c)}_{x}=m_z$, $m^{(c)}_y=0$, and $m^{(c)}_z=100$.
}
\label{Fig4}
\end{figure}

To further determine the Chern number of the equilibrium mean-field topological phases, we consider the features of spin dynamics, which is described by the modified Landau-Lifshitz equation in this interacting system~\cite{gilbert2004phenomenological}
\begin{equation}
\dot{\mathbf{S}}(\mathbf{k},t)=\mathbf{S}(\mathbf{k},t)\times 2\mathbf{h}(\mathbf{k},t)
\end{equation}
with
\begin{equation}
\begin{split}
& S_x(\mathbf{k},t)=[\chi_\mathbf{k}(t)\eta^{*}_\mathbf{k}(t)+\chi^{*}_\mathbf{k}(t)\eta_\mathbf{k}(t)]/2,\\
& S_y(\mathbf{k},t)=i[\chi_\mathbf{k}(t)\eta^{*}_\mathbf{k}(t)-\chi^{*}_\mathbf{k}(t)\eta_\mathbf{k}(t)]/2,\\
& S_z(\mathbf{k},t)=[|\chi_\mathbf{k}(t)|^2-|\eta_\mathbf{k}(t)|^2]/2.\\
 \end{split}
\end{equation}
We show in the following that the topology number $W(t)$ of the time-dependent postquench Hamiltonian $\mathcal{H}_{\mathbf{k}}(t)$ can be determined by the dynamical properties of $S_z$ at four Dirac points $\mathbf{D}_i$. The Chern number $\text{Ch}_1$ for mean-field Hamiltonian $\mathcal{H}^{\mathrm{MF}}_{\mathbf{k}}$ is then obtained by $W$ at the characteristic time $t_s$.

First, for the near fully polarized initial state, the spin at Dirac points $\mathbf{D}_i$ will move around the corresponding polarization direction of $\mathcal{H}_{\mathbf{k}}(t)$ after the quantum quench. When the polarization direction is reversed, e.g., from upward to downward, the motion of spin will be reversed at the same time, e.g., from clockwise to counterclockwise; see Fig.~\ref{Fig4}(a). For this, we can use the motion of spins to determine the polarization direction of $\mathcal{H}_{\mathbf{k}}(t)$ at Dirac points. Second, since the polarization direction is associated with the parity eigenvalue of the occupied eigenstate, the topology of $\mathcal{H}_{\mathbf{k}}(t)$ can be identified from the polarization directions at four Dirac points~\cite{liu2013detecting}, as shown in Fig.~\ref{Fig4}(b). For instance, the polarization directions at four Dirac points are the same for an initial trivial Hamiltonian $\mathcal{H}_{\mathbf{k}}(t)$. Once it enters the topological regime, the motion of spin at $\boldsymbol{\Gamma}$ or $\mathbf{M}$ point will be reversed, manifesting the change of the corresponding polarization direction. This can be captured by the quantity $\text{sgn}[\dot{S}_z(\mathbf{D}_i,t)]$, as shown in Fig.~\ref{Fig5}(d). Based on the above points, we define the following time-dependent dynamical invariant for $\mathcal{H}_{\mathbf{k}}(t)$:
\begin{equation}
(-1)^{\nu(t)}=\prod_{i}\text{sgn}[\dot{S}_z(\mathbf{D}_i,t)]
\end{equation}
with $\nu(t)=1$ for the topological regime $|m^{\text{eff}}_z(t)|<4t_0$ and $\nu(t)=0$ for the trivial regime $|m^{\text{eff}}_z(t)|>4t_0$, respectively. With this, the topological number $W(t)$ of $\mathcal{H}_\mathbf{k}(t)$ and the Chern number of $\mathcal{H}^{\text{MF}}_{\mathbf{k}}$ can be exactly given as
\begin{equation}\label{ch}
W(t)=\frac{\nu(t)}{2}\sum_{i}\text{sgn}[\dot{S}_z(\mathbf{D}_i,t)],~~\text{Ch}_1=W(t_s).
\end{equation}
As an example, the numerical results in Fig.~\ref{Fig4}(c) clearly show the nontrivial topology with $\text{Ch}_1=1$ and the trivial topology with $\text{Ch}_1=0$ for the postquench system with $(m_z,U)=(2,2)$ and $(3,4)$, respectively.

\begin{figure}[t]
\includegraphics[width=\columnwidth]{figure5.pdf}
\caption{(a) Mean-field phase diagram determined by the characteristic time $t_c$ and $t^*$, where the postquench parameters $A$-$F$ (stars) are chosen on the topological phase boundaries with $(m_z,U)=(1.0,7.38)$, $(2.0,4.92)$, $(3.0,2.46)$, $(4.0,0)$, $(5.0,-2.46)$, and $(6.0,-4.92)$. (b) Dynamical evolution of $m^{\text{eff}}_z(t)$ and $|\dot{n}_{\uparrow(\downarrow)}(t)|$ for $A$-$F$, where $t_c=t^{*}=0.180$ (upper) and $t_b=t^{*}=0.180$ (lower) with the same scaling. (c) Dynamical evolution of $|\dot{n}_{\uparrow(\downarrow)}(t)|$, where $t_b=0.165$ for $(m_z,U)=(3.0,4.0)$ and $(5.0,-1.0)$ while $t_b=0.230$ for $(m_z,U)=(2.0,2.0)$ and $(4.0,-4.0)$.
(d) Sign of $\dot{S}_z(\mathbf{D}_i,t)$ at four Dirac points. We have $\text{Ch}_1=1$ for the topological phase with $(m_z,U)=(2.0,2.0)$ due to $t_c=0<t^*$ and $\text{Ch}_1=0$ for the trivial phase with $(m_z,U)=(3.0,4.0)$ due to $t_c=9.485>t^*$. Here we set $t_{\text{so}}=t_0=1$, $m^{(c)}_{x}=m_z$, $m^{(c)}_y=0$, and $m^{(c)}_z=100$.
}
\label{Fig5}
\end{figure}

\section{Experimental detection}\label{sec:Experimental detection}

Based on the above analysis of characteristic time, in this section we propose an experimentally feasible scheme to detect the mean-field topological phase diagram.

In real quench experiments, the characteristic time $t_{c}$ can be directly detected from the spin dynamics by measuring $n_{\uparrow(\downarrow)}(t)$. The challenging part is how to measure the characteristic time $t_{s}$, since it does not have any measurable quantities but only corresponds to the self-consistent results of $\mathcal{H}_\mathbf{k}(t)$. To circumvent this difficulty, we instead choose to compare $t^*$ and $t_{c}$ based on Eq.~(\ref{ch}) to detect the mean-field topological phase diagram; see Fig.~\ref{Fig5}(a). The time $t^*$ can be directly measured by defining an auxiliary time
\begin{equation}\label{eq:7s}
t_b\equiv \text{min}\{t|\ddot{n}_{\uparrow(\downarrow)}(t)=0\},
\end{equation}
at which the short-time evolution of $n_{\uparrow(\downarrow)}(t)$ changes fastestly, i.e., $|\dot{n}_{\uparrow(\downarrow)}(t)|$ is maximal. Although the auxiliary time $t_{b}$ is larger (smaller) than $t^{*}$ in the topological (trivial) regime [see Fig.~\ref{Fig5}(c)], we indeed have $t_{b}=t^*$ on the topological phase boundaries, from which an analytical result for $t^*$ is obtained as
\begin{equation}\label{eq:t*}
t^*\approx \sqrt{\frac{3}{Q_0+Q_1}}
\end{equation}
with $Q_0=\sqrt{-135t_0^2+606t_0^2t_\text{so}^2+57t_\text{so}^4}$ and $Q_1=57t_0^2+15t_\text{so}^2$, consistent with Eq.~(\ref{tst}). Particularly, we have $t^*\approx 0.178$ for $t_0=t_\text{so}=1$. The auxiliary time $t_{b}$ simplifies the detection of the mean-field topological phase diagram.

The dynamical detection scheme is as follows. First, due to the linear scaling of $t^*$ in terms of $U$ and $m_{z}$ [see Fig.~\ref{Fig3}(c)], which is fixed on the mean-field topological phase boundaries, we can determine the special time $t^{*}$ from the noninteracting ($U=0$) quench dynamics. In this case, the Zeeman coupling is quenched from a large $m^{(c)}_{z}$ and a small $m^{(c)}_{x(y)}$ to the postquench parameters on phase boundaries, i.e., $m_z=\pm 4t_0$ and $m_{x(y)}=0$. Then the time-dependent particle number density $n_{\uparrow(\downarrow)}(t)$ in the short-time evolution is measured. By examining $\dot{n}_{\uparrow(\downarrow)}(t)$, we obtain $t^*=t_{b}$. Second, we turn to the interacting quench dynamics with postquench $m_{z}$ to obtain the fully mean-field phase diagram. Here $m^{(c)}_{x(y)}$ remains to be small and $m_{x(y)}=0$. By measuring $t_{c}$ from the quantum dynamics, the mean-field topological phase diagram is obtained by comparing it with $t^*$; see Fig.~\ref{Fig5}(a). Finally, the Chern number of mean-field topological phases can also be obtained via Eq.~(\ref{ch}). Since there is only one topological phase transition point, the topological number $W(t)$ has the same value at $t_s$ and $t^*$. Hence we have
\begin{equation}
\text{Ch}_1\simeq W(t^*).
\end{equation}
With this scheme, the mean-field phase diagram of the interacting Chern insulator can be completely determined, which paves the way to experimentally study topological phases in interacting systems and discover new phases.

\section{Conclusion and discussion}\label{sec:Conclusion}

In conclusion, we have studied the 2D QAH model with a weak-to-intermediate Hubbard interaction by performing quantum quenches. A nonequilibrium detection scheme for the mean-field topological phase diagram is proposed by observing the characteristic time $t_{s}$ and $t_{c}$, which capture the emergence of dynamical self-consistent particle number density and dynamical topological phase transition for the time-dependent postquench Hamiltonian, respectively. Two nontrivial dynamical properties are further uncovered. (i) After quenching the Zeeman coupling from the trivial regime to the topological regime, $t_s>t_c$ ($t_s<t_c$) is observed for the repulsive (attractive) interaction, and $t_s=t_c$ occurs on the topological phase boundaries. (ii) The Chern number of the postquench mean-field topological phase can be determined by the spin dynamics of four Dirac points at characteristic time $t_s$. Our results provide feasible schemes to identify the mean-field topological phases in an interacting Chern insulator and can be applied to quantum simulation experiments in the near future.

The present dynamical scheme can be applied to the Chern insulator with the weak-to-intermediate interaction. On the other hand, when the general strong interactions are considered, the system may host more abundant magnetic phases~\cite{ziegler2020correlated,ziegler2022large,tirrito2022large}. Generalizing these dynamical properties to the related studies and further identifying the rich topological phases would be an interesting and worthwhile work in the future.

\section*{ACKNOWLEDGEMENT}
We acknowledge the valuable discussions with Sen Niu and Ting Fung Jeffrey Poon. This work was supported by National Natural Science Foundation of China (Grants No. 11825401 and No. 11921005), the National Key R\&D Program of China (Project No. 2021YFA1400900), Strategic Priority Research Program of the Chinese Academy of Science (Grant No. XDB28000000), and by the Open Project of Shenzhen Institute of Quantum Science and Engineering (Grant No.SIQSE202003). Lin Zhang also acknowledges support from Agencia Estatal de Investigaci{\'o}n (the R\&D project CEX2019-000910-S, funded by MCIN/AEI/10.13039/501100011033, Plan National FIDEUA PID2019-106901GB-I00, FPI), Fundaci{\'o} Privada Cellex, Fundaci{\'o} Mir-Puig, and Generalitat de Catalunya (AGAUR Grant No. 2017 SGR 1341, CERCA program).


\section{Introduction}

In \cite{bib:AbMu}, Ablowitz and Musslimani introduced the nonlocal
nonlinear Schr\"odinger equation and got its explicit solutions by
inverse scattering. Quite a lot of work were done after that for
this equation and other
equations.\cite{bib:AbMu2,bib:AbMu3,bib:Gadz,bib:HuangLing,bib:Khare,
bib:YJK,bib:LiXu,bib:Lou,bib:Zhu,bib:Sarma}

In \cite{bib:Fokashigh}, Fokas studied high dimensional equations and
introduced the nonlocal Davey-Stewartson I equation
\begin{equation}
   \begin{array}{l}
   \D\I u_t=u_{xx}+u_{yy}+2\sigma u^2\bar u^*+2uw_y,\\
   \D w_{xx}-w_{yy}=2\sigma(u\bar u^*)_{y},\\
   \end{array} \label{eq:EQ_DS}
\end{equation}
where $w$ satisfies $\bar w^*=w$. Here $\bar f(x,y,t)=f(-x,y,t)$ for
a function $f$, $*$ refers to complex conjugation. The solution of
(\ref{eq:EQ_DS}) is PT symmetric in the sense that if
$(u(x,y,t),w(x,y,t))$ is a solution of (\ref{eq:EQ_DS}), then so is
$(u^*(-x,y,-t), w^*(-x,y,-t))$. This leads to a conserved density
$u\bar u^*$, which is invariant under $x\to -x$ together with
complex conjugation.

As is known, the usual Davey-Stewartson I equation does not possess
a Darboux transformation in differential form. Instead, it has a
binary Darboux transformation in integral
form.\cite{bib:Salle,bib:MSbook} However, for the nonlocal
Davey-Stewartson I equation (\ref{eq:EQ_DS}), we can construct a
Darboux transformation in differential form. Like the nonlocal
equations in 1+1 dimensions, the solutions may have singularities.
Starting from the seed solutions which are zero and an exponential
function of $t$, we prove that the derived solutions can be globally
defined and bounded for all $(x,y,t)\in\hr^3$ if the parameters are
suitably chosen. Unlike the usual Davey-Stewartson I equation where
localized solutions are dromion solutions if the seed solution is
zero,\cite{bib:BLP,bib:Herm,bib:Hiet} the derived solutions here are
soliton solutions in the sense that there are $n$ peaks in the
solutions obtained from a Darboux transformation of degree $n$. If
the seed solution is an exponential function of $t$, the norms of
the derived solutions change a lot along some straight lines. We
call them ``line dark soliton'' solutions.

In Section~\ref{sect:LPDT} of this paper, the Lax pair for the
nonlocal Davey-Stewartson I equation is reviewed and its symmetries
are considered. Then the Darboux transformation is constructed and
the explicit expressions of the new solutions are derived. In
Section~\ref{sect:soliton} and Section~\ref{sect:darksoliton}, the
soliton solutions and ``line dark soliton'' solutions are
constructed respectively. The globalness, boundedness and the
asymptotic behaviors of those solutions are discussed.

\section{Lax pair and Darboux transformation}\label{sect:LPDT}

Consider the $2\times 2$ linear system
\begin{equation}
   \begin{array}{l}
   \D\Phi_x=\tau J\Phi_y+\tau
   P\Phi=\tau\left(\begin{array}{cc}
   1&0\\0&-1\end{array}\right)\Phi_y
   +\tau\left(\begin{array}{cc}0&-u\\v&0\end{array}\right)\Phi,\\
   \D\Phi_t=-2\I\tau^2J\Phi_{yy}-2\I\tau^2P\Phi_y+\I Q\Phi\\
   \D\qquad=-2\I\tau^2\left(\begin{array}{cc}
   1&0\\0&-1\end{array}\right)\Phi_{yy}
   -2\I\tau^2\left(\begin{array}{cc}
   0&-u\\v&0\end{array}\right)\Phi_y\\
   \D\qquad+\I\tau\left(\begin{array}{cc}
   -\tau uv-\tau w_y-w_x&u_x+\tau u_y\\
   v_x-\tau v_y&\tau uv+\tau w_y-w_x\end{array}\right)\Phi
   \end{array}\label{eq:LP}
\end{equation}
where $\tau=\pm 1$, $u,v,w$ are functions of $(x,y,t)$. The
compatibility condition $\Phi_{xt}=\Phi_{tx}$ gives the evolution
equation
\begin{equation}
   \begin{array}{l}
   \D\I u_t=u_{xx}+\tau^2u_{yy}+2\tau^2u^2v+2\tau^2uw_y,\\
   \D-\I v_t=v_{xx}+\tau^2v_{yy}+2\tau^2uv^2+2\tau^2vw_y,\\
   \D w_{xx}-\tau^2w_{yy}=2\tau^2(uv)_{y}.\\
   \end{array}\label{eq:eq}
\end{equation}

When $\tau=1$, $v=\sigma\bar u^*$ $(\sigma=\pm 1)$, (\ref{eq:eq})
becomes the nonlocal Davey-Stewartson I equation (\ref{eq:EQ_DS}).
The Lax pair (\ref{eq:eq}) becomes
\begin{equation}
   \begin{array}{l}
   \D\Phi_x=U(\partial)\Phi\overset\triangle
   =J\Phi_y+P\Phi
   =\left(\begin{array}{cc}1&0\\0&-1\end{array}\right)\Phi_y
   +\left(\begin{array}{cc}
   0&-u\\\sigma\bar u^*&0\end{array}\right)\Phi,\\
   \D\Phi_t=V(\partial)\Phi\overset\triangle
   =-2\I J\Phi_{yy}-2\I P\Phi_y+\I Q\Phi\\
   \D\qquad=-2\I\left(\begin{array}{cc}
   1&0\\0&-1\end{array}\right)\Phi_{yy}
   -2\I\left(\begin{array}{cc}
   0&-u\\\sigma\bar u^*&0\end{array}\right)\Phi_y\\
   \D\qquad+\I\left(\begin{array}{cc}
   -\sigma u\bar u^*-w_y-w_x&u_x+u_y\\
   \sigma(\bar u^*)_x-\sigma(\bar u^*)_y
   &\sigma u\bar u^*+w_y-w_x
   \end{array}\right)\Phi
   \end{array}\label{eq:LP_DS}
\end{equation}
where $\D \partial=\frac{\partial}{\partial y}$. Here $U(\partial)$
implies that $U$ is a differential operator with respect to $y$.

The coefficients in the Lax pair (\ref{eq:LP_DS}) satisfies
\begin{equation}
   \bar J^*=-KJK^{-1},\quad
   \bar P^*=-KPK^{-1},\quad
   \bar Q^*=-KQK^{-1}\label{eq:sym_coef}
\end{equation}
where $\D K=\left(\begin{array}{cc}0&\sigma\\1&0\end{array}\right)$.
Here $M^*$ refers to the complex conjugation (without transpose) of
a matrix $M$. (\ref{eq:sym_coef}) implies
\begin{equation}
   \overline{U(\partial)^*}=-KU(\partial)K^{-1},\quad
   \overline{V(\partial)^*}=KV(\partial)K^{-1}.
\end{equation}
Hence we have
\begin{lemma}\label{lemma:sym}
If $\D\Phi=\left(\begin{array}{c}\xi\\\eta\end{array}\right)$ is a
solution of (\ref{eq:LP_DS}), then so is $\D
K\bar\Phi^*=\left(\begin{array}{c}\sigma\bar\eta^*\\\bar
\xi^*\end{array}\right)$.
\end{lemma}

By Lemma~\ref{lemma:sym}, take a solution
$\D\left(\begin{array}{c}\xi\\\eta\end{array}\right)$ of
(\ref{eq:LP_DS}) and let $\D
H=\left(\begin{array}{cc}\xi&\sigma\bar\eta^*\\
\eta&\bar\xi^*\end{array}\right)$, then $G(\partial)=\partial-S$
with $S=H_yH^{-1}$ gives a Darboux
transformation.\cite{bib:ZZX,bib:GHZbook} Written explicitly,
\begin{equation}
   S=\frac 1{\xi\bar\xi^*-\sigma\eta\bar\eta^*}
   \left(\begin{array}{cc}\bar\xi^*\xi_y-\sigma\eta(\bar\eta^*)_y
   &\sigma\xi(\bar\eta^*)_y-\sigma\bar\eta^*\xi_y\\
   \bar\xi^*\eta_y-\eta(\bar\xi^*)_y&\xi(\bar\xi^*)_y-\sigma\bar\eta^*\eta_y.
   \end{array}\right).
\end{equation}
$G(\partial)$ also keeps the symmetries (\ref{eq:sym_coef})
invariant. After the action of $G(\partial)$, $(u,w)$ is transformed
to $(\widetilde u,\widetilde w)$ by
\begin{equation}
   \begin{array}{l}
   G(\partial)U(\partial)+G_x(\partial)
   =\widetilde U(\partial)G(\partial),\quad
   G(\partial)V(\partial)+G_t(\partial)
   =\widetilde V(\partial)G(\partial).
   \end{array} \label{eq:Gtransf}
\end{equation}
That is,
\begin{equation}
   \begin{array}{l}
   \D\widetilde u=u+2\sigma\frac{\bar\eta^*\xi_y-\xi(\bar\eta^*)_y}
   {\bar\xi^*\xi-\sigma\bar\eta^*\eta},\\
   \D\widetilde w=w+2\frac{(\bar\xi^*\xi-\sigma\bar\eta^*\eta)_y}
   {\bar\xi^*\xi-\sigma\bar\eta^*\eta}.
   \end{array}\label{eq:DTuv1}
\end{equation}

The Darboux transformation of degree $n$ is given by a matrix-valued
differential operator
\begin{equation}
   G(\partial)=\partial^n+G_1\partial^{n-1}+\cdots+G_n
\end{equation}
of degree $n$ which is determined by
\begin{equation}
   G(\partial)H_j=0\quad(j=1,\cdots,n)\label{eq:GH=0}
\end{equation}
for $n$ matrix solutions $H_j$ $(j=1,\cdots,n)$ of (\ref{eq:LP}). By
comparing the coefficients of $\partial^j$ in (\ref{eq:Gtransf}),
the transformation of $(P,Q)$ is
\begin{equation}
   \widetilde P=P-[J,G_1],\quad
   \widetilde Q=Q+2[J,G_2]-2[JG_1-P,G_1]+4JG_{1,y}-2nP_y.
   \label{eq:PQ}
\end{equation}

Rewrite (\ref{eq:GH=0}) as
\begin{equation}
   \partial^nH_j+G_1\partial^{n-1}H_j+\cdots+G_nH_j=0\quad
   (j=1,\cdots,n),
\end{equation}
then
\begin{equation}
   \begin{array}{l}
   \D\left(\begin{array}{cccc}G_1&G_2&\cdots&G_n\end{array}\right)
   \left(\begin{array}{cccc}\partial^{n-1}H_1&\partial^{n-1}H_2
   &\cdots&\partial^{n-1}H_n\\
   \partial^{n-2}H_1&\partial^{n-2}H_2&\cdots&\partial^{n-2}H_n\\
   \vdots&\vdots&&\vdots\\
   H_1&H_2&\cdots&H_n\end{array}\right)\\
   \D=\left(\begin{array}{cccc}-\partial^nH_1&-\partial^nH_2
   &\cdots&-\partial^nH_n\end{array}\right).
   \end{array}
\end{equation}
Write $\D H_j=\left(\begin{array}{cc}h_{11}^{(j)}&h_{12}^{(j)}\\
h_{21}^{(j)}&h_{22}^{(j)}\end{array}\right)$. By reordering the rows
and columns, we have
\begin{equation}
   \begin{array}{l}
   \D\left(\begin{array}{cccccc}(G_1)_{11}&\cdots&(G_n)_{11}
   &(G_1)_{12}&\cdots&(G_n)_{12}\\
   (G_1)_{21}&\cdots&(G_n)_{21}
   &(G_1)_{22}&\cdots&(G_n)_{22}\end{array}\right)W=-R\\
   \end{array}\label{eq:G12b}
\end{equation}
where $\D W=(W_{jk})_{1\le j,k\le 2}$, $\D R=(R_{jk})_{1\le j,k\le
2}$,
\begin{equation}
   W_{jk}=\left(\begin{array}{cccc}
   \partial^{n-1}h_{jk}^{(1)}&\partial^{n-1}h_{jk}^{(2)}
   &\cdots&\partial^{n-1}h_{jk}^{(n)}\\
   \partial^{n-2}h_{jk}^{(1)}&\partial^{n-2}h_{jk}^{(2)}
   &\cdots&\partial^{n-2}h_{jk}^{(n)}\\
   \vdots&\vdots&&\vdots\\
   h_{jk}^{(1)}&h_{jk}^{(2)}&\cdots&h_{jk}^{(n)}\\
   \end{array}\right),\label{eq:Wjk}
\end{equation}
\begin{equation}
   R_{jk}=\left(\begin{array}{cccc}
   \partial^nh_{jk}^{(1)}&\partial^nh_{jk}^{(2)}
   &\cdots&\partial^nh_{jk}^{(n)}\end{array}\right)\quad(j,k=1,2).
   \label{eq:Rjk}
\end{equation}
Solving $G$ from (\ref{eq:G12b}), we get the new solution of the
equation (\ref{eq:EQ_DS}) from (\ref{eq:PQ}). Especially,
\begin{equation}
   \widetilde u=u+2(G_1)_{12}.\label{eq:trans_u}
\end{equation}

\section{Soliton solutions}\label{sect:soliton}

\subsection{Single soliton solutions}\label{subsect:soliton1}

Let $u=0$, then $\D\Phi=\left(\begin{array}{c}
\xi\\\eta\end{array}\right)$ satisfies
\begin{equation}
   \begin{array}{l}
   \D\Phi_x=\left(\begin{array}{cc}
   1&0\\0&-1\end{array}\right)\Phi_y,\quad
   \D\Phi_t=-2\I\left(\begin{array}{cc}
   1&0\\0&-1\end{array}\right)\Phi_{yy}.
   \end{array} \label{eq:LP_DSeg}
\end{equation}
Take a special solution
\begin{equation}
   \begin{array}{l}
   \D \xi=\E{\lambda x+\lambda y-2\I\lambda^2t}
   +\E{-\lambda^*x-\lambda^*y-2\I\lambda^{*2}t},\\
   \D \eta=a\E{\lambda x-\lambda y+2\I\lambda^2t}
   +b\E{-\lambda^*x+\lambda^*y+2\I\lambda^{*2}t},
   \end{array}\label{eq:1stnxieta}
\end{equation}
where $\lambda,a,b$ are complex constants. (\ref{eq:DTuv1}) gives
the explicit solution
\begin{equation}
   \begin{array}{l}
   \D\widetilde u
   \D =\frac{2\sigma\lambda_R(a^*-b^*)}{D}
   \E{2\I\lambda_I y-4\I(\lambda_R^2-\lambda_I^2)t}
   \end{array}\label{eq:u1eg}
\end{equation}
of (\ref{eq:EQ_DS}) where
\begin{equation}
   \begin{array}{l}
   D=(2-\sigma|a|^2-\sigma|b|^2)\cosh(2\lambda_R y
   +8\lambda_R\lambda_It)\\
   \qquad+\sigma(|a|^2-|b|^2)\sinh(2\lambda_R y+8\lambda_R\lambda_It)\\
   \qquad+2(1-\sigma\re(ab^*))\cosh(2\lambda_R x)
   -2\I\sigma\im(ab^*)\sinh(2\lambda_Rx).
   \end{array}
\end{equation}
Here $z_R=\re z$ and $z_I=\im z$ for a complex number $z$. Since
$\widetilde w$ is looked as an auxiliary function in
(\ref{eq:EQ_DS}), hereafter we will study mainly the behavior of
$\widetilde u$.

Note that $\widetilde u$ is global if $|a|<1$ and $|b|<1$ since $\re
D>0$ in this case. Moreover, the peak moves in the velocity
$(v_x,v_y)=(0,-4\lambda_I)$.

\begin{remark}
The solution (\ref{eq:u1eg}) may have singularities when the
parameters are not chosen suitably, say, when $\sigma=-1$, $a=1$,
$b=-2$, or $\sigma=1$, $a=2$, $b=1/4$.
\end{remark}

Figure~1 shows a $1$ soliton solution with parameters $\sigma=-1$,
$t=20$, $\lambda=0.07-1.5\I$, $a=0.2$, $b=0.1\I$.

\begin{figure}\begin{center}
\scalebox{1.6}{\includegraphics[150,100]{1soliton.ps}}\label{fig:stn1}
\caption{$|\widetilde u|$ of a $1$ soliton solution.}
\end{center}\end{figure}

\subsection{Multiple soliton solutions}\label{subsect:solitonn}

For an $n\times n$ matrix $M$, define $\D||M||=\sup_{x\in
\smallhc^n,||x||=1}||Mx||$ where $||\cdot||$ is the standard
Hermitian norm in $\hc^n$. The following facts hold obviously.

(i) $||MN||\le ||M||\,||N||$.

(ii) Each entry $M_{jk}$ of $M$ satisfies $|M_{jk}|\le||M||$.

(iii) $|\det M|\le||M||^n$.

(iv) If $||M||<1$, then $\D||(I+M)^{-1}||\le(1-||M||)^{-1}$.

(v) If $||M||<1$, then $\D|\det(I+M)|\ge(1-||M||)^n$.

Now we construct explicit solutions according to (\ref{eq:trans_u}).
As in (\ref{eq:1stnxieta}), take
\begin{equation}
   \begin{array}{l}
   \D\xi_k=\E{\lambda_k(x+y)-2\I\lambda_k^2t}
   +\E{-\lambda_k^*(x+y)-2\I\lambda_k^{*2}t},\\
   \D\eta_k=a_k\E{\lambda_k(x-y)+2\I\lambda_k^2t}
   +b_k\E{-\lambda_k^*(x-y)+2\I\lambda_k^{*2}t},
   \label{eq:Hjeg}
   \end{array}
\end{equation}
then $W_{jk}$'s in (\ref{eq:Wjk}) and $R_{jk}$'s in (\ref{eq:Rjk})
are
\begin{equation}\fl
   \begin{array}{l}
   (W_{11})_{jk}=\lambda_k^{n-j}\ee_{k+}
   +(-\lambda_k^*)^{n-j}\ee_{k+}^{*-1},\quad
   (W_{12})_{jk}=\sigma a_k^*(-\lambda_k^*)^{n-j}\ee_{k+}^{*-1}
   +\sigma b_k^*\lambda_k^{n-j}\ee_{k+},\\
   (W_{21})_{jk}=a_k(-\lambda_k)^{n-j}\ee_{k-}
   +b_k(\lambda_k^*)^{n-j}\ee_{k-}^{*-1},\quad
   (W_{22})_{jk}=(\lambda_k^*)^{n-j}\ee_{k-}^{*-1}
   +(-\lambda_k)^{n-j}\ee_{k-},\\
   (R_{11})_{1k}=\lambda_k^n\ee_{k+}
   +(-\lambda_k^*)^n\ee_{k+}^{*-1},\quad
   (R_{12})_{1k}=\sigma a_k^*(-\lambda_k^*)^n\ee_{k+}^{*-1}
   +\sigma b_k^*\lambda_k^n\ee_{k+},\\
   (R_{21})_{1k}=a_k(-\lambda_k)^n\ee_{k-}
   +b_k(\lambda_k^*)^n\ee_{k-}^{*-1},\quad
   (R_{22})_{1k}=(\lambda_k^*)^n\ee_{k-}^{*-1}
   +(-\lambda_k)^n\ee_{k-}\\
   (j,k=1,\cdots,n)
   \end{array}\label{eq:W22}
\end{equation}
where
\begin{equation}
   \ee_{k\pm}=\E{\lambda_k(x\pm y)\mp 2\I\lambda_k^2t}.\label{eq:ee}
\end{equation}
However, temporarily, we assume $\ee_{k+},\ee_{k-}$ $(k=1,\cdots,n)$
are arbitrary complex numbers rather that (\ref{eq:ee}) holds.

Denote $\D L=\diag((-1)^{n-1},(-1)^{n-2},\cdots,-1,1)$,
\begin{equation}
   F=(\lambda_k^{n-j})_{1\le j,k\le n},\quad
   f=(\lambda_1^n,\cdots,\lambda_n^n),
   \label{eq:V}
\end{equation}
\begin{equation}
   \begin{array}{l}
   A=\diag(a_1,\cdots,a_n),\quad
   B=\diag(b_1,\cdots,b_n),
   \end{array}\label{eq:AB}
\end{equation}
\begin{equation}
   \begin{array}{l}
   \D E_\pm=\diag(\ee_{1\pm},\cdots,\ee_{n\pm}).
   \end{array}
\end{equation}
Then
\begin{equation}
   W=\left(\begin{array}{cc}FE_++LF^*E^{*-1}_{+}
   &\sigma FB^*E_++\sigma LF^*A^*E^{*-1}_{+}\\
   LFAE_-+F^*BE^{*-1}_{-}
   &LFE_-+F^*E^{*-1}_{-}\end{array}\right),
   \label{eq:W}
\end{equation}
\begin{equation}
   R=\left(\begin{array}{cc}fE_++(-1)^nf^*E^{*-1}_{+}
   &\sigma fB^*E_++\sigma(-1)^nf^*A^*E^{*-1}_{+}\\
   (-1)^nfAE_-+f^*BE^{*-1}_{-}
   &(-1)^nfE_-+f^*E^{*-1}_{-}\end{array}\right).
   \label{eq:R}
\end{equation}
By using the identity
\begin{equation}
   \left(\begin{array}{cc}A&B\\C&D\end{array}\right)^{-1}
   =\left(\begin{array}{cc}A^{-1}+A^{-1}B\Delta^{-1}CA^{-1}&-A^{-1}B\Delta^{-1}\\
   -\Delta^{-1}CA^{-1}&\Delta^{-1}\end{array}\right)
\end{equation}
for a block matrix where $\Delta=D-CA^{-1}B$, (\ref{eq:G12b}) gives
\begin{equation}
   \Big((G_1)_{12},\cdots,(G_n)_{12}\Big)=-(RW^{-1})_{12}
   =-(R_{12}-R_{11}W_{11}^{-1}W_{12})\cir W^{-1} \label{eq:G12WR}
\end{equation}
where
\begin{equation}
   \begin{array}{l}
   \D\cir{W}=W_{22}-W_{21}W_{11}^{-1} W_{12}\\
   \D\qquad=L\Big(FE_-+LF^*E^{*-1}_{-}
   -\sigma(FAE_-+LF^*BE^{*-1}_{-})\cdot\\
   \D\qquad\quad\cdot(FE_++LF^*E^{*-1}_{+})^{-1}
   (FB^*E_++LF^*A^*E^{*-1}_{+})\Big).
   \end{array}
   \label{eq:W0}
\end{equation}

\begin{lemma}\label{lemma:lbound}
Suppose $a_j$ and $b_j$ are nonzero complex constants with
$|a_j|<1$, $|b_j|<1$ $(j=1,\cdots,n)$, $\kappa_1,\cdots,\kappa_n$
are nonzero real constants with $|\kappa_j|\ne|\kappa_k|$
$(j,k=1,\cdots,n;\,j\ne k)$, then there exist positive constants
$\delta$, $C_1$ and $C_2$, which depend on $a_j$'s, $b_j$'s and
$\kappa_j$'s, such that $|\det W|\ge C_1$ and
\begin{equation}
   \begin{array}{l}
   \D|(G_1)_{12}|\le C_2\max_{1\le k\le n}
   \frac{|\ee_{k+}|}{1+|\ee_{k+}|^2}
   \max_{1\le k\le n}\frac{|\ee_{k-}|}{1+|\ee_{k-}|^2}
   \end{array}
\end{equation}
hold whenever $|\lambda_j-\I\kappa_j|<\delta$ and $\ee_{j\pm}\in\hc$
$(j=1,\cdots,n)$.
\end{lemma}

\demo Denote $F^{-1}LF^*=I+Z$, then $Z=0$ if
$\lambda_1,\cdots,\lambda_n$ are all purely imaginary. From
(\ref{eq:W}) and (\ref{eq:W0}),
\begin{equation}
   \det W=\det(FE_++LF^*E^{*-1}_{+})\det\cir W,
   \label{eq:WWo}
\end{equation}
\begin{equation}
   \cir{W}=L(FE_-+LF^*E^{*-1}_{-})(I-\sigma\chi_-\chi_+)
   \label{eq:cirW}
\end{equation}
where
\begin{equation}
   \begin{array}{l}
   \chi_+=(FE_++LF^*E^{*-1}_{+})^{-1}(FB^*E_+
   +LF^*A^*E^{*-1}_{+})\\
   \qquad=\Xi_{1+}\Xi_{0+}^{-1}+\Xi_{0+}^{-1}(I+ZE^{*-1}_{+}
   \Xi_{0+}^{-1})^{-1}ZE^{*-1}_{+}
   (A^*-\Xi_{1+}\Xi_{0+}^{-1}),\\
   \chi_-=(FE_-+LF^*E^{*-1}_{-})^{-1}(FAE_-
   +LF^*BE^{*-1}_{-})\\
   \qquad=\Xi_{1-}\Xi_{0-}^{-1}+\Xi_{0-}^{-1}(I+ZE^{*-1}_{-}
   \Xi_{0-}^{-1})^{-1}ZE^{*-1}_{-}
   (B-\Xi_{1-}\Xi_{0-}^{-1}),\\
   \end{array}
\end{equation}
\begin{equation}
   \Xi_{0\pm}=E_\pm+E^{*-1}_{\pm},\quad
   \Xi_{1-}=AE_-+BE^{*-1}_{-},\quad
   \Xi_{1+}=B^*E_++A^*E^{*-1}_{+}.
\end{equation}

Let $\D c_0=\max_{1\le k\le n}\{|a_k|,|b_k|\}<1$. Suppose
$\D||Z||<\frac{1-c_0}2$, then we have the following estimates.
\begin{equation}
   ||A||\le c_0<1,\quad
   ||B||\le c_0<1,\label{eq:esti_AB}
\end{equation}
\begin{equation}
   ||E_{\pm}\Xi_{0\pm}^{-1}||\le 1,\quad
   ||E^{*-1}_{\pm}\Xi_{0\pm}^{-1}||\le 1,
\end{equation}
\begin{equation}
   ||\Xi_{0\pm}||\ge2,\quad
   ||\Xi_{0\pm}^{-1}||\le\frac 12,\quad
   ||\Xi_{1\pm}\Xi_{0\pm}^{-1}||\le c_0<1,
\end{equation}
\begin{equation}
   \begin{array}{l}
   \D||E^{*-1}_{+}(\Xi_{1+}\Xi_{0+}^{-1}-A^*)||
   =\max_{1\le k\le n}\frac{|a_k-b_k|\,
   |\ee_{k+}|}{1+|\ee_{k+}|^2}\le1,\\
   \D||E^{*-1}_{-}(\Xi_{1-}\Xi_{0-}^{-1}-B)||
   =\max_{1\le k\le n}\frac{|a_k-b_k|\,
   |\ee_{k-}|}{1+|\ee_{k-}|^2}\le1,
   \end{array}
\end{equation}
\begin{equation}
   ||(I+Z E^{*-1}_{\pm}\Xi_{0\pm}^{-1})^{-1}||\le||(1-||Z||\,
   ||E^{*-1}_{\pm}\Xi_{0\pm}^{-1}||)^{-1}||
   \le(1-||Z||)^{-1}\le 2.
\end{equation}
Hence $||\chi_\pm-\Xi_{1\pm}\Xi_{0\pm}^{-1}||\le||Z||$,
\begin{equation}
   ||\chi_\pm||\le c_0+||Z||\le\frac{1+c_0}2<1.
   \label{eq:esti_chi}
\end{equation}

Denote
\begin{equation}
   \pi_0=|\det F|\Big|_{\lambda_j=\smallI\kappa_j\atop{j=1,\cdots,n}},\quad
   \pi_1=||F^{-1}||\Big|_{\lambda_j=\smallI\kappa_j\atop{j=1,\cdots,n}},\quad
   \pi_2=||f||\Big|_{\lambda_j=\smallI\kappa_j\atop{j=1,\cdots,n}}.
\end{equation}
Clearly, $\pi_0$, $\pi_1$, $\pi_2$ are all positive since $\D\det
F|_{\lambda_j=\smallI\kappa_j\atop{j=1,\cdots,n}}$ is a Vandermonde
determinant. By the continuity, there exists $\delta>0$ such that
$\D\frac{\pi_0}2\le|\det F|\le 2\pi_0$, $||F^{-1}||\le2\pi_1$,
$||f||\le2\pi_2$, and $\D||F^{-1}LF^*-I||=||Z||<\frac{1-c_0}2$
whenever $|\lambda_j-\I\kappa_j|<\delta$. (\ref{eq:WWo}) and
(\ref{eq:cirW}) lead to
\begin{equation}
   \begin{array}{l}
   |\det W|\!=\!|\det F|^2\,|\det\Xi_{0+}|\,|
   \det(I+ZE^{*-1}_{+}\Xi_{0+}^{-1})|\cdot\\
   \qquad\cdot|\det\Xi_{0-}|\,|\det(I+ZE^{*-1}_{-}\Xi_{0-}^{-1})|\,
   |\det(1-\sigma\chi_-\chi_+)|\\
   \D\qquad\ge\pi_0^2(1-||Z||)^{2n}(1-||\chi_+||\,||\chi_-||)^n
   \ge\pi_0^2\Big(\frac{1+c_0}{2}\Big)^{2n}\Big(1-\Big(\frac{1+c_0}2\Big)^2\Big)^{n},
   \end{array}
\end{equation}
which is a uniform positive lower bound for any $\ee_{j\pm}\in\hc$
$(j=1,\cdots,n)$ when $|\lambda_j-\I\kappa_j|<\delta$.

By (\ref{eq:W}), (\ref{eq:R}), (\ref{eq:G12WR}) and (\ref{eq:W0}),
\begin{equation}
   \begin{array}{l}
   \D((G_1)_{12},(G_2)_{12},\cdots,(G_n)_{12})
   =-(R_{12}-R_{11}W_{11}^{-1}W_{12})\cir W^{-1}\\
   \D\qquad=-\sigma fE_+\Xi_{0+}^{-1}(I+ZE^{*-1}_{+}\Xi_{0+}^{-1})^{-1}
   (I+Z)(B^*-A^*)E^{*-1}_{+}\cir W^{-1}\\
   \qquad\quad-\sigma(-1)^nf^*E^{*-1}_{+}\Xi_{0+}^{-1}
   (I+ZE^{*-1}_{+}\Xi_{0+}^{-1})^{-1}
   (A^*-B^*)E_+\cir W^{-1}\\
   \D\qquad=-\sigma\Big(f-(-1)^nf^*+(fE_+\Xi_{0+}^{-1}+(-1)^nf^*E^{*-1}_{+}\Xi_{0+}^{-1})
   (I+ZE^{*-1}_{+}\Xi_{0+}^{-1})^{-1}Z\Big)\cdot\\
   \qquad\quad\cdot(B^*-A^*)E_+E^{*-1}_{+}\Xi_{0+}^{-1}
   (I-\sigma\chi_-\chi_+)^{-1}\Xi_{0-}^{-1}(I+ZE^{*-1}_{-}
   \Xi_{0-}^{-1})^{-1}F^{-1}L^{-1}.
   \end{array}\label{eq:G12expr}
\end{equation}
Here we have used
$I+Z=(I+ZE^{*-1}_{+}\Xi_{0+}^{-1})+ZE_+\Xi_{0+}^{-1}$. Hence, by
using (\ref{eq:esti_AB})--(\ref{eq:esti_chi}),
\begin{equation}
   \begin{array}{l}
   \D|(G_1)_{12}|\le
   8(||f||+||f^*||)||F^{-1}||\,
   ||(I-\sigma\chi_-\chi_+)^{-1}||\,||\Xi_{0+}^{-1}||\,||\Xi_{0-}^{-1}||\\
   \D\qquad\le 64\pi_1\pi_2\Big(1-\Big(\frac{1+c_0}2\Big)^2\Big)^{-1}
   \max_{1\le k\le n}\frac{|\ee_{k+}|}{1+|\ee_{k+}|^2}
   \max_{1\le k\le n}\frac{|\ee_{k-}|}{1+|\ee_{k-}|^2}.
   \end{array}
\end{equation}
The lemma is proved.

Now we consider the solutions of the nonlocal Davey-Stewartson I
equation. That is, we consider the case where $\ee_{j\pm}$'s are
taken as (\ref{eq:ee}).

\begin{theorem}
Suppose $a_j$ and $b_j$ are nonzero complex constants with
$|a_j|<1$, $|b_j|<1$ $(j=1,\cdots,n)$, $\kappa_1,\cdots,\kappa_n$
are nonzero real constants with $|\kappa_j|\ne|\kappa_k|$
$(j,k=1,\cdots,n;\,j\ne k)$, then there exists a positive constant
$\delta$ such that the following results hold for the derived
solution $\widetilde u=2(G_1)_{12}$ of the nonlocal Davey-Stewartson
I equation when $\re\lambda_j\ne 0$ and
$|\lambda_j-\I\kappa_j|<\delta$ $(j=1,\cdots,n)$.

\noindent(i) $\widetilde u$ is defined globally for $(x,y,t)\in\hr^3$.

\noindent(ii) For fixed $t$, $\widetilde u$ tends to zero
exponentially as $(x,y)\to 0$.

\noindent(iii) Let $y=\widetilde y+vt$ and keep $(x,\widetilde y)$
bounded, then $\widetilde u\to 0$ as $t\to\infty$ if
$v\ne-4\lambda_{kI}$ for all $k$.
\end{theorem}

\demo We have known that $\widetilde u$ is a solution of the
nonlocal Davey-Stewartson equation in Section~\ref{sect:LPDT}.

(i) According to Lemma~\ref{lemma:lbound}, $|\det W|$ has a uniform
positive lower bound. Hence $\widetilde u$ is defined globally.

\vskip6pt(ii) When $x\ge 0$ and $y\ge 0$,
\begin{equation}
   \begin{array}{l}
   |\ee_{k+}|\ge\E{\lambda_{kR}\sqrt{x^2+y^2}+4\lambda_{kR}\lambda_{kI}t}
   \hbox{ if }\lambda_{kR}>0,\\
   |\ee_{k+}|\le\E{-|\lambda_{kR}|\sqrt{x^2+y^2}+4\lambda_{kR}\lambda_{kI}t}
   \hbox{ if }\lambda_{kR}<0.
   \end{array}
\end{equation}
Hence $\D\max_{1\le k\le n}\frac{|\ee_{k+}|}{1+|\ee_{k+}|^2}$ tends
to zero exponentially when $x\ge 0$, $y\ge 0$ and $(x,y)\to\infty$.

Likewise, we have
\begin{equation}
   \begin{array}{l}
   |\ee_{k-}|\le\E{-\lambda_{kR}\sqrt{x^2+y^2}-4\lambda_{kR}\lambda_{kI}t}
   \hbox{ if }x\le 0,\;y\ge 0,\;\lambda_{kR}>0,\\
   |\ee_{k-}|\ge\E{|\lambda_{kR}|\sqrt{x^2+y^2}-4\lambda_{kR}\lambda_{kI}t}
   \hbox{ if }x\le 0,\;y\ge 0,\;\lambda_{kR}<0,\\
   |\ee_{k+}|\le\E{-\lambda_{kR}\sqrt{x^2+y^2}+4\lambda_{kR}\lambda_{kI}t}
   \hbox{ if }x\le 0,\;y\le 0,\;\lambda_{kR}>0,\\
   |\ee_{k+}|\ge\E{|\lambda_{kR}|\sqrt{x^2+y^2}+4\lambda_{kR}\lambda_{kI}t}
   \hbox{ if }x\le 0,\;y\le 0,\;\lambda_{kR}<0,\\
   |\ee_{k-}|\ge\E{\lambda_{kR}\sqrt{x^2+y^2}-4\lambda_{kR}\lambda_{kI}t}
   \hbox{ if }x\ge 0,\;y\le 0,\;\lambda_{kR}>0,\\
   |\ee_{k-}|\le\E{-|\lambda_{kR}|\sqrt{x^2+y^2}-4\lambda_{kR}\lambda_{kI}t}
   \hbox{ if }x\ge 0,\;y\le 0,\;\lambda_{kR}<0.
   \end{array}
\end{equation}
Lemma~\ref{lemma:lbound} implies that $\widetilde u\to 0$
exponentially as $(x,y)\to\infty$.

\vskip6pt(iii)
\begin{equation}
   |\ee_{k\pm}|=\E{\lambda_{kR}(x\pm\widetilde y)
   \pm\lambda_{kR}(v+4\lambda_{kI})t}.
\end{equation}
If $v\ne-4\lambda_{kI}$ for all $k=1,\cdots,n$, then either
$\ee_{k+}\to 0$ or $\ee_{k+}\to\infty$ for all $k=1,\cdots,n$ when
$t\to\infty$. Lemma~\ref{lemma:lbound} implies that $\widetilde u\to
0$ when $t\to\infty$. The theorem is proved.

A $3$ soliton solution is shown in Figure~2 where the parameters are
$\sigma=-1$, $t=20$, $\lambda_1=0.07-1.5\I$, $\lambda_2=0.05+2\I$,
$\lambda_3=0.1+\I$, $a_1=0.2$, $a_2=0.1\I$, $a_3=0.1$, $b_1=0.1\I$,
$b_2=-0.2$, $b_3=-0.2$. The figure of the solution appears similarly
if $\sigma$ is changed to $+1$, although it is not shown here.

\begin{figure}\begin{center}
\scalebox{1.6}{\includegraphics[150,100]{3soliton.ps}}\label{fig:stn3}
\caption{$|\widetilde u|$ of a $3$ soliton solution.}
\end{center}\end{figure}

\section{``Line dark soliton'' solutions}\label{sect:darksoliton}

\subsection{Single ``line dark soliton'' solutions}\label{subsect:darksoliton1}

Now we take
\begin{equation}
   u=\rho\E{-2\I\sigma|\rho|^2t},\quad
   w=0
\end{equation}
as a solution of (\ref{eq:EQ_DS}) where $\rho$ is a complex
constant. The Lax pair (\ref{eq:LP_DS}) has a solution
\begin{equation}
   \begin{array}{l}
   \D\left(\begin{array}{c}\E{\alpha(\lambda)x+\beta(\lambda)y+\gamma(\lambda)t}\\
   \D\frac{\lambda}{\rho}
   \E{\alpha(\lambda)x+\beta(\lambda)y
   +(\gamma(\lambda)+2\I\sigma|\rho|^2)t}\end{array}\right),
   \end{array}\label{eq:LPsndark}
\end{equation}
where
\begin{equation}
   \begin{array}{l}
   \D\alpha(\lambda)=\frac 12\Big(\frac{\sigma|\rho|^2}{\lambda}-\lambda\Big),\quad
   \beta(\lambda)=\frac 12\Big(\frac{\sigma|\rho|^2}{\lambda}+\lambda\Big),\\
   \D\gamma(\lambda)=\I(\alpha(\lambda)^2-2\alpha(\lambda)\beta(\lambda)-\beta(\lambda)^2)
   =\I\lambda^2-\frac\I 2
   \Big(\frac{\sigma|\rho|^2}{\lambda}+\lambda\Big)^2,
   \end{array}
\end{equation}
$\lambda$ is a complex constant. Note that $\alpha(-\lambda^*)=-(\alpha(\lambda))^*$,
$\beta(-\lambda^*)=-(\beta(\lambda))^*$, $\gamma(-\lambda^*)=-(\gamma(\lambda))^*$.

Now take $\D\Phi=\left(\begin{array}{c}\xi\\\eta\end{array}\right)$
where
\begin{equation}
   \begin{array}{l}
   \D \xi=\E{\alpha x+\beta y+\gamma t}+\E{-\alpha^*x-\beta^*y-\gamma^*t},\\
   \D \eta=\frac{\lambda}{\rho}\E{\alpha x+\beta y
   +(\gamma+2\I\sigma|\rho|^2)t}-\frac{\lambda^*}{\rho}
   \E{-\alpha^*x-\beta^*y-(\gamma^*-2\I\sigma|\rho|^2)t}.
   \end{array}\label{eq:darkxieta}
\end{equation}
Here $\alpha=\alpha(\lambda)$, $\beta=\beta(\lambda)$,
$\gamma=\gamma(\lambda)$. This $\Phi$ is a linear combination of the
solutions of form (\ref{eq:LPsndark}). Then (\ref{eq:DTuv1}) gives
the new solution
\begin{equation}
   \widetilde u=\rho\E{-2\I\sigma|\rho|^2t}
   \frac{\D
   \D \frac{\lambda^*}{\lambda}c_1\E{2\beta_Ry+2\gamma_Rt}
   +\frac{\lambda}{\lambda^*}c_1\E{-2\beta_Ry-2\gamma_Rt}-c_2\E{2\alpha_Rx}
   -c_2^*\E{-2\alpha_Rx}}
   {\D c_1(\E{2\beta_Ry+2\gamma_Rt}+\E{-2\beta_Ry-2\gamma_Rt})+c_2\E{2\alpha_Rx}
   +c_2^*\E{-2\alpha_Rx}}
\end{equation}
of the nonlocal Davey-Stewartson I equation where
\begin{equation}
   \begin{array}{l}
   \D c_1=1-\sigma\frac{|\lambda|^2}{|\rho|^2},\quad
   c_2=1+\sigma\frac{\lambda^2}{|\rho|^2}
   \end{array}
\end{equation}
This solution is smooth for all $(x,y,t)\in\hr^3$ if
$|\lambda|<|\rho|$.

Especially, if $\lambda$ is real, then $\gamma_R=0$, so we get a
standing wave solution.

Figure~3 shows a $1$ ``line dark soliton'' solution with parameters
$\sigma=-1$, $t=10$, $\rho=1$, $\lambda=0.3+0.1\I$. The figure on
the right describes the same solution but is upside down.

\begin{figure}\begin{center}
\scalebox{1.24}{\includegraphics[340,100]{1darkstn.ps}}\label{fig:1darkstn}
\caption{$|\widetilde u|$ of a $1$ ``line dark soliton'' solution.}
\end{center}\end{figure}

\subsection{Multiple ``line dark soliton'' solutions}

Now we take $n$ solutions
\begin{equation}
   \begin{array}{l}
   \D \xi_k=\E{\alpha_k x+\beta_k y+\gamma_k t}+\E{-\alpha_k^*x-\beta_k^*y-\gamma_k^*t},\\
   \D \eta_k=\frac{\lambda_k}{\rho}\E{\alpha_k x+\beta_k y
   +(\gamma_k+2\I\sigma|\rho|^2)t}-\frac{\lambda_k^*}{\rho}
   \E{-\alpha_k^*x-\beta_k^*y-(\gamma_k^*-2\I\sigma|\rho|^2)t}
   \end{array}
\end{equation}
of form (\ref{eq:darkxieta}) to get multiple ``line dark soliton''
solutions where
\begin{equation}
   \begin{array}{l}
   \D\alpha_k=\frac 12\Big(\frac{\sigma|\rho|^2}{\lambda_k}-\lambda_k\Big),\quad
   \beta_k=\frac12\Big(\frac{\sigma|\rho|^2}{\lambda_k}+\lambda_k\Big),\\
   \gamma_k=\I(\alpha_k^2-2\alpha_k\beta_k-\beta_k^2).
   \end{array}\label{eq:abc}
\end{equation}
Similar to (\ref{eq:W}) and (\ref{eq:R}), $W$ and $R$ in
(\ref{eq:G12b}) are
\begin{equation}\fl
   W=\left(\begin{array}{cc}FE_++LF^*E^{*-1}_{+}
   &-\sigma\rho^{*-1}\E{-\I\phi}(LF\Lambda E_--F^*\Lambda ^*E^{*-1}_{-})\\
   \rho^{-1}\E{\I\phi}(F\Lambda E_+-LF^*\Lambda^*E^{*-1}_{+})
   &LFE_-+F^*E^{*-1}_{-}
   \end{array}\right),\label{eq:Wdark}
\end{equation}
\begin{equation}\fl
   R=\left(\begin{array}{cc}fE_++(-1)^nf^*E^{*-1}_{+}
   &-\sigma\rho^{*-1}\E{-\I\phi}((-1)^nf\Lambda E_--f^*\Lambda ^*E^{*-1}_{-})\\
   \rho^{-1}\E{\I\phi}(f\Lambda E_+-(-1)^nf^*\Lambda^*E^{*-1}_{+})
   &(-1)^nfE_-+f^*E^{*-1}_{-}
   \end{array}\right),\label{eq:Rdark}
\end{equation}
where
\begin{equation}
   \begin{array}{l}
   E_\pm=\diag(\ee_{k\pm})_{k=1,\cdots,n},\quad
   F=(\beta_k^{n-j})_{1\le j,k\le n},\quad
   f=(\beta_1^n,\cdots,\beta_n^n),\\
   \D\alpha_k=\frac 12\Big(\frac{\sigma|\rho|^2}{\lambda_k}-\lambda_k\Big),\quad
   \beta_k=\frac12\Big(\frac{\sigma|\rho|^2}{\lambda_k}+\lambda_k\Big),\\
   \gamma_k=\I(\alpha_k^2-2\alpha_k\beta_k-\beta_k^2),
   \end{array}\label{eq:EFf}
\end{equation}
$\D \Lambda=\diag(\lambda_k)_{1\le k\le n}$,
$\phi=2\sigma|\rho|^2t$. Moreover, $\ee_{k\pm}=\E{\alpha_kx\pm
\beta_ky\pm\gamma_kt}$. However, as in the soliton case, we suppose
temporarily that $\ee_{k\pm}$'s are arbitrary complex numbers.

\begin{lemma}\label{lemma:lbounddark}
Suppose $\kappa_1,\cdots,\kappa_n$ are distinct nonzero real
numbers, then there exist positive constants $\rho_0$, $\delta$,
$C_1$ and $C_2$, which depend on $\kappa_j$'s, such that $|\det
W|\ge C_1$ and $|(G_1)_{12}|\le C_2$ hold whenever $|\rho|>\rho_0$,
$|\lambda_j-\I\kappa_j|<\delta$ and $e_{j\pm}\in\hc$
$(j=1,\cdots,n)$.
\end{lemma}

\demo Denote $F^{-1}LF^*=I+Z$, then $Z=0$ if
$\lambda_1,\cdots,\lambda_n$ are all purely imaginary. Hence $||Z||$
is small enough if
$|\lambda_1-\I\kappa_1|,\cdots,|\lambda_n-\I\kappa_n|$ are all small
enough.

Let $\D c_0=\max_{1\le k\le n}|\kappa_k|$, $\pi_3=|\det
F|\Big|_{\lambda_k=\smallI\kappa_k\atop k=1,\cdots,n}$,
$\pi_4=||F^{-1} LF||\Big|_{\lambda_k=\smallI\kappa_k\atop
k=1,\cdots,n}$. Then there exists $\delta$ with $0<\delta<c_0$ such
that $\D||Z||\le\frac12$, $\D|\det F|\ge\frac{\pi_3}2$, $||F^{-1}
LF||\le 2\pi_4$ if $|\lambda_k-\I\kappa_k|<\delta$ $(k=1,\cdots,n)$.
In this case, $|\lambda_k|<c_0+\delta<2c_0$.

From (\ref{eq:Wdark}),
\begin{equation}
   \det W=\det(FE_++LF^*E^{*-1}_{+})\det\cir W,
   \label{eq:WWodark}
\end{equation}
where
\begin{equation}
   \begin{array}{l}
   \cir{W}=LFE_-+F^*E^{*-1}_{-}
   +\sigma|\rho|^{-2}(F\Lambda E_+-LF^*\Lambda^*E^{*-1}_{+})\cdot\\
   \qquad\cdot(FE_++LF^*E^{*-1}_{+})^{-1}
   L(F\Lambda E_--LF^*\Lambda^*E^{*-1}_{-})\\
   \qquad=(1+\sigma|\rho|^{-2}F\chi_+F^{-1} LF\chi_-F^{-1} L^{-1})LF
   (I+ZE^{*-1}_{-}\Xi_{0-}^{-1})\Xi_{0-}
   \end{array}
   \label{eq:cirWdark}
\end{equation}
\begin{equation}
   \begin{array}{l}
   \chi_\pm=F^{-1}(F\Lambda E_\pm-LF^*\Lambda^*E^{*-1}_{\pm})
   (FE_\pm+LF^*E^{*-1}_{\pm})^{-1}F\\
   \qquad=\Xi_{1\pm}\Xi_{0\pm}^{-1}
   -(Z\Lambda^*+\Xi_{1\pm}\Xi_{0\pm}^{-1} Z)
   E^{*-1}_{\pm}\Xi_{0\pm}^{-1}(I+ZE^{*-1}_{\pm}\Xi_{0\pm}^{-1})^{-1},
   \end{array}\label{eq:chipm}
\end{equation}
\begin{equation}
   \Xi_{0\pm}=E_\pm+E^{*-1}_{\pm},\quad
   \Xi_{1\pm}=\Lambda E_\pm-\Lambda^*E^{*-1}_{\pm}.
\end{equation}

We have the following estimates:
\begin{equation}
   ||E_{\pm}\Xi_{0\pm}^{-1}||\le 1,\quad
   ||E^{*-1}_{\pm}\Xi_{0\pm}^{-1}||\le 1,\quad
   ||\Xi_{1\pm}\Xi_{0\pm}^{-1}||\le 2c_0,\label{eq:Echi_dark_esti}
\end{equation}
\begin{equation}
   ||\Xi_{0\pm}||\ge 2,\quad
   ||\Xi_{0\pm}^{-1}||\le\frac 12,\quad
   |\det\Xi_{0\pm}|\ge 2^n,
\end{equation}
\begin{equation}
   ||(I+Z E^{*-1}_{\pm}\Xi_{0\pm}^{-1})^{-1}||\le(1-||Z||)^{-1}
   \le 2,\quad
   |\det(I+Z
   E^{*-1}_{\pm}\Xi_{0\pm}^{-1})|\ge(1-||Z||)^n\ge\frac 1{2^n}.
\end{equation}
Hence (\ref{eq:chipm}) implies
\begin{equation}
   \begin{array}{l}
   \D||\chi_\pm||\le 2c_0+8c_0||Z||\le 6c_0,
   \end{array}
\end{equation}
\begin{equation}
   \begin{array}{l}
   \D||\chi_+F^{-1} LF\chi_-F^{-1} L^{-1}F||\le||F^{-1} LF||^2\,||\chi_+||\,||\chi_-||\le
   144c_0^2\pi_4^2.
   \end{array}\label{eq:Echi_dark_esti_end-1}
\end{equation}
By (\ref{eq:WWodark}) and (\ref{eq:cirWdark}),
\begin{equation}
   \begin{array}{l}
   |\det W|=|\det F|^2\,
   |\det\Xi_{0+}|\,|\det\Xi_{0-}|\,|\det(I+ZE^{*-1}_{+}\Xi_{0+}^{-1})|\cdot\\
   \qquad\cdot
   |\det(I+ZE^{*-1}_{-}\Xi_{0-}^{-1})|\,
   |\det(I+\sigma|\rho|^{-2}\chi_+F^{-1} LF\chi_-F^{-1} L^{-1} F)|\\
   \D\ge\frac{\pi_3^2}{4}(1-144c_0^2\pi_4^2|\rho|^{-2})>0
   \end{array}\label{eq:Echi_dark_esti_end}
\end{equation}
if $\D|\rho|>12c_0\pi_4$. Therefore, $|\det W|$ has a uniform
positive lower bound if $\D|\rho|>12c_0\pi_4$ and $|\lambda_k-\I\kappa_k|<\delta$
$(k=1,\cdots,n)$.

By (\ref{eq:G12b}), (\ref{eq:Wdark}), (\ref{eq:Rdark}),
(\ref{eq:WWodark}) and (\ref{eq:cirWdark}),
\begin{equation}
   \begin{array}{l}
   \D((G_1)_{12},(G_2)_{12},\cdots,(G_n)_{12})
   =-(R_{12}-R_{11}W_{11}^{-1}W_{12})\cir W^{-1}\\
   \D=\sigma\rho^{*-1}\E{-\I\phi}(-1)^n\Big(f\Lambda E_--(-1)^nf^*\Lambda^*E^{*-1}_{-}
   -(fE_++(-1)^nf^*E^{*-1}_{+})\cdot\\
   \D\qquad\cdot(-1)^n
   (FE_++LF^*E^{*-1}_{+})^{-1}L(F\Lambda E_--LF^*\Lambda^*E^{*-1}_{-})\Big)\cir W^{-1}\\
   \D=\sigma\rho^{*-1}\E{-\I\phi}(-1)^n
   \Big(f\Lambda E_-\Xi_{0-}^{-1}-(-1)^nf^*\Lambda^*E^{*-1}_{-}\Xi_{0-}^{-1}-\\
   \D\qquad-(-1)^n(fE_+\Xi_{0+}^{-1}+(-1)^nf^*E^{*-1}_{+}\Xi_{0+}^{-1})
   (I+ZE^{*-1}_{+}\Xi_{0+}^{-1})^{-1} F^{-1}LF\cdot\\
   \D\qquad\cdot
   (\Xi_{1-}\Xi_{0-}^{-1}-Z\Lambda^*E^{*-1}_{-}\Xi_{0-}^{-1})\Big)
   (I+ZE^{*-1}_{-}\Xi_{0-}^{-1})^{-1}F^{-1}L^{-1}\cdot\\
   \D\qquad\cdot(1+\sigma|\rho|^{-2}F\chi_+F^{-1} LF\chi_-F^{-1} L^{-1})^{-1}.
   \end{array}
\end{equation}
$(G_1)_{12}$ is bounded when $\D|\rho|>12c_0\pi_4$ and
$|\lambda_k-\I\kappa_k|<\delta$ $(k=1,\cdots,n)$ because of the
estimates (\ref{eq:Echi_dark_esti})--(\ref{eq:Echi_dark_esti_end}).
The lemma is proved.

Now we have the following theorem for the multiply ``line dark
soliton'' solution.

\begin{theorem}
Suppose $\kappa_1,\cdots,\kappa_n$ are distinct nonzero real
numbers, then there exist positive constants $\rho_0$ and $\delta$
such that the following results hold for the derived solution
$\widetilde u=u+2(G_1)_{12}$ of the nonlocal Davey-Stewartson I
equation when $|\rho|>\rho_0$ and $|\lambda_j-\I\kappa_j|<\delta$
$(j=1,\cdots,n)$.

\noindent (i) $\widetilde u$ is globally defined and bounded for
$(x,y,t)\in\hr^3$.

\noindent (ii) Suppose the real numbers $v_x,v_y$ satisfy
$\alpha_{kR}v_x\pm\beta_{kR}v_y\ne 0$ for all $k=1,\cdots,n$ where
$\alpha_k$'s and $\beta_k$'s are given by (\ref{eq:abc}), then
$\D\lim_{s\to+\infty}|\widetilde u|=|\rho|$ along the straight line
$x=x_0+v_xs$, $y=y_0+v_ys$ for arbitrary $x_0,y_0\in\hr$.
\end{theorem}

\demo (i) follows directly from Lemma~\ref{lemma:lbounddark}. Now we
prove (ii).

Since
$|\ee_{k\pm}|=\E{(\alpha_{kR}v_x\pm\beta_{kR}v_y)s
+(\alpha_{kR}x_0\pm\beta_{kR}y_0\pm\gamma_{kR}t)}$
along the straight line $x=x_0+v_xs$, $y=y_0+v_ys$,
$\alpha_{kR}v_x\pm\beta_{kR}v_y\ne 0$ implies that for each $k$,
$e_{k+}\to 0$ or $e_{k+}\to\infty$, and $e_{k-}\to 0$ or
$e_{k-}\to\infty$ as $s\to+\infty$. Let
\begin{equation}
   \begin{array}{l}
   \D \mu_k=\left\{\begin{array}{ll}\lambda_k&\hbox{ if }\alpha_{kR}v_x+\beta_{kR}v_y>0,\\
   -\lambda_k^*&\hbox{ if }\alpha_{kR}v_x+\beta_{kR}v_y<0,\end{array}\right.\\
   \D \nu_k=\left\{\begin{array}{ll}-\lambda_k&\hbox{ if }\alpha_{kR}v_x-\beta_{kR}v_y>0,\\
   \lambda_k^*&\hbox{ if }\alpha_{kR}v_x-\beta_{kR}v_y<0,\end{array}\right.\\
   \D a_k=\left\{\begin{array}{ll}\beta_k&\hbox{ if }\alpha_{kR}v_x+\beta_{kR}v_y>0,\\
   -\beta_k^*&\hbox{ if }\alpha_{kR}v_x+\beta_{kR}v_y<0,\end{array}\right.\\
   \D b_k=\left\{\begin{array}{ll}-\beta_k&\hbox{ if }\alpha_{kR}v_x-\beta_{kR}v_y>0,\\
   \beta_k^*&\hbox{ if }\alpha_{kR}v_x-\beta_{kR}v_y<0,\end{array}\right.\\
   \end{array}
\end{equation}
then
\begin{equation}
   a_k=\frac 12\Big(\frac{\sigma|\rho|^2}{\mu_k}+\mu_k\Big),\quad
   b_k=\frac
   12\Big(\frac{\sigma|\rho|^2}{\nu_k}+\nu_k\Big).\label{eq:akbk}
\end{equation}
Rewrite (\ref{eq:G12b}) as
\begin{equation}\fl
   \begin{array}{l}
   \D\left(\begin{array}{cccccc}(G_1)_{11}&\cdots&(G_n)_{11}
   &\rho^{-1}\E{\I\phi}(G_1)_{12}&\cdots&\rho^{-1}\E{\I\phi}(G_n)_{12}\\
   (G_1)_{21}&\cdots&(G_n)_{21}
   &\rho^{-1}\E{\I\phi}(G_1)_{22}&\cdots&\rho^{-1}\E{\I\phi}(G_n)_{22}\end{array}\right)
   SWS^{-1}=-RS^{-1}\\
   \end{array}\label{eq:G12bb}
\end{equation}
where $\D
S=\left(\begin{array}{cc}I_n\\&\rho\E{-\I\phi}I_n\end{array}\right)$,
$I_n$ is the $n\times n$ identity matrix.

Applying Cramer's rule to (\ref{eq:G12bb}) and using
(\ref{eq:akbk}), we have
\begin{equation}
   \lim_{s\to+\infty}\widetilde u=\lim_{s\to+\infty}(\rho\E{-\I\phi}+2(G_1)_{12})
   =\rho\E{-\I\phi}\Big(1-2\frac{\det W_1}{\det W_0}\Big)
   =\rho\E{-\I\phi}\frac{\det W_2}{\det W_0}
\end{equation}
where
\begin{equation}\fl
   W_0=\left(\begin{array}{cccccc}
   a_1^{n-1}&\cdots&a_n^{n-1}
   &\sigma|\rho|^{-2}b_1^{n-1}\nu_1&\cdots
   &\sigma|\rho|^{-2}b_n^{n-1}\nu_n\\
   a_1^{n-2}&\cdots&a_n^{n-2}
   &\sigma|\rho|^{-2}b_1^{n-2}\nu_1&\cdots
   &\sigma|\rho|^{-2}b_n^{n-2}\nu_n\\
   \vdots&&\vdots&\vdots&&\vdots\\
   a_1&\cdots&a_n
   &\sigma|\rho|^{-2}b_1\nu_1&\cdots
   &\sigma|\rho|^{-2}b_n\nu_n\\
   1&\cdots&1&\sigma|\rho|^{-2}\nu_1&\cdots&\sigma|\rho|^{-2}\nu_n\\
   a_1^{n-1}\mu_1&\cdots&a_n^{n-1}\mu_n
   &b_1^{n-1}&\cdots&b_n^{n-1}\\
   a_1^{n-2}\mu_1&\cdots&a_n^{n-2}\mu_n
   &b_1^{n-2}&\cdots&b_n^{n-2}\\
   \vdots&&\vdots&\vdots&&\vdots\\
   a_1\mu_1&\cdots&a_n\mu_n
   &b_1&\cdots&b_n\\
   \mu_1&\cdots&\mu_n&1&\cdots&1\\
   \end{array}\right),
\end{equation}
$W_1$ is obtained from $W_0$ by replacing the $(n+1)$-th row with
\begin{equation}\fl
   \left(\begin{array}{cccccc}a_1^n&\cdots&a_n^n
   &\sigma|\rho|^{-2}b_1^n\nu_1&\cdots
   &\sigma|\rho|^{-2}b_n^n\nu_n\end{array}\right),
\end{equation}
and
\begin{equation}\fl
   W_2=\left(\begin{array}{cccccc}
   a_1^{n-1}&\cdots&a_n^{n-1}
   &\sigma|\rho|^{-2}b_1^{n-1}\nu_1&\cdots
   &\sigma|\rho|^{-2}b_n^{n-1}\nu_n\\
   a_1^{n-2}&\cdots&a_n^{n-2}
   &\sigma|\rho|^{-2}b_1^{n-2}\nu_1&\cdots
   &\sigma|\rho|^{-2}b_n^{n-2}\nu_n\\
   \vdots&&\vdots&\vdots&&\vdots\\
   a_1&\cdots&a_n
   &\sigma|\rho|^{-2}b_1\nu_1&\cdots
   &\sigma|\rho|^{-2}b_n\nu_n\\
   1&\cdots&1&\sigma|\rho|^{-2}\nu_1&\cdots&\sigma|\rho|^{-2}\nu_n\\
   -\sigma|\rho|^2a_1^{n-1}\mu_1^{-1}&\cdots&-\sigma|\rho|^2a_n^{n-1}\mu_n^{-1}
   &-\sigma|\rho|^{-2}b_1^{n-1}\nu_1^2&\cdots&-\sigma|\rho|^{-2}b_n^{n-1}\nu_n^2\\
   a_1^{n-2}\mu_1&\cdots&a_n^{n-2}\mu_n
   &b_1^{n-2}&\cdots&b_n^{n-2}\\
   \vdots&&\vdots&\vdots&&\vdots\\
   a_1\mu_1&\cdots&a_n\mu_n
   &b_1&\cdots&b_n\\
   \mu_1&\cdots&\mu_n&1&\cdots&1\\
   \end{array}\right).
\end{equation}
Denote $\row_k$ and $\col_k$ to be the $k$-th row and $k$-th column
of $W_2$ respectively. The elementary transformations
\begin{equation}
   \begin{array}{l}
   \row_{n+k+1}-2\cdot\row_k\to\row_{n+k+1}\quad(k=1,\cdots,n-1),\\
   \mu_k\cdot\col_k\to\col_k\quad(k=1,\cdots,n),\\
   \sigma|\rho|^2\nu_k^{-1}\cdot\col_{n+k}\to\col_{n+k}\quad(k=1,\cdots,n),\\
   -\sigma|\rho|^{-2}\cdot\row_{n+k}\to\row_{n+k}\quad(k=1,\cdots,n),\\
   \row_k\leftrightarrow\row_{n+k}\quad(k=1,\cdots,n)
   \end{array}
\end{equation}
transform $W_2$ to $W_0$. Hence $\D\det
W_2=\prod_{k=1}^n\frac{\nu_k}{\mu_k}\det W_0$. This leads to
$\D\lim_{s\to+\infty}|\widetilde u|=|\rho|$ since
$\D\prod_{k=1}^n|\mu_k|=\prod_{k=1}^n|\nu_k|=\prod_{k=1}^n|\lambda_k|$.
The theorem is proved.

A $2$ ``line dark soliton'' solution is shown in Figure~4 where the
parameters are $\sigma=-1$, $t=10$, $\rho=1$, $\lambda_1=0.8+0.1\I$,
$\lambda_2=-0.6-0.3\I$. The figure on the right describes the same
solution but is upside down.

\begin{figure}\begin{center}
\scalebox{1.25}{\includegraphics[340,100]{2darkstn.ps}}\label{fig:2darkstn}
\caption{$|\widetilde u|$ of a $2$ ``line dark soliton'' solution.}
\end{center}\end{figure}

\section*{Acknowledgements}

This work was supported by the Natural Science Foundation of
Shanghai (No.\ 16ZR\-1402600) and the Key Laboratory of Mathematics
for Nonlinear Sciences of Ministry of Education of China. The author
is grateful to Prof.\ S.Y.Lou for helpful discussion.