\begin{theorem}[\textbf{Dynamical stability of $n$-solitons}]\label{thm1.1}
Given $ n\in \mathbb{N} $, $ n\geq 1 $, a collection of wave speeds $\textbf{c} = (c_{1},\dots,c_{n})$ with $ 0 < c_{1} < \dots < c_{n} $ and a collection of space transitions $\textbf{x} = (x_{1},\dots,x_{n})\in \mathbb{R}^{n}$, let $ U^{(n)}(\cdot,\cdot;\textbf{c},\textbf{x}) $ be the corresponding $n$-soliton solutions of (\ref{ILW}). Then for any $ \epsilon > 0 $, there exists $ \delta > 0 $ such that for any $ u_{0}\in H^{\frac{n}{2}}(\mathbb{R}) $, the following stability property holds, if

\begin{align}
	&\bigl\lVert u_{0} - U^{(n)}(0,\cdot;\textbf{c},\textbf{x}) \bigr\rVert_{H^{\frac{n}{2}}} < \delta,
\end{align}
then for any $t \in \mathbb{R}$, the corresponding solution of (\ref{ILW}) verifies
\begin{align}
\inf_{\tau \in \mathbb{R},\  \textbf{y} \in \mathbb{R}^{n}} \bigl\lVert u(t) - U^{(n)}(\tau,\cdot;\textbf{c},\textbf{y}) \bigr\rVert_{H^{\frac{n}{2}}} < \epsilon.
\end{align}
\end{theorem}

\label{sec.1}
In this work, we consider the stability of $n$-soliton solutions for the intermediate long wave (ILW) equation\cite{J,KAS,KKD}
\begin{align}
	u_t+\frac{1}{\delta}u_x+2uu_x+T^\delta u_{xx}=0,\quad  (t,x)\in \mathbb{R} \times \mathbb{R},\label{ILW}
\end{align}
where $u=u(t,x)\in\mathbb{R}$ is a real-valued function, $T^\delta $ is the inverse of Tilbert  transform $T$ with a large scale parameter $\delta>0$. These two operators are defined as
\begin{align}
	(T^\delta f)(x)&=\frac{1}{2\delta}\text{P.V.}\int_{\mathbb{R}}\coth \frac{\pi (y-x)}{2\delta}f(y)dy,\\\nonumber
		(Tf)(x)&=\frac{1}{2\delta}\text{P.V.}\int_{\mathbb{R}}\mathrm{cosech} \frac{\pi (y-x)}{2\delta}f(y)dy,
\end{align}
and P.V. indicates that the integral is to be computed in the principle value sense. Moreover, $T^\delta$ is a zero Fourier multiplier, in the sense that $\partial_x T^\delta$ is the multiplier with symbol
\begin{align}
	\sigma(\partial_x T^\delta)=\widehat{\partial_x T^\delta}=-2\pi\xi\coth(2\pi\delta \xi).
\end{align}

The ILW equation (\ref{ILW}) describes weakly nonlinear internal wave propagation in stratified fluids of finite depth $\delta$. First derived by Kubota, Ko and Dobbs \cite{KKD}, this model captures essential nonlinear and dispersive characteristics of wave dynamics. Joseph \cite{J} subsequently formalized the derivation of the ILW equation by integrating the linear dispersion relation from \cite{P} within Whitham's nonlocal framework \cite{W}, and explicitly constructed its solitary wave solution. This represents a key advancement in establishing the integrability of the equation. Furthermore, the ILW equation itself arises in the context of the two-layer internal wave system \cite{BLS, CGK}. A rigorous derivation of its system form in one or two spatial dimensions, including the case with a free upper surface, can be found in \cite{Xu}.

In addition, the ILW equation (\ref{ILW}) establishes a fundamental connection between two canonical models in water wave theory: the Korteweg-de Vries (KdV) equation for shallow water and the Benjamin-Ono (BO) equation for deep water \cite{ABFS, AFS,HKO,SAF}, with shared initial data in appropriate Sobolev spaces. Specifically, the asymptotic behavior of the ILW solution is governed by the depth parameter $\delta$. Let $u(t,x)$ be the solution of the ILW equation (\ref{ILW}). Then, under the scaling transformation
\begin{align}
	v = \frac{3}{\delta} u\left( \frac{3}{\delta}t,x\right), \label{T1}
\end{align}
the rescaled function $v$ converges (in a suitable functional sense, typically distributional or weak convergence) to the solution of the KdV equation
\begin{align}
	u_t + 2uu_x + \frac{\delta}{3}u_{xxx} = 0, \label{KdV}
\end{align}
in the shallow-water limit ($\delta \to 0$) and in the deep-water limit ($\delta \to \infty$), (\ref{T1}) will converge to the the solution of the BO equation
\begin{align}
	u_t + 2uu_x + Hu_{xx} = 0, \label{BO}
\end{align}
where $H$ denotes the Hilbert transform, defined by the principal value integral
\begin{align*}
	(Hf)(x)=\frac{1}{\pi}\text{P.V.}\int_{\mathbb{R}}\frac{f(y)}{y-x}\mathrm{d}y.
\end{align*}

As a fundamental integrable system \cite{AC,FA,KS}, the ILW equation (\ref{ILW}) possesses an infinite-dimensional completely integrable Hamiltonian structure \cite{KAS, Sa}. This integrability is manifested in the existence of an infinite hierarchy of conservation laws and a Lax-pair formulation.
A key distinction between the ILW equation and the KdV equation lies in their dispersion operators \cite{MPS,Sa,Xu}. Specifically, the ILW equation incorporates a singular nonlocal integro-differential operator $T^\delta$, which fundamentally alters the asymptotic properties of its solutions. In particular, the soliton solutions of the ILW equation exhibit algebraic decay, in contrast to the exponential decay characteristic of KdV solitons. This difference in decay rates represents a key qualitative difference between the two models. Formally, the following quantities are conserved along the flow of the ILW equation \cite{LR}
\begin{align}
	H_0(u)&=:\int_{\mathbb{R}}u\text{d}x, \\
    H_1(u)&=:\frac{1}{2}\int_{\mathbb{R}}u^2\text{d}x, \\
	H_2(u)&=:-\int_{\mathbb{R}}\frac{1}{3}u^3+\frac{1}{2}uT^\delta u_x +\frac{u^2}{2\delta}\text{d}x,\\
	H_3(u)&=:\int_{\mathbb{R}}\frac{1}{4}u^4+\frac{3}{4}u^2T^\delta u_x+\frac{3}{8}(T^\delta u_x)^2+\frac{1}{3\delta}u^3+\frac{1}{2\delta}uT^\delta u_x+\frac{u^2}{8\delta^2}\text{d}x.
\end{align}

It is noteworthy that the energy space for the ILW equation, defined as the domain of the Hamiltonian $H_2(u)$, is $H^{\frac{1}{2}}(\mathbb{R})$.
This functional framework provides the natural setting for investigating solution properties.
In the context of weak solutions, Ginibre and Velo \cite{GV} established the existence of global solutions satisfying
$u \in L^\infty(\mathbb{R}, H^1(\mathbb{R})) \cap L^2_{\mathrm{loc}}(\mathbb{R}, H^{\frac{3}{2}}_{\mathrm{loc}}(\mathbb{R}))$.
For strong solutions, the first global well-posedness result in $H^s(\mathbb{R})$ was obtained by \cite{ABFS} for $s > \frac{3}{2}$.
Subsequent developments progressively lowered the regularity threshold: Molinet and Vento \cite{MV} extended well-posedness to $s > \frac{1}{2}$
(where unconditional uniqueness holds), while Molinet, Pilod, and Vento \cite{MPV} advanced it to $s > \frac{1}{4}$.
A significant breakthrough in the sharp low-regularity theory was achieved by Ifrim and Saut \cite{IS}, who proved global well-posedness
in $L^2(\mathbb{R})$ for small initial data. Most recently, Gassot and Laurens \cite{GAL} established sharp global well-posedness
for the ILW equation in $H^s(\mathbb{T})$ for all $s > -\frac{1}{2}$, thereby
completing the well-posedness theory on the torus. Additionally, they proved the
continuous convergence of solutions to those of the Benjamin-Ono equation in the
deep-water limit $\delta \to \infty$.

Based upon the facts above, the (\ref{ILW}) equation can naturally be viewed as a Hamiltonian system of the form \cite{KSA, LR}
\begin{align}
	u_t=\mathcal{J}\frac{\delta H_2(u)}{\delta u},
\end{align}
where $\mathcal{J}=\partial_x$, and $\frac{\delta H_2(u)}{\delta u}$ (subsequently simplified to $H_2'(u)$) refers to the variational derivative of $H_2(u)$ can be written as
\begin{align*}
	(\frac{\partial}{\partial \epsilon}H_2(u+\epsilon f))\Big|_{\epsilon=0}=\int_{\mathbb{R}}\frac{\delta H_{2}}{\delta u}(x)f(x)\text{d}x.
\end{align*}

Moreover, the ILW equation possesses a bi-Hamiltonian structure \cite{Sa}. However, unlike the KdV equation, the bi-Hamiltonian structure of the ILW equation is highly nontrivial due to its formulation involving both the spatial derivative operator $\partial_x$ and the singular integral operator $T^\delta$. This operator composition aligns the ILW equation with structural features characteristic of completely integrable systems in two spatial dimensions.

To formalize this connection, let subscripts $1, 2$ denote dependence on the independent variables $x:= x_1$ and $x_2$. For arbitrary functions $f_{12}$ and $g_{12}$, we define the associated bilinear form by
\begin{align}
	\langle f_{12},g_{12}\rangle
	&:= \int_{\mathbb{R}^{2}}f_{12}g_{12}^{*} \mathrm{d}x_{1}\mathrm{d}x_{2},\label{1.9}
\end{align}
where the asterisk superscript $^*$ denotes complex conjugation. Define operators in $L^{2}(\mathbb{R}^{2},\mathbb{C})$ (with domain $H^{1}(\mathbb{R}^{2},\mathbb{C})$)
\begin{align}
	\theta_{12}^{\pm}
	&:= u_{1}\pm u_{2} + i(\partial_{x_{1}} \mp \partial_{x_{2}}), \quad u_{j}=u(x_{j},t), \quad j=1,2.\label{1.10}
\end{align}
Then the two compatible Hamiltonian operators for the ILW equation are given by
\begin{align}
	\mathcal{J}_{12}^{(1)}
	:= \theta_{12}^{-}, \quad
	\mathcal{J}_{12}^{(2)}
	:= (i\theta_{12}^{-}T^\delta_{12} - \theta_{12}^{+})\theta_{12}^{-},
\end{align}
where $T^\delta_{12}$ is an extended operator defined as
\begin{align}
	(T^\delta_{12}f_{12})(x_{1},x_{2}) =\frac{1}{2\delta}\mathrm{P.V.}\int_{\mathbb{R}}\coth \left(\frac{\pi}{2\delta}[\xi-(x_1+x_2)]\right)F(\xi,x_1-x_2)\mathrm{d}\xi,\label{1.13}
\end{align}
and
\begin{align}
	f(x_1,x_2)=F(x_1+x_2,x_1-x_2).
\end{align}
Then the ILW hierarchy can be expressed in the form
\begin{align}\label{1.14}
	u_t&=\frac{i}{2^n}\int_{\mathbb{R}}\delta(x_1-x_2)(\mathcal{R}_{12}^\star)^n\theta_{12}^-\cdot1\mathrm{d}x_2\\\nonumber
	&=\frac{i}{2^n}\int_{\mathbb{R}}\delta(x_1-x_2)\theta_{12}^-\mathcal{R}_{12}^n\cdot1\mathrm{d}x_2=\mathcal{J}\frac{\delta H_n(u)}{\delta u},\quad \text{for all} ~n\in \mathbb{N},
\end{align}
where $\star$ denotes the adjoint with respect to the bilinear form (\ref{1.9}). The recursion operator $\mathcal{R}_{12}$ and its adjoint $\mathcal{R}_{12}^{\star}$ are defined by
\begin{equation}
	\mathcal{R}_{12} := (\mathcal{J}_{12}^{(1)})^{-1}\mathcal{J}_{12}^{(2)},\quad
	\mathcal{R}_{12}^{\star} := \mathcal{J}_{12}^{(2)}(\mathcal{J}_{12}^{(1)})^{-1} = i\theta_{12}^{-}T^\delta_{12} - \theta_{12}^{+},\label{1.15}
\end{equation}
and in view of (\ref{1.15}), they satisfy the compatibility condition
\begin{equation*}
	\mathcal{R}_{12}^{\star}\mathcal{J}_{12}^{(1)} = \mathcal{J}_{12}^{(1)}\mathcal{R}_{12}.
\end{equation*}
Consequently, the first few equations of the ILW hierarchy are
\begin{align*}
	&u_t-u_x=0, \quad \text{for}\quad n=1, \\
	&u_t+\frac{1}{\delta}u_x+2uu_x+T^\delta u_{xx}=0,\quad \text{for}\quad n=2;\\
	&u_t-\left(\frac{1}{4}({T^{\delta}}^2-1)u_{xx}+ u^3+\frac{3}{2}(uT^\delta u_x+T^\delta uu_{xx})\right)_x=0, \quad \text{for}\quad n=3.
\end{align*}

In common with the classical KdV and BO equations, between which the ILW equation (\ref{ILW}) serves as a model-theoretical bridge and admits a family of unique exact solitary-wave solutions \cite{AT,SA,SAK} of the form
\begin{align}
	u(x,t)=Q_c(x-ct-x_0), \quad Q_c(s)=\frac{a\sin(a\delta)}{\cosh(as)+\cos(a\delta)},\quad c>0,~ x_0\in \mathbb{R}, \label{1.16}
\end{align}
where $c$ represents the wave speed and $a$ is the unique solution of the transcendental equation
\begin{align*}
	a\delta \cot(a\delta)=1-c\delta, \quad a\in \left(0, \frac{\pi}{\delta}\right).
\end{align*}
Substituting (\ref{1.16}) into equation (\ref{ILW}) yields
\begin{align}
	T^\delta \partial_x Q_c +\left(\frac{1}{\delta}-c\right)Q_c+Q_c^2=0, \quad c>0.\label{1.17}
\end{align}
Beyond these fundamental solitary waves, the ILW equation (\ref{ILW}) also supports more complex solutions, including multi-solitons \cite{JE} that admit parametric representations similar to the single-soliton case \cite{AT,LP,M,PD}.
The ILW $n$-soliton solution $U^{(n)}_\mathbf{c}$ is characterized by a collection of wave speeds $\mathbf{c}=(c_1, c_2, \cdots, c_n)$ and initial positions $\mathbf{x}=(x_1, x_2, \cdots, x_n)$, where the wave speeds satisfy $c_j>0$ with $c_j\neq c_k$ for $j\neq k$ ($j,k=1, 2, \cdots, n$). Moreover, in the long-time limit, these $n$-soliton solutions decompose into a superposition of $n$ individual solitons as follows
\begin{align}
	\lim_{|t| \to +\infty} \left\| U^{(n)}(t,\cdot;\mathbf{c},\mathbf{x}) - \sum_{j=1}^{n} Q_{c_{j}}(\cdot - c_{j}t - x_{j}) \right\|_{H^{s}(\mathbb{R})} = 0, \quad s \in \mathbb{N}.\label{resolution}
\end{align}

In recent years, the stability theory for solitary waves and multi-solitons has emerged as a prominent research direction. Within soliton theory, stability concepts are systematically categorized into four distinct types based on analytical methodology and robustness guarantees: (i) linear (spectral) stability, concerning the eigenvalue distribution of linearized operators; (ii) Lyapunov (dynamical) stability, established via positive definiteness of the second variation of Lyapunov functionals at soliton solutions; (iii) orbital (nonlinear) stability, which requires that solutions remain within a neighborhood of the soliton orbit under finite-amplitude disturbances; and (iv) asymptotic stability, demanding convergence to specific soliton profiles.

As fundamental models in integrable systems, the stability of solitary waves for both KdV and BO equations has been extensively investigated. For instance, Maddocks and Sachs \cite{MS} established orbital stability of KdV multi-solitons in the energy space defined by conservation laws. Killip and Visan \cite{KV} proved orbital stability of KdV multi-solitons in $H^{-1}(\mathbb{R})$ using low-regularity conservation laws in \cite{BP}. For the BO equation, Wang \cite{LW} and Matsuno \cite{Mat} studied dynamical stability of multi-soliton solutions. Additionally, Badreddine, Killip and Visan \cite{BKP} proved that the multi-soliton solutions to the BO equation are uniformly orbitally stable in $H^s(\mathbb{R})$ with $-\frac{1}{2}<s<\frac{1}{2}$.

In contrast to the extensive results for KdV and BO equations, stability analysis for solitary wave solutions of the ILW equation remains comparatively underdeveloped. Existing studies primarily focus on orbital stability. Albert and Bona \cite{AB} employed a rigorous variational approach combined with spectral analysis to establish orbital stability of ILW solitary waves in its energy space $\mathcal{X} = \{g \in L^2(\mathbb{R})| \int_\mathbb{R}(1+\alpha(k))|\hat{g}|^2\mathrm{d}k < \infty\}$. Subsequent work \cite{A} extended these results to spaces $H^{\frac{1}{2}}(\mathbb{R})$, while \cite{PCA} addressed linear and orbital stability of periodic wave solutions in periodic Sobolev spaces.

Considering the growing importance of understanding coherent structures in nonlocal dispersive equations, a systematic analysis of soliton and multi-soliton stability for the ILW equation represents a significant open research direction. Hence, in this work, we aim to establish dynamical stability results of $n$-soliton solutions to the ILW equation (\ref{ILW}) in an appropriate Sobolev space.

Our strategy is to adapt the methods of Maddocks and Sachs \cite{MS}.
In accordance with the ideas in \cite{MS}, we construct an appropriate  Lyapunov functional $\mathcal{S}_{n}$ of the ILW $n$-solitons is given by
\begin{align}
	\mathcal{S}_{n}(u) = H_{n+1}(u) + \sum_{m=1}^{n} \mu_{m} H_{m}(u),\label{lagrange}
\end{align}
and $\mu_{m}$ are Lagrange multipliers which will be expressed in terms of the elementary symmetric functions of $c_{1},c_{2},\dots,c_{n}$. Using (\ref{lagrange}), this condition can be written as the following Euler-Lagrange equation
\begin{align}
	\frac{\delta H_{n+1}(u)}{\delta u} + \sum_{m=1}^{n} \mu_{m} \frac{\delta H_{m}(u)}{\delta u} = 0,\quad \text{at } u = U^{(n)}.\label{1.20}
\end{align}
The dynamical stability of $U^{(n)}$ is implied by the fact that $U^{(n)}(x)$ is a minimizer of the functional $H_{n+1}$ under the following $n$ constraints
\begin{align}
	H_{m}(u) = H_{m}\left(U^{(n)}\right),\quad m = 1,2,\dots,n,
\end{align}
which requires that the self-adjoint second variation of the operator $\mathcal{S}_n$,
\begin{align}
	\mathcal{L}_n:=\mathcal{S}''_{n}(U^{(n)}), \label{VL}
\end{align}
is strictly positive if one modulates the directions given by the constraints.

The demonstration of dynamical stability for the ILW equation relies on the recursion operators and the construction of a Lyapunov functional. This approach utilizes the trace formula derived from the inverse scattering transform (IST) framework to forge a link with conservation laws, in a manner analogous to techniques applied in the study of the Camassa-Holm equation \cite{AYP, CGI, DKS}. The stability proof centers on a spectral analysis of the second variation $\mathcal{L}_{n} = \mathcal{S}_{n}''(U^{(n)})$ evaluated at a smooth $n$-soliton $U^{(n)}$, coupled with an eigenvalue computation for the Hessian matrix $D = \left\{\frac{\partial^2 \mathcal{S}_{n}}{\partial \mu_{i} \partial \mu_{j}} \right\}$.

The spectral characterization of the linearized operator $\mathcal{L}_{n}$ around the $n$-soliton $U^{(n)}$ is central to this endeavor. In the case of a single soliton, the operator $L_n = H''_{n+1}(Q_c) + c H''_{n}(Q_c)$ is analyzed. The inertia index $\mathrm{in}(L_n)$ is determined by constructing the derivative operators $\mathcal{J}L_n$ and $L_n\mathcal{J}$ and leveraging IST theory, specifically employing Jost solutions and square eigenfunctions. For the general $n$-soliton, the recursive operator $\mathcal{R}$ originating from the bi-Hamiltonian structure is applied to compute $\mathrm{in}(\mathcal{L}_n)$.

Finally, we conclude by proving the orbital stability of double solitons. This final result is obtained through a spectral analysis of the linearized operator $\mathcal{L}_{2}$ around the double soliton $U^{(2)}$, supported by a contradictory argument. Our analysis shows that $\mathcal{L}_{2}$ has one simple negative eigenvalue and one double eigenvalue at zero.

In what follows, we will present our main results. The first is the dynamical stability of the multi-solitons to the ILW equation (\ref{ILW}).

To formalize this connection, let subscripts $1, 2$ denote dependence on the independent variables $x:= x_1$ and $x_2$. For arbitrary functions $f_{12}$ and $g_{12}$, we define the associated bilinear form by
\begin{align}
 \langle f_{12},g_{12}\rangle
 &:= \int_{\mathbb{R}^{2}}f_{12}g_{12}^{*} \mathrm{d}x_{1}\mathrm{d}x_{2},\label{1.9}
\end{align}
where the asterisk superscript $^*$ denotes complex conjugation. Define operators in $L^{2}(\mathbb{R}^{2},\mathbb{C})$ (with domain $H^{1}(\mathbb{R}^{2},\mathbb{C})$)
\begin{align}
 \theta_{12}^{\pm}
 &:= u_{1}\pm u_{2} + i(\partial_{x_{1}} \mp \partial_{x_{2}}), \quad u_{j}=u(x_{j},t), \quad j=1,2.\label{1.10}
\end{align}
Then the two compatible Hamiltonian operators for the ILW equation are given by
\begin{align}
 \mathcal{J}_{12}^{(1)}
 := \theta_{12}^{-}, \quad
 \mathcal{J}_{12}^{(2)}
 := (i\theta_{12}^{-}T^\delta_{12} - \theta_{12}^{+})\theta_{12}^{-},
\end{align}
where $T^\delta_{12}$ is an extended operator defined as
\begin{align}
 (T^\delta_{12}f_{12})(x_{1},x_{2}) =\frac{1}{2\delta}\mathrm{P.V.}\int_{\mathbb{R}}\coth \left(\frac{\pi}{2\delta}[\xi-(x_1+x_2)]\right)F(\xi,x_1-x_2)\mathrm{d}\xi,\label{1.13}
\end{align}
and
\begin{align}
 f(x_1,x_2)=F(x_1+x_2,x_1-x_2).
\end{align}
Then the ILW hierarchy can be expressed in the form
\begin{align}\label{1.14}
 u_t&=\frac{i}{2^n}\int_{\mathbb{R}}\delta(x_1-x_2)(\mathcal{R}_{12}^\star)^n\theta_{12}^-\cdot1\mathrm{d}x_2\\\nonumber
 &=\frac{i}{2^n}\int_{\mathbb{R}}\delta(x_1-x_2)\theta_{12}^-\mathcal{R}_{12}^n\cdot1\mathrm{d}x_2=\mathcal{J}\frac{\delta H_n(u)}{\delta u},\quad \text{for all} ~n\in \mathbb{N},
\end{align}
where $\star$ denotes the adjoint with respect to the bilinear form (\ref{1.9}). The recursion operator $\mathcal{R}_{12}$ and its adjoint $\mathcal{R}_{12}^{\star}$ are defined by
\begin{equation}
 \mathcal{R}_{12} := (\mathcal{J}_{12}^{(1)})^{-1}\mathcal{J}_{12}^{(2)},\quad
 \mathcal{R}_{12}^{\star} := \mathcal{J}_{12}^{(2)}(\mathcal{J}_{12}^{(1)})^{-1} = i\theta_{12}^{-}T^\delta_{12} - \theta_{12}^{+},\label{1.15}
\end{equation}
and in view of (\ref{1.15}), they satisfy the compatibility condition
\begin{equation*}
 \mathcal{R}_{12}^{\star}\mathcal{J}_{12}^{(1)} = \mathcal{J}_{12}^{(1)}\mathcal{R}_{12}.
\end{equation*}
Consequently, the first few equations of the ILW hierarchy are
\begin{align*}
 &u_t-u_x=0, \quad \text{for}\quad n=1, \\
 &u_t+\frac{1}{\delta}u_x+2uu_x+T^\delta u_{xx}=0,\quad \text{for}\quad n=2;\\
 &u_t-\left(\frac{1}{4}({T^{\delta}}^2-1)u_{xx}+ u^3+\frac{3}{2}(uT^\delta u_x+T^\delta uu_{xx})\right)_x=0, \quad \text{for}\quad n=3.
\end{align*}

In common with the classical KdV and BO equations, between which the ILW equation (\ref{ILW}) serves as a model-theoretical bridge and admits a family of unique exact solitary-wave solutions \cite{AT,SA,SAK} of the form
\begin{align}
 u(x,t)=Q_c(x-ct-x_0), \quad Q_c(s)=\frac{a\sin(a\delta)}{\cosh(as)+\cos(a\delta)},\quad c>0,~ x_0\in \mathbb{R}, \label{1.16}
\end{align}
where $c$ represents the wave speed and $a$ is the unique solution of the transcendental equation
\begin{align*}
 a\delta \cot(a\delta)=1-c\delta, \quad a\in \left(0, \frac{\pi}{\delta}\right).
\end{align*}
Substituting (\ref{1.16}) into equation (\ref{ILW}) yields
\begin{align}
 T^\delta \partial_x Q_c +\left(\frac{1}{\delta}-c\right)Q_c+Q_c^2=0, \quad c>0.\label{1.17}
\end{align}
Beyond these fundamental solitary waves, the ILW equation (\ref{ILW}) also supports more complex solutions, including multi-solitons \cite{JE} that admit parametric representations similar to the single-soliton case \cite{AT,LP,M,PD}.
The ILW $n$-soliton solution $U^{(n)}_\mathbf{c}$ is characterized by a collection of wave speeds $\mathbf{c}=(c_1, c_2, \cdots, c_n)$ and initial positions $\mathbf{x}=(x_1, x_2, \cdots, x_n)$, where the wave speeds satisfy $c_j>0$ with $c_j\neq c_k$ for $j\neq k$ ($j,k=1, 2, \cdots, n$). Moreover, in the long-time limit, these $n$-soliton solutions decompose into a superposition of $n$ individual solitons as follows
\begin{align}
 \lim_{|t| \to +\infty} \left\| U^{(n)}(t,\cdot;\mathbf{c},\mathbf{x}) - \sum_{j=1}^{n} Q_{c_{j}}(\cdot - c_{j}t - x_{j}) \right\|_{H^{s}(\mathbb{R})} = 0, \quad s \in \mathbb{N}.\label{resolution}
\end{align}

Our strategy is to adapt the methods of Maddocks and Sachs \cite{MS}.
In accordance with the ideas in \cite{MS}, we construct an appropriate Lyapunov functional $\mathcal{S}_{n}$ of the ILW $n$-solitons is given by
\begin{align}
 \mathcal{S}_{n}(u) = H_{n+1}(u) + \sum_{m=1}^{n} \mu_{m} H_{m}(u),\label{lagrange}
\end{align}
and $\mu_{m}$ are Lagrange multipliers which will be expressed in terms of the elementary symmetric functions of $c_{1},c_{2},\dots,c_{n}$. Using (\ref{lagrange}), this condition can be written as the following Euler-Lagrange equation
\begin{align}
 \frac{\delta H_{n+1}(u)}{\delta u} + \sum_{m=1}^{n} \mu_{m} \frac{\delta H_{m}(u)}{\delta u} = 0,\quad \text{at } u = U^{(n)}.\label{1.20}
\end{align}
The dynamical stability of $U^{(n)}$ is implied by the fact that $U^{(n)}(x)$ is a minimizer of the functional $H_{n+1}$ under the following $n$ constraints
\begin{align}
 H_{m}(u) = H_{m}\left(U^{(n)}\right),\quad m = 1,2,\dots,n,
\end{align}
which requires that the self-adjoint second variation of the operator $\mathcal{S}_n$,
\begin{align}
 \mathcal{L}_n:=\mathcal{S}''_{n}(U^{(n)}), \label{VL}
\end{align}
is strictly positive if one modulates the directions given by the constraints.

Finally, we conclude by proving the orbital stability of double solitons. This final result is obtained through a spectral analysis of the linearized operator $\mathcal{L}_{2}$ around the double soliton $U^{(2)}$, supported by a contradictory argument. Our analysis shows that $\mathcal{L}_{2}$ has one simple negative eigenvalue and one double eigenvalue at zero.

In what follows, we will present our main results. The first is the dynamical stability of the multi-solitons to the ILW equation (\ref{ILW}).

Then we will provide a more precise description of the orbital stability of the double solitons to the ILW equation.

\begin{theorem}[\textbf{Orbital stability of double solitons}]\label{thm1.2}
 The double solitons $U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)$ of ILW equation (\ref{ILW}) with $0<c_1<c_2$ are orbitally stable in $H^1(\mathbb{R})$ in the following sense.
There exist parameters $\epsilon_0$ and $A_0$, depending on $c_1$ and $c_2$, if there exists $\epsilon \in (0,\epsilon_0)$ such that for any $u_0\in H^1(\mathbb{R})$,
\begin{align}\label{u02}
 \bigl\lVert u_0 - U^{(2)}_{c_1,c_2}(0;0,0) \bigr\rVert_{H^1(\mathbb{R})} < \epsilon,
\end{align}
then there exist $x_1(t), x_2(t) \in \mathbb{R}$, such that the corresponding solution $u(t,x)$ of the ILW equation (\ref{ILW}) with the initial data $u(0)=u_0$ satisfies $u_t \in C([0,\infty),H^1(\mathbb{R}))$ and
\begin{align}\label{1.22}
 \sup_{t\in (0,\infty)}|\!|u(t)-U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)|\!|_{H^1(\mathbb{R})}<A_0\epsilon,
\end{align}
with
\begin{align}
 \sup_{t\in (0,\infty)}(|x_1'(t)|+|x_2'(t)|)\leq CA_0\epsilon.
\end{align}
\end{theorem}

\begin{align*}
 (\mathcal{R}^\star(u))^m u_x &= \mathcal{J} (\mathcal{R} (u))^m H_1'(u) = \mathcal{J} (\mathcal{R} (u))^m u, \quad m \in \mathbb{N}.
\end{align*}
For subsequent applications, establishing the uniqueness and differentiability of $\mathcal{R}(u)$ becomes essential. In contrast, the Korteweg-de Vries system ($\delta = 3$) exhibits explicit structure. For Schwartz-class functions $u \in \mathcal{S}(\mathbb{R})$, its recursion operator is $\mathcal{R}_K(u) = -\partial_x^2 - \tfrac{2}{3}u - \tfrac{2}{3}\partial_x^{-1} u \partial_x$, then $\mathcal{R}_K'(u) = -\tfrac{2}{3} - \tfrac{2}{3}\partial_x^{-1}(\cdot\partial_x)$.
\begin{prop}\label{prop2.2}
 Given $u \in H^{k+1}(\mathbb{R})$ with $k \geq 0$, there exists a unique linear operator
 \[
 \mathcal{R}(u): H^{k+1}(\mathbb{R}) \to H^{k}(\mathbb{R}),
 \]
 such that relations (\ref{2.39}) and (\ref{2.41}) hold true. Moreover, $\mathcal{R}(u)$ is differentiable with respect to $u$.
\end{prop}
\begin{proof}
 The fundamental approach involves establishing a connection between the recursion operators $\mathcal{R}(u)$ and $\mathcal{R}_{12}$ (\ref{1.15}). Assuming $u \in \mathcal{S}(\mathbb{R})$, it follows from (\ref{1.14}) and (\ref{2.39}) that
 \begin{align*}
 \mathcal{S}(\mathbb{R}) \ni H_{m+1}^{\prime}(u) & = \frac{i}{2^{m+1}}\mathcal{J}^{-1}\int_{\mathbb{R}} \delta(x_{1} - x_{2}) \theta_{12}^{-} \mathcal{R}_{12}^{m+1} \cdot 1 \mathrm{d}x_{2} \\
 & = \mathcal{R}(u)H_{m}^{\prime}(u) \\
 & = \mathcal{R}(u) \frac{i}{2^m}\mathcal{J}^{-1}\int_{\mathbb{R}} \delta(x_{1} - x_{2}) \theta_{12}^{-} \mathcal{R}_{12}^{m} \cdot 1 \mathrm{d}x_{2}.
 \end{align*}
 The uniqueness of $\mathcal{R}(u)$ follows by an induction argument over $m$. Furthermore, the asymptotic behavior of the operator is characterized by
 \[
 \mathcal{R}(u) \sim -T^\delta\partial_{x} + L(u),
 \]
 where $L(u): \mathcal{S}(\mathbb{R}) \mapsto \mathcal{S}(\mathbb{R})$ represents a differentiable higher-order remainder term. By standard density argument, $\mathcal{R}(u)$ itself is differentiable and satisfies $\mathcal{R}^{\prime}(u) \sim L^{\prime}(u)$.
\end{proof}
\begin{re}
The propositions above will be shown in Section \ref{sec.3} that understanding the spectral information of the (adjoint) recursion operators $\mathcal{R} (u)$ and $\mathcal{R}^\star(u)$ is essential in proving the (spectral) stability of the ILW multi-solitons.
\end{re}

Next, we will proceed to prove Theorem \ref{th1.3}.
\subsection{The negative eigenvalues of $\mathcal{L}_n$ :Proof of Theorem \ref{th1.3}}
In this subsection, our goal is to prove the Theorem \ref{th1.3}.
\begin{proof}[\textbf{Proof of Theorem \ref{th1.3}}]
 The linearized operators around the $n$-solitons $\mathcal{L}_{n} = S_{n}^{\prime\prime}(U^{(n)})$ possess $\left[\frac{n+1}{2}\right]$ negative eigenvalues, as previously verified in (\ref{3.47}). We now prove the expression (\ref{negetive}) for these eigenvalues. As $t\to \infty$, the spectrum $\sigma(\mathcal{L}_{n}(t))$ converges to the union of the spectra $\sigma(\mathcal{L}_{n,j})$ where $\mathcal{L}_{n,j} = S_{n}^{\prime\prime}(Q_{c_{j}})$, that is
 \begin{align}
 \sigma(\mathcal{L}_{n}(t)) \rightarrow \bigcup_{j=1}^{n} \sigma(\mathcal{L}_{n,j}), \quad \text{as} \quad t \rightarrow +\infty.
 \end{align}
 Since the operators $\mathcal{L}_{n}(t)$ are isoinertial for each $n$, the spectrum $\sigma(\mathcal{L}_{n}(t))$ is independent of $t$. Thus, the negative eigenvalues of $\mathcal{L}_{n}$ match those of $\mathcal{L}_{n,j}$ for all $j = 1, 2, \dots, n$. According to Lemma \ref{lem3.5}, $\mathcal{L}_{n,j}$ has negative eigenvalues when $j = 2k-1$ and $1 \leq k \leq \left[\frac{n+1}{2}\right]$. Specially, when $n=1$, one has $\mathcal{L}_1=L_1$. Therefore, (\ref{n=1}) indicates that $\mathcal{L}_1$ has the negative $\lambda_1<0$. With the methods in \cite{BBSSB}, we could compute $\lambda_1$, which is
 \begin{align}\label{1.30}
 \lambda_1=-\frac{c-\frac{1}{\delta}}{2a^2\sin^2(a\delta)}<0,
 \end{align}
 Then for $n=2$, we have
 \begin{align}\label{lamb}
 \mathcal{L}_{2,1}=(\mathcal{R}(Q_{c_1})+c_2)\mathcal{S}_1''(Q_{c_1}).
 \end{align}
In terms of Lemma \ref{lem3.2}, the operator $(\mathcal{R}(Q_{c_1}) + c_{2})$ has an eigenvalue $c_{2} - c_{1} > 0$, with continuous spectrum $(c_2, +\infty)$ whose generalized eigenfunctions are not in $L^{2}(\mathbb{R})$. Therefore, the unique negative eigenvalues of $\mathcal{L}_{2,1}$ is $(c_2-c_1)\lambda_1<0$.\\
Assuming the result holds for $n = p$, namely, the $\left[\frac{p+1}{2}\right]$-th negative eigenvalue of $\mathcal{L}_{p}$ is
 \begin{align}\label{4.48}
 \lambda_{k}^{(p)} = -M^\delta (c_{2k-1}-\frac{1}{\delta}) \prod_{j \neq 2k-1}^{p} (c_{j} - c_{2k-1}), \quad k = 1, 2, \dots, \left[\frac{p+1}{2}\right].
 \end{align}
Then for $n = p+1$ is even, we have $\left[\frac{p+1}{2}\right] = \left[\frac{p+2}{2}\right]$ and for $k = 1, 2, \dots, \left[\frac{p+1}{2}\right]$, we have
 \begin{align}
 \mathcal{L}_{p+1,2k-1} =(\mathcal{R}(Q_{c_{2k-1}}) + c_{p+1}) \mathcal{S}_{p}^{\prime\prime}(Q_{c_{2k-1}}).
 \end{align}
 By Lemma \ref{lem3.2}, the operator $(\mathcal{R}(Q_{c_{2k-1}}) + c_{p+1})$ has an eigenvalue $c_{p+1} - c_{2k-1} > 0$, with continuous spectrum $(c_{p+1}, +\infty)$ whose generalized eigenfunctions are not in $L^{2}(\mathbb{R})$. Therefore, the $\left[\frac{p+2}{2}\right]$-th negative eigenvalues of $\mathcal{L}_{p+1,2k-1}$ is
 \begin{align}\label{4.50}
 \lambda_{k}^{(p+1)}= -M^\delta (c_{2k-1}-\frac{1}{\delta}) \prod_{j \neq 2k-1}^{p+1} (c_{j} - c_{2k-1}), \quad k = 1, 2, \dots, \left[\frac{p+1}{2}\right].
 \end{align}
For the case where $n = p+1$ is odd, we have $\left[\frac{p+1}{2}\right] + 1 = \left[\frac{p+2}{2}\right]$. The first $\left[\frac{p+1}{2}\right]$ negative eigenvalues of $\mathcal{L}_{p+1}$ are given by expression (\ref{4.50}). We now compute the final negative eigenvalue. Consider
\begin{align}
 \mathcal{L}_{p+1,p+1} =\left( \mathcal{R}(Q_{c_{p+1}}) + c_{j} \right) \tilde{S}_{p}^{\prime\prime}(Q_{c_{p+1}})
\end{align}
where $\tilde{S}_{p}$ denotes the action with wave speed $c_j$ in $S_p$ replaced by $c_{p+1}$ for some $1 \leq j \leq p$. By assumption (\ref{4.48}), the discrete eigenvalue of $\tilde{S}_{p}^{\prime\prime}(Q_{c_{p+1}})$ is
\begin{align}
 -M^\delta (c_{p+1}-\frac{1}{\delta})\prod_{l \neq j}^{p} (c_l - c_{p+1}).
\end{align}
The operator $\mathcal{R}(Q_{c_{p+1}}) + c_j$ has eigenvalue $c_j - c_{p+1} < 0$ by Lemma \ref{lem3.2}, with continuous spectrum $(c_j, +\infty)$ whose generalized eigenfunctions are not in $L^2(\mathbb{R})$. Therefore, the last negative eigenvalue of $\mathcal{L}_{p+1}$ is
\begin{align*}
 \lambda_{\left[\frac{p+2}{2}\right]}^{(p+1)} = -M^\delta (c_{p+1}-\frac{1}{\delta}) \prod_{l=1}^{p} (c_l - c_{p+1}).
\end{align*}
The proof of Theorem \ref{th1.3} is completed by combining the eigenvalue expressions.
\end{proof}

Since the stability result in \cite{MS} coincides with orbital stability theorem in \cite{GSS} for $n=1$, we obtain the following corollary regarding the orbital stability of ILW soliton solutions.
\begin{cor}
The single soliton solution of the ILW equation is orbital stable in $H^{\frac{1}{2}}(\mathbb{R})$.
\end{cor}
\begin{re}
For $n \geq 2$, it should be noted that the stability result in \cite{MS} aligns with the notion of orbital stability in \cite{GSS} when the stability framework of \cite{MS} is suitably extended. In Hamiltonian systems, invariant functionals (i.e., integrals of motion) generate Hamiltonian flows that commute with the original time evolution.
\end{re}

Then we will provide a more precise description of the orbital stability of the double solitons to the ILW equation.

\begin{theorem}[\textbf{Orbital stability of double solitons}]\label{thm1.2}
	The double solitons $U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)$ of ILW equation (\ref{ILW}) with $0<c_1<c_2$ are orbitally stable in $H^1(\mathbb{R})$ in the following sense.
There exist parameters $\epsilon_0$ and $A_0$, depending on $c_1$ and $c_2$, if there exists $\epsilon \in (0,\epsilon_0)$ such that for any $u_0\in H^1(\mathbb{R})$,
\begin{align}\label{u02}
	\bigl\lVert u_0 - U^{(2)}_{c_1,c_2}(0;0,0) \bigr\rVert_{H^1(\mathbb{R})} < \epsilon,
\end{align}
then there exist $x_1(t), x_2(t) \in \mathbb{R}$, such that the corresponding solution $u(t,x)$ of the ILW equation (\ref{ILW}) with the initial data $u(0)=u_0$ satisfies $u_t \in C([0,\infty),H^1(\mathbb{R}))$ and
\begin{align}\label{1.22}
	\sup_{t\in (0,\infty)}|\!|u(t)-U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)|\!|_{H^1(\mathbb{R})}<A_0\epsilon,
\end{align}
with
\begin{align}
	\sup_{t\in (0,\infty)}(|x_1'(t)|+|x_2'(t)|)\leq CA_0\epsilon.
\end{align}
\end{theorem}

Moreover, something is interesting is that one can express the negative eigenvalues of the isoinertial operator $\mathcal{L}_n$ in terms of the wave speeds $\{c_j\}^n_{j=1}$ as follows.
\begin{theorem}[\textbf{The negative eigenvalues of $\mathcal{L}_n$}]\label{th1.3}
	The linearized operator $\mathcal{L}_{n}$ about the $n$-soliton solution possesses exactly $\left\lfloor\frac{n+1}{2}\right\rfloor$ negative eigenvalues $\lambda_{k}$, where $k=1,2,\cdots,\left\lfloor\frac{n+1}{2}\right\rfloor$ and $\lfloor x\rfloor$ denotes the floor function. Furthermore, the negative eigenvalues of $\mathcal{L}_n(n\geq 1$)  have the following expression
	\begin{align}\label{negetive}
		\lambda_{k} = -M^\delta_k (c_{2k-1}-\frac{1}{\delta})\prod_{ j\neq 2k-1}^{n}(c_{j}-c_{2k-1}),\quad k=1,2,\cdots,\left\lfloor\frac{n+1}{2}\right\rfloor,
	\end{align}
	where $M^\delta_k$ are positive constants independent of the wave speeds $c_1,\cdots, c_n$.
\end{theorem}
\begin{re}
	Since the ILW equation serves as the bridge connecting KdV equation and BO equation. The stability results above are also valid for both equations. The specific results can be found in \cite{WA} and \cite{LW}, respectively.
\end{re}

The proof of our main stability result, Theorem \ref{thm1.1}, proceeds by transforming the stability problem into a spectral analysis of an explicitly constructed linearized operator. Through the IST method, we establish a fundamental connection between the Hamiltonians and scattering data. Similarly, the proof of Theorem \ref{thm1.2} on orbital stability is reduced to a detailed spectral analysis of the linearized operator $\mathcal{L}_n$ arising from the second variation of the Lyapunov functional. Finally, the proof of Theorem \ref{th1.3} relies on the application of spectral analysis results. The proof strategy unfolds as follows.

In Section \ref{sec.2}, we begin by establishing the properties of the singular integral operators $T^\delta$ and $T^\delta_{12}$. We then introduce the IST framework for the ILW equation, from which the $n$-soliton solutions are formally derived.

Section \ref{sec.3} is devoted to a detailed spectral analysis of the linearized operator $\mathcal{L}_n = \mathcal{S}_n''(U^{(n)})$. We demonstrate that its inertia index is $\left( \left\lfloor \frac{n+1}{2} \right\rfloor, n \right)$, thereby satisfying the crucial stability condition.

In Section \ref{sec.4}, we apply the framework of constrained variational principles to establish the dynamical stability result in Theorem \ref{thm1.1}. We then prove Theorem \ref{thm1.2} on orbital stability of double solitons ($n=2$) through a refined spectral analysis of the linearized operator $\mathcal{L}_2$, yielding a direct proof of orbital stability. Finally, we establish Theorem \ref{th1.3} based on the spectral analysis results from Section \ref{sec.3}.

The proof of Theorem~\ref{thm1.1} “proceeds by transforming the stability problem into a spectral analysis of an explicitly constructed linearized operator.” “Through the IST method, we establish a fundamental connection between the Hamiltonians and scattering data.”

More concretely, the stated strategy is organized by sections as follows:
\begin{itemize}
\item \emph{Section~\ref{sec.2}:} establish “the properties of the singular integral operators $T^\delta$ and $T^\delta_{12}$,” and “introduce the IST framework for the ILW equation, from which the $n$-soliton solutions are formally derived.”
\item \emph{Section~\ref{sec.3}:} carry out “a detailed spectral analysis of the linearized operator $\mathcal{L}_n = \mathcal{S}_n''(U^{(n)})$,” showing “its inertia index is $\left( \left\lfloor \frac{n+1}{2} \right\rfloor, n \right)$, thereby satisfying the crucial stability condition.”
\item \emph{Section~\ref{sec.4}:} “apply the framework of constrained variational principles to establish the dynamical stability result in Theorem~\ref{thm1.1}.”
\end{itemize}

(Additionally, the introduction notes that the orbital stability theorem (Theorem~\ref{thm1.2}) is “reduced to a detailed spectral analysis of the linearized operator $\mathcal{L}_n$ arising from the second variation of the Lyapunov functional,” and that Theorem~\ref{th1.3} “relies on the application of spectral analysis results.”)

The proof of the main theorem proceeds by transforming the stability problem into a spectral analysis of an explicitly constructed linearized operator. Through the IST method, we establish a fundamental connection between the Hamiltonians and scattering data.

More concretely, the stated strategy is organized by sections as follows:
\begin{itemize}
\item \emph{Section~\ref{sec.2}:} establish “the properties of the singular integral operators $T^\delta$ and $T^\delta_{12}$,” and “introduce the IST framework for the ILW equation, from which the $n$-soliton solutions are formally derived.”
\item \emph{Section~\ref{sec.3}:} carry out “a detailed spectral analysis of the linearized operator $\mathcal{L}_n = \mathcal{S}_n''(U^{(n)})$,” showing “its inertia index is $\left( \left\lfloor \frac{n+1}{2} \right\rfloor, n \right)$, thereby satisfying the crucial stability condition.”
\item \emph{Section~\ref{sec.4}:} “apply the framework of constrained variational principles to establish the dynamical stability result in the main theorem.”
\end{itemize}

Additionally, the introduction notes that the orbital stability theorem
\begin{theorem}[\textbf{Orbital stability of double solitons}]\label{thm1.2}
	The double solitons $U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)$ of ILW equation (\ref{ILW}) with $0<c_1<c_2$ are orbitally stable in $H^1(\mathbb{R})$ in the following sense.
There exist parameters $\epsilon_0$ and $A_0$, depending on $c_1$ and $c_2$, if there exists $\epsilon \in (0,\epsilon_0)$ such that for any $u_0\in H^1(\mathbb{R})$,
\begin{align}\label{u02}
	\bigl\lVert u_0 - U^{(2)}_{c_1,c_2}(0;0,0) \bigr\rVert_{H^1(\mathbb{R})} < \epsilon,
\end{align}
then there exist $x_1(t), x_2(t) \in \mathbb{R}$, such that the corresponding solution $u(t,x)$ of the ILW equation (\ref{ILW}) with the initial data $u(0)=u_0$ satisfies $u_t \in C([0,\infty),H^1(\mathbb{R}))$ and
\begin{align}\label{1.22}
	\sup_{t\in (0,\infty)}|\!|u(t)-U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)|\!|_{H^1(\mathbb{R})}<A_0\epsilon,
\end{align}
with
\begin{align}
	\sup_{t\in (0,\infty)}(|x_1'(t)|+|x_2'(t)|)\leq CA_0\epsilon.
\end{align}
\end{theorem}

is “reduced to a detailed spectral analysis of the linearized operator $\mathcal{L}_n$ arising from the second variation of the Lyapunov functional,” and that the following theorem “relies on the application of spectral analysis results”:
\begin{theorem}[\textbf{The negative eigenvalues of $\mathcal{L}_n$}]\label{th1.3}
	The linearized operator $\mathcal{L}_{n}$ about the $n$-soliton solution possesses exactly $\left\lfloor\frac{n+1}{2}\right\rfloor$ negative eigenvalues $\lambda_{k}$, where $k=1,2,\cdots,\left\lfloor\frac{n+1}{2}\right\rfloor$ and $\lfloor x\rfloor$ denotes the floor function. Furthermore, the negative eigenvalues of $\mathcal{L}_n(n\geq 1$)  have the following expression
	\begin{align}\label{negetive}
		\lambda_{k} = -M^\delta_k (c_{2k-1}-\frac{1}{\delta})\prod_{ j\neq 2k-1}^{n}(c_{j}-c_{2k-1}),\quad k=1,2,\cdots,\left\lfloor\frac{n+1}{2}\right\rfloor,
	\end{align}
	where $M^\delta_k$ are positive constants independent of the wave speeds $c_1,\cdots, c_n$.
\end{theorem}

[\textbf{Dynamical stability of $n$-solitons}]\label{thm1.1}
Given $ n\in \mathbb{N} $, $ n\geq 1 $, a collection of wave speeds $\textbf{c} = (c_{1},\dots,c_{n})$ with $ 0 < c_{1} < \dots < c_{n} $ and a collection of space transitions $\textbf{x} = (x_{1},\dots,x_{n})\in \mathbb{R}^{n}$, let $ U^{(n)}(\cdot,\cdot;\textbf{c},\textbf{x}) $ be the corresponding $n$-soliton solutions of
\begin{align}
\tu_t+\frac{1}{\delta}u_x+2uu_x+T^\delta u_{xx}=0,\quad  (t,x)\in \mathbb{R} \times \mathbb{R},\label{ILW}
\end{align}
Then for any $ \epsilon > 0 $, there exists $ \delta > 0 $ such that for any $ u_{0}\in H^{\frac{n}{2}}(\mathbb{R}) $, the following stability property holds, if

\begin{align}
	&\bigl\lVert u_{0} - U^{(n)}(0,\cdot;\textbf{c},\textbf{x}) \bigr\rVert_{H^{\frac{n}{2}}} < \delta,
\end{align}
then for any $t \in \mathbb{R}$, the corresponding solution of the equation above verifies
\begin{align}
\inf_{\tau \in \mathbb{R},\  \textbf{y} \in \mathbb{R}^{n}} \bigl\lVert u(t) - U^{(n)}(\tau,\cdot;\textbf{c},\textbf{y}) \bigr\rVert_{H^{\frac{n}{2}}} < \epsilon.
\end{align}

Question:
Let \(n\in\mathbb N\) with \(n\ge 1\), let \(\mathbf c=(c_1,\dots,c_n)\) satisfy \(0<c_1<\cdots<c_n\), and let \(\mathbf x=(x_1,\dots,x_n)\in\mathbb R^n\). Let \(U^{(n)}(t,x;\mathbf c,\mathbf x)\) be the corresponding \(n\)-soliton solution of the ILW equation
\[
u_t+\frac{1}{\delta}u_x+2uu_x+T^\delta u_{xx}=0,\qquad (t,x)\in\mathbb R\times\mathbb R,
\]
where \(T^\delta\) is the ILW nonlocal operator. Suppose \(u(t)\) is the solution with initial data \(u(0)=u_0\in H^{n/2}(\mathbb R)\). Which of the following conclusions holds about the orbit of the \(n\)-soliton family \(\{U^{(n)}(\tau,\cdot;\mathbf c,\mathbf y):\tau\in\mathbb R,\ \mathbf y=(y_1,\dots,y_n)\in\mathbb R^n\}\) in the \(H^{n/2}(\mathbb R)\) norm?

For every \(\epsilon>0\), there exists \(\eta>0\) such that if
\[
\bigl\lVert u_0-U^{(n)}(0,\cdot;\mathbf c,\mathbf x)\bigr\rVert_{H^{n/2}}<\eta,
\]
then the corresponding solution satisfies
\[
\inf_{\tau\in\mathbb R,\ \mathbf y\in\mathbb R^n}\bigl\lVert u(t)-U^{(n)}(\tau,\cdot;\mathbf c,\mathbf y)\bigr\rVert_{H^{n/2}}<\epsilon
\quad\text{for every } t\in\mathbb R.
\]
Thus the \(n\)-soliton family is orbitally stable in \(H^{n/2}(\mathbb R)\).

Choices:
B.
For every \(\epsilon>0\), there exists \(\eta>0\) such that if
\[
\bigl\lVert u_0-U^{(n)}(0,\cdot;\mathbf c,\mathbf x)\bigr\rVert_{H^{n/2}}<\eta,
\]
then the corresponding solution satisfies
\[
\inf_{\mathbf y\in\mathbb R^n}\bigl\lVert u(t)-U^{(n)}(t,\cdot;\mathbf c,\mathbf y)\bigr\rVert_{H^{n/2}}<\epsilon
\quad\text{for every } t\in\mathbb R.
\]
Thus the \(n\)-soliton family is orbitally stable in \(H^{n/2}(\mathbb R)\) with the time parameter fixed.
C.
For every \(\epsilon>0\), there exists \(\eta>0\) such that if
\[
\bigl\lVert u_0-U^{(n)}(0,\cdot;\mathbf c,\mathbf x)\bigr\rVert_{H^{n/2}}<\eta,
\]
then the corresponding solution satisfies
\[
\inf_{\mathbf y\in\mathbb R^n}\bigl\lVert u(t)-U^{(n)}(t,\cdot;\mathbf c,\mathbf y)\bigr\rVert_{H^{n/2}}<\epsilon
\quad\text{for every } t\in\mathbb R.
\]
In particular, the orbit remains close to the \(n\)-soliton family in \(H^{n/2}(\mathbb R)\).
D.
There exists \(\eta>0\) such that for every \(\epsilon>0\), if
\[
\bigl\lVert u_0-U^{(n)}(0,\cdot;\mathbf c,\mathbf x)\bigr\rVert_{H^{n/2}}<\eta,
\]
then the corresponding solution satisfies
\[
\inf_{\tau\in\mathbb R,\ \mathbf y\in\mathbb R^n}\bigl\lVert u(t)-U^{(n)}(\tau,\cdot;\mathbf c,\mathbf y)\bigr\rVert_{H^{n/2}}<\epsilon
\quad\text{for every } t\in\mathbb R.
\]
Thus the \(n\)-soliton family is orbitally stable in \(H^{n/2}(\mathbb R)\) with a uniform stability radius independent of \(\epsilon\).
E.
For every \(\epsilon>0\), there exists \(\eta>0\) such that if
\[
\bigl\lVert u_0-U^{(n)}(0,\cdot;\mathbf c,\mathbf x)\bigr\rVert_{H^{n/2}}<\eta,
\]
then the corresponding solution satisfies
\[
\bigl\lVert u(t)-U^{(n)}(t,\cdot;\mathbf c,\mathbf x)\bigr\rVert_{H^{n/2}}<\epsilon
\quad\text{for every } t\in\mathbb R.
\]
Thus the given \(n\)-soliton solution is asymptotically stable in \(H^{n/2}(\mathbb R)\).

quantifier order between \(\epsilon\) and neighborhood size

orbital stability modulo symmetries replaced by fixed-profile/asymptotic stability


\begin{thm}\label{thm:main} Let $F$ be a totally real field,
and $F'$ be a real quadratic extension of $F$. Assume that $p$ is
unramified in both $F$ and $F'$. Let $f$ be a Hilbert newform of
parallel even weight $k$ over $F$ with trivial central character and
level $\mathfrak{n}_f$. Let $f'$ be the base change of $f$ to $F'$.
Let $\mathfrak{n}_{f'}$ be the level of $f'$.

Let $p$ be a prime number satisfying $p\geq\mathrm{max}(k+2,7)$. We
assume that $\mathfrak{n}_f$ is prime to $p$ and $f$ is ordinary at
each prime above $p$. When $[F:\mathbb{Q}]$ is odd, we assume that
there exists at least one prime $\mathfrak{q}$ such that
$\mathfrak{q}|| \mathfrak{n}_f$, and $\mathfrak{q}$ is split in
$F'$.

Suppose the following conditions hold.
\begin{enumerate}
\item The restrictions of $\bar{\rho}_f$ to $G_{F'(\xi_p)}$ and $G_{F'(
\sqrt{ p^*})}$ are absolutely irreducible, where
$p^*=(-1)^{\frac{p-1}{2}}p$.
\item In the case of $k=2$, for each place $v$ of $F$ above $p$ we
have  $a_v^2(f)\ {\backslash\hskip -10pt \equiv } 1 \ (\mathrm{mod}
\ p)$; similarly, for each place $v'$ of $F'$ above $p$ we have
$a_{v'}^2(f')\ {\backslash\hskip -10pt \equiv } 1 \ (\mathrm{mod} \
p)$.
\item For each $\mathfrak{l}|\mathfrak{n}$, if
$\bar{\rho}_f|_{I_{F_\mathfrak{l}}}$ is absolutely irreducible, then
$\mathrm{N}(\mathfrak{l})\equiv  {\hskip -10pt /}-1 \ (\mathrm{mod
}\ p)$. Similarly, for each $\mathfrak{l}'|\mathfrak{n}'$, if
$\bar{\rho}_{f'}|_{I_{F'_{\mathfrak{l}'}}}$ is absolutely
irreducible, then $\mathrm{N}(\mathfrak{l}')\equiv  {\hskip -10pt
/}-1 \ (\mathrm{mod }\ p)$.
\item $\rho_f$ is a minimal modular lifting of $\bar{\rho}_f$.
\end{enumerate} Then $$
\Omega_{f'}^\mathrm{can}=(\Omega_f^\mathrm{can})^2 $$ up to a
$p$-adic unit.
\end{thm}

\subsection{Special value formula, Gross period and Hida canonical period}

In \cite{Gro87} Gross provided a formula expressing the special
values of $L$-functions via Heegner points. To describe his result,
let $f$ be a newform on $\Gamma_0(N)$  of even weight $k$, and let $K$ be an imaginary
quadratic field of discriminant $D$ such that $D$ is prime to $N$.

Assume $k=2$ and $N$ is a prime inert in $K$. Let $B$ be the definite quaternion algebra that is ramified exactly at $N$. Fix a prime number $p$. Let $H_n$ be the ring class field of $K$ of conductor $p^n$. When $\chi$ is a character of the Galois group $\mathrm{Gal}(H_n/K)$, one forms the Rankin-Selberg $L$-function $L(f,\chi,s)$.
Gross \cite{Gro87} showed that there exists a period $\Omega_{f,K}$
depending on $f$ and $K$ but independent of $n$ and $\chi$, called Gross
period, such that $\frac{L(f,\chi,1)}{\Omega_{f,K}}$ are algebraic
and they satisfy
$$\frac{L(f,\chi,1)}{\Omega_{f,K}}\cdot\sqrt{D}p^{n}=|\sum\limits_{\sigma\in\mathrm{Gal}(H_n/K)}\chi(\sigma)\psi(P^{\sigma})|^{2},$$
where $P$ is the Heegner point on the Shimura set attached to $B$.

Gross formula is generalized by Shouwu Zhang \cite[Theorem 7.1]{Zha04} for $k=2$ and general $N$, and by Haiping Yuan \cite{Yua05} for $k>2$.

In \cite{BD07}  Bertolini and Darmon used Gross' special value
formula to construct anticyclotomic $p$-adic $L$-functions, i.e.
$p$-adic $L$-functions for anticyclotomic $\BZ_p$-extension instead
of the cyclotomic $\BZ_p$-extension. In the same paper they proved
one divisibility for Iwasawa main conjecture for elliptic curves in
the setting of anticyclotomic $\BZ_p$-extension (\cite[Theorem
1]{BD07}), and also gave new evidence for Birch and Swinnerton-Dyer
conjecture (\cite[Corollary 4]{BD07}).

Gross period serves as a bridge between the complex $L$-function and
the $p$-adic $L$-function. But it depends on the choice of an imaginary
quadratic extension $K$.

Hida \cite{Hid81, Hid81.2} introduced a period that is independent
of $K$ and called the {\it canonical period}. Attached to $f$ there
exists a homomorphism $\lambda_f:T\longrightarrow \mathcal{O}_f$
from the Hecke algebra $T$ of level $N$ to a discrete valuation ring
$\mathcal{O}_f$ that is finite over $\BZ_p$. Let $\eta $ be the
congruence number for $\lambda_f$ defined by
$$\eta =\lambda_f(\mathrm{Ann}(\mathrm{Ker}(\lambda_f))).$$ Then the
canonical period is defined by
$$\Omega_{f}^{\mathrm{can}}=\frac{\langle f, f
\rangle_{\mathrm{Pet}}}{\eta }.$$ The reader may consult
\cite[Section 2]{Vat03} about more knowledge on Gross period and the
canonical period.

Vatsal \cite{Vat03} showed that
$\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}$ lies in
$\mathcal{O}_f$. Pollack and Weston \cite{PW11} pointed out that
$\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}\in\mathcal{O}_f^{\times}$
is equivalent to the freeness of spaces of modular forms on $B$ over
the associated Hecke algebra and the vanishing of certain local
Tamagawa components. In \cite{CH18} Chida and Hsieh actually proved
that $\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}$ is a $p$-adic
unit, i.e.
$\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}\in\mathcal{O}_f^{\times}$
under a condition ($\mathrm{CR^{+}}$). Wang \cite{Wang} generalized Chida and Hsieh's result to the setting of Hilbert
modular forms under similar hypothesis.
There are two normalizations of anticyclotomic $p$-adic
$L$-functions depending on which period one uses. However, in the
present paper we assume those hypothesis in \cite{Wang} in order
that there is essentially no difference between these two
normalizations.

\subsection{Comparison of periods under base change}

The goal of the present paper is to compare the canonical period of
a Hilbert modular form and that of its base change to a (totally)
real quadratic extension.

To make it precise, let $F$ be a totally real number field, and  $f$
be a Hilbert newform of parallel even weight over $F$. Let $F'$ be a
totally real quadratic extension of $F$, and $f'$ be the base change
of $f$ to $F'$. One may expect the following holds.

\begin{conj} We have
$$ \Omega_{f'}^\mathrm{can}=(\Omega_f^\mathrm{can})^2 $$ up to a
$p$-adic unit.
\end{conj}

The main result of this paper is the following theorem. Let $\rho_f$ be $p$-adic Galois representation attached to $f$ (see \cite{Wil88,Tay}), and let $\bar{\rho}_f$ denote the residue representation of $\rho_f$.

Assume $k=2$ and $N$ is a prime inert in $K$. Let $B$ be the definite quaternion algebra that is ramified exactly at $N$. Fix a prime number $p$. Let $H_n$ be the ring class field of $K$ of conductor $p^n$. When $\chi$ is a character of the Galois group $\mathrm{Gal}(H_n/K)$, one forms the Rankin-Selberg $L$-function $L(f,\chi,s)$.
Gross \cite{Gro87} showed that there exists a period $\Omega_{f,K}$
depending on $f$ and $K$ but independent of $n$ and $\chi$, called Gross
period, such that $\frac{L(f,\chi,1)}{\Omega_{f,K}}$ are algebraic
and they satisfy
$$\frac{L(f,\chi,1)}{\Omega_{f,K}}\cdot\sqrt{D}p^{n}=|\sum\limits_{\sigma\in\mathrm{Gal}(H_n/K)}\chi(\sigma)\psi(P^{\sigma})|^{2},$$
where $P$ is the Heegner point on the Shimura set attached to $B$.

Hida \cite{Hid81, Hid81.2} introduced a period that is independent
of $K$ and called the {\it canonical period}. Attached to $f$ there
exists a homomorphism $\lambda_f:T\longrightarrow \mathcal{O}_f$
from the Hecke algebra $T$ of level $N$ to a discrete valuation ring
$\mathcal{O}_f$ that is finite over $\BZ_p$. Let $\eta $ be the
congruence number for $\lambda_f$ defined by
$$\eta =\lambda_f(\mathrm{Ann}(\mathrm{Ker}(\lambda_f))).$$ Then the
canonical period is defined by
$$\Omega_{f}^{\mathrm{can}}=\frac{\langle f, f
\rangle_{\mathrm{Pet}}}{\eta }.$$ The reader may consult
\cite[Section 2]{Vat03} about more knowledge on Gross period and the
canonical period.

Vatsal \cite{Vat03} showed that
$\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}$ lies in
$\mathcal{O}_f$. Pollack and Weston \cite{PW11} pointed out that
$\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}\in\mathcal{O}_f^{\times}$
is equivalent to the freeness of spaces of modular forms on $B$ over
the associated Hecke algebra and the vanishing of certain local
Tamagawa components. In \cite{CH18} Chida and Hsieh actually proved
that $\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}$ is a $p$-adic
unit, i.e.
$\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}\in\mathcal{O}_f^{\times}$
under a condition ($\mathrm{CR^{+}}$). Wang \cite{Wang} generalized Chida and Hsieh's result to the setting of Hilbert
modular forms under similar hypothesis.
There are two normalizations of anticyclotomic $p$-adic
$L$-functions depending on which period one uses. However, in the
present paper we assume those hypothesis in \cite{Wang} in order
that there is essentially no difference between these two
normalizations.

To make it precise, let $F$ be a totally real number field, and $f$
be a Hilbert newform of parallel even weight over $F$. Let $F'$ be a
totally real quadratic extension of $F$, and $f'$ be the base change
of $f$ to $F'$. One may expect the following holds.

\begin{conj} We have
$$ \Omega_{f'}^\mathrm{can}=(\Omega_f^\mathrm{can})^2 $$ up to a
$p$-adic unit.
\end{conj}

The main result of this paper is the following theorem. Let $\rho_f$ be $p$-adic Galois representation attached to $f$ (see \cite{Wil88,Tay}), and let $\bar{\rho}_f$ denote the residue representation of $\rho_f$.

The main result of this paper is the following theorem. Let $\rho_f$ be $p$-adic Galois representation attached to $f$ (see \cite{Wil88,Tay}), and let $\bar{\rho}_f$ denote the residue representation of $\rho_f$.

We choose an imaginary quadratic extension $K$ of $F$ that is
inertia at $\mathfrak{q}$ when $[F:\mathbb{Q}]$ is odd. Denote $KF'$
by $K'$. Fix a place $\mathfrak{p}$ of $F$ above $p$. Let $K_\infty$ be the $\mathfrak{p}$-anticyclotomic
$\mathbb{Z}_p$-extension of $K$.

\begin{lem}\label{lem:fitt} Let $S$ be the set of primes dividing $\mathfrak{n}^-$. Then
$$ \mathrm{Fitt}(\prod_{\mathfrak{l}|
\mathfrak{n}^- }H^1((\mathcal{K}_\infty)_\mathfrak{l},
F^-_\mathfrak{l}A_\rho)^\vee) =\left(\frac{L^S_p (\mathcal{K}_\infty,
f)}{L_p (\mathcal{K}_\infty, f)}\right).
$$
\end{lem} This should be well known.
However, the authors do not have a reference about it.
\begin{proof}
By definition (see Section \ref{ss:selmer}) $H^{1}((\mathcal{K}_{\infty})_{\mathfrak{l}},F_{\mathfrak{l}}^{-}A_{\rho})$ is the inductive limit $\lim\limits_{\overrightarrow{\;\;L\;\;}} H^{1}(L_{\mathfrak{l}},F_{\mathfrak{l}}^{-}A_{\rho}),$ where $L$ runs through all finite extensions of $K$ contained in $\mathcal{K}_\infty$. For each $L$, $H^{1}(L_{\mathfrak{l}},F_{\mathfrak{l}}^{-}A_{\rho})$ is isomorphic to
$\bigoplus_{\mathfrak{L}|\mathfrak{l}} H^1(G_{L_\mathfrak{L}}, F_{\mathfrak{l}}^{-}A_{\rho})$ where $\mathfrak{L}$ runs over all places of $L$ above $\mathfrak{l}$,
and sits in the following exact sequence
{\small $$ \bigoplus_{\mathfrak{L}|\mathfrak{l}} H^1(G_{L_\mathfrak{L}}/I_{L_\mathfrak{L}}, (F_{\mathfrak{l}}^{-}A_{\rho})^{I_{L_\mathfrak{L}}})\rightarrowtail \bigoplus_{\mathfrak{L}|\mathfrak{l}} H^1(G_{L_\mathfrak{L}}, F_{\mathfrak{l}}^{-}A_{\rho})\twoheadrightarrow \bigoplus_{\mathfrak{L}|\mathfrak{l}} H^1(I_{L_\mathfrak{L}}, F_{\mathfrak{l}}^{-}A_{\rho})^{G_{L_\mathfrak{L}}/I_{L_\mathfrak{L}}} . $$}
As $G_{L_{\mathfrak{L}}}$ acts trivially on $F_\mathfrak{l}^-A_\rho$, we have
$$ H^1(G_{L_\mathfrak{L}}/I_{L_\mathfrak{L}}, (F_{\mathfrak{l}}^{-}A_{\rho})^{I_{L_\mathfrak{L}}}) \cong F_{\mathfrak{l}}^{-}A_{\rho}. $$ If $L'$ is a finite extension of $L$ contained in $\mathcal{K}_\infty$, and $\mathfrak{L}'$ is a place of $L'$ above $\mathfrak{L}$, we have the following commutative diagram
\[ \xymatrix{ H^1(G_{L_\mathfrak{L}}/I_{L_\mathfrak{L}}, (F_{\mathfrak{l}}^{-}A_{\rho})^{I_{L_\mathfrak{L}}})\ar[rr]^{\hskip45pt \simeq}\ar[d] && F_{\mathfrak{l}}^{-}A_{\rho} \ar[d]^{\times[G_{L_\mathfrak{L}}/I_{L_\mathfrak{L}}:G_{L'_{\mathfrak{L}'}}/I_{L'_{\mathfrak{L}'}}]} \\ H^1(G_{L'_{\mathfrak{L}'}}/I_{L'_{\mathfrak{L}'}}, (F_{\mathfrak{l}}^{-}A_{\rho})^{I_{L'_{\mathfrak{L}'}}})\ar[rr]^{\hskip45pt\simeq} && F_{\mathfrak{l}}^{-}A_{\rho}, } \] where the left vertical arrow is the restriction map. Since both $L_{\mathfrak{L}}$ and $L'_{\mathfrak{L}'}$ are unramified over $K_\mathfrak{l}$, $$[G_{L_\mathfrak{L}}/I_{L_\mathfrak{L}}:G_{L'_{\mathfrak{L}'}}/I_{L'_{\mathfrak{L}'}}]
=|\mathrm{Gal}(L'_{\mathfrak{L}'}/L_{\mathfrak{L}})|=[L'_{\mathfrak{L}'}:L_{\mathfrak{L}}]$$ is a power of $p$. Thus taking limit we have $$\lim\limits_{\overrightarrow{\;\;L\;\;}}\bigoplus_{\mathfrak{L}: \text{places of } L \text{ above }\mathfrak{l}} H^1(G_{L_\mathfrak{L}}/I_{L_\mathfrak{L}}, (F_{\mathfrak{l}}^{-}A_{\rho})^{I_{L_\mathfrak{L}}})=0.$$

\begin{proof} The representation $\bar{\rho}_f$ factors through
$\mathrm{Gal}(F_1/F)$ for a finite Galois extension $F_1$ of $F$.
Take a finite place $\mathfrak{q}\nmid p\mathfrak{n}$. For each
$\mathfrak{l}|p\mathfrak{n}$ take a unit $u_\mathfrak{l}$ of
$\mathcal{O}_{F_{\mathfrak{l}}}$ such that the image
$u_\mathfrak{l}$ in $k_{\mathfrak{l}}$ by the natural homomorphism
$$ \mathcal{O}_{F_\mathfrak{l}}\rightarrow
\mathcal{O}_{F_\mathfrak{l}}/\mathfrak{l}\cong k_\mathfrak{l} $$
generates the multiplicative group $k_\mathfrak{l}^\times$. Take a
uniformizing element $\omega_\mathfrak{q}$ of
$\mathcal{O}_{F_{\mathfrak{q}}}$. By the weak approximation theorem
there exists a totally positive element $a$ of $F$ such that
$a_{\mathfrak{l}}-u_{\mathfrak{l}} \in
\mathfrak{l}\mathcal{O}_{F_\mathfrak{l}}$ for each
$\mathfrak{l}|p\mathfrak{n}$, and
$a_\mathfrak{q}-\omega_\mathfrak{q}\in
\mathfrak{q}^2\mathcal{O}_{F_\mathfrak{q}}$. Put $F'=F(\sqrt{a})$.
Then each place dividing $p\mathfrak{n}$ splits in $F'$. As $a$ is
totally positive, $F'$ is totally real. Let $f'$ be the base change
of $f$ to $F'$.

\begin{thm}\label{thm:Iw-main} Assume $f$ satisfies
$(\mathrm{CR}^+)$, $(\mathfrak{n}^+\text{-}\mathrm{DT})$,
$(\mathrm{PO})$ and $(\mathrm{Fuji}1$-$4)$.
\begin{enumerate}
\item\label{it:Iw-main-a}
We have \begin{equation}\label{eq:char} \mathrm{char}\ \mathrm{Sel}
(K_\infty, A_{\rho})^\vee = (L_p(K_\infty, f)). \end{equation}
\item\label{it:Iw-main-b} Let $\mathcal{L}$ be $\mathcal{K}_\infty,
\mathcal{K}^-_\infty$ or $K^-_J$. If $[F:\BQ]$ is even, then
$$ \mathrm{char}\ \mathrm{Sel}(\mathcal{L}, A_\rho)^\vee = (L_p(\mathcal{L}, f))
$$ in $\mathcal{O}_\mathfrak{P}[[\mathrm{Gal}(\mathcal{L}/K)]]_E$.
\end{enumerate}
\end{thm}
\begin{proof} Let $F'$ be a totally real quadratic extension of $F$, as in Lemma \ref{lem:quad-ext}. Then $[F':\BQ]$ is even.
Let $\chi_{F'/F}$ be the quadratic character corresponding to the
extension $F'/F$. Put $K'=F'K$. Let $\mathcal{K}'^-_\infty$ be the
maximal abelian anticyclotomic $\BZ_p$-extension of $K'$ unramified
outside $p$. Put $\mathcal{K}'^+_\infty=F_\infty K'$,
$\mathcal{K}'_\infty=\mathcal{K}'^+_\infty\mathcal{K}'^-_\infty$ and
$\Gamma_{K'}=\mathrm{Gal}(\mathcal{K}'_\infty/K')$. Then
$K'\mathcal{K}_\infty=F'\mathcal{K}_\infty$ is contained in
$\mathcal{K}'_\infty$.

\begin{thm}\label{thm:main-stronger} Let $F$ be a totally real field,
and $F'$ be a real quadratic extension of $F$. Assume that $p$ is
unramified in both $F$ and $F'$. Let $f$ be a Hilbert newform of
parallel even weight $k$ over $F$ with trivial central character and
level $\mathfrak{n}_f$. Let $f'$ be the base change of $f$ to $F'$.
Let $\mathfrak{n}_{f'}$ be the level of $f'$.

We assume that $\mathfrak{n}_f$ is prime to $p$ and $f$ is ordinary
at each place above $p$. When $[F:\mathbb{Q}]$ is odd, we assume
that there exists at least one prime $\mathfrak{q}$ such that
$\mathfrak{q}|| \mathfrak{n}_f$, $\mathfrak{q}$ is split in $F'$,
and $\bar{\rho}_{f}$ is ramified at $\mathfrak{q}$. Suppose that, if
$\mathfrak{l}'||\mathfrak{n}_{f'}$ and
$\mathrm{N}(\mathfrak{l}')\equiv 1 \ (\mathrm{mod} \ p)$, then
$\bar{\rho}_{f'}$ is ramified at $\mathfrak{l}'$.

\begin{conj}\label{conj-mu} Let $F$ be a totally real number field, $K$ a quadratic imaginary
extension of $F$. Let $F'$ be a finite totally real extension of
$F$, and $f_{F'}$ be a Hilbert modular form over $F'$ that is
ordinary at each place of $F'$ above $p$. Then
$$\mu(L_p(F'K_\infty, f_{F'}))=0.$$
\end{conj}
Corollary \ref{thm:mu} says that Conjecture \ref{conj-mu} holds when
$F'$ is a quadratic real extension of $F$ and $f_{F'}$ is the base
change of a Hilbert modular form over $F$ (that satisfies some
conditions). It seems impossible to use the method in \cite{CH18,
Hung} to prove Conjecture \ref{conj-mu}, though it works when
$F'=F$.

We choose an imaginary quadratic extension $K$ of $F$ that is
inertia at $\mathfrak{q}$ when $[F:\mathbb{Q}]$ is odd. Denote $KF'$
by $K'$. Fix a place $\mathfrak{p}$ of $F$  above $p$. Let $K_\infty$ be the $\mathfrak{p}$-anticyclotomic
$\mathbb{Z}_p$-extension of $K$.

Using the canonical periods one attaches to $f$,
$f\otimes\chi_{F'/F}$ and $f'$ the $p$-adic $L$-functions
$$L_{p}(K_{\infty} ,f), \
L_{p}(K_{\infty} ,f\otimes\chi_{F'/F})  \ \text{ and }
L_{p}(K'K_{\infty} ,f').$$ All of them can be considered as elements
of $\mathcal{O}_{\mathfrak{P}}[[\mathrm{Gal}(K_{\infty} /K)]]$. The
ring $\mathcal{O}_{\mathfrak{P}}\supset \mathcal{O}_f$ is some
coefficient ring that is clearly defined in our context. To prove
Theorem \ref{thm:main} we only need to show
\begin{equation}\label{eq:main}
(L_{p}(K'K_{\infty} ,f'))=(L_{p}(K_{\infty} ,f))\cdot
(L_{p}(K_{\infty} ,f\otimes\chi_{F'/F}))  \end{equation} in
$\mathcal{O}_{\mathfrak{P}}[[\mathrm{Gal}(K_{\infty} /K)]].$

Our strategy is to use Iwasawa main conjecture that provides an
equality between a quantity measuring Selmer groups and $p$-adic
$L$-functions. Actually what we need is Iwasawa main conjecture for
Hilbert modular forms in the anticyclotomic setting.

Its proof is divided into two parts, one part proving one
divisibility by Ribet's method, and the other proving the converse
divisibility by Euler systems. The former divisibility was proved by
Skinner and Urban \cite{S-U} for elliptic modular forms, and was
proved by Wan \cite{Wan} for Hilbert modular forms. When
$[F:\mathbb{Q}]$ is odd, Wan needs the condition that Ihara Lemma
for Shimura curves holds. The latter divisibility was proved by
Bertolini and Darmon \cite{BD07} for elliptic curves, and by Chida
and Hsieh \cite{CH15} for elliptic modular forms. It was proved by
Longo \cite{Longo} and Wang \cite{Wang} for Hilbert modular forms
assuming Ihara Lemma for Shimura curves. This condition was removed
by the second author \cite{Xie}.

Combining results in \cite{Wan} and \cite{Xie} we obtain the
following theorem. Remark that we do not need Ihara Lemma for
Shimura curves even when $[F:\mathbb{Q}]$ is odd, though such a
condition is needed in \cite{Wan}.

\begin{thm}\label{thm:Iw-main-int}
Assume that $f$ satisfies conditions $(\mathrm{CR}^+)$,
$(\mathfrak{n}^+\text{-}\mathrm{DT})$, $(\mathrm{PO})$ and $(\mathrm{Fuji}1$-$4)$.
Then we have $$(L_p(K_\infty, f))=\mathrm{char}\ \mathrm{Sel}
(K_\infty, A_{\rho})^\vee .$$
\end{thm} See our context for Conditions $(\mathrm{CR}^+)$,
$(\mathfrak{n}^+\text{-}\mathrm{DT})$, $(\mathrm{PO})$ and
$(\mathrm{Fuji}1$-$4)$.

By Theorem \ref{thm:Iw-main-int} we have $$
(L_{p}(K_{\infty},f))=\mathrm{char}\
\mathrm{Sel}(K_{\infty},A_{\rho})^{\vee} $$ and
    $$  (L_{p}(K_{\infty},f\otimes\chi_{F'/F}))=\mathrm{char}\ \mathrm{Sel}(K_{\infty},A_{\rho\otimes\chi_{F'/F}})^{\vee}    .$$
Using the relation
$$\mathrm{Sel}(K'K_{\infty},A_{\rho})=\mathrm{Sel}(K_{\infty},A_{\rho})\bigoplus\mathrm{Sel}(K_{\infty},A_{\rho\otimes\chi_{F'/F}})$$
we obtain \begin{equation}\label{eq:Lp-eq} (L_{p}(K_{\infty},f)\cdot
L_{p}(K_{\infty},f\otimes\chi_{F'/F}))= \mathrm{char}\
\mathrm{Sel}(K'K_{\infty},A_{\rho})^{\vee}. \end{equation}

Let $\mathfrak{p}'$ be a place of $F'$ above $\mathfrak{p}$. Let
$K'_\infty$ be the $\mathfrak{p}'$-anticyclotomic
$\mathbb{Z}_p$-extension of $K'$.    With $K'_\infty$ instead of
$K_\infty$ we also have $$ (L_{p}(K_{\infty}',f'))=\mathrm{char}\
\mathrm{Sel}(K_{\infty}',A_{\rho})^{\vee},$$ from which we deduce
that \begin{equation}\label{eq:con}(L_{p}(K'K_{\infty},f'))\supseteq
\mathrm{char}\ \mathrm{Sel}(K'K_{\infty},A_{\rho})^{\vee}.
\end{equation} Combining (\ref{eq:Lp-eq}) and (\ref{eq:con}) we
obtain
\begin{equation}\label{eq:contain} (L_{p}(K'K_{\infty},f'))\supseteq (L_{p}(K_{\infty},f)\cdot
L_{p}(K_{\infty},f\otimes\chi_{F'/F})) . \end{equation} As
(\ref{eq:main}) holds in
$\mathcal{O}_{\mathfrak{P}}[[\mathrm{Gal}(K_{\infty}
/K)]][\frac{1}{p}]$, and as the $\mu$-invariants of
$L_{p}(K_{\infty} ,f)$ and $L_{p}(K_{\infty} ,f\otimes\chi_{F'/F})$
are zero, (\ref{eq:contain}) implies (\ref{eq:main}).

The paper is organized as follows. In Section \ref{sec:a} we collect
some basic facts about automorphic forms on definite quaternion
algebras. In Section \ref{sec:b} we recall the construction of
anticyclotomic $p$-adic $L$-functions via Theta elements. In Section
\ref{sec:d}, we combine Wan's result in \cite{Wan} with the result
in \cite{Xie} to prove Iwasawa main conjecture for Hilbert modular
form in the anticyclotomic setting (under certain conditions).
Finally we prove Theorem \ref{thm:main} in Section \ref{sec:e}.

In Sections \ref{sec:a}-\ref{sec:d}, the imaginary quadratic
extension $K$ of $F$ is fixed. In Section \ref{sec:e}, we will
choose a suitable $K$.

The authors thank B.H. Gross for his helpful advice.

\subsection*{Notations}

We fix a totally real number field \(F\) over \(\mathbb{Q}\). Let
\(\Sigma_{F}\) be the set of all real embeddings of \(F\) and let
\(\Sigma_{p}\) be the set of all places of \(F\) above \(p\).

For a number field \(L\), and each place \(v\) of \(L\) above a
prime number \(l\), let \(|\cdot|_{v}\) or \(|\cdot|_{L_{v}}\) be
the absolute value on \(L_{v}\) defined by
\(|x|_{v}=|N_{L_{v}/\mathbb{Q}_{l}}(x)|_{l}\), and  let
\(\varpi_{v}\) be a uniformizer of \(\mathcal{O}_{L_{v}}\).

Let $E$ be a finite extension of $\mathbb{Q}$, $\mathcal{O}$ be the ring of
integers in $E$ and $\mathfrak{P}$ be a prime  of $\mathcal{O}$
above $p$ such that $\mathcal{O}_\mathfrak{P}$ contains
$\mathcal{O}_f$. Let $\omega$ be a uniformizer of
$\mathcal{O}_\mathfrak{P}$. For each positive integer $n$ we put
$\mathcal{O}_{n}=\mathcal{O}_\mathfrak{P}/\omega^n$. We consider
$E_\mathfrak{P}$, $\mathcal{O}_\mathfrak{P}$ and $\mathcal{O}_n$ as
coefficient rings, and let $G_F=\mathrm{Gal}(\overline{F}/F)$ act trivially on them.

Let $\epsilon$ be the $p$-adic cyclotomic character of $G_F$.

Choose an imaginary quadratic extension $K/F$ (inert at $\mathfrak{q}$ when $[F:\mathbb{Q}]$ is odd) and put $K'=KF'$. Fix $\mathfrak{p}\mid p$ of $F$ and let $K_\infty$ be the $\mathfrak{p}$-anticyclotomic $\mathbb{Z}_p$-extension of $K$. Using the canonical periods, attach $p$-adic $L$-functions $L_p(K_\infty,f)$, $L_p(K_\infty,f\otimes\chi_{F'/F})$ and $L_p(K'K_\infty,f')$ in $\mathcal{O}_{\mathfrak{P}}[[\mathrm{Gal}(K_\infty/K)]]$. To prove Theorem~\ref{thm:main} it suffices to show the ideal identity
\[\tag{\ref{eq:main}} (L_{p}(K'K_{\infty} ,f'))=(L_{p}(K_{\infty} ,f))\cdot (L_{p}(K_{\infty} ,f\otimes\chi_{F'/F}))\]
inside $\mathcal{O}_{\mathfrak{P}}[[\mathrm{Gal}(K_\infty/K)]]$.

The strategy is to use the anticyclotomic Iwasawa main conjecture, i.e. an equality between $p$-adic $L$-functions and characteristic ideals of Selmer groups. By Theorem~\ref{thm:Iw-main-int},
\[(L_p(K_\infty,f))=\mathrm{char}\,\mathrm{Sel}(K_\infty,A_{\rho})^\vee,\qquad (L_p(K_\infty,f\otimes\chi_{F'/F}))=\mathrm{char}\,\mathrm{Sel}(K_\infty,A_{\rho\otimes\chi_{F'/F}})^\vee.
\]
Using the Selmer decomposition
\[\mathrm{Sel}(K'K_{\infty},A_{\rho})=\mathrm{Sel}(K_{\infty},A_{\rho})\bigoplus\mathrm{Sel}(K_{\infty},A_{\rho\otimes\chi_{F'/F}}),\]
one gets
\[\tag{\ref{eq:Lp-eq}} (L_p(K_\infty,f)\cdot L_p(K_\infty,f\otimes\chi_{F'/F}))=\mathrm{char}\,\mathrm{Sel}(K'K_\infty,A_{\rho})^\vee.
\]
Then, letting $\mathfrak{p}'\mid\mathfrak{p}$ in $F'$ and $K'_\infty$ be the $\mathfrak{p}'$-anticyclotomic $\mathbb{Z}_p$-extension of $K'$, one also has $(L_p(K'_\infty,f'))=\mathrm{char}\,\mathrm{Sel}(K'_\infty,A_{\rho})^\vee$, which implies the containment
\[\tag{\ref{eq:con}} (L_p(K'K_\infty,f'))\supseteq \mathrm{char}\,\mathrm{Sel}(K'K_\infty,A_{\rho})^\vee.
\]
Combining \eqref{eq:Lp-eq} and \eqref{eq:con} yields
\[\tag{\ref{eq:contain}} (L_{p}(K'K_{\infty},f'))\supseteq (L_{p}(K_{\infty},f)\cdot L_{p}(K_{\infty},f\otimes\chi_{F'/F})).
\]
Finally, since \eqref{eq:main} holds after inverting $p$ and the $\mu$-invariants of $L_p(K_\infty,f)$ and $L_p(K_\infty,f\otimes\chi_{F'/F})$ are zero, the containment \eqref{eq:contain} implies the equality \eqref{eq:main}, hence Theorem~\ref{thm:main}.

Choose an imaginary quadratic extension $K/F$ (inert at $\mathfrak{q}$ when $[F:\mathbb{Q}]$ is odd) and put $K'=KF'$. Fix $\mathfrak{p}\mid p$ of $F$ and let $K_\infty$ be the $\mathfrak{p}$-anticyclotomic $\mathbb{Z}_p$-extension of $K$. Using the canonical periods, attach $p$-adic $L$-functions $L_p(K_\infty,f)$, $L_p(K_\infty,f\otimes\chi_{F'/F})$ and $L_p(K'K_\infty,f')$ in $\mathcal{O}_{\mathfrak{P}}[[\mathrm{Gal}(K_\infty/K)]]$. To prove the main theorem, it suffices to show the ideal identity
\begin{equation}\label{eq:main}
(L_{p}(K'K_{\infty} ,f'))=(L_{p}(K_{\infty} ,f))\cdot
(L_{p}(K_{\infty} ,f\otimes\chi_{F'/F}))  \end{equation}
inside $\mathcal{O}_{\mathfrak{P}}[[\mathrm{Gal}(K_\infty/K)]]$.

The strategy is to use the anticyclotomic Iwasawa main conjecture, i.e. an equality between $p$-adic $L$-functions and characteristic ideals of Selmer groups. We first use the following theorem.
\begin{thm}\label{thm:Iw-main-int}
Assume that $f$ satisfies conditions $(\mathrm{CR}^+)$,
$(\mathfrak{n}^+\text{-}\mathrm{DT})$, $(\mathrm{PO})$ and $(\mathrm{Fuji}1$-$4)$.
Then we have $$(L_p(K_\infty, f))=\mathrm{char}\ \mathrm{Sel}
(K_\infty, A_{\rho})^\vee .$$
\end{thm}
Thus
\[(L_p(K_\infty,f))=\mathrm{char}\,\mathrm{Sel}(K_\infty,A_{\rho})^\vee,\qquad (L_p(K_\infty,f\otimes\chi_{F'/F}))=\mathrm{char}\,\mathrm{Sel}(K_\infty,A_{\rho\otimes\chi_{F'/F}})^\vee.
\]
Using the Selmer decomposition
\[\mathrm{Sel}(K'K_{\infty},A_{\rho})=\mathrm{Sel}(K_{\infty},A_{\rho})\bigoplus\mathrm{Sel}(K_{\infty},A_{\rho\otimes\chi_{F'/F}}),\]
one gets
\begin{equation}\label{eq:Lp-eq} (L_{p}(K_{\infty},f)\cdot
L_{p}(K_{\infty},f\otimes\chi_{F'/F}))= \mathrm{char}\
\mathrm{Sel}(K'K_{\infty},A_{\rho})^{\vee}. \end{equation}
Then, letting $\mathfrak{p}'\mid\mathfrak{p}$ in $F'$ and $K'_\infty$ be the $\mathfrak{p}'$-anticyclotomic $\mathbb{Z}_p$-extension of $K'$, one also has $(L_p(K'_\infty,f'))=\mathrm{char}\,\mathrm{Sel}(K'_\infty,A_{\rho})^\vee$, which implies the containment
\begin{equation}\label{eq:con}(L_{p}(K'K_{\infty},f'))\supseteq
\mathrm{char}\ \mathrm{Sel}(K'K_{\infty},A_{\rho})^{\vee}.
\end{equation}
Combining the equation above and the containment above yields
\begin{equation}\label{eq:contain} (L_{p}(K'K_{\infty},f'))\supseteq (L_{p}(K_{\infty},f)\cdot
L_{p}(K_{\infty},f\otimes\chi_{F'/F})) . \end{equation}
Finally, since the displayed ideal identity above holds after inverting $p$ and the $\mu$-invariants of $L_p(K_\infty,f)$ and $L_p(K_\infty,f\otimes\chi_{F'/F})$ are zero, the containment above implies the ideal identity
\begin{equation}\label{eq:main}
(L_{p}(K'K_{\infty} ,f'))=(L_{p}(K_{\infty} ,f))\cdot
(L_{p}(K_{\infty} ,f\otimes\chi_{F'/F}))  \end{equation}
and this completes the proof of the main theorem.

Question:
Let $F$ be a totally real field and let $F'/F$ be a real quadratic extension. Let $f$ be a Hilbert newform over $F$ of parallel even weight $k$, trivial central character, and level $\mathfrak{n}_f$, and let $f'$ be the base change of $f$ to $F'$ with level $\mathfrak{n}_{f'}$. Let $p$ be a prime such that $p\ge \max(k+2,7)$ and $p$ is unramified in both $F$ and $F'$. Assume that $\mathfrak{n}_f$ is prime to $p$ and that $f$ is ordinary at every prime of $F$ above $p$. If $[F:\mathbb{Q}]$ is odd, assume there exists a prime $\mathfrak{q}$ such that $\mathfrak{q}\parallel \mathfrak{n}_f$ (that is, $\mathfrak{q}$ divides $\mathfrak{n}_f$ exactly once) and $\mathfrak{q}$ is split in $F'$. Let $\rho_f$ be the $p$-adic Galois representation attached to $f$, and let $\bar\rho_f$ and $\bar\rho_{f'}$ denote the residual Galois representations attached to $f$ and $f'$, respectively. Suppose moreover that:

1. the restrictions of $\bar\rho_f$ to $G_{F'(\xi_p)}$ and to $G_{F'(\sqrt{p^*})}$ are absolutely irreducible, where $p^*=(-1)^{\frac{p-1}{2}}p$;
2. if $k=2$, then for every place $v$ of $F$ above $p$, $a_v(f)^2\not\equiv 1\pmod p$, and for every place $v'$ of $F'$ above $p$, $a_{v'}(f')^2\not\equiv 1\pmod p$;
3. for every prime $\mathfrak{l}\mid \mathfrak{n}_f$, if $\bar\rho_f|_{I_{F_{\mathfrak{l}}}}$ is absolutely irreducible, then $\mathrm N(\mathfrak{l})\not\equiv -1\pmod p$, and for every prime $\mathfrak{l}'\mid \mathfrak{n}_{f'}$, if $\bar\rho_{f'}|_{I_{F'_{\mathfrak{l}'}}}$ is absolutely irreducible, then $\mathrm N(\mathfrak{l}')\not\equiv -1\pmod p$;
4. $\rho_f$ is a minimal modular lifting of $\bar\rho_f$.

For a modular form $g$, write its canonical period as $\Omega_g^{\mathrm{can}}=\langle g,g\rangle_{\mathrm{Pet}}/\eta_g$, where $\langle g,g\rangle_{\mathrm{Pet}}$ is the Petersson inner product and $\eta_g$ is the associated congruence number. Under these hypotheses, which statement about the canonical periods of $f$ and $f'$ is valid?

The canonical periods satisfy
$$\Omega_{f'}^{\mathrm{can}}=(\Omega_f^{\mathrm{can}})^2$$
up to a $p$-adic unit; equivalently, the ratio $\Omega_{f'}^{\mathrm{can}}/(\Omega_f^{\mathrm{can}})^2$ is a $p$-adic unit.

Choices:
B.
The canonical periods satisfy
$$\Omega_{f'}^{\mathrm{can}}=(\Omega_f^{\mathrm{can}})^2$$
exactly in the coefficient ring, i.e. the ratio $\Omega_{f'}^{\mathrm{can}}/(\Omega_f^{\mathrm{can}})^2$ is equal to $1$.
C.
The canonical periods satisfy
$$\Omega_{f'}^{\mathrm{can}}=u\, (\Omega_f^{\mathrm{can}})^2$$
for some nonzero element $u$ in the coefficient field; in particular, $\Omega_{f'}^{\mathrm{can}}$ and $(\Omega_f^{\mathrm{can}})^2$ agree up to multiplication by a scalar.
D.
The canonical periods satisfy
$$\Omega_{f'}^{\mathrm{can}}=(\Omega_f^{\mathrm{can}})^2$$
up to a $p$-adic unit whenever the restrictions of $\bar\rho_f$ to $G_{F(\xi_p)}$ and to $G_{F(\sqrt{p^*})}$ are absolutely irreducible; no hypothesis involving the corresponding restrictions over $F'$ is needed.
E.
The canonical periods satisfy
$$\Omega_{f'}^{\mathrm{can}}=(\Omega_f^{\mathrm{can}})^2$$
up to multiplication by an arbitrary $p$-power; equivalently, the ratio $\Omega_{f'}^{\mathrm{can}}/(\Omega_f^{\mathrm{can}})^2$ is merely $p$-adically integral.

unit-ambiguity from ideal identity and period normalization

dropped the requirement that the scalar be a p-adic unit

base-change irreducibility hypotheses over F'

mu=0 used to upgrade containment to equality up to unit


\label{main}
    Let $C\subset\mathbb CP^2$ be an algebraic plane curve of degree at most five. Then the fundamental group $\pi_1(\mathbb CP^2\setminus C)$ is linear and virtually polyfree. In particular, $\pi_1(\mathbb CP^2\setminus C)$ is residually finite.

Let $C$ be an algebraic curve in the projective plane $\mathbb CP^2$. We consider the following question.

\begin{ques}
\label{q1}
 Is the fundamental group $\pi_1(\mathbb CP^2\setminus C)$ of an algebraic plane curve residually finite?
\end{ques}

This question has been open for a long time. Zariski had already posed the question of the existence of non-residually finite groups, cf. § 1 and Appendix 1) of Chapter VIII of \cite{Zar} (see also Libgober \cite[Problem 3.1] {Lib}).
Even in the low-degree cases (degree up to five), such a question was still unknown (see the survey article of Artal Bartolo-Cogolludo Agustın-Tokunaga \cite{bartolo2008survey}, page 3). Toledo \cite{Toledo} constructed a smooth projective variety with a non-residually finite fundamental group, answering in the negative a question of Serre. However, it is not clear whether one can choose the example as the complement of an algebraic curve with a non-residually finite fundamental group. Koberda and Suciu \cite[Problem 1.9]{KS} asked Question \ref{q1} for affine line arrangements.

The purpose of this article is to answer Question \ref{q1} positively for algebraic curves of degree at most five. Recall that a group $G$ is linear if it can be embedded into a general linear group $\mathrm{GL}(n,F)$ for some field $F$ and positive integer $n$. A linear group is residually finite. A group $G$ is called poly-free if there is a finite
subnormal sequence\begin{equation*}
1<G_{1}\trianglelefteq G_{2}\trianglelefteq \cdots \trianglelefteq G_{n}=G
\end{equation*}such that the quotient $G_{i}/G_{i+1}$ is free for each $i$. Polyfree groups have many nice properties, like left-orderable, having Tits alternative, and so on. We say a group $G$ virtually has property P, if there is a finite-index subgroup $H<G$ that has the property P. The following is our main result.

\begin{ques}
\label{q1}
    Is the fundamental group $\pi_1(\mathbb CP^2\setminus C)$ of an algebraic plane curve residually finite?
\end{ques}

\begin{abstract}
We prove that for any algebraic plane curve $C$ of degree at most $5$, the fundamental group $\pi_1(\mathbb CP^2\setminus C)$ is linear and virtually polyfree. As a consequence, we answer positively the open question on the residual finiteness of these groups for all plane curves of degree at most $5$.
\end{abstract}

\begin{ques}
\label{q1}
 Is the fundamental group $\pi_1(\mathbb CP^2\setminus C)$ of an algebraic plane curve residually finite?
\end{ques}

The purpose of this article is to answer Question \ref{q1} positively for algebraic curves of degree at most five. Recall that a group $G$ is linear if it can be embedded into a general linear group $\mathrm{GL}(n,F)$ for some field $F$ and positive integer $n$. A linear group is residually finite. A group $G$ is called poly-free if there is a finite
subnormal sequence\begin{equation*}
1<G_{1}\trianglelefteq G_{2}\trianglelefteq \cdots \trianglelefteq G_{n}=G
\end{equation*}such that the quotient $G_{i}/G_{i+1}$ is free for each $i$. Polyfree groups have many nice properties, like left-orderable, having Tits alternative, and so on. We say a group $G$ virtually has property P, if there is a finite-index subgroup $H<G$ that has the property P. The following is our main result.

It is well-known (cf.\cite[2.3.2]{cogolludo2011braid}) that for any plane curve $C\subset\mathbb CP^2$ of degree $\le3$, the fundamental group $\pi_1(\mathbb CP^2\setminus C)$ is either finitely generated abelian or a free group $\mathbb Z\ast\mathbb Z$. Thus $\pi_1(\mathbb CP^2\setminus C)$ is linear. For degree-$4$ curves, we applied the results due to Nori \cite{nori1983zariski} and Cogolludo-Agustin and Elduque \cite{cogolludo2025quasi} to conclude that
that $\pi_1(\mathbb{C}P^2\setminus C)$ must be one of the following groups: a (virtually) finitely generated abelian group, a finite group, the braid group on the sphere $B_3(S^2)$, the free group $F_3$, the braid group $B_3$, the free product $\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z}$, or the semi-direct product $F_2 \rtimes \mathbb{Z}$, (see Subsection \ref{degree4} for details), which are all virtually poly-free and linear. For degree-$5$ curves, Degtyarev \cite{degtyarev1987topology} obtained presentations of all the $\pi_1(\mathbb{C}P^2\setminus C)$ when $\deg C=5$. In Section \ref{degree5}, we will review the presentations of these groups from Degtyarev and prove the virtual polyfreeness and linearity of these fundamental groups. Our key observation is that most of the non-abelian groups are virtual surface groups or (triangle or right-angled) Artin groups.

\subsection{Curves of degree $\le3$}
\noindent\\
It is well-known (cf.\cite[2.3.2]{cogolludo2011braid}) that for any plane curve $C\subset\mathbb CP^2$ of degree $\le3$, $\pi_1(\mathbb CP^2\setminus C)$ is either finitely generated abelian or a free group $\mathbb Z\ast\mathbb Z$. Thus $\pi_1(\mathbb CP^2\setminus C)$ is linear and virtually polyfree.

\subsection{Quartic curves}\label{degree4}
\noindent\\
If $C\subset \mathbb CP^2$ is an irreducible quartic curve that is not a 3-cuspidal quartic, then the fundamental group $\pi_1(\mathbb CP^2\setminus C)$ is abelian (cf. \cite[p.130, 4.3]{dimca2012singularities}). When $C$ is a 3-cuspidal quartic $\pi_1(\mathbb CP^2\setminus C) \cong B_3(S^2)$, the sphere braid group (cf. \cite{zariski1936poincare}), a finite group of order 12 and thus linear (see also\cite[4.8, page 131]{dimca2012singularities}). Since any finitely generated abelian group is linear, we know that for any irreducible quartic curve $C$, $\pi_1(\mathbb CP^2\setminus C)$ is linear. Also, these groups are known to be virtually polyfree.

\begin{cor}
 If $C$ is a plane curve of degree $\le 4$, then $\pi_1(\mathbb CP^2\setminus C)$ is linear and virtually polyfree.
\end{cor}

\section{Quintic}\label{degree5}
The presentation of fundamental groups of $\mathbb{C}P^2 \setminus\ C$ for degree-5 algebraic curves $C$ has been classified by Degtyarev \cite{Deg}. Based on this classification, we will prove Theorem \ref{main} for these curves by studying the presentations.
\subsection{Presentation of fundamental groups}
In this subsection, we will review the results proved by Degtyarev \cite{Deg}. Before introducing the list of Degtyarev, we need some notations.
\begin{enumerate}
\item $F_n$ is the free group of rank $n$, and $B_n$ is the braid group with $n$-strands.
\item $T_{p,q}$ is the fundamental group of a toric link of type $(p, q)$. In particular, if $p = 2$, then
$T_{2,2r}=\langle a,b| (ab)^r=(ba)^r\rangle$; if $\text{gcd}(p,q)=1$ then $T_{p,q}=\langle a,b| a^p=b^q\rangle$.
\item $G(T)$ and $G_p(T)$, where $T\in\mathbb Z[t]$ is an integral polynomial, are the extensions
$$1\to \mathbb Z[t]/T\to G(T)\to \mathbb Z\to 1$$ and
$$1\to (\mathbb Z/{p\mathbb Z})[t]/T\to G(T)\to \mathbb Z\to 1,$$ where the conjugation action of the generator of the quotient $\mathbb Z$ on the kernel is the multiplication by $t$.
\item $\text{Gr}\langle p,q,r \rangle :=\langle a,b,c|a^p=b^q=c^r=abc\rangle$.
\end{enumerate}

\begin{ques}
\label{q1}
    Is the fundamental group $\pi_1(\mathbb CP^2\setminus C)$ of an algebraic plane curve residually finite?
\end{ques}

It is well-known (cf.\cite[2.3.2]{cogolludo2011braid}) that for any plane curve $C\subset\mathbb CP^2$ of degree $\le3$, the fundamental group $\pi_1(\mathbb CP^2\setminus C)$ is either finitely generated abelian or a free group $\mathbb Z\ast\mathbb Z$. Thus $\pi_1(\mathbb CP^2\setminus C)$ is linear. For degree-$4$ curves, we applied the results due to Nori \cite{nori1983zariski} and Cogolludo-Agustin and Elduque  \cite{cogolludo2025quasi} to conclude that
that $\pi_1(\mathbb{C}P^2\setminus C)$ must be one of the following groups: a (virtually) finitely generated abelian group, a finite group, the braid group on the sphere $B_3(S^2)$, the free group $F_3$, the braid group $B_3$, the free product $\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z}$, or the semi-direct product $F_2 \rtimes \mathbb{Z}$, (see Subsection \ref{degree4} for details), which are all virtually poly-free and linear. For degree-$5$ curves, Degtyarev \cite{degtyarev1987topology} obtained presentations of all the $\pi_1(\mathbb{C}P^2\setminus C)$ when $\deg C=5$. In Section \ref{degree5}, we will review the presentations of these groups from Degtyarev and prove the virtual polyfreeness and linearity of these fundamental groups. Our key observation is that most of the non-abelian groups are virtual surface groups or (triangle or right-angled) Artin groups.

\subsection*{Acknowledgments}
We would like to thank Anatoly Libgober, Jose Ignacio Cogolludo, Corey Bregman for useful discussions and Alex Degtyarev for pointing us the results from his thesis and book.

To prove Theorem~\ref{main}, the argument is by degree.

- For $\deg C\le 3$: it is "well-known" that $\pi_1(\mathbb CP^2\setminus C)$ is "either finitely generated abelian or a free group $\mathbb Z\ast\mathbb Z$"; hence it is linear.

- For $\deg C=4$: using results of Nori and Cogolludo-Agustin--Elduque, one concludes that $\pi_1(\mathbb CP^2\setminus C)$ is one of an explicit list ("a (virtually) finitely generated abelian group, a finite group, the braid group on the sphere $B_3(S^2)$, the free group $F_3$, the braid group $B_3$, the free product $\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z}$, or the semi-direct product $F_2 \rtimes \mathbb{Z}$"), and these groups are "all virtually poly-free and linear."

- For $\deg C=5$: Degtyarev "obtained presentations" of all groups $\pi_1(\mathbb{C}P^2\setminus C)$ when $\deg C=5$; the plan is to "review the presentations" and "prove the virtual polyfreeness and linearity" of these groups, with the "key observation" that "most of the non-abelian groups are virtual surface groups or (triangle or right-angled) Artin groups."

To prove the main theorem, the argument is by degree.

- For $\deg C\le 3$: it is "well-known" that $\pi_1(\mathbb CP^2\setminus C)$ is "either finitely generated abelian or a free group $\mathbb Z\ast\mathbb Z$"; hence it is linear.

- For $\deg C=4$: using results of Nori and Cogolludo-Agustin--Elduque, one concludes that $\pi_1(\mathbb CP^2\setminus C)$ is one of an explicit list ("a (virtually) finitely generated abelian group, a finite group, the braid group on the sphere $B_3(S^2)$, the free group $F_3$, the braid group $B_3$, the free product $\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z}$, or the semi-direct product $F_2 \rtimes \mathbb{Z}$"), and these groups are "all virtually poly-free and linear."

- For $\deg C=5$: Degtyarev "obtained presentations" of all groups $\pi_1(\mathbb{C}P^2\setminus C)$ when $\deg C=5$; the plan is to "review the presentations" and "prove the virtual polyfreeness and linearity" of these groups, with the "key observation" that "most of the non-abelian groups are virtual surface groups or (triangle or right-angled) Artin groups."

Question: Let \(C\subset \mathbb{CP}^2\) be an algebraic plane curve of degree at most \(5\). Recall that a group is called linear if it embeds into \(\mathrm{GL}(n,F)\) for some field \(F\) and some positive integer \(n\), and that a group is called virtually polyfree if it has a finite-index subgroup that is poly-free, meaning it admits a finite subnormal series with free successive quotients. Under these assumptions, which statement about the fundamental group \(\pi_1(\mathbb{CP}^2\setminus C)\) holds?

The group \(\pi_1(\mathbb{CP}^2\setminus C)\) is linear and virtually polyfree. In particular, it is residually finite.

Choices:
B. The group \(\pi_1(\mathbb{CP}^2\setminus C)\) is residually finite and virtually polyfree, but linearity is only guaranteed when \(\deg C\le 4\).
C. The group \(\pi_1(\mathbb{CP}^2\setminus C)\) is residually finite.
D. The group \(\pi_1(\mathbb{CP}^2\setminus C)\) has a finite-index subgroup that is linear and poly-free. In particular, \(\pi_1(\mathbb{CP}^2\setminus C)\) is residually finite.
E. The group \(\pi_1(\mathbb{CP}^2\setminus C)\) is linear and poly-free. In particular, it is residually finite.

dropped linearity and virtual polyfreeness, keeping only the stated consequence

moved linearity from the whole group to only a finite-index subgroup

replaced virtually polyfree by poly-free