[Structure of $f(R)$ gravity in spherical symmetry]
\label{main-theo}
Consider the field equations in $f(R)$ gravity \eqref{Eq1-14}, coupled to a
possibly massive, real-valued scalar field $\phi$. Suppose that the defining function $f= f(R)$ and the matter potential $U=U(\phi)$ satisfy the following conditions:
\bel{hypo-Einstein}
\aligned
& (1)
\, &  \phi \, U'(\phi) &\geq&& 0,
\\
& (2)
\, &  U(\phi) &\geq&& 0,
\\
& (3)
\, & f'(R) &>&&0,
\\
& (4)
\, &
f(R) &\leq&& R \, f'(R).
\endaligned
\ee
Then, the field equations of $f(R)$ gravity for spherically symmetric spacetimes can be reduced to an \emph{integro-differential system} consisting of two first-order coupled, nonlinear hyperbolic equations, stated in Proposition~\ref{main-propo-first-order}, below.
Indeed, this system is equivalent to the full Einstein equations in the class of $C^1$ solutions that are suitably regular at the center in the sense of Definition~\ref{definition}, below. Furthermore, the Hawking mass is future non-decreasing along radial directions and non-increasing along null directions. In the limit ${f(R) \to R}$ and
$U(\phi)\to 0$, one recovers Christodoulou's formulation of the Einstein-massless scalar field system.

\label{section---1}

\subsection{Purpose of this paper}

\paragraph{Theories of modified gravity}

In recent years, new observational data suggest that extensions of Einstein's field equations may be relevant for explaining the accelerated expansion of the Universe and certain galactic-scale instabilities. Many of the proposed physical theories of modified gravity exhibit undesirable features, including lack of strong hyperbolicity in commonly used gauges, or non-uniqueness of solutions, or changes in the character of the evolution equations that may alter the system's causal structure. The development of robust numerical simulations in nonlinear, physically relevant regimes is hampered by the lack of mathematically rigorous formulations.
Indeed, only a few studies address the formulation and well-posedness of these theories \cite{Avalos,Lehner2,Lehner,Cocke,Hilditch1,Felice,Figueras,LeFlochMa17a,Reall,Reall2,Pretorius,Salgado}.
More recently, alternative, well-posed theories have been proposed; cf.~\cite{Brady-Figueras,Figueras} (and the references therein).

Among the various extensions of general relativity, $f(R)$ gravity is widely regarded as a natural and physically viable alternative to Einstein's theory \cite{Felice,Ferreira}. The $f(R)$ field equations are considerably more involved than the Einstein equations: in addition to the second-order Ricci curvature terms, they contain \emph{fourth-order} metric derivatives (in particular, second-order derivatives of the scalar curvature). This partly explains why, despite their physical importance, rigorous mathematical results to date encompass only local well-posedness~\cite{Felice,LeFlochMa17a} and the global nonlinear stability in the near-Minkowski regime \cite{LeFlochMa23}.

\paragraph{Evolution in spherical symmetry.}

In this paper, we initiate the study of the global evolution of a (massive) scalar field in $f(R)$ gravity in spherical symmetry. Christodoulou \cite{Chr,Chr2,Chr3} developed, within general relativity, a framework to analyze a massless scalar field in spherical symmetry using Bondi-type coordinates. This framework was also employed to study black-hole formation via numerical methods; cf.~\cite{Goldwirth-Piran} and the references therein. These developments inspired further investigations of gravitational collapse~\cite{Brady-etal,Hilditch2,Zhang-Lu, Michel-Moss}, as well as additional mathematical results, cf.~\cite{Chae,Costa-Mena,Costa-Duarte-Mena} (classical solutions) and \cite{LeFloch-Mena} (generalized solutions). Building on Christodoulou's subsequent work using a double-null foliation~\cite{Chr-bv,Chr-naked}, many studies have also addressed the gravitational collapse toward black holes in spherical symmetry. Next, with the growing interest in modified gravity, subsequent works include \cite{Quo-Joshi, Zhang-etal-2016, Chow}, which mostly consider quadratic curvature corrections to the Einstein--Hilbert action. Despite these advances, rigorous mathematical results on the \emph{global geometry} of Cauchy developments in $f(R)$ gravity are still lacking.

\paragraph{Aim of this paper.}

Our aim here is to advance the mathematical analysis of $f(R)$ gravity by building on the work of Christodoulou  \cite{Chr,Chr2,Chr3}.
We are also relying on the formulation proposed in LeFloch and Ma \cite{LeFlochMa17a,LeFlochMa23} to deal with the equations of $f(R)$ gravity in the near-Minkowski regime. In the present paper, we focus on spacetimes containing a (possibly massive) scalar field evolving in spherical symmetry and, drawing on Christodoulou's  insights, we provide a framework suited to addressing the global causal structure of such spacetimes.
Our companion paper \cite{LeFloch-Mena} considered the evolution of a self-gravitating massive scalar field and provided a first step toward understanding the role of a massive scalar field on the global spacetime geometry. While in \cite{LeFloch-Mena} we considered generalized solutions, in the present paper we consider solutions of class $C^1$ that satisfy regularity conditions at the center and are asymptotically Euclidean. Following Christodoulou, we use a generalization of the Bondi--Sachs coordinates and formulate the characteristic initial value problem with data imposed on a future light cone.

The new system is significantly more involved than the Einstein system, and a remarkable mathematical structure is uncovered here by introducing  an augmented formulation in which the metric and its scalar curvature are regarded as independent unknowns. We thus obtain a new integro-differential system of two coupled, first-order, nonlinear hyperbolic equations whose unknowns are the spacetime scalar curvature and, in this case, the scalar field. We then prove several consistency and regularity properties for the new system and its solutions as well as monotonicity properties of the Hawking mass in the $f(R)$ setting. In the process we also recover the general relativity limit (when ${f(R) \to R}$), as well as the special case of a massless scalar field.

\subsection{A first-order formulation of $f(R)$ gravity}

\paragraph{Action of $f(R)$ modified gravity.}

Recall first that Einstein's theory is based on the Einstein--Hilbert action
\be
\Acal_{\text{EH}}[\phi, g] := \int_M \Big( {R_g  \over 16 \pi} + L[\phi,g] \Big) \, dV_g,
\ee
associated with a $(1+3)$--dimensional spacetime $({\Mcal},g)$ with signature $(-, +,+,+)$.
Hence, the functional $\Acal_{\text{EH}}[\phi, g] $ is determined from the scalar curvature $R_g$ of the metric $g$ representing the geometry of the spacetime and
 a Lagrangian $L[\phi,g]$ describing the matter content and represented by certain fields $\phi$ defined on $\Mcal$. Here, $dV_g$ denotes the canonical volume form associated with $g$.
As is well-known, the action $\Acal_{\text{EH}}[\phi, g]$ is (formally) critical at metrics $g$
satisfying Einstein's field equations
\be
\label{Eq1-01}
G := \Ric - {R_g \over 2} \, g = 8 \pi \, T[\phi,g]
\ee
in which the right-hand side
\bel{Eq:12}
T_{ab}[\phi,g] := -2 \, {\delta L \over \delta g^{ab}} [\phi,g]  + g_{ab}\, L[\phi,g]
\ee
is the stress-energy tensor of the matter system (Latin indices $a,b,..= 0, \ldots, 3$ denoting spacetime components).
Our emphasis in this paper is on massive scalar fields $\phi$ described by
\be
\label{stress-energy}
T_{ab} = \nabla_a \phi \nabla_b \phi- \big( {1 \over 2} \nabla^c \phi \nabla_c \phi + U(\phi) \Big) g_{ab},
\ee
in which the potential function is $U=U(\phi)$, so that the field $\phi$ satisfies the Klein-Gordon equation
\be
\Box_g \phi = U'(\phi).
\end{equation}
We study an extension of Einstein's theory, defined as follows. A function $f: \RR \to \RR$ being given, we consider the modified gravity action for the $f(R)$ theory
$$
\Acal_{\text{MG}}[\phi,g] =: \int_M \Big( {f(R) \over 16 \pi} + L[\phi, g]\Big) \, dV,
$$
whose critical points satisfy the following \emph{field equations of modified gravity}
\bel{Eq1-14}
E_{ab} :=
f'(R) \, G_{ab} - \frac{1}{2} \Big( f(R) - R f'(R) \Big) g_{ab}
+  \big( g_{ab} \, \Box_g   - \nabla_a\nabla_b \big)  \big( f'(R) \big) = 8 \pi \, T_{ab}[\phi,g].
\ee
The right-hand side is still given by \eqref{Eq:12}. The modified gravity tensor $E_{ab}$ replaces
the Einstein tensor $G_{ab}$ and satisfies $\nabla^a E_{ab} = 0$,
 so that $T_{ab}$ is also divergence-free.

\paragraph{Main result.}

In local coordinates, \eqref{Eq1-14} consists of a nonlinear system of fourth-order partial differential equations, while Einstein theory in \eqref{Eq1-01} leads to second-order equations. We investigate
how to include the effect of these fourth-order terms in techniques developed earlier for the Einstein equations.
The condition $f'(R) >0$ in \eqref{hypo-Einstein}, below, is fundamental throughout the $f(R)$ theory and, for instance, we use it to be able to introduce the \emph{conformal metric} $\gt$ and the \emph{augmented variable} $\rho$ (in \eqref{equa-233}, below):
\be
\gt_{ab} := e^{\kappa \rho} g_{ab},
\qquad
\rho := {1 \over \kappa} \log f'(R).
\ee
Here, $\kappa>0$ is a parameter  for the definition of $\rho$, so that the general relativity limit corresponds to $f(R)\to R$ when ${\kappa \to 0}$. We summarize our results as follows.

\vskip.15cm

Our aim here is to advance the mathematical analysis of $f(R)$ gravity by building on the work of Christodoulou \cite{Chr,Chr2,Chr3}.
We are also relying on the formulation proposed in LeFloch and Ma \cite{LeFlochMa17a,LeFlochMa23} to deal with the equations of $f(R)$ gravity in the near-Minkowski regime. In the present paper, we focus on spacetimes containing a (possibly massive) scalar field evolving in spherical symmetry and, drawing on Christodoulou's insights, we provide a framework suited to addressing the global causal structure of such spacetimes.
Our companion paper \cite{LeFloch-Mena} considered the evolution of a self-gravitating massive scalar field and provided a first step toward understanding the role of a massive scalar field on the global spacetime geometry. While in \cite{LeFloch-Mena} we considered generalized solutions, in the present paper we consider solutions of class $C^1$ that satisfy regularity conditions at the center and are asymptotically Euclidean. Following Christodoulou, we use a generalization of the Bondi--Sachs coordinates and formulate the characteristic initial value problem with data imposed on a future light cone.

The new system is significantly more involved than the Einstein system, and a remarkable mathematical structure is uncovered here by introducing an augmented formulation in which the metric and its scalar curvature are regarded as independent unknowns. We thus obtain a new integro-differential system of two coupled, first-order, nonlinear hyperbolic equations whose unknowns are the spacetime scalar curvature and, in this case, the scalar field. We then prove several consistency and regularity properties for the new system and its solutions as well as monotonicity properties of the Hawking mass in the $f(R)$ setting. In the process we also recover the general relativity limit (when ${f(R) \to R}$), as well as the special case of a massless scalar field.

Recall first that Einstein's theory is based on the Einstein--Hilbert action
\be
\Acal_{\text{EH}}[\phi, g] := \int_M \Big( {R_g \over 16 \pi} + L[\phi,g] \Big) \, dV_g,
\ee
associated with a $(1+3)$--dimensional spacetime $({\Mcal},g)$ with signature $(-, +,+,+)$.
Hence, the functional $\Acal_{\text{EH}}[\phi, g] $ is determined from the scalar curvature $R_g$ of the metric $g$ representing the geometry of the spacetime and
 a Lagrangian $L[\phi,g]$ describing the matter content and represented by certain fields $\phi$ defined on $\Mcal$. Here, $dV_g$ denotes the canonical volume form associated with $g$.
As is well-known, the action $\Acal_{\text{EH}}[\phi, g]$ is (formally) critical at metrics $g$
satisfying Einstein's field equations
\be
\label{Eq1-01}
G := \Ric - {R_g \over 2} \, g = 8 \pi \, T[\phi,g]
\ee
in which the right-hand side
\bel{Eq:12}
T_{ab}[\phi,g] := -2 \, {\delta L \over \delta g^{ab}} [\phi,g] + g_{ab}\, L[\phi,g]
\ee
is the stress-energy tensor of the matter system (Latin indices $a,b,..= 0, \ldots, 3$ denoting spacetime components).
Our emphasis in this paper is on massive scalar fields $\phi$ described by
\be
\label{stress-energy}
T_{ab} = \nabla_a \phi \nabla_b \phi- \big( {1 \over 2} \nabla^c \phi \nabla_c \phi + U(\phi) \Big) g_{ab},
\ee
in which the potential function is $U=U(\phi)$, so that the field $\phi$ satisfies the Klein-Gordon equation
\be
\Box_g \phi = U'(\phi).
\end{equation}
We study an extension of Einstein's theory, defined as follows. A function $f: \RR \to \RR$ being given, we consider the modified gravity action for the $f(R)$ theory
$$
\Acal_{\text{MG}}[\phi,g] =: \int_M \Big( {f(R) \over 16 \pi} + L[\phi, g]\Big) \, dV,
$$
whose critical points satisfy the following \emph{field equations of modified gravity}
\bel{Eq1-14}
E_{ab} :=
f'(R) \, G_{ab} - \frac{1}{2} \Big( f(R) - R f'(R) \Big) g_{ab}
+ \big( g_{ab} \, \Box_g - \nabla_a\nabla_b \big) \big( f'(R) \big) = 8 \pi \, T_{ab}[\phi,g].
\ee
The right-hand side is still given by \eqref{Eq:12}. The modified gravity tensor $E_{ab}$ replaces
the Einstein tensor $G_{ab}$ and satisfies $\nabla^a E_{ab} = 0$,
 so that $T_{ab}$ is also divergence-free.

In local coordinates, \eqref{Eq1-14} consists of a nonlinear system of fourth-order partial differential equations, while Einstein theory in \eqref{Eq1-01} leads to second-order equations. We investigate
how to include the effect of these fourth-order terms in techniques developed earlier for the Einstein equations.
The condition $f'(R) >0$ in \eqref{hypo-Einstein}, below, is fundamental throughout the $f(R)$ theory and, for instance, we use it to be able to introduce the \emph{conformal metric} $\gt$ and the \emph{augmented variable} $\rho$ (in \eqref{equa-233}, below):
\be
\gt_{ab} := e^{\kappa \rho} g_{ab},
\qquad
\rho := {1 \over \kappa} \log f'(R).
\ee
Here, $\kappa>0$ is a parameter for the definition of $\rho$, so that the general relativity limit corresponds to $f(R)\to R$ when ${\kappa \to 0}$. We summarize our results as follows.

\vskip.15cm

Our aim here is to advance the mathematical analysis of $f(R)$ gravity by building on the work of Christodoulou \cite{Chr,Chr2,Chr3}.
We are also relying on the formulation proposed in LeFloch and Ma \cite{LeFlochMa17a,LeFlochMa23} to deal with the equations of $f(R)$ gravity in the near-Minkowski regime. In the present paper, we focus on spacetimes containing a (possibly massive) scalar field evolving in spherical symmetry and, drawing on Christodoulou's insights, we provide a framework suited to addressing the global causal structure of such spacetimes.
Our companion paper \cite{LeFloch-Mena} considered the evolution of a self-gravitating massive scalar field and provided a first step toward understanding the role of a massive scalar field on the global spacetime geometry. While in \cite{LeFloch-Mena} we considered generalized solutions, in the present paper we consider solutions of class $C^1$ that satisfy regularity conditions at the center and are asymptotically Euclidean. Following Christodoulou, we use a generalization of the Bondi--Sachs coordinates and formulate the characteristic initial value problem with data imposed on a future light cone.

\vskip.15cm

\vskip.15cm

In the modified field equations, there are terms that contain derivatives of the scalar
curvature, that is, third order derivatives of the metric functions $\nu$ and $\lambda$. This leads to an essential difficulty in dealing with the equations.
Following LeFloch and Ma~\cite{LeFlochMa17a}, we introduce an \emph{augmented system}
 where the relation
\be
\rho = \frac{1}{\kappa}\ln f'(R) \quad \text{(conformal formulation)}
\ee
is no longer imposed but $\rho$
 is regarded as an independent variable.
To clarify the notation, we introduce a new independent variable denoted by
\be
\rhoh \quad \text{(augmented conformal formulation)}
\ee
which plays the role of $\rho$ and will coincides with $\rho$ once the constraint below is enforced.
In view of Proposition~\ref{prop essential f(R) Bondi}, this leads us to the following system
\bel{Bondi system diff augmented1.5}
\aligned
\del_r(\nu+\lambda)
& = \frac{3}{4}\kappa^{2}r|\del_r\rhoh|^2 + 4\pi r e^{- \kappa\rhoh} \, |\del_r\phi|^2,
\\
\del_r\big(re^{\nu- \lambda}\big)
 & = \big(1 - r^{2}e^{-2\kappa\rhoh} \big( \Vstar(\rhoh) + 8\pi U(\phi) \big) \big)e^{\nu+\lambda},
 \\
D(\del_r(r\phi)) + {r \over 2} e^{\nu+\lambda- \kappa\rhoh}U'(\phi)
& = \frac{1}{2}e^{\nu+\lambda}\Big(1 - e^{-2\lambda} - r^2e^{-2\kappa\rhoh}\big(\Vstar(\rhoh)+8\pi U(\phi)\big)\Big)\del_r\phi \\
& \quad + \frac{\kappa r}{2}\big(D\rhoh \del_r\phi + D\phi \del_r\rhoh\big),
\\
D\big(\del_r(r\rhoh)\big) + {r \over \kappa} e^{\nu+\lambda} \Wstar(\rhoh)
& = \frac{1}{2}e^{\nu+\lambda}\Big(1 - e^{-2\lambda} - r^2e^{-2\kappa\rhoh}\big(\Vstar(\rhoh)+8\pi U(\phi)\big)\Big)\del_r\rhoh
\\
& - \frac{8\pi r}{3\kappa} e^{- \kappa\rhoh}\del_r\phi D\phi
 + {r \over \kappa} e^{\nu+\lambda} \frac{16\pi}{3}e^{-2\kappa\rhoh}U(\phi).
\endaligned
\ee
Observe that, as before, the last equation of \eqref{Bondi system diff augmented1.5} is equivalent to
\be
\label{box-tilde-rho}
\Box_{\gt}\rhoh - \frac{2}{\kappa} \Wstar(\rhoh)
= - {8\pi \over 3\kappa}e^{-2\kappa\rhoh}\big(\sigma+4 \, U(\phi)\big).
\ee
Now that we have defined the augmented system, the next question is whether a solution of \eqref{Bondi system diff augmented1.5} (with a certain regularity) is also a solution of the original system. We emphasize that to be a solution of the full set of field equations, a solution of \eqref{equa-three-equa}, \eqref{Bondi scalar}, and \eqref{eq:599}
must additionally satisfy the condition
\bel{Bondi system diff augmented1.5 constraint}
f'(R) = e^{\kappa\rhoh},
\ee
which we regard as a nonlinear differential constraint on the solutions.
 Conversely, any classical solution of the original $f(R)$ system with $\rho=\frac{1}{\kappa}\ln f'(R)$ satisfies \eqref{Bondi system diff augmented1.5} upon setting $\rhoh=\rho$; thus the two formulations are equivalent in the admissible class once \eqref{Bondi system diff augmented1.5 constraint} holds on the initial cone (and is then propagated by the evolution)

Under the regularity conditions at the center and the asymptotic flatness condition, we can reduce the field equations (cf.~Appendix~\ref{section---A2})) to the the following integro-differential system with the main unknowns $\phi$ and $\rho$:
\bel{notresysteme}
\aligned
D(\del_r(r\phi))
& = \frac{1}{2}e^{\nu+\lambda}\Big(1 - e^{-2\lambda} - r^2e^{-2\kappa\rho}\big(\Vstar(\rho)+8\pi U(\phi)\big)\Big)\del_r\phi
\\
 & \quad + \frac{\kappa r}{2}\big(D\rho \del_r\phi + D\phi \del_r\rho\big)-{r \over 2} e^{\nu+\lambda- \kappa\rho}U'(\phi),
\\
D\big(\del_r(r\rho)\big)
& = \frac{1}{2}e^{\nu+\lambda}\Big(1 - e^{-2\lambda} - r^2e^{-2\kappa\rho}\big(\Vstar(\rho)+8\pi U(\phi)\big)\Big)\del_r\rho
\\
& \quad - \frac{8\pi r}{3\kappa} e^{- \kappa\rho}\del_r\phi D\phi- {r \over \kappa} e^{\nu+\lambda}\Big(\Wstar(\rho) - \frac{16\pi}{3}e^{-2\kappa\rho}U(\phi)\Big),
\endaligned
\ee
in which the metric coefficients $\nu, \lambda$ are given by
\bel{eq:403}
\aligned
\nu + \lambda
& = - \frac{3}{4}\kappa^{2} \int_r^{+ \infty}s|\del_r\rho|^2ds - 4\pi\int_r^{+ \infty} e^{- \kappa\rho} \, s|\del_r\phi|^2 \, ds,
\\
e^{\nu- \lambda}
& = {1 \over r} \int_0^r\Big(1 - s^{2}e^{-2\kappa\rho} \big( \Vstar(\rho) + 8\pi U(\phi) \big)\Big)e^{\nu+\lambda} \, ds.
\endaligned
\ee
We refer to \eqref{notresysteme} as the \emph{augmented conformal system} of modified gravity.

Suppose now that $h$ and $l$ are solutions to the system \eqref{Bondi system int augmented3} defined on $[0,u_0]\times [0,+ \infty)$. If
\bel{center.condition.2}
h,l\in C^1\big([0,u_0]\times [0,+ \infty)\big),
\ee
then the first and second regularity conditions at the center hold. In fact, it follows from the system \eqref{Bondi system int augmented3} that
\be
\label{more-conditions-center}
\hb,\,\lb,\,\nu,\,\lambda\in C^1\big([0,u_0]\times [0,+ \infty)\big).
\ee
In view of \eqref{eq fR components} and \eqref{eq Bondi components T}, we get
\be
\aligned
& r^2{e^{-\kappa\lb}}\big(E-8\pi T\big)(D,D)
\\
& = r{e^{\nu- \lambda}}\Big(\frac{e^{\nu- \lambda}}{2}\del_r(\nu+\lambda) - 2\del_u\lambda\Big)
- \frac{3r^2\kappa^{2}{ } }{2}(D\lb)^2-8\pi r^2{ }(D\hb)^2.
\endaligned
\ee
Hence, from the regularity conditions, and taking into account \eqref{more-conditions-center}, we deduce that
\be
\lim_{r\to0}{\Big(r^2e^{-\kappa\rho}\big(E-8\pi T\big)(D,D)\Big)}= 0,
\ee
since $\nu,\lambda=O(r)$ and $D\hb,D\lb$ are bounded near $r=0$. For the second regularity condition at the center, we compute
\be
r\del_r\lb = l- \lb, \qquad r\del_r\hb = h- \hb.
\ee
 This shows that $h=\partial_r(r\phi)$ and $l=\partial_r(r\rho)$ are $C^1$ on $[0,u_0]\times[0,+\infty)$, as claimed. Conversely, if the two center regularity conditions hold and $(h,l)\in C^0$, the right-hand sides of \eqref{Bondi system int augmented3} are continuous; integrating along characteristics then yields $(h,l)\in C^1$, so the two formulations are equivalent in the admissible class.

\vskip.15cm

We point out that our assumptions are quite natural since they ensure positivity and monotonicity properties that also arise in the massless case. The condition $\phi \, U'(\phi) \geq 0$ is imposed since it guarantees that the Klein-Gordon energy is \emph{defocusing}, so that the forward evolution will not be limited by the matter model.
The conditions ${U(\phi) \geq 0}$ and ${f(R) \leq R \, f'(R)}$ are required to prove that the Hawking-mass is non-negative.

\subsection{Outline of this paper.}

In Section~\ref{section---2}, we introduce Bondi coordinates and express the field equations of $f(R)$ gravity for a scalar field. We then identify the essential equations, which imply the full set of $f(R)$ field equations, provided a mild regularity condition is assumed at the center. In Section~\ref{section---A2}, we analyze the regularity at the center. In Section~\ref{section---3}, we introduce a first-order formulation and establish that it is equivalent to the full set of $f(R)$ equations. Finally, Section~\ref{section--- 4} is devoted to the study of the Hawking mass and its monotonicity properties.

In the discussion after Theorem~\ref{main-theo}, the paper indicates the following argument structure. First, introduce Bondi coordinates and write the spherically symmetric $f(R)$-scalar field equations; then “identify the essential equations, which imply the full set of $f(R)$ field equations, provided a mild regularity condition is assumed at the center.” Next, “analyze the regularity at the center.” Then “introduce a first-order formulation and establish that it is equivalent to the full set of $f(R)$ equations,” yielding the claimed integro-differential system. Finally, “study of the Hawking mass and its monotonicity properties,” using the assumptions (notably $U(\phi)\ge 0$ and $f(R)\le R f'(R)$) which are said to be “required to prove that the Hawking-mass is non-negative,” and $\phi U'(\phi)\ge 0$ to ensure the Klein--Gordon energy is “defocusing.”

In the discussion after the main theorem, the paper indicates the following argument structure. First, introduce Bondi coordinates and write the spherically symmetric $f(R)$-scalar field equations; then “identify the essential equations, which imply the full set of $f(R)$ field equations, provided a mild regularity condition is assumed at the center.” Next, “analyze the regularity at the center.” Then “introduce a first-order formulation and establish that it is equivalent to the full set of $f(R)$ equations,” yielding the claimed integro-differential system. Finally, “study of the Hawking mass and its monotonicity properties,” using the assumptions (notably $U(\phi)\ge 0$ and $f(R)\le R f'(R)$) which are said to be “required to prove that the Hawking-mass is non-negative,” and $\phi U'(\phi)\ge 0$ to ensure the Klein--Gordon energy is “defocusing.”

[Structure of $f(R)$ gravity in spherical symmetry]
\label{main-theo}
Consider the field equations in $f(R)$ gravity \eqref{Eq1-14}, coupled to a
possibly massive, real-valued scalar field $\phi$. Suppose that the defining function $f= f(R)$ and the matter potential $U=U(\phi)$ satisfy the following conditions:
\bel{hypo-Einstein}
\aligned
& (1)
\, &  \phi \, U'(\phi) &\geq&& 0,
\\
& (2)
\, &  U(\phi) &\geq&& 0,
\\
& (3)
\, & f'(R) &>&&0,
\\
& (4)
\, &
f(R) &\leq&& R \, f'(R).
\endaligned
\ee
Then, the field equations of $f(R)$ gravity for spherically symmetric spacetimes can be reduced to an \emph{integro-differential system} consisting of two first-order coupled, nonlinear hyperbolic equations. We first state the following proposition.

\begin{proposition}[A first-order formulation of $f(R)$ gravity]
\label{main-propo-first-order}
\bse\label{Bondi system int augmented3}
The equations of $f(R)$ gravity take the form of a first-order system, in which $h$ and $l$ are the main unknowns:
\be
\aligned
& Dh = \frac{1}{2r}\big(g- \gb\big)(h- \hb) +
\frac{\kappa}{2}\big(q(h- \hb) + p(l- \lb)\big)-{r \over 2} e^{\nu+\lambda- \kappa\lb}U'(\hb),
\\
& Dl = \frac{1}{2r}\big(g- \gb\big)(l- \lb)
- \frac{8\pi}{3\kappa}e^{- \kappa\lb}(h- \hb) p-{r \over \kappa} e^{\nu+\lambda}\Big(\Wstar(\lb)- \frac{16\pi}{3}e^{-2\kappa\lb}U(\hb)\Big),
\endaligned
\ee
in which the metric coefficient $\nu + \lambda$ is computed by
\be
\aligned
& \nu + \lambda = - \frac{3\kappa^{2}}{4}\int_r^{+ \infty}\frac{|l- \lb|^2}{s} \, ds - 4\pi\int_r^{+ \infty}\frac{|h- \hb|^2}{se^{\kappa\lb}} \, ds,
\endaligned
\ee
and the auxiliary variables $p$ and $q$ are defined explicitly by
\bel{p,q}
\aligned
&p = \frac{\kappa}{2r}\int_0^r\Big((l- \lb)p + (h- \hb)q\Big) \, ds +\frac{1}{2r}\int_0^r\frac{e^{\nu- \lambda}(h- \hb)}{s} \, ds
- \frac{1}{2r}\int_0^rse^{\nu+\lambda- \kappa\lb}U'(\hb) \, ds,
\\
& \kappa q
= - \frac{8\pi}{3 r}\int_0^r\frac{(h- \hb)p}{e^{\kappa\lb}} \, ds+ \frac{\kappa}{2r}\int_0^r{\frac{e^{\nu- \lambda}(l- \lb)}{s} \, ds}
 - \frac{1}{r}\int_0^r se^{\nu+\lambda}\Big (\Wstar(\lb)-
\frac{16\pi}{3} e^{-2\kappa\lb}U(\hb)\Big) \, ds.
\endaligned
\ee
\ese
\end{proposition}

Indeed, this system is equivalent to the full Einstein equations in the class of $C^1$ solutions that are suitably regular at the center in the following sense.

\begin{definition}
\label{definition}
\emph{The \emph{first regularity condition at the center\footnote{In view of \eqref{eq fR components} and \eqref{eq Bondi components T}, this condition can be expressed in terms of first-order derivatives of the unknowns.}}} is defined as
\be
\label{regularity}
\lim_{r\to0} r^2e^{-\kappa\rho}\big(E - 8\pi T\big)(D,D) = 0.
\ee
The \emph{second regularity condition at the center} is defined as
\be
\label{regularity2}
\del_r(r\phi) \text{ and }\del_r(r\rho) \text{ are } C^1\text{ on } [0,u_0]\times [0,+ \infty).
\ee
\end{definition}

Furthermore, the Hawking mass is future non-decreasing along radial directions and non-increasing along null directions. In the limit ${f(R) \to R}$ and
$U(\phi)\to 0$, one recovers Christodoulou's formulation of the Einstein-massless scalar field system.,

Implication
Biconditional or Equivalence

Question: Consider the modified-gravity field equations E_{ab}=8π T_{ab} on a spherically symmetric spacetime, where E_{ab}:=f'(R)G_{ab}−(1/2)(f(R)−R f'(R))g_{ab}+(g_{ab}□_g−∇_a∇_b)(f'(R)) and T_{ab}=∇_aφ∇_bφ−( (1/2)∇^cφ∇_cφ+U(φ) )g_{ab} for a real-valued scalar field φ. Assume that φ U'(φ)≥0, U(φ)≥0, f'(R)>0, and f(R)≤R f'(R). Introduce ρ:=(1/κ)log f'(R) with κ>0, and in Bondi-type coordinates let h:=∂_r(rφ) and l:=∂_r(rρ). Which conclusion about the resulting spherically symmetric f(R) system holds under these assumptions?

For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. In the class of C^1 solutions satisfying the two center regularity conditions lim_{r→0} r^2 e^{−κρ}(E−8πT)(D,D)=0 and ∂_r(rφ), ∂_r(rρ) ∈ C^1([0,u_0]×[0,∞))—where D is the characteristic Bondi derivative—this first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations. Moreover, the Hawking mass of the symmetry spheres is future non-decreasing along radial directions and non-increasing along null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system.

Choices:
B. For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. This first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations for every C^1 solution, without any additional regularity assumptions at the center. Moreover, the Hawking mass of the symmetry spheres is future non-decreasing along radial directions and non-increasing along null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system.
C. For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. Moreover, in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system.
D. For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. In the class of C^1 solutions satisfying the two center regularity conditions lim_{r→0} r^2 e^{−κρ}(E−8πT)(D,D)=0 and ∂_r(rφ), ∂_r(rρ) ∈ C^1([0,u_0]×[0,∞))—where D is the characteristic Bondi derivative—this first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations. Moreover, the Hawking mass of the symmetry spheres is future non-decreasing both along radial directions and along null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system.
E. For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. In the class of C^1 solutions satisfying the two center regularity conditions lim_{r→0} r^2 e^{−κρ}(E−8πT)(D,D)=0 and ∂_r(rφ), ∂_r(rρ) ∈ C^1([0,u_0]×[0,∞))—where D is the characteristic Bondi derivative—this first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations. Moreover, the Hawking mass of the symmetry spheres is non-negative and therefore constant along radial directions and null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system.

center regularity needed for equivalence

equivalence and Hawking-mass monotonicity clauses removed

direction/sign of Hawking-mass monotonicity along null directions

nonnegative Hawking mass incorrectly upgraded to constancy


\setcounter{equation}{0}

We consider the following diffusive prey-predator model with nonlocal dispersal
\begin{equation}\label{pp}
\begin{cases}
U_t(x,t)= \bN_1[U(\cdot,t)](x)+[aU(1-U)-V](x,t), \; x\in\bR,\, t>0,\\
V_t(x,t)=d \bN_2[V(\cdot,t)](x)+[bV(1-V/U)](x,t), \; x\in\bR,\, t>0,
\end{cases}
\end{equation}
where the unknown functions $U,V$ stand for the population densities of prey and predator species at position $x$ and time $t$, respectively, $d,a,b$ are positive constants such that $1,d$ are diffusion coefficients and $a,b$ are intrinsic growth rates of species $U,V$, respectively. The functional response of predator to prey is normalized to be $1$. The prey obeys the logistic growth and its carrying capacity is normalized to be 1. However, the density of predator follows a logistic dynamics with a varying carrying capacity proportional to the density of prey. Moreover, for $i=1,2$, $\bN_i$ formulates the spatial nonlocal dispersal of individuals and is defined by
\beaa
\bN_i[u(\cdot,t)](x):=\int_{\mathbb{R}}J_i(y)u(x-y,t)dy-u(x,t),\; u=U,V,
\eeaa
where $J_i$ is a probability density function satisfying the following conditions:
\begin{enumerate}
\item[(H1)]\,  $J_i$ is a nonnegative continuous function defined in $\bR$;
\item[(H2)]\, $\int_{\mathbb{R}}J_i(y)dy=1$ and $J_i(y)=J_i(-y)$ for all $y\in \mathbb{R}$;
\item[(H3)] there exists $\hat{\lambda}_i\in(0,\infty]$ such that  $\int_{\mathbb{R}}J_i(y)e^{\lambda y} dy < \infty$ for any $\lambda \in (0, \hat{\lambda_i})$ and $\int_{\mathbb{R}}J_i(y)e^{\lambda y} dy\to\infty$ as $\lambda \uparrow \hat{\lambda}_i$.
\end{enumerate}
Note that there is always the predator-free state $(1,0)$, and there is the unique constant co-existence state $(a^*,a^*)$, $a^*:=1-1/a\in(0,1)$, when $a>1$.

When the nonlocal dispersal in system \eqref{pp} is replaced by the classical diffusion, the dynamical behaviors of the corresponding system was investigated in the survey paper \cite{CGS21}. In fact, system \eqref{pp} without diffusion (the ODE system) arises in the control of introduced rabbits to protect native birds from introduced cat predation in an island (cf. \cite{CLS}), when we consider the case without rabbits and control. For the detailed biological background of the full ODE system including the rabbits and control, we refer the reader to \cite{CLS,CS}. However, it is reasonable and more realistic to take into account the influence of spatial movements of birds and cats, namely, the effect of diffusion. On the other hand, to model the long range movements and nonadjacent interactions of individuals it is more realistic to consider the nonlocal dispersal instead of the random movement with classical diffusion. This motivates us to study system \eqref{pp}.

The main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \eqref{pp} is called a traveling wave solution of \eqref{pp}, if there exist a constant $s\in\R$ ({\it the wave speed}) and a function $(\phi,\psi)$ ({\it the wave profile}) of class $C^1(\R)$ such that
$$(U,V)(x,t)=(\phi,\psi)( z),\ \ z:=x-st.$$
We are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\{s,\phi,\psi\}$ satisfies the following system of equations:
\begin{equation}\label{TWS}
\left\{
\begin{aligned}
&   \bN_1[\phi](z)+ s\phi'(z) +a\phi(z)\left[1-\phi(z)\right]-\psi(z)    =0, &\; z\in \mathbb{R},\\
&  d \bN_2[\psi](z)+ s\psi'(z) +b\psi(z)\left[1-\frac{\psi(z)}{\phi(z)}\right] =0, & \; z\in \mathbb{R},
\end{aligned}
\right.
\end{equation}
together with the following asymptotic boundary conditions
\begin{equation}\label{BC}
(\phi,\psi)(\infty) = (1,0),\quad (\phi,\psi)(-\infty) = (a^{*},a^{*}).
\end{equation}

For convenience, for $i=1,2$, we set
\beaa
I_i(\lambda):=\int_\bR J_i(y)e^{\lambda y}dy,\ \ \lambda\in\R,
\eeaa
with $I_i(\lambda)\in(0,\infty)$ if $0\le\lambda<\hat{\lambda}_i$ and $I_i(\lambda)=\infty$ if $\lambda\ge\hat{\lambda}_i$. Note that, due to the symmetry of $J_i$, the function $I_i$ is even, and it is also strictly convex in $(-\hat{\lambda}_i,\hat{\lambda}_i)$. Also, we introduce the quantity
\beaa
s^*:=\inf_{\lambda\in(0,\hat{\lambda}_2)}\frac{d[I_2(\lambda)-1]+b}{\lambda}.
\eeaa
Note that $s^*$ is well-defined, the infimum is reached, and $s^*>0$.

We now state our main theorem as follows.

We consider the following diffusive prey-predator model with nonlocal dispersal
\begin{equation}\label{pp}
\begin{cases}
U_t(x,t)= \bN_1[U(\cdot,t)](x)+[aU(1-U)-V](x,t), \; x\in\bR,\, t>0,\\
V_t(x,t)=d \bN_2[V(\cdot,t)](x)+[bV(1-V/U)](x,t), \; x\in\bR,\, t>0,
\end{cases}
\end{equation}
where the unknown functions $U,V$ stand for the population densities of prey and predator species at position $x$ and time $t$, respectively, $d,a,b$ are positive constants such that $1,d$ are diffusion coefficients and $a,b$ are intrinsic growth rates of species $U,V$, respectively. The functional response of predator to prey is normalized to be $1$. The prey obeys the logistic growth and its carrying capacity is normalized to be 1. However, the density of predator follows a logistic dynamics with a varying carrying capacity proportional to the density of prey. Moreover, for $i=1,2$, $\bN_i$ formulates the spatial nonlocal dispersal of individuals and is defined by
\beaa
\bN_i[u(\cdot,t)](x):=\int_{\mathbb{R}}J_i(y)u(x-y,t)dy-u(x,t),\; u=U,V,
\eeaa
where $J_i$ is a probability density function satisfying the following conditions:
\begin{enumerate}
\item[(H1)]\, $J_i$ is a nonnegative continuous function defined in $\bR$;
\item[(H2)]\, $\int_{\mathbb{R}}J_i(y)dy=1$ and $J_i(y)=J_i(-y)$ for all $y\in \mathbb{R}$;
\item[(H3)] there exists $\hat{\lambda}_i\in(0,\infty]$ such that $\int_{\mathbb{R}}J_i(y)e^{\lambda y} dy < \infty$ for any $\lambda \in (0, \hat{\lambda_i})$ and $\int_{\mathbb{R}}J_i(y)e^{\lambda y} dy\to\infty$ as $\lambda \uparrow \hat{\lambda}_i$.
\end{enumerate}
Note that there is always the predator-free state $(1,0)$, and there is the unique constant co-existence state $(a^*,a^*)$, $a^*:=1-1/a\in(0,1)$, when $a>1$.

When the nonlocal dispersal in system \eqref{pp} is replaced by the classical diffusion, the dynamical behaviors of the corresponding system was investigated in the survey paper \cite{CGS21}. In fact, system \eqref{pp} without diffusion (the ODE system) arises in the control of introduced rabbits to protect native birds from introduced cat predation in an island (cf. \cite{CLS}), when we consider the case without rabbits and control. For the detailed biological background of the full ODE system including the rabbits and control, we refer the reader to \cite{CLS,CS}. However, it is reasonable and more realistic to take into account the influence of spatial movements of birds and cats, namely, the effect of diffusion. On the other hand, to model the long range movements and nonadjacent interactions of individuals it is more realistic to consider the nonlocal dispersal instead of the random movement with classical diffusion. This motivates us to study system \eqref{pp}.

The main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \eqref{pp} is called a traveling wave solution of \eqref{pp}, if there exist a constant $s\in\R$ ({\it the wave speed}) and a function $(\phi,\psi)$ ({\it the wave profile}) of class $C^1(\R)$ such that
$$(U,V)(x,t)=(\phi,\psi)( z),\ \ z:=x-st.$$
We are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\{s,\phi,\psi\}$ satisfies the following system of equations:
\begin{equation}\label{TWS}
\left\{
\begin{aligned}
& \bN_1[\phi](z)+ s\phi'(z) +a\phi(z)\left[1-\phi(z)\right]-\psi(z) =0, &\; z\in \mathbb{R},\\
& d \bN_2[\psi](z)+ s\psi'(z) +b\psi(z)\left[1-\frac{\psi(z)}{\phi(z)}\right] =0, & \; z\in \mathbb{R},
\end{aligned}
\right.
\end{equation}
together with the following asymptotic boundary conditions
\begin{equation}\label{BC}
(\phi,\psi)(\infty) = (1,0),\quad (\phi,\psi)(-\infty) = (a^{*},a^{*}).
\end{equation}

For convenience, for $i=1,2$, we set
\beaa
I_i(\lambda):=\int_\bR J_i(y)e^{\lambda y}dy,\ \ \lambda\in\R,
\eeaa
with $I_i(\lambda)\in(0,\infty)$ if $0\le\lambda<\hat{\lambda}_i$ and $I_i(\lambda)=\infty$ if $\lambda\ge\hat{\lambda}_i$. Note that, due to the symmetry of $J_i$, the function $I_i$ is even, and it is also strictly convex in $(-\hat{\lambda}_i,\hat{\lambda}_i)$. Also, we introduce the quantity
\beaa
s^*:=\inf_{\lambda\in(0,\hat{\lambda}_2)}\frac{d[I_2(\lambda)-1]+b}{\lambda}.
\eeaa
Note that $s^*$ is well-defined, the infimum is reached, and $s^*>0$.

We now state our main theorem as follows.

\begin{equation}\label{TWS}
\left\{
\begin{aligned}
&   \bN_1[\phi](z)+ s\phi'(z) +a\phi(z)\left[1-\phi(z)\right]-\psi(z)    =0, &\; z\in \mathbb{R},\\
&  d \bN_2[\psi](z)+ s\psi'(z) +b\psi(z)\left[1-\frac{\psi(z)}{\phi(z)}\right] =0, & \; z\in \mathbb{R},
\end{aligned}
\right.
\end{equation}

The main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \eqref{pp} is called a traveling wave solution of \eqref{pp}, if there exist a constant $s\in\R$ ({\it the wave speed}) and a function $(\phi,\psi)$ ({\it the wave profile}) of class $C^1(\R)$ such that
$$(U,V)(x,t)=(\phi,\psi)( z),\ \ z:=x-st.$$
We are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\{s,\phi,\psi\}$ satisfies the following system of equations:
\begin{equation}\label{TWS}
\left\{
\begin{aligned}
& \bN_1[\phi](z)+ s\phi'(z) +a\phi(z)\left[1-\phi(z)\right]-\psi(z) =0, &\; z\in \mathbb{R},\\
& d \bN_2[\psi](z)+ s\psi'(z) +b\psi(z)\left[1-\frac{\psi(z)}{\phi(z)}\right] =0, & \; z\in \mathbb{R},
\end{aligned}
\right.
\end{equation}
together with the following asymptotic boundary conditions
\begin{equation}\label{BC}
(\phi,\psi)(\infty) = (1,0),\quad (\phi,\psi)(-\infty) = (a^{*},a^{*}).
\end{equation}

We now state our main theorem as follows.

Hereafter, a function $(\phi,\psi)$ is positive if $\phi,\psi>0$ in $\bR$. A corresponding theorem to Theorem~\ref{th:main}, under the condition $a\ge 4$, on the traveling waves in the classical diffusion case was derived in \cite{CGS21}. Theorem~\ref{th:main} characterizes the minimal speed $s^*$ in the nonlocal dispersal case, but it needs the extra condition $d<b$ in the construction of lower and upper solutions to~\eqref{TWS}.

The existence of wave profiles $(\phi,\psi)$ to \eqref{TWS}-\eqref{BC} for all speeds $s>s^*$ is based on the construction of lower and upper solutions, which is new for this singular nonlocal system. The case $s=s^*$ is more involved, it requires a special care and uses the boundedness of the support of $J_2$. The characterization of the limiting state behind the front, that is, $(\phi,\psi)(-\infty)=(a^*,a^*)$ is carried out with a squeezing method and an argument by contradiction. The proof of the nonexistence of wave profiles for all speeds $s<s^*$ is also done by contradiction. Section~\ref{sec2} is devoted to the existence part in Theorem~\ref{th:main}, while the nonexistence is shown in Section~\ref{sec3}.

\begin{proposition}\label{le6}
Let $s>0$ be given. Let $(\overline{\phi },\overline{\psi })$ and $(\underline{\phi},\underline{\psi})$ be a pair of bounded upper and lower solutions of \eqref {TWS}. Then system \eqref {TWS} admits a positive solution $(\phi,\psi)$ of class $C^1(\R)$ such that
\begin{equation*}
\underline{\phi }( z )\leq \phi(z)\leq \overline{\phi}( z ),\; \upsi(z)\leq \psi(z)\leq\overline{\psi}( z ),\; z \in \mathbb{R}.
\end{equation*}
\end{proposition}

For a given $s>s^*$, we consider the quantity
\beaa
A(\lambda)=A(\lambda;s):=d[I_2(\lambda)-1]-s\lambda+b,\; \lambda>0.
\eeaa
It follows from the definition of $s^*$ and the strict convexity of $A(\lambda)$ that there are two positive constants $\lambda_1<\lambda_2<\hat{\ld}_2$ such that
$$A(\lambda_1)=A(\ld_2)=0$$
and $A(\lambda)<0$ for all $\lambda\in(\lambda_1,\lambda_2)$. Also, set
$$B(\ld)=B(\ld;s):=[I_1(\ld)-1]-s\ld.$$
Since $B(0)=0$ and $B'(0)=-s<0$, we can choose a constant $\ld_0\in(0,\min\{\ld_1,\hat{\ld}_1\})$ small enough such that
$$B(\ld_0)<0.$$
Now, for a fixed constant $\mu\in(1,\min{\{ \lambda_2/ \lambda_1,2\}})$, we choose
\be\label{q0}
q>\max\Big\{1,\frac{2b}{-A(\mu\ld_1)}\Big\}.
\ee
Then, the function $f(z) := e^{ -\lambda_1z}-qe^{-\mu \lambda_1z}$ has exactly one zero $z_0>0$ and exactly one maximum point $z_M\in(z_0,\infty)$, and there holds $f(z_M)>0$. Thus, using $a\ge4>1$ and $d<b$, we can choose $\delta$ such that
$$0<\delta<\min\Big\{f(z_M),\underbrace{1-\frac{1}{a}}_{=a^*},\frac12\Big(1-\frac{d}{b}\Big)\Big\},$$
and a point $z_1\in(z_0,z_M)$ such that $f(z_1)=\delta$. Note that
\be\label{delta}
d<b(1-2\delta).
\ee
Furthermore, the inequalities $z_1>z_0>0$, $\ld_1>0$ and $\mu>1$ imply
\be\label{q}
{e^{(\mu-1)\ld_1z_1}>e^{(\mu-1)\ld_1z_0}=q.}
\ee
With these choices of $\mu,q,\delta$, we finally choose $\e$ such that
\be\label{ep1}
0<\e<\min{\left\{\frac{\delta}{1+s\ld_1+a},\frac{e^{(\mu-1)\lambda_1z_1}-q}{(1+s\lambda_1+a)e^{(\mu-1)\lambda_1z_1}} \right\}}.
\ee
Note that the constant $\e$ is admissible, due to \eqref{q}, and that $0<\e<\delta<1/2$.

\begin{lemma}\label{la:upper-lower2}
Assume that $a\ge 4$, $d<b$, and that $J_2$ has a compact support such that its support is contained in $[-S,S]$ for some $S\in(0,\infty)$. For $s=s^{*}$, the functions $(\overline{\phi},\overline{\psi})$ and~$(\underline{\phi},\underline{\psi})$ defined in \eqref{uplosol2} are a pair of upper and lower solutions of \eqref{TWS}.
\end{lemma}

First, for any $s\ge s^*$, the existence of a $C^1(\R)$ solution $(\phi,\psi)$ to~\eqref{TWS} such that $1/2\le\underline{\phi}\le\phi\le\overline{\phi}<1$ and $0<\underline{\psi}\le\psi\le\overline{\psi}\le1$ in $\R$ follows from Lemmas~\ref{la:upper-lower1}-\ref{la:upper-lower2} and Proposition~\ref{le6}. Moreover, by the construction of upper and lower solutions, it is clear that $(\phi,\psi)(\infty)=(1,0)$.

\begin{equation}\label{TWS}
\left\{
\begin{aligned}
&   \bN_1[\phi](z)+ s\phi'(z) +a\phi(z)\left[1-\phi(z)\right]-\psi(z)    =0, &\; z\in \mathbb{R},\\
&  d \bN_2[\psi](z)+ s\psi'(z) +b\psi(z)\left[1-\frac{\psi(z)}{\phi(z)}\right] =0, & \; z\in \mathbb{R},
\end{aligned}
\right.
\end{equation}

\begin{equation}\label{pp}
\begin{cases}
U_t(x,t)= \bN_1[U(\cdot,t)](x)+[aU(1-U)-V](x,t), \; x\in\bR,\, t>0,\\
V_t(x,t)=d \bN_2[V(\cdot,t)](x)+[bV(1-V/U)](x,t), \; x\in\bR,\, t>0,
\end{cases}
\end{equation}

\begin{theorem}\label{th:main}
Let $a$, $b$ and $d$ be given positive constants such that $a\ge 4$ and $d<b$. Then there exists a positive solution $(\phi,\psi)$ of \eqref{TWS}-\eqref{BC} for each $s > s^{*}$. This existence also holds for $s=s^*$, if we further assume that $J_2$ has a compact support. Moreover, there exist no positive solutions of \eqref{TWS}-\eqref{BC} for $s < s^{*}$.
\end{theorem}

Hereafter, a function $(\phi,\psi)$ is positive if $\phi,\psi>0$ in $\bR$. A corresponding theorem to Theorem~\ref{th:main}, under the condition $a\ge 4$, on the traveling waves in the classical diffusion case was derived in \cite{CGS21}. Theorem~\ref{th:main} characterizes the minimal speed $s^*$ in the nonlocal dispersal case, but it needs the extra condition $d<b$ in the construction of lower and upper solutions to~\eqref{TWS}.

It is worth to note that there are some difficulties in the study of system \eqref{pp}. It is a non-monotone system which is lack of comparison principle and a singularity may occur in the $V$-equation when $U$ reaches zero in a finite time which has been confirmed numerically in \cite{CLS}. Thus system \eqref{pp} is named as a {\it singular} prey-predator model due to the latter fact. This is in contrast to the system with $U$-equation in \eqref{pp} replaced by
\beaa
U_t(x,t)= \bN_1[U(\cdot,t)](x)+[aU(1-U)-kUV](x,t), \; x\in\bR,\, t>0,
\eeaa
in which the functional response of predation is a linear function of prey. In this case, the strong maximum principle for the scalar equation gives that $U>0$ for all $t>0$. Hence no singularity can occur for the nonlinear term $V/U$ in the $V$-equation.

The existence of wave profiles $(\phi,\psi)$ to \eqref{TWS}-\eqref{BC} for all speeds $s>s^*$ is based on the construction of lower and upper solutions, which is new for this singular nonlocal system. The case $s=s^*$ is more involved, it requires a special care and uses the boundedness of the support of $J_2$. The characterization of the limiting state behind the front, that is, $(\phi,\psi)(-\infty)=(a^*,a^*)$ is carried out with a squeezing method and an argument by contradiction. The proof of the nonexistence of wave profiles for all speeds $s<s^*$ is also done by contradiction. Section~\ref{sec2} is devoted to the existence part in Theorem~\ref{th:main}, while the nonexistence is shown in Section~\ref{sec3}.

The post-theorem introduction outlines the proof of Theorem~\ref{th:main} as follows:

- **Existence for $s>s^*$:** “based on the construction of lower and upper solutions,” which the authors note “is new for this singular nonlocal system,” and the extra condition $d<b$ is needed “in the construction of lower and upper solutions to~\eqref{TWS}.”

- **Existence at $s=s^*$:** “more involved,” requiring “special care” and using “the boundedness of the support of $J_2$” (i.e., compact support).

- **Limiting state behind the front:** the claim “$(\phi,\psi)(-\infty)=(a^*,a^*)$” is proved “with a squeezing method and an argument by contradiction.”

- **Nonexistence for $s<s^*$:** proved “by contradiction.”

- **Organization:** “Section~\ref{sec2} is devoted to the existence part in Theorem~\ref{th:main}, while the nonexistence is shown in Section~\ref{sec3}.”

The post-theorem introduction outlines the proof strategy used to establish the main theorem as follows:

- **Existence for $s>s^*$:** “based on the construction of lower and upper solutions,” which the authors note “is new for this singular nonlocal system,” and the extra condition $d<b$ is needed “in the construction of lower and upper solutions to
\begin{equation}\label{TWS}
\left\{
\begin{aligned}
&   \bN_1[\phi](z)+ s\phi'(z) +a\phi(z)\left[1-\phi(z)\right]-\psi(z)    =0, &\; z\in \mathbb{R},\\
&  d \bN_2[\psi](z)+ s\psi'(z) +b\psi(z)\left[1-\frac{\psi(z)}{\phi(z)}\right] =0, & \; z\in \mathbb{R},
\end{aligned}
\right.
\end{equation}.”

- **Existence at $s=s^*$:** “more involved,” requiring “special care” and using “the boundedness of the support of $J_2$” (i.e., compact support).

- **Limiting state behind the front:** the claim “$(\phi,\psi)(-\infty)=(a^*,a^*)$” is proved “with a squeezing method and an argument by contradiction.”

- **Nonexistence for $s<s^*$:** proved “by contradiction.”

- **Organization:** Next they prove the existence part in the main theorem, while the nonexistence is shown later.

\label{th:main}
Let $a$, $b$ and $d$ be given positive constants such that $a\ge 4$ and $d<b$. Then there exists a positive solution $(\phi,\psi)$ of
\begin{equation}\label{TWS}
\left\{
\begin{aligned}
&   \bN_1[\phi](z)+ s\phi'(z) +a\phi(z)\left[1-\phi(z)\right]-\psi(z)    =0, &\; z\in \mathbb{R},\\
&  d \bN_2[\psi](z)+ s\psi'(z) +b\psi(z)\left[1-\frac{\psi(z)}{\phi(z)}\right] =0, & \; z\in \mathbb{R},
\end{aligned}
\right.
\end{equation}
with boundary conditions
\begin{equation}\label{BC}
(\phi,\psi)(\infty) = (1,0),\quad (\phi,\psi)(-\infty) = (a^{*},a^{*}).
\end{equation}
for each $s > s^{*}$. This existence also holds for $s=s^*$, if we further assume that $J_2$ has a compact support. Moreover, there exist no positive solutions of the system \eqref{TWS} with boundary conditions \eqref{BC} for $s < s^{*}$.

Question:
Let \(a,b,d>0\) with \(a\ge 4\) and \(d<b\), and set \(a^*=1-\frac1a\). For \(i=1,2\), define the nonlocal dispersal operator
\[
\mathcal N_i[u](z)=\int_{\mathbb R}J_i(y)u(z-y)\,dy-u(z),
\]
where each kernel \(J_i\) is a nonnegative continuous symmetric probability density on \(\mathbb R\) satisfying the exponential-moment condition: there exists \(\hat\lambda_i\in(0,\infty]\) such that \(\int_{\mathbb R}J_i(y)e^{\lambda y}\,dy<\infty\) for \(0<\lambda<\hat\lambda_i\), and this integral tends to \(\infty\) as \(\lambda\uparrow \hat\lambda_i\). Let
\[
I_2(\lambda)=\int_{\mathbb R}J_2(y)e^{\lambda y}\,dy,
\qquad
s^*=\inf_{\lambda\in(0,\hat\lambda_2)}\frac{d\,[I_2(\lambda)-1]+b}{\lambda}.
\]
Consider the traveling-wave profile system
\[
\begin{cases}
\mathcal N_1[\phi](z)+s\phi'(z)+a\phi(z)(1-\phi(z))-\psi(z)=0, & z\in\mathbb R,\\
 d\,\mathcal N_2[\psi](z)+s\psi'(z)+b\psi(z)\left(1-\dfrac{\psi(z)}{\phi(z)}\right)=0, & z\in\mathbb R,
\end{cases}
\]
with boundary conditions
\[
(\phi,\psi)(+\infty)=(1,0),\qquad (\phi,\psi)(-\infty)=(a^*,a^*).
\]
Which statement holds for positive solutions \((\phi,\psi)\) of this boundary-value problem?

A positive solution exists for every wave speed \(s>s^*\). It also exists for the critical speed \(s=s^*\) provided that \(J_2\) has compact support. For every \(s<s^*\), no positive solution exists.

Choices:
B. A positive solution exists for every wave speed \(s\ge s^*\). Moreover, for every \(s<s^*\), no positive solution exists.
C. For every \(s>s^*\), there exists a positive solution. In addition, for every \(s<s^*\), no positive solution exists.
D. A positive solution exists for every wave speed \(s>s^*\), and also for \(s=s^*\) without any additional assumption on \(J_2\). For every \(s<s^*\), no positive solution exists.
E. A positive solution exists for every wave speed \(s>s^*\). It also exists for the critical speed \(s=s^*\) provided that both \(J_1\) and \(J_2\) have compact support. For every \(s<s^*\), no positive solution exists.

critical-speed compact-support hypothesis at s=s^*

dropped the critical-speed existence clause

dependence of critical-speed existence on compact support of J_2

which kernel requires compact support at the critical speed


Question:
Let $a,b,d>0$ with $a\ge 4$ and $d<b$. For $i=1,2$, define the nonlocal dispersal operators
\[
\mathcal N_i[u](z)=\int_{\mathbb R}J_i(y)u(z-y)\,dy-u(z),
\]
where $J_i$ are the dispersal kernels from the model, and set
\[
a^*=1-\frac1a,
\qquad
I_2(\lambda)=\int_{\mathbb R}J_2(y)e^{\lambda y}\,dy,
\qquad
s^*=\inf_{\lambda\in(0,\hat\lambda_2)}\frac{d\,[I_2(\lambda)-1]+b}{\lambda}.
\]
Consider positive profile pairs $(\phi,\psi)$, meaning $\phi(z)>0$ and $\psi(z)>0$ for all $z\in\mathbb R$, satisfying
\[
\begin{cases}
\mathcal N_1[\phi](z)+s\phi'(z)+a\phi(z)(1-\phi(z))-\psi(z)=0, & z\in\mathbb R,\\
 d\,\mathcal N_2[\psi](z)+s\psi'(z)+b\psi(z)\left(1-\dfrac{\psi(z)}{\phi(z)}\right)=0, & z\in\mathbb R,
\end{cases}
\]
with asymptotic boundary conditions
\[
(\phi,\psi)(\infty)=(1,0),\qquad (\phi,\psi)(-\infty)=(a^*,a^*).
\]
Which statement holds for the existence and nonexistence of such positive solutions as the wave speed $s$ varies?

For every $s>s^*$, there exists a positive solution $(\phi,\psi)$ of the above system with the stated boundary conditions. The same existence conclusion also holds for $s=s^*$ if, in addition, $J_2$ has compact support. For every $s<s^*$, there is no positive solution of the system with those boundary conditions.

Choices:
B. For every $s\ge s^*$, there exists a positive solution $(\phi,\psi)$ of the above system with the stated boundary conditions, without any additional assumption on $J_2$. For every $s<s^*$, there is no positive solution of the system with those boundary conditions.
C. For every $s>s^*$, there exists a positive solution $(\phi,\psi)$ of the above system with the stated boundary conditions. For every $s<s^*$, there is no positive solution of the system with those boundary conditions.
D. There exists a positive solution $(\phi,\psi)$ of the above system with the stated boundary conditions for every $s>s^*$, and there is no positive solution for $s\le s^*$.
E. If $J_2$ has compact support, then there exists a positive solution $(\phi,\psi)$ of the above system with the stated boundary conditions for every $s\ge s^*$. Moreover, for every $s<s^*$ there is no positive solution only for some choice of the parameters $a,b,d$ with $a\ge 4$ and $d<b$.

compact-support requirement at the critical speed

exceptional critical case $s=s^*$ with compact support

universal quantifier in the nonexistence statement


\label{THM:main}
Let $u(t,x)$ be solutions to \eqref{weakNLS1} with initial data $u_0^\omega$ given in \eqref{random_data}.
For any $t = \mathcal O ( \varepsilon^{-1} |\log \varepsilon|)$ and $z_0 > 0$, we have that
\begin{align}
\label{THM_LDP4}
\lim_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P} \Big( \|u(t,x) \|_{\mathcal{F} L^1 (\mathbb{T})} > z_0\varepsilon^{-1/2} \Big) = - \frac{z_0^2}{\sum_k c_k^2},
\end{align}

\noindent
holds if and only if $(c_k) \in \ell^1$.

\begin{proposition}
\label{PROP:linear2}
Consider the linear Schr\"odinger equation on the torus $\T = [0,2\pi]$ as in \eqref{linear}, with random initial data $u_0^\o$ given by \eqref{random_data}.
Then,
\begin{align}
\label{LDP_linear2}
\lim_{\eps\to 0^+} \eps \log \PP \big( \|u(t)\|_{L^\infty (\T)} \ge z_0 \eps^{-1/2} \big) = - \frac{z_0^2}{\sum_{k \in \Z} c_k^2},
\end{align}

\noi
provided $(c_k) \in \l^1$.
\end{proposition}

\begin{theorem}
\label{THM:main}
Let $u(t,x)$ be solutions to \eqref{weakNLS1} with initial data $u_0^\o$ given in \eqref{random_data}.
For any $t = \mathcal O ( \eps^{-1} |\log \eps|)$ and $z_0 > 0$, we have that
\begin{align}
\label{THM_LDP4}
\lim_{\eps \to 0^+} \eps \log \PP \Big( \|u(t,x) \|_{\FL^1 (\T)} > z_0\eps^{-1/2} \Big) = - \frac{z_0^2}{\sum_k c_k^2},
\end{align}

\noi
holds if and only if $(c_k) \in \l^1$.
\end{theorem}

In this paper, we consider the following Cauchy problem of the one-dimensional cubic nonlinear Schr\"odinger equation (NLS) on the torus $\mathbb{T}= \mathbb{R} / (2 \pi \mathbb{Z})$:
\begin{equation}
\label{NLS}
\begin{cases}
i\pa_t u + \partial_x^2 u = \pm \,|u|^2\, u,\\
u(t,x)|_{t=0}=u_0.
\end{cases}
\end{equation}

\noindent
The equation \eqref{NLS} is fundamental in theoretical physics and applied mathematics, for describing the evolution of wave packets in various physical systems, such as nonlinear optics, fluids, and plasmas; see \cite{SS99} for a review.
Bourgain \cite{BO93} proved \eqref{NLS} is deterministically global well-posed in $L^2 (\mathbb{T})$.
The probabilistic study of \eqref{NLS} has emerging importance since the seminal work of Bourgain on invariant measures \cite{BO94,BO96}.

More recently, probabilistic analyses of \eqref{NLS} have also found applications in oceanography, particularly in modelling rare extreme events such as the formation of {\it rogue waves} in deep-sea dynamics~\cite{DGV18,DGV19,GGKS21}. In particular, Garrido et al.~\cite{GGKS21} established a large deviations principle (LDP) describing the probability of observing a wave of unusually large height in the weakly nonlinear regime. Their result relied on a strong exponential decay assumption on the Fourier coefficients of the random initial data, which plays a crucial role in their analysis.
In this paper, we improve the exponential decay condition on coefficients of initial data in \cite{GGKS21} to a $\ell^1$ decay condition.

We consider
random initial data of the form:
\begin{equation}\label{random_data}
u_0 (x)= \sum_{k\in\mathbb{Z}} c_k g_k e^{ikx},
\end{equation}
where $c_k \in \mathcal{C}$ and $\{g_k\}_{k\in\mathbb{Z}}$ are independent, identically distributed, complex Gaussian random variables with $\mathbb{E} g_k=0$, $\mathbb{E} g_k g_j=0$ and $\mathbb{E} g_k \overline{g}_j =\delta_{kj}$ for $k,j\in\mathbb{Z}$.
Without loss of any generality, we only consider nonnegative real coefficients $c_k \ge 0$, due to the rotation invariance of complex Gaussian random variables.

The study of the large deviation principle for solutions to \eqref{NLS} was initiated by \cite{DGV18,DGV19}, where the authors conjectured that the existence of an LDP can be used to predict the formation of rogue waves.
In particular, they considered \eqref{NLS} with random initial data $u_0^N$ of the form
\begin{align}
\label{random1}
u_0^N (x) = \sum_{|k| \le N} \theta_k e^{ikx},
\end{align}

\noindent
where the initial data is parametrized by a random vector $\theta = (\theta_k)_{|k| \le N} \in \mathbb C^{2N+1}$.
Here $(\theta_k)$ are independent complex Gaussian random variables with $\mathbb{E} \theta_k = \mathbb{E} \theta_k^2 = 0$ and $\mathbb{E} |\theta_k|^2 = c_k^2$, for some fast-decaying $c_k > 0$.
Then, the set of initial data that generates a rouge wave of height at least $z>0$ at time $t>0$ is given by
\begin{align}
\label{Dtz}
\mathcal D (t,z) : = \bigg\{ (\theta_k)_{|k| \le N} \in \mathcal{C}^{2N+1} \Big| \sup_x |u (t,x| \theta)| > z\bigg\},
\end{align}

\noindent
where $u$ is the solution to NLS \eqref{NLS} with the truncated initial data \eqref{random1}.
Dematteis et al, then proposed a theoretical framework of a large deviations principle (LDP) to quantify the likelihood of $\mathcal D (t,z)$.
In particular, consider the minimization problem
\begin{align}\label{minp}
\theta^* (z) = {\rm argmin}\,\, \mathcal I (\theta)
\end{align}

\noindent
where
\[
\mathcal I (\theta) = \max_{y \in \mathcal{C}^{2N+1}} [\langle y,\theta \rangle - S(y)] \quad \textup{ and } \quad S(y) = \log \mathbb{E} e^{\langle y, \theta \rangle}.
\]

\noindent
Then Dematteis et al. claim that for $t >0$
\begin{align}
\label{LDP1}
\log \mathbb{P} (\mathcal D(t,z)) = - \mathcal I (\theta^* (z)) + o(1),
\end{align}

\noindent
as $z \to \infty$, provided
the minimization problem \eqref{minp} has a unique solution.

Garrido et al.~\cite{GGKS21} rigorously proved the conjectured LDP~\eqref{LDP1} locally in time for the nonlinear Schr\"odinger equation with weak nonlinearity.
A major difficulty in analysing the LDP problem~\eqref{LDP1} lies in estimating the gradient
$\nabla_{\theta} \sup_{x} |u(t,x|\theta)|$
and in verifying the convexity of the set $\mathcal{D}(t,z)$,
both of which are extremely challenging to establish in practice.
To overcome these obstacles, Garrido et al.~\cite{GGKS21} introduced a new formulation and considered the following LDP problem:
\begin{align}
\label{LDP2}
\log \mathbb{P} \Big( \sup_{x} |u(t,x)| > z \Big) = - I(z) + o(1), \quad z \to \infty,
\end{align}
where $I$ denotes the corresponding rate function.
The advantage of this reformulation is that it circumvents the need to solve the minimization problem in~\eqref{minp}.
Within this framework, they established~\eqref{LDP2} for initial data of the form~\eqref{random1} with infinitely many Fourier modes ($N = \infty$) satisfying a \emph{strong decay condition} on the coefficients.

In particular, consider the weakly nonlinear Schr\"odinger equation on the circle $\mathbb{T} = [0,2\pi]$:
\begin{align}
\label{weakNLS1}
i\pa_t u + \Delta u = \varepsilon^{2} |u|^2 u.
\end{align}
It was proved in~\cite{GGKS21} that
\begin{align}
\label{THM_LDP3}
\lim_{\varepsilon \to 0^+}
\varepsilon \log \mathbb{P} \Big( \sup_{x} |u(t,x)| > z_0 \varepsilon^{-1/2} \Big)
= - \frac{z_0^2}{\sum_k c_k^2},
\end{align}
for $0 < t \lesssim \varepsilon^{-1}$ and $z_0 > 0$,
where $u(t,x)$ denotes the solution to~\eqref{weakNLS1} with random initial data $u_0^\omega$ given by~\eqref{random_data} and coefficients satisfying
\begin{align}\label{decay}
c_k = a e^{-b|k|}
\quad \text{or} \quad
c_k = a e^{-b|k|^2},
\end{align}
for some fixed $a,b > 0$.

The proof of~\eqref{THM_LDP3} is based on the G\"artner--Ellis theorem combined with resonant approximation techniques.
In their analysis, the rapid decay condition~\eqref{decay} plays a crucial role in both the linear and nonlinear estimates.
They further observed that alternative families of coefficients $(c_k)$ might also yield~\eqref{LDP2}, and posed an \textit{open question} concerning the identification of the optimal decay condition under which~\eqref{LDP2} holds; see~\cite[Remark~1.3]{GGKS21}.
One of the principal aims of the present work is to address this question.

To this purpose, we consider a LDP under Fourier-Lebesgue norm, which is stronger than the $L^\infty (\mathbb{T})$ norm used in \eqref{THM_LDP3}.
Our conventions for the Fourier transform are as follows:
\[
\widehat f (n) = \frac1{{2\pi}} \int_0^{2\pi} f(x) e^{ix n}dx \quad \textup{ and } \quad f(x) = \sum_{n\in \mathbb{Z}} \widehat f(n) e^{ixn}
\]

\noindent
for functions on the circle $\mathbb{T} = \mathbb{R}/(2\pi \mathbb{Z})$.
With the above notations, we define the Fourier-Lebesgue norm by
\[
\begin{split}
\|f\|_{\mathcal{F} L^1 (\mathbb{T})} = \sum_{n \in \mathbb{Z}} |\widehat f(n)|.
\end{split}
\]

\noindent
It is easy to see that
\[
\sup_{x \in \mathbb{T}} |f(x)| \le \|f\|_{\mathcal{F} L^1 (\mathbb{T})}.
\]

\noindent
We note that $L^\infty$ and $\mathcal{F} L^1$ scales the same, and furthermore, the LDPs under $L^\infty$ and $\mathcal{F} L^1$ share the same rate function.
In particular, for $t \lesssim \varepsilon^{-1}$, we have
\begin{align}\label{obser}
\lim_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P} \Big( \sup_{x} |u(t,x) | > z_0\varepsilon^{-1/2} \Big) = \lim_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P} \Big( \|u(t,x) \|_{\mathcal{F} L^1} > z_0\varepsilon^{-1/2} \Big),
\end{align}

\noindent
where $u(t,x)$ is the solution to \eqref{weakNLS1} with initial data \eqref{random1} in $\mathcal{F}L^1$.
This motivates us to consider the LDP under the $\mathcal{F} L^1$ norm.
We are ready to state our first main result.

We consider
random initial data of the form:
\begin{equation}\label{random_data}
u_0 (x)= \sum_{k\in\mathbb{Z}} c_k g_k e^{ikx},
\end{equation}
where $c_k \in \mathcal{C}$ and $\{g_k\}_{k\in\mathbb{Z}}$ are independent, identically distributed, complex Gaussian random variables with $\mathbb{E} g_k=0$, $\mathbb{E} g_k g_j=0$ and $\mathbb{E} g_k \overline{g}_j =\delta_{kj}$ for $k,j\in\mathbb{Z}$.
Without loss of any generality, we only consider nonnegative real coefficients $c_k \ge 0$, due to the rotation invariance of complex Gaussian random variables.

\noindent
where the initial data is parametrized by a random vector $\theta = (\theta_k)_{|k| \le N} \in \mathbb C^{2N+1}$.
Here $(\theta_k)$ are independent complex Gaussian random variables with $\mathbb{E} \theta_k = \mathbb{E} \theta_k^2 = 0$ and $\mathbb{E} |\theta_k|^2 = c_k^2$, for some fast-decaying $c_k > 0$.
Then, the set of initial data that generates a rouge wave of height at least $z>0$ at time $t>0$ is given by
\begin{align}
\label{Dtz}
\mathcal D (t,z) : = \bigg\{ (\theta_k)_{|k| \le N} \in \mathcal{C}^{2N+1} \Big| \sup_x |u (t,x| \theta)| > z\bigg\},
\end{align}

Garrido et al.~\cite{GGKS21} rigorously proved the conjectured LDP~\eqref{LDP1} locally in time for the nonlinear Schr\"odinger equation with weak nonlinearity.
A major difficulty in analysing the LDP problem~\eqref{LDP1} lies in estimating the gradient
$\nabla_{\theta} \sup_{x} |u(t,x|\theta)|$
and in verifying the convexity of the set $\mathcal{D}(t,z)$,
both of which are extremely challenging to establish in practice.
To overcome these obstacles, Garrido et al.~\cite{GGKS21} introduced a new formulation and considered the following LDP problem:
\begin{align}
\label{LDP2}
\log \mathbb{P} \Big( \sup_{x} |u(t,x)| > z \Big) = - I(z) + o(1), \quad z \to \infty,
\end{align}
where $I$ denotes the corresponding rate function.
The advantage of this reformulation is that it circumvents the need to solve the minimization problem in~\eqref{minp}.
Within this framework, they established~\eqref{LDP2} for initial data of the form~\eqref{random1} with infinitely many Fourier modes ($N = \infty$) satisfying a \emph{strong decay condition} on the coefficients.

In particular, consider the weakly nonlinear Schr\"odinger equation on the circle $\mathbb{T} = [0,2\pi]$:
\begin{align}
\label{weakNLS1}
i\pa_t u + \Delta u = \varepsilon^{2} |u|^2 u.
\end{align}
It was proved in~\cite{GGKS21} that
\begin{align}
\label{THM_LDP3}
\lim_{\varepsilon \to 0^+}
\varepsilon \log \mathbb{P} \Big( \sup_{x} |u(t,x)| > z_0 \varepsilon^{-1/2} \Big)
= - \frac{z_0^2}{\sum_k c_k^2},
\end{align}
for $0 < t \lesssim \varepsilon^{-1}$ and $z_0 > 0$,
where $u(t,x)$ denotes the solution to~\eqref{weakNLS1} with random initial data $u_0^\omega$ given by~\eqref{random_data} and coefficients satisfying
\begin{align}\label{decay}
c_k = a e^{-b|k|}
\quad \text{or} \quad
c_k = a e^{-b|k|^2},
\end{align}
for some fixed $a,b > 0$.

\noindent
We note that $L^\infty$ and $\mathcal{F} L^1$ scales the same, and furthermore, the LDPs under $L^\infty$ and $\mathcal{F} L^1$ share the same rate function.
In particular, for $t \lesssim \varepsilon^{-1}$, we have
\begin{align}\label{obser}
\lim_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P} \Big( \sup_{x} |u(t,x) | > z_0\varepsilon^{-1/2} \Big) = \lim_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P} \Big( \|u(t,x) \|_{\mathcal{F} L^1} > z_0\varepsilon^{-1/2} \Big),
\end{align}

\noindent
where $u(t,x)$ is the solution to \eqref{weakNLS1} with initial data \eqref{random1} in $\mathcal{F}L^1$.
This motivates us to consider the LDP under the $\mathcal{F} L^1$ norm.
We are ready to state our first main result.

\begin{align}
\label{LDP1}
\log \PP (\mathcal D(t,z)) = - \mathcal I (\theta^* (z)) + o(1),
\end{align}

\begin{align}
\label{LDP2}
\log \PP \Big( \sup_{x} |u(t,x)| > z \Big) = - I(z) + o(1), \quad z \to \infty,
\end{align}

\begin{align}\label{minp}
\theta^* (z) = {\rm argmin}\,\, \mathcal I (\theta)
\end{align}

\begin{align}
\label{random1}
u_0^N (x) = \sum_{|k| \le N} \theta_k e^{ikx},
\end{align}

\begin{equation}\label{random_data}
u_0 (x)= \sum_{k\in\Z} c_k g_k e^{ikx},
\end{equation}

In particular, consider the weakly nonlinear Schr\"odinger equation on the circle $\T = [0,2\pi]$:
\begin{align}
\label{weakNLS1}
i\pa_t u + \Delta u = \varepsilon^{2} |u|^2 u.
\end{align}
It was proved in~\cite{GGKS21} that
\begin{align}
\label{THM_LDP3}
\lim_{\varepsilon \to 0^+}
\varepsilon \log \PP \Big( \sup_{x} |u(t,x)| > z_0 \varepsilon^{-1/2} \Big)
= - \frac{z_0^2}{\sum_k c_k^2},
\end{align}
for $0 < t \lesssim \varepsilon^{-1}$ and $z_0 > 0$,
where $u(t,x)$ denotes the solution to~\eqref{weakNLS1} with random initial data $u_0^\omega$ given by~\eqref{random_data} and coefficients satisfying
\begin{align}\label{decay}
c_k = a e^{-b|k|}
\quad \text{or} \quad
c_k = a e^{-b|k|^2},
\end{align}
for some fixed $a,b > 0$.

\noi
We note that $L^\infty$ and $\FL^1$ scales the same, and furthermore, the LDPs under $L^\infty$ and $\FL^1$ share the same rate function.
In particular, for $t \les \eps^{-1}$, we have
\begin{align}\label{obser}
\lim_{\eps \to 0^+} \eps \log \PP \Big( \sup_{x} |u(t,x) | > z_0\eps^{-1/2} \Big) = \lim_{\eps \to 0^+} \eps \log \PP \Big( \|u(t,x) \|_{\FL^1} > z_0\eps^{-1/2} \Big),
\end{align}

\begin{remark}
\rm
We remark that the time scale $t = \mathcal O ( \eps^{-1} |\log \eps|)$ is critical;
see \cite[Remark 1.7]{GGKS21} for more discussion on critical time.
In the limit $\eps \to 0^+$, the large deviation principle \eqref{THM_LDP4} may hold for any fixed time $t > 0$.
\end{remark}

\begin{corollary}\label{COR:main}
Let $u(t,x)$ be solutions to \eqref{weakNLS1} with initial data $u_0^\o$ given in \eqref{random_data}.
For any $t = \mathcal O ( \eps^{-1} |\log \eps|)$ and $z_0 > 0$, we have that \eqref{THM_LDP3} holds provided $(c_k) \in \l^1$.
\end{corollary}

\begin{proposition}
\label{PROP:linear1}
Consider the linear Schr\"odinger equation on the torus $\T = [0,2\pi]$ as in \eqref{linear}, with random initial data $u_0^\o$ given by \eqref{random_data}.
Then,
\begin{align}
\label{LDP_linear1}
\lim_{\eps\to 0^+} \eps \log \P \big( \|u(t)\|_{\FL^1(\T)} \ge z_0 \eps^{-1/2} \big) = - \frac{z_0^2}{\sum_{k \in \Z} c_k^2},
\end{align}

We turn to the summation in \eqref{main} over $k \in B$.
We note that
\begin{align}
\label{main2}
\begin{split}
\eps & \sum_{k \in B} \log \left( 1+ \sqrt{\pi}\varepsilon^{-1/2}\, c_k\, \exp\left(\frac{\varepsilon^{-1} \, c_k^2}{4}\right) \right) \\
& \le \eps \sum_{k \in B} \log \left( \big( 1+ \sqrt{\pi}\varepsilon^{-1/2}\, c_k\, \big) \exp\left(\frac{\varepsilon^{-1} \, c_k^2}{4}\right) \right) \\
& \le \eps \sum_{k \in B} \log \big( 1+ \sqrt{\pi}\varepsilon^{-1/2}\, c_k\, \big) + \eps \sum_{k \in B} \left(\frac{\varepsilon^{-1} \, c_k^2}{4}\right) \\
& \le \sqrt{\pi}\varepsilon^{1/2} \sum_{k \in B} \, c_k\, + \frac14 \sum_{k \in B} c_k^2,
\end{split}
\end{align}
which is again uniformly bounded as $(c_k) \in \l^1 \subset \l^2$, where we used $\eps^{1/2} \le c_k$ for $k \in B$.
\end{proof}

\subsection{Linear LDP II - \texorpdfstring{$L^\infty$}{Lg} norm}
In this subsection,
we prove an LDP for the linear solution to \eqref{linear} under $L^\infty$ norm.
In particular, we shall prove
\begin{proposition}
\label{PROP:linear2}
Consider the linear Schr\"odinger equation on the torus $\T = [0,2\pi]$ as in \eqref{linear}, with random initial data $u_0^\o$ given by \eqref{random_data}.
Then,
\begin{align}
\label{LDP_linear2}
\lim_{\eps\to 0^+} \eps \log \PP \big( \|u(t)\|_{L^\infty (\T)} \ge z_0 \eps^{-1/2} \big) = - \frac{z_0^2}{\sum_{k \in \Z} c_k^2},
\end{align}

\noi
where $\mathcal B_\eps$ is defined in \eqref{Beps}.
We first note, see also \cite[Proposition 4.1]{GGKS21},\footnote{Please note that \cite[Proposition 4.1]{GGKS21} only requires that $\|c_k\|_{\l^1_k} < \infty$.} that
\begin{align}
\label{pp400}
\lim_{\eps \to 0^+} \eps \log \PP \big( \| u_{\rm app} (t) \|_{L^\infty (\T)} > z_0 (\eps^{-1/2} + \eps^{-1/2 + \dl}) \big) = - \frac{z_0^2}{\sum_{k \in \Z} c_k^2}.
\end{align}

\begin{align}
\label{THM_LDP4}
\lim_{\eps \to 0^+} \eps \log \PP \Big( \|u(t,x) \|_{\FL^1 (\T)} > z_0\eps^{-1/2} \Big) = - \frac{z_0^2}{\sum_k c_k^2},
\end{align}

\begin{equation}\label{random_data}
u_0 (x)= \sum_{k\in\Z} c_k g_k e^{ikx},
\end{equation}

\begin{remark}
\rm
We remark that the time scale $t = \mathcal O ( \varepsilon^{-1} |\log \varepsilon|)$ is critical;
see \cite[Remark 1.7]{GGKS21} for more discussion on critical time.
In the limit $\varepsilon \to 0^+$, the large deviation principle \eqref{THM_LDP4} may hold for any fixed time $t > 0$.
\end{remark}

\begin{remark}
\rm
To prove Theorem~\ref{THM:main}, we first develop a novel approach to establish
the LDP for the linear Schr\"odinger equation under the sole assumption that
the initial data lies in $\mathcal{F} L^1$. We then carry out the nonlinear analysis,
together with a perturbative argument, entirely within the $\mathcal{F} L^1$ framework.
\end{remark}

As a consequence of Theorem \ref{THM:main} and the observation \eqref{obser}, we can improve \eqref{THM_LDP3} in terms of the decay of $c_k$.

\begin{corollary}\label{COR:main}
Let $u(t,x)$ be solutions to \eqref{weakNLS1} with initial data $u_0^\omega$ given in \eqref{random_data}.
For any $t = \mathcal O ( \varepsilon^{-1} |\log \varepsilon|)$ and $z_0 > 0$, we have that \eqref{THM_LDP3} holds provided $(c_k) \in \ell^1$.
\end{corollary}

\begin{remark}
\rm
We note that Corollary \ref{COR:main} improves \cite[Theorem 1.1]{GGKS21} in the following sense.
The coefficients $(c_k)$ in \cite[Theorem 1.1]{GGKS21} is required to decay condition \eqref{decay}.
In Proposition \ref{PROP:linear2}, it requires only $(c_k) \in \ell^1$.
It is expected that \eqref{THM_LDP3} holds for initial data $u_0 \in L^2 \cap L^\infty$, where $\mathcal{F} L^1 \subset L^2 \cap L^\infty$.
However, our current argument can only handle $\mathcal{F} L^1$ data.
\end{remark}

\begin{remark}
\rm
Recently, in \cite{BGMS25}, the authors studied rogue waves and large deviations
for two-dimensional pure-gravity deep-water waves, assuming initial data with
exponentially decaying Fourier coefficients; see \cite[(1.14)]{BGMS25}. It is
expected that our argument also applies in their setting and allows one to
relax this assumption to merely $\ell^1$ decay of the Fourier coefficients.
We plan to pursue this in future work.
\end{remark}

To prove Theorem~\ref{THM:main}, the authors say they (i) “first develop a novel approach to establish the LDP for the linear Schr\"odinger equation under the sole assumption that the initial data lies in $\mathcal{F} L^1$,” and then (ii) “carry out the nonlinear analysis, together with a perturbative argument, entirely within the $\mathcal{F} L^1$ framework.”

To prove the main theorem, the authors say they (i) “first develop a novel approach to establish the LDP for the linear Schr\"odinger equation under the sole assumption that the initial data lies in $\mathcal{F} L^1$,” and then (ii) “carry out the nonlinear analysis, together with a perturbative argument, entirely within the $\mathcal{F} L^1$ framework.”,

\label{THM:main}
Let $u(t,x)$ be solutions to \begin{align}
\label{weakNLS1}
i\pa_t u + \Delta u = \varepsilon^{2} |u|^2 u.
\end{align} with initial data $u_0^\omega$ given in \begin{equation}\label{random_data}
u_0 (x)= \sum_{k\in\Z} c_k g_k e^{ikx},
\end{equation}.
For any $t = \mathcal O ( \varepsilon^{-1} |\log \varepsilon|)$ and $z_0 > 0$, we have that
\begin{align}
\label{THM_LDP4}
\lim_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P} \Big( \|u(t,x) \|_{\mathcal{F} L^1 (\mathbb{T})} > z_0\varepsilon^{-1/2} \Big) = - \frac{z_0^2}{\sum_k c_k^2},
\end{align}

\noindent
holds if and only if $(c_k) \in \ell^1.\,

Biconditional or Equivalence
Asymptotic or Limit

Question:
Let $u(t,x)$ solve the weakly nonlinear Schr\"odinger equation on the torus $\mathbb T$,
\[
i\partial_t u+\Delta u=\varepsilon^2|u|^2u,
\]
with random initial data
\[
u_0(x)=\sum_{k\in\mathbb Z} c_k g_k e^{ikx},
\]
where $\{g_k\}_{k\in\mathbb Z}$ are independent identically distributed complex Gaussian random variables and $(c_k)$ is a sequence of coefficients. For any $t=\mathcal O(\varepsilon^{-1}|\log\varepsilon|)$ and any $z_0>0$, suppose one considers the large-deviation asymptotic
\[
\lim_{\varepsilon\to0^+}\varepsilon\log\mathbb P\Big(\|u(t,\cdot)\|_{\mathcal F L^1(\mathbb T)}>z_0\varepsilon^{-1/2}\Big)
= -\frac{z_0^2}{\sum_k c_k^2}.
\]
Which of the following statements is equivalent to this asymptotic formula?

The coefficient sequence satisfies $(c_k)\in \ell^1$.

Choices:
B. The coefficient sequence satisfies $(c_k)\in \ell^2$.
C. The coefficient sequence satisfies $\sum_k c_k^2<\infty$.
D. For each fixed $z_0>0$, there exists a time range $t=\mathcal O(\varepsilon^{-1}|\log\varepsilon|)$ on which the asymptotic formula holds whenever $(c_k)\in \ell^1\cap \ell^2$.
E. The asymptotic formula holds if and only if the coefficient sequence satisfies $(c_k)\in \mathcal F L^1(\mathbb T)$.

sharp \ell^1 threshold replaced by weaker square-summability

dropped the full equivalence and kept only the weaker consequence implied by $(c_k)\in\ell^1$

replaced the iff criterion by a merely sufficient mixed assumption with altered quantifier structure

confused coefficient summability on $(c_k)$ with function-space membership notation


The study of equilateral polygons within lattices combines elements of geometry, number theory, and discrete mathematics. In particular, the problem of characterizing which polygons can be embedded in given planar lattices has drawn sustained interest. A central focus has been on determining for which values of $n$ there exists a convex equilateral $n$-gon whose vertices lie in a specified lattice.

Recent results by Maehara \cite{maehara2019planar} established that every planar integral lattice contains convex equilateral $n$-gons for all even $n \geq 4$, and for certain odd values of $n$, depending on a number-theoretic invariant of the lattice. In particular, Maehara showed that a planar integral lattice $L$ contains some equilateral polygon with an odd number of sides if and only if the square-free part of the square of the determinant of $L$, denoted $\nu(L)$, satisfies $\nu(L) \equiv 3 \pmod{4}$.

Building on this, Iino and Sakiyama \cite{iino2025planarlatticesequilateraloddgons} studied rectangular lattices of the form $\Lambda(m) = L[(1,0),(0,\sqrt{m})]$ for square-free integers $m \equiv 3 \pmod{4}$, and provided both necessary and sufficient conditions for the existence of equilateral polygons with a given odd number of sides. Among their key findings is that for such lattices, if a convex equilateral $n$-gon exists, then $n$ must be at least as large as every prime dividing $m$. They also proved that this is a sufficient condition when the largest prime factor of $m$ is less than 29 by a brute-force computer search.

In this paper, we extend the classification of equilateral polygons in rectangular lattices of the form $\Lambda(m)$, with $m \equiv 3 \pmod{4}$.

\begin{abstract}
We study the existence of equilateral polygons in planar integer lattices. Maehara showed that it's sufficient to work with rectangular lattices $\Lambda(m) = L[(1,0),(0,\sqrt{m})]$ with $m \equiv 3 \pmod{4}$. Building on results of Maehara and of Iino and Sakiyama, we show that for every such $m$ there exists $N$ such that for all $n \geq N$, the lattice $\Lambda(m)$ contains an equilateral $n$-gon. This extends previous classifications of equilateral polygons in planar lattices.
\end{abstract}

Recent results by Maehara \cite{maehara2019planar} established that every planar integral lattice contains convex equilateral $n$-gons for all even $n \geq 4$, and for certain odd values of $n$, depending on a number-theoretic invariant of the lattice. In particular, Maehara showed that a planar integral lattice $L$ contains some equilateral polygon with an odd number of sides if and only if the square-free part of the square of the determinant of $L$, denoted $\nu(L)$, satisfies $\nu(L) \equiv 3 \pmod{4}$.

Building on this, Iino and Sakiyama \cite{iino2025planarlatticesequilateraloddgons} studied rectangular lattices of the form $\Lambda(m) = L[(1,0),(0,\sqrt{m})]$ for square-free integers $m \equiv 3 \pmod{4}$, and provided both necessary and sufficient conditions for the existence of equilateral polygons with a given odd number of sides. Among their key findings is that for such lattices, if a convex equilateral $n$-gon exists, then $n$ must be at least as large as every prime dividing $m$. They also proved that this is a sufficient condition when the largest prime factor of $m$ is less than 29 by a brute-force computer search.

In this paper, we extend the classification of equilateral polygons in rectangular lattices of the form $\Lambda(m)$, with $m \equiv 3 \pmod{4}$.

We will give an existence proof in \Cref{Section: Existence} followed by a explicit construction in \Cref{Section: Construction}.

For any $m \equiv 3 \pmod{4}$ does there exist $N$ such that for all $n \geq N$ there exists a \textbf{convex} equilateral polygon of size n?

\begin{open}
For any $m \equiv 3 \pmod{4}$ does there exist an equilateral polygon of size $n'$, where $n'$ is the largest prime factor of m?
\end{open}

For any $m \equiv 3 \pmod{4}$ does there exist a \textbf{convex} equilateral polygon of size $n'$, where $n'$ is the largest prime factor of m?

The post-theorem introduction indicates the proof strategy: it will present "an existence proof" (in \Cref{Section: Existence}) and then an "explicit construction" (in \Cref{Section: Construction}).

\label{thm: new theorem}
For any $m \equiv 3 \pmod{4}$ there exists $N$ such that for all $n \geq N$ there exists an equilateral polygon of size n.,

Question: Let \(\Lambda(m)=L[(1,0),(0,\sqrt m)]=\{(a,b\sqrt m):a,b\in \mathbb Z\}\) be the planar rectangular lattice associated to an integer \(m\equiv 3\pmod 4\). An equilateral polygon of size \(n\) means an \(n\)-gon whose vertices lie in \(\Lambda(m)\) and whose side lengths are all equal. Which statement holds for such lattices?

For every integer \(m\equiv 3\pmod 4\), there exists an integer \(N\) such that for every integer \(n\ge N\), the lattice \(\Lambda(m)\) contains an equilateral polygon of size \(n\).

Choices:
B. For every integer \(m\equiv 3\pmod 4\), there exists an integer \(N\) such that for every integer \(n\ge N\), the lattice \(\Lambda(m)\) contains a convex equilateral polygon of size \(n\).
C. For every integer \(m\equiv 3\pmod 4\), there exists an integer \(n\) such that the lattice \(\Lambda(m)\) contains an equilateral polygon of size \(n\).
D. There exists an integer \(N\) such that for every integer \(m\equiv 3\pmod 4\) and every integer \(n\ge N\), the lattice \(\Lambda(m)\) contains an equilateral polygon of size \(n\).
E. For every integer \(m\equiv 3\pmod 4\), there exists an integer \(N\) such that for every odd integer \(n\ge N\), the lattice \(\Lambda(m)\) contains an equilateral polygon of size \(n\), and one may take \(N\) to be the largest prime factor of \(m\).