\section{Introduction}
\label{sec_intro}
In the last few decades, research on layered magnetic superlattices has shown that they can have low density, transparency and mechanical strength, which find potential applications in magnetic recording, information storage and magneto-resistive sensors \cite{Barbic}. Amongst them, few-layered \textit{ferrimagnetic} materials are often found to have physical properties very different from the bulk. Though ferrimagnetism was discovered in 1948 \cite{Cullity}, the experimental interest in ferrimagnetism has grown up rapidly with the discovery of thin film growth techniques, like, metalorganic chemical vapour deposition (MOCVD) \cite{Stringfellow}, molecular-beam epitaxy (MBE) \cite{Herman}, pulsed laser deposition (PLD) \cite{Singh}, and atomic layer deposition (ALD) \cite{George}. Such experimental advancements have made the growth of bilayered \cite{Stier}, trilayered \cite{Leiner}, and multilayered \cite{Sankowski,Pradhan,Maitra} systems with desired characteristics a reality. Expectedly, theoretical and computational studies of layered magnets have also gained momentum. For a multilayered ferrimagnet, magnetizations of each of the monolayers may evolve differently with temperature. Combination of such different magnetic behaviours in specific cases, exhibit \textit{compensation}. The \textit{Compensation point} for layered magnets is that specific temperature, lower than the critical temperature, where the total magnetization of the system vanishes but individual layers remain magnetically ordered \cite{Cullity}.\\

The temperature dependence of total magnetization of layered magnets with antiferromagnetic interlayer coupling, exhibiting a ferrimagnetic ground state, may show magnetic compensation. Compensation is not related to the criticality of the system but the magnetic coercivity shows singularity at the compensation point \cite{Connell,Ostorero} for some ferrimagnetic materials. Strong temperature dependence of the coercive field around the compensation point and compensation point about the room temperature, make such ferrimagnets useful for thermomagnetic recording \cite{Connell}. At the compensation point, a small driving field can reverse the
sign of magnetization. That is why, the Magnetocaloric Effect in the vicinity of compensation temperature is studied in \cite{Ma}. So the control and manipulation of the compensation phenomenon becomes an important topic of research from the point of view of theoreticians and experimentalists. A few related examples, in this direction, follow. In \cite{Mathoniere}, it has been shown that, Molecular magnets, e.g., $AFe^{II} Fe^{III} (C_{2}O_{4})_{3} [A=N(n-C_{n}H_{2n+1})_{4},n=3-5]$ have compensation temperatures near $30$ K depending on the type of cation $A^{+}$ . This particular kind of system was simulated by Monte Carlo Simulation with a mixed spin model of spin-$2$ and spin-$5/2$ on a layered honeycomb structure with nearest neighbour interactions to clarify the effects of interlayer interactions and single-ion anisotropies on compensation \cite{Nakamura}. In the last decade, the ferrimagnetic trilayered structure,\\ $Fe_{3}O_{4} (25nm)/Mn_{3}O_{4} (50nm)/Fe_{3}O_{4} (25nm)$ , prepared by \textit{oxygen-plasma-assisted} MBE, was shown to have magnetic compensation due to the formation of domain-wall-like configurations,
mainly in $Fe_{3}O_{4}$ \cite{Lin}. \\

To examine compensation, numerical studies on the equilibrium (field-free and in the presence of static fields) properties of layered Ising ferrimagnets on various lattice geometries have been performed \cite{Oitmaa,Lv,Fadil,Diaz1,Diaz2,Chandra1,Chandra2,Chandra3}. The trilayered ferrimagnetic spin-1/2 Ising superlattices on square sublayers of ABA and AAB type [Figure \ref{fig_lattice_structure}] in the current study have an advantage as, in field-free cases they show compensation effect \cite{Diaz1,Diaz2,Chandra1,Chandra2,Chandra3}, even without site-dilution or mixed-spin structures. So, they are among the simplest systems to display compensation. The magnetic description provided by traditional Monte Carlo Simulation is in good agreement with the description provided by Inverse absolute of reduced residual magnetisation (IARRM) and Temperature interval between Critical and Compensation temperatures (TICCT) \cite{Chandra1,Chandra2,Chandra3} for both types of \textit{sandwiched} configurations with square and triangular monolayers. The equilibrium studies are now well established.\\

However, real systems cannot preserve the pristine character, and disorder is almost unavoidable in the description of any spin model. A few sources of time-dependent disorder for the type of systems studied in this work are \cite{Chandra4}: (a) time varying interaction strength between pairs of spins; (b) the number of interacting spins for a particular site may be changing with time; (c) Even the nature of the spins (magnitude of spin, different magnetic atoms etc.) in the ordered structure may vary with time. As a result, modelling such temporally variable compositional and morphological disorders is extremely difficult. In an attempt in this direction \cite{Chandra4}, the trilayered spin$-1/2$ superlattices has been subjected to a uniform random external magnetic field with spatio-temporal variations. So the dynamic Hamiltonian for these systems then is basically a Random Field Ising Model (RFIM). RFIM was developed by Larkin in 1970 \cite{Larkin} and is historically used to model many remarkable static and dynamic behaviours in disordered systems \cite{Belanger}. Prominent examples of experimentally observed Random field type phenomenology in disordered systems range from disorder-induced frustration and electronic transport in disordered insulators to melting of intercalates in layered compounds e.g. $TiS_{2}$, \cite{Efros,Childress,Maher,Pastor,Kirkpatrick,Fisher1,Fisher2,Suter} to name a few. The simulational results and subsequent analyses in \cite{Sethna} and references therein explain how RFIM may describe various types of noises in magnets.\\

For a large variety of random field distributions, the critical exponents of the power laws are independent of the particular choice \cite{Sethna}. But it is yet to be established how a change in the nature of the continuous random external field, from uniform random \cite{Chandra4} to Gaussian random, affects the compensation phenomenon in the superlattices of Figure \ref{fig_lattice_structure}. We know, uniform random field has a lower and upper cut-off whereas the Gaussian random field admits all the possible real values of the field. Evidently, only one physical measure i.e. the standard deviation of these continuous random field distributions is common to both of these and provides us with a description of randomness, irrespective of the nature of the distribution. Thus the objective of the current study is to examine the influence of the Gaussian random external magnetic field (or, more specifically, the standard deviation of the Gaussian distribution) with certain spatiotemporal variation on the compensation phenomenon associated with a trilayered spin-$1/2$ Ising ferrimagnet with square monolayers.
The plan of the paper follows. The layered magnetic model and the dynamic Hamiltonian are described, in detail, in Section \ref{sec_model}. The simulational details are described in Section \ref{sec_simulation}. Section \ref{sec_results} contains the numerical results and associated discussions. In Section \ref{sec_summary}, the summary of the work is presented.

\section{Outline of the Model}
\label{sec_model}

The ferrimagnetic Ising superlattice in this study is similar to the one used in \cite{Chandra4}. Each site has spin value, $s=1/2$, and contains three magnetic sub-layers on square lattice. Each alternate layer is exhaustively composed of by either A or B type of atoms. The magnetic atoms on the top and bottom layers do not interact. [Fig.-\ref{fig_lattice_structure}]. The magnetic interaction between the like atoms (A-A and B-B) is ferromagnetic and between dislike atoms (A-B) is antiferromagnetic. Additionally to the cooperative interactions, the $z$-component of spins, $S_{i}^{z}$ at each site couples with a Gaussian random external magnetic field, $h_{i}(t)$. At a particular site, this external field varies in time and at any time instant, the values of this field are different over the lattice sites.

The spins interact Ising-like, in-plane as well as inter-plane. The time dependent Hamiltonian for such a trilayered ferrimagnetic system is:
\begin{eqnarray}
\nonumber
& &H(t) = -
J_{11}\sum_{<t,{t}^{\prime }>}S_{t}^{z}S_{{t}^{\prime }}^{z} - J_{22}\sum_{<m,{m}^{\prime }>}S_{m}^{z}S_{{m}^{\prime }}^{z} \\\nonumber
& - &
J_{33} \sum_{<b,{b}^{\prime }>}S_{b}^{z}S_{{b}^{\prime }}^{z} -
J_{12} \sum_{<t,m>}S_{t}^{z}S_{m}^{z} -
J_{23} \sum_{<m,b>}S_{m}^{z}S_{b}^{z} \\
\label{eq_Hamiltonian}
&-& \sum_{i}h_{i}(t)S_{i}^{z}
\end{eqnarray}
$\langle t,{t}^{\prime }\rangle$, $\langle m,{m}^{\prime }\rangle$, $\langle b,{b}^{\prime }\rangle$ are nearest-neighbor pairs in the top, mid and bottom layers respectively and $\langle t,m\rangle$, $\langle m,b\rangle$ are, respectively, pairs of nearest-neighbor sites in, top \& mid and mid \& bottom layers. The first three terms are for the intra-planar ferromagnetic interactions. The fourth and fifth terms are for the inter-planar nearest neighbour interactions, between top and mid layers and mid and bottom layers, respectively. The sixth term denotes the \textit{spin-field interaction term} of all the spins to the external Gaussian random magnetic field, at time instant $t$. Because of the interactions: $J_{AA}>0$ , $J_{BB}>0$, and $J_{AB}<0$. For an ABA type system, $J_{11}=J_{33}=J_{AA}$; $J_{22}=J_{BB}$ and $J_{12}=J_{23}=J_{AB}$. For an AAB type system, $J_{11}=J_{22}=J_{12}=J_{AA}$; $J_{33}=J_{BB}$ and $J_{23}=J_{AB}$.

\begin{figure*}[!htb]
	\begin{center}
		\begin{tabular}{c}
			\resizebox{7.5cm}{!}{\includegraphics[angle=0]{1a_ABA_sq_trilayer.eps}}
			{\hskip 1.5cm}
			\resizebox{7.5cm}{!}{\includegraphics[angle=0]{1b_AAB_sq_trilayer.eps}}
		\end{tabular}
		\caption{ (Colour Online) Miniaturised versions ($3\times4\times4$) of (a) ABA and (b) AAB square trilayered ferrimagnet with two types of theoretical atoms, $A$ and $B$. Each of the sublattices of the ferrimagnetic systems are formed on square lattice. The actual simulation is carried out on a system with $N_{sites}=3\times100\times100$ . Courtesy: \cite{Chandra4}}
		\label{fig_lattice_structure}
	\end{center}
\end{figure*}

\section{Simulation Protocol}
\label{sec_simulation}
\subsection{Monte Carlo Simulation}
\label{subsec_MC Sim}
The Metropolis single spin-flip algorithm \cite{Landau, Binder} is employed for simulation of the system. The three square monolayers has $L^{2}$ sites with $L=100$. The $z$-components of spin projections of nearest neighbours, $S_{i}^{z}$ $(S_{i}^{z}=\pm1)$ contribute to the cooperative and spin-field interactions. At each site $i$, a \textit{local, time-varying, Gaussian random field} $h_{i}$ couples with each spin. In \cite{Diaz2}, Compensation temperature has been found to be practically constant for $L\geqslant60$, for the systems of this study. So the system sizes in this study is statistically reliable for simulational investigation. For both the configurations, the systems are initiated at a high temperature paramagnetic phase, with randomly selected half of the total spin projections being "UP" (with $S_{i}^{z}=+1$) and the rest being "DOWN" (with $S_{i}^{z}=-1$) (Using $1$ for $1/2$ fixes up the energy scale). At a fixed temperature $T$, the spin flipping is governed by the Metropolis rate \cite{Metropolis, Newman}, of Equation [\ref{eq_metropolis}]:
\begin{equation}
\label{eq_metropolis}
P(S_{i}^{z} \to -S_{i}^{z}) = \text{min} \{1, \exp (-\Delta E/k_{B}T)\}
\end{equation}
where the associated change in internal energy in flipping the $i$-th spin projection from $S_{i}^{z}$ to $-S_{i}^{z}$, is $\Delta E$ . Similar $3L^{2}$ individual, random single-spin updates constitute One Monte Carlo sweep (MCS) of the entire system. This \textit{one MCS} is the unit of time in this study. \textit{Periodic boundary conditions in-plane and Open boundary conditions along the vertical are employed.}

The systems are kept for $10^{5}$ MCS at every temperature step. The last configuration at the previous \textit{higher} temperature acts as the starting configuration at a new \textit{lowere} temperature. In a field-free environment, for the first $5\times10^{4}$ MCS the system is allowed to \textit{equilibrate} (the equilibration time is sufficient [Refer to Figure \ref{fig_field_mag_time} and discussions therein]). After that the external field is switched on and kept switched on for the next $5\times10^{4}$ MCS. So for the systems, the \textit{exposure time interval} in the field, $\delta$ is $5\times10^{4}$. The temperatures are measured in units of $J_{BB}/k_{B}$. For each of the fixed standard deviation of the Gaussian random field, the system is observed for seven equidistant values of $J_{AA}/J_{BB}$, from $0.04$ to $1.0$ with an interval of $0.16$. For each fixed value of $J_{AA}/J_{BB}$, $J_{AB}/J_{BB}$ is decreased from $-0.04$ to $-1.0$ with a step of $-0.16$.\\
\indent For any combination of $J_{AA}/J_{BB}$ and $J_{AB}/J_{BB}$, and a fixed sd of the external field $\sigma$, the time averages of the following quantities are calculated after equilibration at any temperature, $(T)$ in the following manner \cite{Robb,Chandra4}:\\
\textbf{(1) Sublattice magnetisations} are calculated at time instant say, $t$, by:
\begin{equation}
M_{q}(T,t)=\frac{1}{L^{2}}\sum_{x,y=1}^{L} \left( S_{q}^{z}(T,t)\right)_{xy}
\end{equation}
The time averaged sublattice magnetizations is calculated by:
\begin{equation}
\langle M_{q}(T)\rangle =\frac{1}{\delta}\int_{t_{0}}^{t_{0}+\delta} M_{q}(T,t)dt
\end{equation}
where $q$ is to be replaced by $t,m\text{ or }b$ for top, mid and bottom layers.\\
\textbf{(2) The order parameter}, $O_{T}$, for the trilayer at temperature, $T$ is defined as:
\begin{equation}
O(T)=\frac{1}{3}(\langle M_{t}(T)\rangle+\langle M_{m}(T)\rangle+\langle M_{b}(T)\rangle)
\end{equation}
\\
\textbf{(3) Fluctuation of the order parameter,} $\Delta O(T)$ at temperature, $T$ as follows:
\begin{equation}
{\Delta O}(T)=\sqrt{\dfrac{1}{\delta} \int_{t_{0}}^{t_{0}+\delta} \left[M(T,t)-O(T)\right]^{2}dt }
\end{equation}
where $M(T,t)$ is the total magnetisation of the whole system, at temperature, $T$, calculated at the (\textit{t-th time instant}). \\
\textbf{(4) The time averaged value of cooperative energy per site},  $\langle E(T) \rangle$, at temperature, $T$, is determined by:
\begin{eqnarray}
\nonumber
& \langle E (T)\rangle_{ABA} & = \dfrac{-1}{3L^{2}\delta}  \int_{t_{0}}^{t_{0}+\delta} dt [ J_{AA}(\sum_{<t,{t}^{\prime }>}S_{t}^{z}S_{{t}^{\prime }}^{z}
\\\nonumber
&+& \sum_{<b,{b}^{\prime }>}S_{b}^{z}S_{{b}^{\prime }}^{z}) + J_{BB} \sum_{<m,{m}^{\prime }>}S_{m}^{z}S_{{m}^{\prime }}^{z}
\\
&+& J_{AB} (\sum_{<t,m>}S_{t}^{z}S_{m}^{z}+\sum_{<m,b>}S_{m}^{z}S_{b}^{z})]
\end{eqnarray}
and
\begin{eqnarray}
\nonumber
& \langle E (T)\rangle_{AAB} & = \dfrac{-1}{3L^{2}\delta}  \int_{t_{0}}^{t_{0}+\delta} dt [ J_{AA}(\sum_{<t,{t}^{\prime }>}S_{t}^{z}S_{{t}^{\prime }}^{z}
\\\nonumber
&+& \sum_{<m,{m}^{\prime }>}S_{m}^{z}S_{{m}^{\prime }}^{z} \sum_{<t,m>}S_{t}^{z}S_{m}^{z} ) \\
&+& J_{BB} \sum_{<b,{b}^{\prime }>}S_{b}^{z}S_{{b}^{\prime }}^{z} + J_{AB} \sum_{<m,b>}S_{m}^{z}S_{b}^{z} ]
\end{eqnarray}
\textbf{(5) The fluctuation of the cooperative energy per site} at temperature, $T$, by:
\begin{equation}
{\Delta E}(T)=\sqrt{\dfrac{1}{\delta} \int_{t_{0}}^{t_{0}+\delta} \left[E(T,t)-\langle E (T)\rangle\right]^{2}dt }
\end{equation}
where $E(T,t)$ is the instantaneous cooperative energy per site, for the system at time $t$ and temperature, $T$, within the exposure interval, $\delta$.

At the pseudo-critical temperatures, the fluctuations peak. Around this temperature close range simulations were performed to narrow down the position of \textit{the reported critical temperatures} with an accuracy of, $\Delta T_{crit}=0.04$ . Compensation temperature ($<T_{crit}$), where the total magnetisation again becomes zero, is determined by linear interpolation from the two neighbouring points across the zero of magnetization in the plots of order parameter vs. temperature [e.g. Figure \ref{fig_mag_fr_afr}(a)]. The Jackknife method \cite{Newman} is used to provide an estimate of the errors with the magnetizations and fluctuations.
\subsection{Characteristics of the External field}
\label{subsec_char_field}
\indent The local, Gaussian random external magnetic field values $h_{i}(t)$ at any site, $i$ at time instant $t$, is drawn from the following probability distribution:
\begin{equation}
\label{eq_prob_dist_Gaussian}
P_{Gaussian}(h_{i}(t)) =
\frac{1}{\sqrt{2\pi \sigma^{2}}} \exp \left( \frac{-h_{i}^{2}(t)}{2\sigma^{2}}\right)
\end{equation}

with the standard deviation (sd) of the Gaussian distribution being, $\sigma$. To implement, Box-Muller algorithm \cite{Box} with the following formula is used to get a Gaussian distribution $G_{0}$ of sd, $\sigma$ and zero mean:

\begin{equation}
G_{0}=\sigma \sqrt{-2 ln(U_{1})} cos(2\pi U_{2})
\end{equation}

Here $U_{1}$ and $U_{2}$ are two uniform random distributions between $\left[0,1\right]$ .

\indent A few additional characteristics are also added to the external field distribution \cite{Chandra4}:
\begin{itemize}
	\item[(a)] At different lattice sites, the values of the external field are uncorrelated at any time instant. Again at a lattice site, the values of the external field are uncorrelated for different time instants. So :  $h_{m}(t)h_{n}(t^{\prime})=a(t)\text{ }\delta_{mn}\text{ }\delta (t-t^{\prime})$, where $m,n$ are two different lattice sites and $t,t^{\prime}$ are two different time instants.

	\item[(b)] The following conditions are also satisfied:
	\begin{itemize}
		\item[(i)] After the field is switched ON, the bulk average of the Gaussian field at a time instant $t$, is zero: $$\sum_{m} h_{m}(t) =0$$.\\
		So, $$ \sum_{m,n} h_{m}(t)h_{n}(t)\delta_{mn} =3L^{2}\sigma^{2}$$.

		\item[(ii)] At the $m$-th site, the temporal mean of the local field over the exposure interval, $\delta$, is \textit{zero}: $\langle h_{m}(t)\rangle=\dfrac{1}{\delta}\int_{t_{0}}^{t_{0}+\delta}  h_{m}(t)dt=0$ .

	\end{itemize}

\end{itemize}

\indent At a few randomly chosen time instants within the exposure interval, the implementation of the desired field distribution is checked by the Cumulative Distribution Function (CDF) \cite{Deisenroth}, the Kernel Density Estimate (KDE) \cite{Rosenblatt-Parzen} and the Histogram.

\section{Results and discussions}
\label{sec_results}
\subsection{Thermodynamic Response}
\label{subsec_response}
\subsubsection{Magnetization versus temperature:}
\label{subsubsec_magvtemp}

For a \textit{fixed standard deviation} of the external field with characteristics of Section \ref{subsec_char_field}, in the few selective cases in Figure \ref{fig_mag_fr_afr} we see \textit{the compensation and critical temperatures shift as we increase the magnitude of any of the coupling strengths}. As we increase the magnitude of either of the coupling strengths, we can identify the nature of the magnetization curves by the N\`{e}el classification scheme. A detailed discussion on the classification schemes (e.g. P-type, N-type, R-type etc.) can be found out \cite{Neel,Chikazumi,Strecka}. \textbf{For the ABA configuration}: \textbf{\textit{in Figure \ref{fig_mag_fr_afr} (a)}}  with $J_{AA}/J_{BB}=0.20$ and $\sigma=0.60$: for $J_{AB}/J_{BB}=-0.04$ we see a P-type magnetization; all the intermediate curves are of N-type and for $J_{AB}/J_{BB}=-1.00$ we see a R-type magnetization; and \textbf{\textit{in Figure \ref{fig_mag_fr_afr} (b)}} with $J_{AB}/J_{BB}=-0.20$ and $\sigma=0.60$: for $J_{AA}/J_{BB}=0.04$ we see a P-type magnetization; the intermediate curves up to are of N-type and for $J_{AB}/J_{BB}=1.00$ we see a Q-type magnetization. The L-type, within braces and not explicitly shown, would be encountered while moving from the former type to the latter. For the weakest combination of coupling strengths, we witness the field-driven vanishing of compensation. \textbf{For the AAB configuration}: \textbf{\textit{in Figure \ref{fig_mag_fr_afr} (c)}}  with $J_{AA}/J_{BB}=0.20$ and $\sigma=0.76$: for $J_{AB}/J_{BB}=-0.04$ we see a P-type magnetization and for $J_{AB}/J_{BB}=-0.20$ we see an L-type magnetization and all other curves are of N-type and \textbf{\textit{in Figure \ref{fig_mag_fr_afr} (d)}} with $J_{AB}/J_{BB}=-0.20$ and $\sigma=0.76$: for $J_{AA}/J_{BB}=0.04$ we see a P-type magnetization; for $J_{AA}/J_{BB}=0.20$ we see an L-type; the $J_{AA}/J_{BB}=0.36,0.52$ curves are of N-type; the $J_{AA}/J_{BB}=0.68,0.84$ curves are of Q-type and for $J_{AB}/J_{BB}=1.00$ we see a P-type magnetization again. For the two weakest combinations of coupling strengths, we again witness the field-driven vanishing of compensation in the AAB configuration. \textit{The magnetic response under the influence of Gaussian random magnetic field with spatio-temporal variation is quite similar to what we see for the spatio-temporally varying uniform random magnetic field in} \cite{Chandra4}.\\

\begin{figure*}[!htb]
	\begin{center}
		\begin{tabular}{c}

			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{2a_abasq_f020_mag_gau060.eps}}
			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{2b_abasq_af020_mag_gau060.eps}}\\

			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{2c_aabsq_f020_mag_gau076.eps}}
			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{2d_aabsq_af020_mag_gau076.eps}}

		\end{tabular}
		\caption{ (Colour Online) Plots of Order parameter versus reduced temperature for a $3\times 100 \times 100$ system of type: (a) ABA with $J_{AA}/J_{BB}=0.20$ and variable $J_{AB}/J_{BB}$ for $\sigma=0.60$; (b) ABA with $J_{AB}/J_{BB}=-0.20$ and variable $J_{AA}/J_{BB}$ for $\sigma=0.60$; (c) AAB with $J_{AA}/J_{BB}=0.20$ and variable $J_{AB}/J_{BB}$ for $\sigma=0.76$; (d) AAB with $J_{AB}/J_{BB}=-0.20$ and variable $J_{AA}/J_{BB}$ for $\sigma=0.76$ . Shift of Compensation and Critical temperatures towards higher temperature end are witnessed with increase in any of the coupling ratios. The field-driven vanishing of compensation is witnessed for the weakest combination of coupling strengths.}
		\label{fig_mag_fr_afr}
	\end{center}
\end{figure*}

Now we will focus on the effects of the randomness of the external Gaussian random field has on the magnetic response. For \textit{any fixed combination of the coupling strengths, an increase in the value of the standard deviation of the external Gaussian random field decreases the compensation and critical temperatures for both the ABA and AAB type of sandwiched structures} [Figures \ref{fig_mag_response_ABA} and \ref{fig_mag_response_AAB}]. Similar to the uniform random field \cite{Chandra4}, as we increase the randomness of the external Gaussian random field the decrement for the compensation temperatures is much more visible than the decrement of critical temperature, with or without compensation. The \textit{field driven absence of compensation phenomenon} is also present in Figure \ref{fig_mag_response_ABA}(a) for ABA and Figures \ref{fig_mag_response_AAB}(a)\&(b) for AAB configration. Like the uniform random external field, we can see and identify the nature of ferrimagnetic curves and field-driven transitions among them in Figures \ref{fig_mag_response_ABA} and \ref{fig_mag_response_AAB}. According to the classification schemes of references \cite{Neel,Chikazumi,Strecka}: \textbf{(A) For ABA}, in Figure \ref{fig_mag_response_ABA}(a), the magnetic response changes from type-$N$ $(\sigma ={0,0.20})$ to type-$L$ $(\sigma =0.40)$ to type-$P$ $(\sigma ={0.60,0.76,1.00})$; In Figure \ref{fig_mag_response_ABA}(b), for all the fields only type-$N$ response is witnessed; and in Figures \ref{fig_mag_response_ABA}(c)\&(d), we see only type-$Q$ response for all the fields. \textbf{(B) For AAB}, in Figure \ref{fig_mag_response_AAB}(a), the transition happens from type-$N$ $(\sigma={0.00,0.20})$ to type-$P$ $(\sigma={0.40,0.60,0.76,1.00})$ via type-$L$; Similar transitions are witnessed in Figure \ref{fig_mag_response_AAB}(b); in Figure \ref{fig_mag_response_AAB}(c) \& (d), all the magnetic responses are of type-$Q$.

\begin{figure*}[!htb]
	\begin{center}
		\begin{tabular}{c}

			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{3a_abasq_004_020_mag.eps}}
			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{3b_abasq_004_084_mag.eps}}\\

			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{3c_abasq_084_020_mag.eps}}
			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{3d_abasq_084_084_mag.eps}}\\

		\end{tabular}
		\caption{ (Colour Online) Magnetic response of the ABA trilayered ($3\times100\times100$) system. The shift of both, the compensation (where it is present) and critical temperatures towards the low temperature ends and shift of the magnetic behaviours between N,L,P,Q etc. type of ferrimagnetism, with increase in standard deviation of the uniform random external magnetic field, are clearly visible in all these plots. The type L within brackets is explicitly not seen in the plots but encountered in-transition.  Where, the errorbars are not visible, they are smaller than the area of the point-markers. }
		\label{fig_mag_response_ABA}
	\end{center}
\end{figure*}

\begin{figure*}[!htb]
	\begin{center}
		\begin{tabular}{c}

			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{4a_aabsq_004_020_mag.eps}}
			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{4b_aabsq_004_084_mag.eps}}\\

			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{4c_aabsq_084_020_mag.eps}}
			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{4d_aabsq_084_084_mag.eps}}
		\end{tabular}
		\caption{ (Colour Online) Magnetic response of the AAB trilayered ($3\times100\times100$) system. The shift of both, the compensation (where it is present) and critical temperatures towards the low temperature ends and shift of the magnetic behaviours between N,L,P,Q etc. type of ferrimagnetism, with increase in standard deviation of the uniform random external magnetic field, are clearly visible in all these plots. The type L within brackets is explicitly not seen in the plots but encountered in-transition. Where, the errorbars are not visible, they are smaller than the area of the point-markers.}
		\label{fig_mag_response_AAB}
	\end{center}
\end{figure*}

\subsubsection{Fluctuations versus temperature:}
\label{subsubsec_flucvtemp}
To better understand \textit{(a) the shifts of compensation and critical temperatures and (b) the reason behind the field driven vanishing of compensation}, we will now observe both the fluctuations: fluctuation of the \textit{order parameter} and fluctuation of the \textit{cooperative energy per site}, as functions of temperature while sd of the external Gaussian random field acts as the parameter. We witness a plateau with a smeared peak in the vicinity of compensation point for both the fluctuations of order parameter and energy in Figures \ref{fig_fluc_mageng_ABA} and \ref{fig_fluc_mageng_AAB} for \textit{the ABA and AAB configurations}. We clearly see the compensation and critical temperatures moving towards lower temperature values, with the increase in the standard deviation(sd) of the Gaussian field. Even the smeared peaks at the low temperature segments flatten out as the sd is increased in steps, which signifies the vanishing of compensation. So we again witness a field-driven vanishing of compensation. At the lower parts of the temperature axis, the increase in both the fluctuations imply that magnetic ordering is gradually decreasing with the increase of the sd of the external field.

\begin{figure*}[!htb]
	\begin{center}
		\begin{tabular}{c}

			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{5a_abasq_004_020_flucmag.eps}}

			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{5b_abasq_004_020_fluceng.eps}}\\

			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{5c_abasq_004_084_flucmag.eps}}

			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{5d_abasq_004_084_fluceng.eps}}\\

		\end{tabular}
		\caption{ (Colour Online) Temperature dependence of Fluctuation of order parameter, $\Delta O (T)$ and Fluctuation of cooperative energy per site, $\Delta E (T)$, for ABA type $(3\times100\times100)$ configuration in (a)-(d) with ${J_{AA}}/{J_{BB}}=0.04$ and ${J_{AB}}/{J_{BB}}=-0.20$ and with ${J_{AA}}/{J_{BB}}=0.04$ and ${J_{AB}}/{J_{BB}}=-0.84$. Where, the errorbars are not visible, they are smaller than the dimension of the point-markers. The nature of the curves prominently shows the shift of critical temperatures and even reason for absence of compensation can be understood from the low temperature segment of the curves.}
		\label{fig_fluc_mageng_ABA}
	\end{center}
\end{figure*}

\begin{figure*}[!htb]
	\begin{center}
		\begin{tabular}{c}

			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{6a_aabsq_004_020_flucmag.eps}}

			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{6b_aabsq_004_020_fluceng.eps}}\\

			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{6c_aabsq_004_084_flucmag.eps}}

			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{6d_aabsq_004_084_fluceng.eps}}

		\end{tabular}
		\caption{ (Colour Online) Temperature dependence of Fluctuation of order parameter, $\Delta O (T)$ and Fluctuation of cooperative energy per site, $\Delta E (T)$, for AAB type $(3\times100\times100)$ configuration in (a)-(d) with ${J_{AA}}/{J_{BB}}=0.04$ and ${J_{AB}}/{J_{BB}}=-0.20$ and with ${J_{AA}}/{J_{BB}}=0.04$ and ${J_{AB}}/{J_{BB}}=-0.84$. Where, the errorbars are not visible, they are smaller than the dimension of the point-markers. The nature of the curves prominently shows the shift of critical temperatures and even reason for absence of compensation can be understood from the low temperature segment of the curves.}
		\label{fig_fluc_mageng_AAB}
	\end{center}
\end{figure*}

\subsubsection{Magnetization versus time:}
\label{subsubsec_magvtime}
\indent The section is devoted to the behaviour of sublattice magnetisations with time as the field is switched ON, at a suitably very low temperature $T=0.01$. We have chosen three distinct combinations of the coupling ratios, $J_{AA}/J_{BB}$ and $J_{AB}/J_{BB}$, as (0.04,-0.20), (0.20,-0.20), (1.00,-0.52) for the two values of the standard deviation, $\sigma=$\{0.20,0.60\} of the external Gaussian random field for both the ABA and AAB configurations [Figures \ref{fig_ABA_field_mag_time} and \ref{fig_AAB_field_mag_time}]. At $T=0.01$, till $t= 5\times 10^{4}$ MCS, the Hamiltonian only has the associative part and the sublattice and total magnetisation of the system remains in equilibrium. Just after $t=t_{0}=5 \times 10^{4}$ MCS, the external field is switched ON and the sublayer and total magnetisations \textit{start to react}. As in the case with uniform random field \cite{Chandra4}, both the systems, ABA and AAB, reach the steady state very quickly. Conclusive features are unraveled in these cases.

\begin{figure*}[!htb]
	\begin{center}
		\begin{tabular}{c}

			\resizebox{9.5cm}{!}{\includegraphics[angle=0]{7aa_abasq_004_020_h020_magtime.eps}}
			\resizebox{9.5cm}{!}{\includegraphics[angle=0]{7ab_abasq_004_020_h060_magtime.eps}}\\

			\large {\textbf{(A) Top panel: $J_{AA}/J_{BB}=0.04$ and $J_{AB}/J_{BB}=-0.20$}}\\

			\resizebox{9.5cm}{!}{\includegraphics[angle=0]{7ba_abasq_020_020_h020_magtime.eps}}
			\resizebox{9.5cm}{!}{\includegraphics[angle=0]{7bb_abasq_020_020_h060_magtime.eps}}\\

			\large {\textbf{(B) Middle panel: $J_{AA}/J_{BB}=0.20$ and $J_{AB}/J_{BB}=-0.20$}}\\

			\resizebox{9.5cm}{!}{\includegraphics[angle=0]{7ca_abasq_100_052_h020_magtime.eps}}
			\resizebox{9.5cm}{!}{\includegraphics[angle=0]{7cb_abasq_100_052_h060_magtime.eps}}\\

			\large {\textbf{(C) Bottom panel: $J_{AA}/J_{BB}=1.00$ and $J_{AB}/J_{BB}=-0.52$}}\\
		\end{tabular}
		\caption{ (Colour Online) Plots of Magnetisations for square monolayers (sublattices) and total magnetisation of the bulk versus time in MCS, \textbf{for the \textit{ABA} configuration}, where $M_{t}(t)$: Magnetization of the top layer; $M_{m}(t)$: Magnetization of the mid layer; $M_{b}(t)$: Magnetization of the bottom layer are all functions of time,$t$, in units of MCS. The magnetisation curves for the surface layers (orange and green) of ABA configuration overlap for the most of the times.}
		\label{fig_ABA_field_mag_time}
	\end{center}
\end{figure*}

\textbf{For the \textit{ABA configuration}}, we see \textbf{in the top panel: Figure \ref{fig_ABA_field_mag_time}(A)}, with $J_{AA}/J_{BB}=0.04$ and $J_{AB}/J_{BB}=-0.20$, both the surface A-layers react magnetically for both the sds of the external Gaussian field. The reason is, the per site associative energy of the A-layers is comparable to the spin-field energy. But the mid B-layer with dominant in-plane coupling, remains in its equilibrium magnetic state. So it is evident that the reduction (or, destruction) of magnetic order in the sublayers is the result of the competition between the relevant energy scales: associative and spin-field. The similar inference is valid for \textbf{the middle panel: Figure \ref{fig_ABA_field_mag_time} (B)}, with the $\sigma=0.60$, where per site spin-field energies are comparable to the associative part of the Hamiltonian for the surface A-layers. In \textbf{the bottom panel: Figure \ref{fig_ABA_field_mag_time} (C)}, with the $\sigma=0.20$ and $\sigma=0.60$, all the three sublayers don't react. Here, the associative part of the Hamiltonian for the surface A-layers dominates the steady state per site spin-field energies. The magnetization curves for the top and bottom layers overlap for most of the times as they have identical interacting magnetic neighborhood. In \textbf{Figure \ref{fig_ABA_field_mag_time}(A)} with $J_{AA}/J_{BB}=0.04$, $J_{AB}/J_{BB}=-0.20$ and $\sigma=0.60$ , we see the \textit{detectable} randomisation of the spins (consequently, reduction in magnetization) of the surface layers in presence of the external field leads to the vanishing of compensation even in the lowest possible temperature in simulation. Even in the lowest simulational temperature, the cumulative steady state value of magnetization of the surface A-layers becomes smaller than even the steady state value of magnetization of the middle B-layer (that is why the total magnetisation remains positive, same signature as the B-layer, in the presence of the field).  Even in the lowest temperature, $$|\langle M_{t}(T) + M_{b}(T) \rangle| < |\langle M_{m}(T)\rangle|$$ which causes compensation to disappear in the presence of the field. This is the reason behind all the instances with field-driven vanishing of compensation.

\begin{figure*}[!htb]
	\begin{center}
		\begin{tabular}{c}

			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{8aa_aabsq_004_020_h020_magtime.eps}}
			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{8ab_aabsq_004_020_h060_magtime.eps}}\\

			\large {\textbf{(A) Top panel: $J_{AA}/J_{BB}=0.04$ and $J_{AB}/J_{BB}=-0.20$}}\\

			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{8ba_aabsq_020_020_h020_magtime.eps}}
			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{8bb_aabsq_020_020_h060_magtime.eps}}\\

			\large {\textbf{(B) Middle panel: $J_{AA}/J_{BB}=0.20$ and $J_{AB}/J_{BB}=-0.20$}}\\

			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{8ca_aabsq_100_052_h020_magtime.eps}}
			\resizebox{8.5cm}{!}{\includegraphics[angle=0]{8cb_aabsq_100_052_h060_magtime.eps}}\\

			\large {\textbf{(C) Bottom panel: $J_{AA}/J_{BB}=1.00$ and $J_{AB}/J_{BB}=-0.52$}}\\
		\end{tabular}
		\caption{ (Colour Online) Plots of Magnetisations for square monolayers (sublattices) and total magnetisation of the bulk versus time in MCS, \textbf{for the \textit{AAB} configuration}, where $M_{t}(t)$: Magnetization of the top layer; $M_{m}(t)$: Magnetization of the mid layer; $M_{b}(t)$: Magnetization of the bottom layer are all functions of time,$t$, in units of MCS.}
		\label{fig_AAB_field_mag_time}
	\end{center}
\end{figure*}

\textbf{For the \textit{AAB configuration}}, the influence of the bottom B-layer is limited to the middle A-layer (because of the nearest neighbour Ising interactions). So the top A-layer gets much more affected by the external field than the middle A-layer. In Figure \ref{fig_AAB_field_mag_time}, we can understand it by simply noticing the orange line for the magnetisation of the top A-layer. \textbf{In the top panel: Figure \ref{fig_AAB_field_mag_time}(A)}, with  $J_{AA}/J_{BB}=0.04$, $J_{AB}/J_{BB}=-0.20$ and $\sigma={0.60,1.00}$, the spin-field energies per site dominates the cooperative energy per site of the A-layers. Consequuently, the A layers lose much of the magnetic ordering at even the lowest temperature, the extent of randomisation is much more prominent for the top A-layer and \textit{the top A-layer is almost completely randomised when $\sigma=0.60$}. So the combined magnetisation of A-layers is \textit{not enough to null the magnetisation of the bottom B-layer}, leading to vanishing of compensation. \textbf{In Figures \ref{fig_AAB_field_mag_time}(B) and (C)}, we can explain the behaviour in light of the discussions above. Such behaviour is qualitatively similar to what we have seen with uniform random field \cite{Chandra4}.

This is another example of \textit{dynamic field-driven vanishing of compensation} in the Ising spin-1/2 trilayers, driven by Gaussian random external field with spatiotemporal variation. The bottom panel with $J_{AA}/J_{BB}=1.00$ and $J_{AB}/J_{BB}=-0.52$ for both the configurations supports that vanishing of compensation is a result of the competition between the associative and spin-field energies.

\subsection{Phase Diagram and Scaling}

\indent In the Figure \ref{fig_phasecurve_2d}, a few examples of phase diagrams, under the influence of the Gaussian random external field, in the Hamiltonian parameter space, for both the configurations are shown. The features revealed here are qualitatively similar to the ones in \cite{Chandra4}. For a given standard deviation of the field, compensation temperature merges with the critical temperature for higher values of $J_{AA}/J_{BB}$ when $|J_{AB}/J_{BB}|$ is fixed or vice-versa, just like in the zero field case. In the diagrams, compensation is \textit{present} (marked by P) within the orange areas and absence of compensation is marked by white areas (marked by A). The phase diagram is drawn following the procedures of zero-field cases \cite{Diaz1,Diaz2,Chandra3}.
The extra feature is the enclave or islands with no-compensation within the phase area where parameters support compensation, like the one witnessed in the case with uniform random external field. In Figure \ref{fig_phasecurve_2d}, the areas coloured in orange are where compensation is \textit{present} (marked by P) and the area(s) in white are the regions where compensation is absent (marked by A). The qualitative features of the phase separation curves are same like that in the field-free cases. One of these curves divide the entire area of the parameter space into roughly two halves: one contains ferrimagnetic phases with compensation and the others contain ferrimagnetic phases without compensation. But the presence of the external Gaussian field [From $\sigma$=0.2, onwards], makes compensation disappear for a certain range of values of the coupling strengths [Refer to Figure \ref{fig_phasecurve_2d}]. In the phase diagrams, the range of values of relative coupling strengths for which compensation is absent are confined in areas within the region where compensation is present. These closed areas may be called No-Compensation Islands (NCI) after Ref. \cite{Chandra4}. Similar to the case with uniform random magnetic field, the magnitude of areas of such an island also grows as the randomness of the external field increases.

\begin{figure*}[!htb]
	\begin{center}
		\begin{tabular}{c}
			\textbf{(a)}
			\resizebox{8cm}{!}{\includegraphics[angle=0]{9a_aba_phasecurve_h040.eps}}
			\hfill
			\textbf{(b)}
			\resizebox{8cm}{!}{\includegraphics[angle=0]{9b_aba_phasecurve_h100.eps}}\\

			\textbf{(c)}
			\resizebox{8cm}{!}{\includegraphics[angle=0]{9c_aab_phasecurve_h040.eps}}
			\hfill
			\textbf{(d)}
			\resizebox{8cm}{!}{\includegraphics[angle=0]{9d_aab_phasecurve_h100.eps}}

		\end{tabular}
		\caption{ (Colour Online) Phase diagram for the: ABA trilayered ferrimagnetic system when: (a) $\sigma=0.40$; (b) $\sigma=1.00$ and AAB trilayered ferrimagnetic system when: (c) $\sigma=0.40$; (d) $\sigma=1.00$, in presence of the uniform random external magnetic field. A: Compensation is absent; P: Compensation is present. With increase in the standard deviation of the external field, the magnitude of the area of the no-compensation island have grown. The blue segment of the phase separation curves are obtained via linear extrapolation. All these plots are obtained for a system of $3\times100\times100$ sites. Where the errorbars are not visible, they are smaller than the point markers.}
		\label{fig_phasecurve_2d}
	\end{center}
\end{figure*}

In Figure \ref{fig_area_slope_field}, we see the plots of absolute area and rate of increase of absolute area versus the sd of the applied field. With reference to Figure \ref{fig_mag_fr_afr}, to get a closed curve for the boundary of the NCI, linear interpolation/extrapolation is employed between two adjacent points/at the boundary (coloured in blue). Next, the fractional area is obtained by Monte Carlo Integration \cite{Krauth}. Central difference formula is employed to find out the rate of increase of the area of NCIs to determine the nature of curve of\textit{ the area of NCI vs. sd of the external field}.

\begin{figure*}[!htb]
	\begin{center}
		\begin{tabular}{c}
			\textbf{(a)}
			\resizebox{8.5cm}{!}{\includegraphics[angle=-90]{10a_aba_field_area_rate.eps}}
			\hfill
			\textbf{(b)}
			\resizebox{8.5cm}{!}{\includegraphics[angle=-90]{10b_aab_field_area_rate.eps}}

		\end{tabular}
		\caption{ (Colour Online) Plots of: Magnitude of the area of the no-compensation islands versus standard deviation of the field (in RED) and the rate of increase in the magnitude of the area of the no-compensation islands versus standard deviation of the field (in BLUE) for (a) ABA and (b) AAB configurations for a system of $3\times100\times100$ sites.}
		\label{fig_area_slope_field}
	\end{center}
\end{figure*}

To analyse the nature of the curves of absolute area versus the sd, we note, the curve (in RED), comes out to be a mixture of \textit{superlinear and sublinear} in nature, \textbf{for the ABA configuration}. From the BLUE curve of slope versus sd of the field in Figure \ref{fig_area_slope_field}(a), in the vicinity of $\sigma=0.60$  we find the curve follows a sublinear (almost linear) response. Again at and after $\sigma=0.88$, the area changes in a prominent sublinear manner. At the low random fields around $\sigma=0.20$ and around moderately high randomness around $\sigma=0.70$, the response is superlinear. \textbf{For the AAB configuration}, the area of NCIs increase superlinearly on average till $\sigma=0.64$. After that the response becomes almost linear. Now the scaling between the magnitude of the area of NCIs and the sd of the field can be performed by the following scaling function \cite{Chandra4}:
\begin{equation}
\label{eq_scaling}
	f(A(\sigma), \sigma)= \sigma^{-b}A(\sigma)
\end{equation}
The scaling exponents come out to be: for ABA: $b_{ABA}=1.913\pm0.137$ and for AAB: $b_{AAB}=1.625\pm0.066$. A faithful estimate of error in the values of the exponents is obtained by the standard deviation among all the sets of data.
\section{Summary and Conclusion}
\label{sec_summary}

In the Ising model, the coupling constants are traditionally taken to be translationally invariant. Along with that, competing ferromagnetic and antiferromagnetic interactions in the systems of Figure \ref{fig_lattice_structure} throw up exciting bulk behaviour e.g. Compensation. The equilibrium studies on these systems \cite{Diaz1,Diaz2,Chandra1,Chandra2} have shown us the prevalent complexity in deriving conditions for the existence of compensation. Now, in the current article, a Metropolis Monte Carlo study has been performed on the magnetic and thermodynamic responses of the systems of Figure \ref{fig_lattice_structure} under the influence of a Gaussian Random external field with spatio-temporal variations.

It is time to discuss the implications of the current work. From \cite{Chandra4}, we have a fair idea about how such systems react under the influence of uniform random magnetic field. From Section \ref{subsubsec_magvtemp} we find that the magnetic response is qualitatively similar to the reponses under the uniform random field. The compensation and critical temperatures shift towards the low temperature ends and even results in the vanishing of compensation in proper cases as we increase the sd of the external Gaussian random magnetic field. Similar inference can be drawn from the thermodynamic behaviour of the suitably defined fluctuations of magnetization and associative energy [Refer to Section \ref{subsubsec_flucvtemp}]. The effect of the time-dependent part of the Hamiltonian is established in Section \ref{subsubsec_magvtime}. We observe that the systems react very quickly after switching ON the external field and the dynamics is governed by the competition between spin-field energies and associative energies. \textit{Thus the Gaussian random field-driven vanishing of compensation, observed in this work, is also a dynamic phenomenon} like it was in \cite{Chandra4} with a uniform random external field. The phase diagrams in Figure \ref{fig_phasecurve_2d}, for both the ABA and AAB configurations, have similar kind of appearance with No-Compensation Islands engraved within the phase area with compensation. As we investigate the plot of the magnitude of the area of NO-Compensation Islands versus the sd of the external field and find out the scaling exponent according to the Equation \ref{eq_scaling}, we find the responses are qualitatively similar for both the continuous field distributions: Uniform and Gaussian. A quick comparison follows: $b_{ABA}^{Uniform} = 1.958 \pm 0.122$ and
$b_{ABA}^{Gaussian} = 1.913 \pm 0.137$ (\textbf{For ABA}) and $b_{AAB}^{Uniform} = 1.783 \pm 0.118$ and
$b_{AAB}^{Gaussian} = 1.625 \pm 0.066$ (\textbf{For AAB}), where data for the uniform random field are taken from \cite{Chandra4}. So, there exists a very good agreement for the scaling exponents as they overlap within the statistical interval of one-another. So the dynamic response of the trilayered Ising spin-1/2 square ferrimagnnetic systems, in this article, are quite similar under the influence of Gaussian random external magnetic field with spatiotemporal variations listed in Section \ref{subsec_char_field} to the Uniform random external field \cite{Chandra4}. The exact nature of the external \textit{continuous} field distribution, for these two types of distributions, does not show a distinguished effect on the qualitative and quantitative features of such systems. But the results also pose a difficulty for technological applications. We know, it is difficult to create a source of purely static magnetic field, as some kind of ripple may exist. That ripple or time dependent part, following uniform or Gaussian distribution with characteristics described here or in \cite{Chandra4}, may shift the compensation and critical temperatures from designated values.

Still we have unanswered questions. For example, how would the system behave under the influence of an external magnetic field following a Simpson distribution \cite{Wentzel,Alder}, with spatio-temporal variation of similar kind of the present work? The results would definitely help us comment strongly on the behaviour of the scaling exponent, $b$, under a wide variety of continuous field distributions for both the ABA and AAB type of trilayered stackings. The responses under the discrete distributions are also yet to be reported. these are planned for the future. In real magnetic systems, impurities, compositional disorder, lattice dislocations etc. modify the Hamiltonian to a translationally non-invariant kind. Such a complexity, with Ising mechanics, may be described by a dynamic Hamiltonian such as Equation \ref{eq_Hamiltonian} with a time dependent part, where \textit{the external field} is characterized by a probability distribution.
\section*{Acknowledgements}
The author acknowledges the financial assistance from the University Grants Commission, India in the form of Research Fellowship and extends his thanks to Dr. Tamaghna Maitra for feedback and technical assistance. Useful comments and suggestions provided by the anonymous referees are also gratefully acknowledged.
\newpage

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Robb D. T., Rikvold P. A., Berger A., and Novotny M. A., Phys. Rev. E \textbf{ 76}, 021124 (2007).

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Deisenroth M. P., Aldo Faisal A., and Ong C. S., Mathematics for Machine Learning (Cambridge University Press, New York, 2020).

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See, e.g., Rosenblatt M., Ann. Math. Statist. 27, 832 (1956); Parzen E., Ann. Math. Statist. 33, 1065 (1962).

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N\'{e}el M. L., Ann. de Phys. \textbf{ 12}, 137 (1948).

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\end{enumerate}

\end{document}


\section{Problem formulation}
Consider a waveguide composed of two one-dimensional layers weakly coupled with each other.
The subsystems (the layers) are described by unknown functions
$u_1(t, x)$ and $u_2(t, x)$ (here $t$ is time, and $x$ is longitudinal coordinate).
Assume that the waveguide is described by a {\em matrix Klein-Gordon equation\/}
of dimension~2
(another naming for this equation is WFEM or WaveFEM):
\begin{equation}
\label{MKGE}
\left[ \begin{pmatrix} c_1^2 \quad 0 \\ 0 \quad c_2^2 \end{pmatrix} \partial_x^2 + \begin{pmatrix} -\Omega_1^2 \qquad \mu \\ \mu \qquad -\Omega_2^2
\end{pmatrix} -
\begin{pmatrix} 1 \quad 0 \\ 0 \quad 1 \end{pmatrix} \partial_t^2 \right] \begin{pmatrix} u_1(t,x) \\ u_2(t,x) \end{pmatrix} =
\begin{pmatrix} f_1 \\ f_2 \end{pmatrix} \delta(t) \delta(x)
\end{equation}
$\begin{pmatrix} f_1 , f_2 \end{pmatrix}^T$ is a constant vector describing the excitation. The
excitation is a short pulse localized at~$x = 0$. Indeed, we are looking for a causal
solution equal to zero for~$t < 0$.
For simplicity, we take
\[
\left( \begin{array}{cc} f_1  \\ f_2 \end{array} \right) =
\left( \begin{array}{cc} 1  \\ 0 \end{array} \right).
\]

Parameter
$\mu$ is a small value describing a coupling between the subsystems.
A system with $\mu=0$ is {\em unperturbed}. Matrix coefficients
of the unperturbed system are diagonal,
and the waves in the subsystems
propagate
independently of each other.
Each subsystem is described by a scalar Klein--Gordon equation
\[
( c_j^2 \partial_x^2 - \partial_t^2 - \Omega_j^2) u_j (t, x) = f_j \, \delta(t) \, \delta(x).
\]
The parameters $\Omega_j$ are
cut-off frequencies of the unperturbed subsystems.

Our aim is to obtain
a formal solution of \eqref{MKGE} (see Section~\ref{FSsection})
and to get
asymptotic estimations of the wave field for different values of $t$ and~$x$.
A classical way to do this is to apply a
{\em  saddle point method\/} (see Section~\ref{SPsection}). This method allows one to describe the wave field for $x \rightarrow \infty$ and $t \rightarrow \infty$. However, for values of $x$ and $t$ that are moderately large,
one needs some more
sophisticated asymptotics. Section~\ref{ZDsection} is devoted to them.

\section{Formal solution of the matrix Klein--Gordon equation}
\label{FSsection}
\subsection{Double integral solution}

Apply Fourier transform with respect to $x$ and Laplace transform with respect to $t$ to
\eqref{MKGE}. As a result, obtain a double integral representation of the waveguide field:
\begin{equation}
\label{Double_int_repr}
\begin{pmatrix} u_1(t,x)\\u_2(t,x)\end{pmatrix} =
\int\limits_{-\infty + i\varepsilon}^{\infty + i \varepsilon}\int\limits_{-\infty}^{\infty}
\frac{{\rm A}(\omega,k)}{D(\omega,k)} e^{-i\omega t + i k x}  dk \, d\omega  ,
\end{equation}
where
\begin{equation}
\label{A_det}
{\rm A}(\omega,k)= \begin{pmatrix} -k^2 c_2^2-\Omega_2^2 + \omega^2 \\ -\mu \end{pmatrix},
\end{equation}
\begin{equation}
\label{D_det}
D(\omega,k) = (-k^2c_1^2 + \omega^2 -\Omega_1^2)(-k^2c_2^2 + \omega^2 -\Omega_2^2)-\mu^2 ,
\end{equation}
$\varepsilon$ is an arbitrary  positive real number (it provides causality of the solution).
We assume that $\epsilon$ is small.

\subsection{Series--integral solution}

Let be $x > 0$.
\eqref{Double_int_repr} can be transformed into a sum of single integrals.
For that purpose, one has to close the contour of integration with respect to $k$ in the upper half plane of $k$ (it is possible due to Jordan's lemma) and apply the residue method. As a result, obtain the following representation of the solution
represented by \eqref{Double_int_repr}:
\begin{equation}
\label{Ser_int_repr}
\begin{pmatrix} u_1(t,x)\\u_2(t,x)\end{pmatrix} = 2\pi i
\sum _{m=1}^{2}\int\limits_{-\infty + i\varepsilon}^{\infty + i \varepsilon}
\frac{{\rm A}(\omega,k_m(\omega))}{
\partial_k D(\omega,k_m(\omega))} e^{i k_m(\omega) x - i\omega t } d\omega  ,
\end{equation}
where $\partial_k D$ is a partial derivative of function $D$ with respect to the second argument.
Functions $k_m(\omega)$, $m = 1,2$ are two of four roots of the {\em dispersion equation}
\begin{equation}
\label{DE}
D(\omega,k)=0.
\end{equation}
solved for a fixed $\omega$.
Among four roots of this equation, we select two roots with ${\rm Im}[k_m (\omega)] > 0$
as follows.
Four roots of \eqref{DE} can be split into pairs: $\pm k_m(\omega)$, $m = 1,2$.
We denote
by $k_1(\omega)$ and $k_2(\omega)$ the solutions having positive group velocities
\[
v_{\rm gr}(\omega) \equiv \left( \partial_{\omega} k \right) ^{-1}.
\]
Solutions $k_1(\omega)$ and $k_2(\omega)$ have ${\rm Im}[k]>0$ for ${\rm Im}[\omega] = \varepsilon > 0$. This can be shown by the Taylor's series:
\begin{equation}
\label{eq0}
{\rm Im} [k(\omega' + i\varepsilon)] \approx {\rm Im}[k(\omega') + \frac{dk(\omega')}{d\omega} \cdot i\varepsilon] = \varepsilon \frac{dk(\omega')}{d\omega}, \quad \omega' = {\rm Re}[\omega].
\end{equation}

\subsection{Dispersion diagram}

A {\em dispersion diagram\/} is a graphical representation of the roots of \eqref{DE}.
We will use a real dispersion diagram, which is the set of all real points $(\omega,k)$
obeying \eqref{DE}, and a complex dispersion diagram, which is the set of complex
$(\omega,k)$ obeying \eqref{DE}.

In the real dispersion diagram,
we assign indices 1 or 2 to the branches,
depending on their asymptotics as $\omega \rightarrow \infty$.
Namely, we assume that  $k_1(\omega) \rightarrow \omega/c_1$, $k_2(\omega) \rightarrow \omega/c_2$.

\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.8]{DD_Sh.pdf}
\includegraphics[scale=0.8]{vgr_full2.pdf}
\end{center}
	\caption{
    Real dispersion diagram (left) and group velocities (right)
    for the system described by \eqref{MKGE}}
\label{DD}
\end{figure}

For a working example throughout this paper,
let us choose some values of the waveguide parameters in such a way that an {\em exchange pulse} is observed \cite{Shanin_2018}. In our case $c_1 > c_2$, $\Omega_1 <\Omega_2$. Dispersion diagram of such a system looks like in \figurename~\ref{DD}~left, black.

Dispersion diagram of the corresponding unperturbed system is
presented by
two intersecting red lines in \figurename~\ref{DD}~left. The point of intersection
$(\omega_{\rm sh},k_{\rm sh})$ is a referred to as a
Shestopalov's singular point \cite{Shestopalov1996}:
\begin{equation}
\label{Shest_Point}
\omega_{\rm sh} = \sqrt{(c_1^2 \Omega_2^2 - c_2^2 \Omega_1^2)/(c_1^2 - c_2^2)},
\qquad
k_{\rm sh} = \sqrt{(\Omega_2^2 - \Omega_1^2)/(c_1^2 - c_2^2)}.
\end{equation}
One can see that the perturbed real dispersion diagram displays an {\em avoiding crossing\/}
near the Shestopalov's point.

In \figurename~\ref{DD}~right,
the group velocities  for the system with $\mu \neq 0$ (black) and with $\mu = 0$ (red)
are displayed.
Let
$v_1'$ and $v_2'$ be a local maximum and a local minimum
of the group velocity. The corresponding frequencies are $\omega_1^e$ and $\omega_2^e$, respectively. Obviously, these points are inflection points of the real dispersion diagram.

Let $v_1$ and $v_2$ be
the values of the group velocity of the unperturbed system taken at $\omega_{\rm sh}$.
They can be easily computed:
\begin{equation}
\label{Vgr_Shest_form}
v_1 = c_1 \sqrt{\omega_{\rm sh}^2 - \Omega_{1}^2} \, / \omega_{\rm sh}, \qquad
v_2 = c_2 \sqrt{\omega_{\rm sh}^2 - \Omega_{2}^2} \, / \omega_{\rm sh}.
\end{equation}
For the systems, which we chose, $v_1 > v_2$. Obviously, $v_1 < c_1$ and $v_2 < c_2$. It can be $v_1 > c_2$ or $v_1 < c_2$ or even $v_1 = c_2$. We take $v_1 < c_2$ for definiteness.

\begin{figure}[!h]
\centering
\includegraphics[scale = 0.9]{surface_AV-eps-converted-to}
\caption{Scheme of the Riemann surface of $k(\omega)$.
Blue lines are cuts. Equal numbers denote the sides that should be attached to each other. Integration contours $\gamma_1$ and $\gamma_2$ are the red and green lines, respectively}
\label{RS}
\end{figure}

Instead of four functions $\pm k_1(\omega)$, $\pm k_2(\omega)$, we consider a multivalued function $k(\omega)$ on the complex plane $\omega$. A scheme of Riemann surface of
the function $k(\omega)$ is shown in \figurename~\ref{RS}. This
Riemann surface has four sheets. Denote the sheet corresponding to $k_1(\omega)$ by ``Sheet~1'', the sheet for $k_2(\omega)$ by ``Sheet~2'', the sheet for $-k_1(\omega)$ by ``Sheet~3'', and the sheet for $-k_2(\omega)$ by ``Sheet~4''.

The Riemann surface has branch points of two types:
{\em cut-off branch points\/} and {\em exchange branch points}.
The cut-off branch points, are solutions of the equation
$
D(\omega,k=0) = 0.
$
These branch points
connect sheets corresponding to the solutions of the dispersion equation with opposite signs
(i.~e.\ $k_1(\omega)$ with $-k_1(\omega)$, and $k_2(\omega)$ with $-k_2(\omega)$).
The cut-off frequencies belong to the real axis of~$\omega$.

The rest of the branch points are the exchange branch points.
These points exist due to interaction between the subsystems. They connect
the sheets corresponding to different modes. In \figurename~\ref{RS} the
exchange branch points correspond to the values of $\omega$ that are not real.

One can consider \eqref{Ser_int_repr} as a contour integral of a multivalued function,
whose Riemann surface is the same of the Riemann surface of $k(\omega)$. The contour of integration is $\gamma_1 + \gamma_2$, where $\gamma_1$
and $\gamma_2$ are shown in \figurename~\ref{RS}. These contours are preimages of
the real axis of $\omega$ on Sheet~1 and Sheet~2,
slightly shifted into the upper half plane.

\section{Analysis of solutions by saddle point method (far zone)} \label{SPsection}

\subsection{Saddle point approach}

According to the Cauchy's theorem, the contour of integration
in \eqref{Ser_int_repr}
can be homotopically deformed in the area of analyticity of the integrand.
Using this fact, one can obtain some asymptotical assessments of  \eqref{Ser_int_repr}. Particularly, the saddle point method allows to estimate
the wave field in a far zone.

Introduce a {\em  formal velocity\/}  $V$ by
\[
V = x/t.
\]
Let $V$ be fixed, and let be $x \to \infty$. Indeed, this means $t \to \infty $ as well.

As it was stated above, \eqref{Ser_int_repr} can be rewritten as follows:
\begin{equation}
\label{SIR2}
u_j(t,x) = \int \limits_{\gamma_1 + \gamma_2} h_j(\omega,k(\omega)) \exp\{ ixg(\omega) \}
\, d\omega,
\quad
h_j(\omega,k) = \frac{A_j(\omega,k)}{\partial_k D(\omega,k)},
\quad j = 1,2,
\end{equation}
where $A_j$ are components of the vector ${\rm A}$, and
\begin{equation}
\label{g_det}
g(\omega) \equiv k(\omega) - \omega / V.
\end{equation}

Since the exponential factor $\exp\{ixg\}$ contains the large parameter $x$,
while $h_j$ has not, the exponential factor mainly
determines the magnitude of the integrand. Namely,  the
integrand is {\em exponentially large\/} if ${\rm Im} [g] < 0$, and is
{\em exponentially small\/} if ${\rm Im}[g] > 0$.

Such integrals can be asymptotically estimated using the saddle point method \cite{Borovikov_1994}. Since the integration is held on the Riemann surface of a multivalued function, one has to use a multi-contour version of the saddle point method
\cite{Shanin_2018,Shanin_2017}.

According to the saddle point method,
the contour $\gamma_1 + \gamma_2$ can be deformed into a sum of several {\em saddle point contours} $\gamma_{\ast m}$:
\begin{equation}
\label{SPsum}
u_j(t,x) = \sum_m I_{j,\ast m},
\quad
I_{j,\ast m} = \int \limits_{\gamma_{\ast m}} h_j(\omega,k(\omega)) \exp\{ ixg(\omega) \}d\omega.
\end{equation}
Each saddle point contour $\gamma_{j,\ast m}$ passes through {\em a saddle point\/}
$\omega_{\ast m}$.
A saddle point $\omega_{\ast m}$ is a root of the first derivative of $g$
with an additional condition:
\begin{equation}
\label{SPdet}
\partial_{\omega} g(\omega_{\ast m}) = 0, \quad \partial_{\omega}^2 g(\omega_{\ast m}) \neq 0.
\end{equation}
After substitution of \eqref{g_det} into \eqref{SPdet} one obtains
\begin{equation}
\label{SPeq}
\frac{dk(\omega_{\ast m})}{d \omega} = V^{-1}
\quad \mbox{or} \quad
v_{\rm gr} (\omega_{\ast m}) = V.
\end{equation}

The contours $\gamma_{\ast m}$ are the contours of steepest growth of ${\rm Im}[g]$.
Consequently, the magnitude of the integrand is significant only in some vicinity of
$\omega_{\ast m}$.

Note that not all the saddle points satisfying the \eqref{SPeq} are passed by saddle point contours obtained by an appropriate deformation of $\gamma_1 + \gamma_2$.

\subsection{Wave component due to a saddle point}

Since the saddle point integration is held over some vicinity of $\omega_{\ast m}$,
$h_j(\omega,k(\omega))$ and $g(\omega)$
can be approximated by a constant $h_j(\omega_{\ast m}, k(\omega_{\ast m}))$
and by a quadratic function
\begin{equation}
\label{gTaylor}
g(\omega) \approx g(\omega_{\ast m}) + \frac{1}{2} \frac{d^2 g (\omega_{\ast m}) }{d \omega^2} (\omega - \omega_{\ast m})^2.
\end{equation}
Using \eqref{g_det} and \eqref{gTaylor},
one can obtain an approximation of the integral $I_{j,\ast m}$:
\begin{equation}
\label{SPint1}
I_{j,\ast m} = h_j(\omega_{\ast m}, k(\omega_{\ast m}))
\sqrt{\frac{2 \pi}{|\alpha| x}}
\exp\{ ik(\omega_{\ast m})x - i \omega_{\ast m} t + \mbox{sign}(\alpha) \cdot i\pi/4\}, \quad \alpha = \partial_{\omega}^2k(\omega_{\ast m}).
\end{equation}

\subsection{Merging of two saddle points near an inflection point.  Airy wave}

Consider an inflection point of the real dispersion diagram, for example
the local minimum of the group velocity (a local maximum is described in
a similar way). Such a point is shown in
\figurename~\ref{DD},~right, as the point $\omega_2^e$.
If $V$ is slightly higher than $v_2'$, there should be two real saddle points
near~$\omega_2^e$. If $V = v_2'$ these two saddle points merge with each other.
An asymptotics of wave field with $V \approx \omega_2^e$ is described by an Airy function.

Approximate function $k(\omega)$ as follows:
\begin{equation}
\label{gTaylor2}
k(\omega) \approx k(\omega^e) + \frac{\omega - \omega^e}{v'_2} - \frac{1}{3}
\alpha (\omega - \omega^e_2)^3,
\quad
\alpha = - \frac{1}{2}\partial_{\omega}^3 k(\omega_2^e),
\quad
\alpha > 0.
\end{equation}

The integral over some vicinity of $\omega^e_2$ can be approximated by an Airy function \cite{Borovikov_1994}, \cite{Brekhovskih_1980}:
\begin{equation}
\label{AiryInt1}
I_j(V, x) \approx h_j(\omega_2^e, k (\omega_2^e))\, \frac{2 \pi}{(x \alpha)^{1/3}} \,
\mbox{Ai}\left( \frac{x^{2/3}}{\alpha^{2/3}} \left(V^{-1} - (v'_2)^{-1}\right) \right) .
\end{equation}

\subsection{An overview of the saddle points  evolution}

Let us list the {\em real\/} roots of \eqref{SPeq} for different $V$:
\begin{itemize}

\item
$V>c_1$: there are no roots. This case is shown by an orange line in \figurename~\ref{SPnot}.
Moreover, for $V>c_1 > c_2$ the wave field is equal to zero identically.
This can be shown by closing the integration contours in the upper half-plane.

\item
$c_1 > V > c_2$: there is a single real root $\omega_{\ast1}$.
The case is shown by red lines in \figurename~\ref{SPnot}.

\item
$c_2 > V > v_1'$: there are two real roots $\omega_{\ast1}$ and $\omega_{\ast4}$
shown by blue lines  in \figurename~\ref{SPnot}.

\item
$v_1' > V > v_2'$: there are four
real roots $\omega_{\ast1}$, $\omega_{\ast2}$, $\omega_{\ast3}$, $\omega_{\ast4}$.
They are shown by magenta lines in \figurename~\ref{SPnot}.

\item
$v_2' > V > 0$: there are two real roots $\omega_{\ast1}$ and $\omega_{\ast2}$. They are  shown
by  green lines in \figurename~\ref{SPnot}.
\end{itemize}

\begin{figure}[!h]
\centering
\includegraphics[scale = 0.8]{SP.pdf}
\caption{Solving \eqref{SPeq} graphically for real $\omega$}
\label{SPnot}
\end{figure}

After deformation of the contour $\gamma_1 + \gamma_2$,
some
steepest descend contours pass through each of the listed saddle points.
There also exists a set of negative saddle points denoted as follows:
\[
\omega_{\ast -n}, \quad n=1,\dots,6, \quad \omega_{\ast -n} = - \omega_{\ast n}.
\]
They should be taken into account in just a similar way.

\begin{figure}[!h]
	\centering
	\begin{subfigure}{0.4\textwidth}
		\includegraphics[width=\textwidth]{w2w3_2.pdf}
		\caption{Vicinity of $V = v_1'$}
	\end{subfigure}
	\vspace{0.1cm}
    \hspace{8ex}
	\begin{subfigure}{0.4\textwidth}
		\includegraphics[width=\textwidth]{w3w4_2.pdf}
		\caption{Vicinity of $V = v_2'$}
	\end{subfigure}
	\caption{Evolution of the saddle points in vicinities of local extrema of the group velocity. Black circles: points of an extremum, green circles: saddle points passed by steepest descend contours, red circles: saddle points not passed by steepest descend contours. The arrows
	correspond to growth of~$V$}
\label{VgrExtr}
\end{figure}

The analysis made above, together with \eqref{SPint1} and
\eqref{AiryInt1}, displays the basic structure of the field for very large~$x$.
The list contains the description of the zones, in which there exists a different number
of asymptotic terms. The formulae for the terms are given by \eqref{SPint1}.
There are also two intermediate zones ($V \approx v_1'$ and $V \approx v_2'$)
described by the Airy asymptotics of the form of \eqref{AiryInt1}.

All the wave components described above do not decay exponentially as $x$ grows.
There exist also saddle points with complex $\omega_{\ast m}$ possessing an exponential
growth. Sometimes, corresponding terms cannot be ignored (if $x$ is not large, decay is
not fast, or the decaying wave is the only wave observed for corresponding $x$ and~$t$).
Let us consider evolution of saddle points position for $V$ changing in some vicinity of $v_1'$. As $V$ increases from $v_1' - \nu$ to $v_1' + \nu$, where $\nu$ is a small real positive number, saddle points $\omega_{\ast 2}$ and $\omega_{\ast 3}$ evolve into two complex saddle points (\figurename~\ref{VgrExtr}~(a)). Among the saddle points for $V > v_1'$, only saddle point
$\omega_{\ast 5}$ is passed by deformed a contour because the initial contour of integration is shifted in the direction of growth of ${\rm Im}[g]$.

The case when $V$ grows near $v_2'$ is analogous. However, for $V < v_2'$ the deformed contour passes through the saddle point $\omega_{\ast 6}$ with negative imaginary part (\figurename~\ref{VgrExtr}~(b)).

Thus, for $v_1'<V$ and for $V<v_2'$ there may exist a complex saddle point, which gives
a decaying saddle point asymptotics.

\section{Zone diagrams} \label{ZDsection}
\subsection{The concept of zone diagram}

The saddle point method does not provide a correct asymptotics of the wave field
if $x$ is not large enough. Each saddle point has a domain of influence (DOI)
introduced in \cite{Borovikov_1994},
which is a spot on the saddle point contour on which the
integrand is not negligibly small. Typically, the size of such a spot in the
$\omega$-plane has size $\sim x^{-1/2}$. if $x$ is not very large, it may happen
that the DOI of two or more saddle points overlap. In this case the saddle point
approximation is not adequate.

An example of such a configuration is the Airy wave mentioned above, which is
a result of merging of two saddle points.
Depending on $V$ and $x$, there can be more sophisticated asymptotics.

The concept of merging of DOI of saddle points is formalized as follows.
 Let $(\omega_{\ast},k_{\ast})$ be a saddle point, and let $\gamma_{\ast}$ be the corresponding steepest descend contour. Introduce the length coordinate $l$ directed along $\gamma_{\ast}$. Let $l = 0$ correspond to the saddle point, and the coordinate direction coincide with the contour direction. The integral along $\gamma_{\ast}$ can be represented as
\begin{equation}
\label{eq1}
I_{\ast}(x) = \int \limits _{-\infty}^{\infty} h^*(l) \exp \{ ix g^*(l) \} \, dl
\end{equation}
where $h^*$ and $g^*$ are restrictions of functions
$h \, d\omega / dl$ and $g$ onto the contour.

The imaginary part of $g^*(l)$ has minimum at $l = 0$, and grows monotonically as $|l|$ grows. The DOI is a segment $[a_1,a_2]$ on the $l$-axis, such that $a_1 < 0 < a_2$, and
\begin{equation}
\label{InflDom}
x({\rm Im}[g^*(a_1)]-{\rm Im}[g^*(0)]) = x({\rm Im}[g^*(a_2)]-{\rm Im}[g^*(0)]) = {S}.
\end{equation}
Here and below,
${S}$ is some arbitrary positive moderately large real number of order 1
(a good choice is ${S} = 3$).
We find convenient to write $a > S$ instead of $a \gg 1$.

The segment $[a_1,a_2]$ is, indeed, just an estimation of the DOI.
One can see that
the size of the DOI depends on $x$, and the domain becomes smaller as $x$ grows.

One needs an Airy representation (\eqref{AiryInt1})
when the DOIs of the saddle points $\omega_{\ast 2}$ and
$\omega_{\ast 3}$  (or $\omega_{\ast 3}$ and $\omega_{\ast 4}$) overlap. Note that the Airy function ${\rm Ai}(z)$ has two asymptotics: for $z \gg 1$ and for $-z \gg 1$. One of these asymptotics correspond to two real saddle points, while another corresponds to a single
complex saddle point.

Evaluate the phase difference for any pair of
{\em neighbouring} saddle points $\omega_{\ast m}$ and $\omega_{\ast n}$:
\begin{equation}
\label{Phase_deff}
\Delta_{m,n} \equiv |(k(\omega_{\ast m})x - \omega_{\ast m} t) -
(k(\omega_{\ast n})x - \omega_{\ast n} t) |
=
x\, |\, g^*(\omega_{\ast m}) -  g^*(\omega_{\ast n}) \,|.
\end{equation}
If $\Delta_{m,n}$ is bigger than ${S}$ then
$\omega_{\ast m}$ and $\omega_{\ast n}$ can be considered as separate saddle points.
If  $\Delta_{m,n} < {S}$
then one should introduce some more sophisticated asymptotics.

Let us say when we consider saddle points $\omega_{\ast m}$ and $\omega_{\ast n}$ being neighboring. This is when there exists a path between $\omega_{\ast m}$ and $\omega_{\ast n}$ along the Riemann surface of $k(\omega)$, on which the phase $xg(\omega)$
does not change strongly (for more than ${S}$).

The asymptotics of a waveguide can be organized in {\em a zone diagram}.
We plot the zone diagram in the coordinates $(t,V) = (t , x/t)$,
The diagram is branched, i.~e.\ over each point $(t, V)$ there may exist several sheets
of the zone diagram. Each sheet corresponds to a separate asymptotic term.

There are parent / child relations between the asymptotics on the diagram. The
``children'' are obtained by taking some asymptotics of the ``parents''.

\subsection{A simple example: a zone diagram for a scalar Klein--Gordon equation}

Consider a scalar inhomogeneous Klein--Gordon equation
\begin{equation}
(c^2 \partial_x^2 - \partial_t^2 - \Omega^2) u(t, x) = \delta(t) \, \delta(x).
\label{SKGE}
\end{equation}
A causal solution of this equation can be easily obtained:
\begin{equation}
u(t, x) = \left\{ \begin{array}{ll}
-(2c)^{-1} J_0 (\Omega \sqrt{t^2 - x^2 / c^2}) & t > x /c \\
0                                              & 0 < t < x /c
\end{array} \right.
\label{SKGEsol}
\end{equation}
where $J_0$ is the Bessel function.

One can see that there are two asymptotics of the solution:
if the argument of the Bessel function is much greater than~1 (we prefer to say that it is greater than
${S}$), then one can represent the Bessel function in terms of two exponential functions:
\begin{equation}
u(t, x) \approx -\frac{1}{2c \sqrt{2 \pi z}}
(\exp \{i z - i \pi / 4 \} + \exp \{-i z + i \pi / 4 \}) ,
\quad
z = \Omega \sqrt{t^2 - x^2 / c^2},
\quad
z > {S}.
\label{SKGEas1}
\end{equation}
This is the ``far field'' for the waveguide.

If the argument is small, there is another asymptotics:
\begin{equation}
u(t, x) \approx - (2c)^{-1},
\quad
z < {S}^{-1}.
\label{SKGEas2}
\end{equation}
This is the ``near field''.

A zone diagram for the scalar Klein--Gordon equation is plot in
\figurename~\ref{SKGEdiagram}. The left part of the figure is the diagram projected onto
the $(t, V)$-plane, and the left part shows the branching of the diagram.
One can see that the zone of the diagram labeled as ``far field'' is represented by two sheets.
These sheets denote two terms of \eqref{SKGEas1}. Indeed, these terms can be treated as
the saddle point contributions of a well-known integral representation
for the Bessel function.

\begin{figure}[!h]
	\centering
	\begin{subfigure}{0.4\textwidth}
		\includegraphics[width=0.8 \textwidth]{figB5201-eps-converted-to}
		\caption{Zone diagram in the $(t, V)$-plane}
	\end{subfigure}
	\vspace{0.1cm}
    \hspace{8ex}
	\begin{subfigure}{0.4\textwidth}
		\includegraphics[width=0.8 \textwidth]{figB5202-eps-converted-to}
		\caption{Branching of the diagram}
	\end{subfigure}
	\caption{Zone diagram for the scalar Klein--Gordon equation}
\label{SKGEdiagram}
\end{figure}

The central part of the diagram
denoted by $J_0$ is the domain where one cannot use any asymptotics, and should use the
initial expression given by \eqref{SKGEsol}.
In the ``near field'' zone the field is the constant given by \eqref{SKGEas2}.

The green arrows show the parent / child relations between the zones.

\subsection{Zone diagram for 2D matrix Klein--Gordon equation}

Building of a zone diagram for the matrix Klein--Gordon equation (\eqref{MKGE}) is a  sophisticated task. The reason is that there are four saddle points
($\omega_{\ast 1} , \dots , \omega_{\ast 4}$) that can merge.
As a result, the zone diagram becomes complicated. Here we do not build the
complete zone diagram, namely, we don't show the near field zones.

A sketch of the boundaries between the zones is shown in \figurename~\ref{ZD1}.
Corresponding DOIs overlap if the point $(t , V)$ is located to the left of a boundary.

\begin{figure}[!h]
\centering
\includegraphics[scale = 1.1]{ZD1-eps-converted-to}
\caption{Boundaries between zones for \eqref{MKGE}}
\label{ZD1}
\end{figure}

The central part of the zone diagram is shown in more details
in \figurename~\ref{ZD2}.
The letter notations describe the asymptotic terms that exist
in corresponding zones.

Note that for the sake of compact notations, only the terms produced by
saddle points located in the right half-plane of $\omega$ are indicated.
In a complete description of terms, our notations should be doubled.

Let us list the types of the asymptotics.

\begin{figure}[!h]
\centering
\includegraphics[scale = 1]{ZD2-eps-converted-to}
\caption{Central part of the zone diagram}
\label{ZD2}
\end{figure}

{\bf SP}: A term due to a real saddle point (see \eqref{SPint1}).

{\bf SPe}: A term due to a saddle point with complex $\omega_{\ast m}$. Such a term has
a structure close to  \eqref{SPint1}, but the exponential factors decays as $x$ grows.

{\bf Ai}: A term having Airy-type asymptotics of the form (\eqref{AiryInt1}).
The diagram indicates that the ``Ai'' term degenerates into
two ``SP'' terms or into a single ``SPe'' term.

{\bf J}: Such a term can be observed in the zone where the DOIs of the saddle points
$\omega_{\ast 1}$ and $\omega_{\ast 3}$ merge, while the saddle points
$\omega_{\ast 2}$ and $\omega_{\ast 4}$ are isolated.
\figurename~\ref{cont_J}
shows the integration contour $\gamma_1 + \gamma_2$
after an appropriate deformation
corresponding to this point. Bold parts of the contours show the points of the complex plane
$\omega$, where the magnitude of the integrand is not exponentially small. The points
$\omega_{\ast 2}$ and $\omega_{\ast 4}$ produce standard saddle point terms.
Another significant part of the integral is computed  along the contours encircling the cuts. This term is the J asymptotics.
We should note that this term
describes the exchange pulse \cite{Shanin_2018}.

\begin{figure}[!h]
\centering
\includegraphics[scale = 1]{cont_J-eps-converted-to}
\caption{The deformed contour of integration for the points
$(t, V)$, for which the asymptotics J exists}
\label{cont_J}
\end{figure}

Let us compute the J-type asymptotics of the field.
Introduce new coordinates
\begin{equation}
\label{new_var}
\omega' = \omega - \omega_{\rm sh}, \quad k' = k - k_{\rm sh}.
\end{equation}
We assume that
\[
\frac{k'}{k_{\rm sh}} \sim \frac{\omega'}{\omega_{\rm sh}} << 1.
\]
Applying the perturbation method one can obtain the following representation of the dispersion function $D(\omega,k)$ (\eqref{D_det}):
\begin{equation}
\label{D_as}
D \approx 4c_1^2 c_2^2 k_{\rm sh}^2 \left( k' - \frac{\omega'}{v_1} \right)\left( k' - \frac{\omega'}{v_2} \right) - \mu^2.
\end{equation}
Substituting \eqref{D_as} into \eqref{DE} one can obtain the following representation of the function $k(\omega)$:
\begin{equation}
\label{k_asympt}
k(\omega) \approx k_{\rm sh} + \frac{v_1^{-1} + v_2^{-1}}{2}\omega' \pm \sqrt{\frac{(v_2^{-1} - v_1^{-1})^2 (\omega')^2}{4} + \frac{\mu^2}{4c_1^2 c_2^2 k_{\rm sh}^2}}.
\end{equation}
Introduce a stretched variable
\begin{equation}
\label{coord}
\tau = \xi \omega' ,
\end{equation}
where
\[
\xi = \frac{c_1 c_2 k_{\rm sh} (v_2^{-1} - v_1^{-1})}{\mu}.
\]
The
J-asymptotics can be expressed as an integral
\begin{equation}
\label{J_int}
I_j \approx -\frac{i \pi}{2 c_1^2 c_2^2 k_{\rm sh}^2 (v_2^{-1} - v_1^{-1})}
A_j(\omega_{\rm sh},k_{\rm sh})
\,
\exp\{ ik_{\rm sh}x -i \omega_{\rm sh} t  \}
\,\times
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\end{equation}
\[
\qquad \qquad \qquad \qquad \qquad
\int \limits_{\sigma}
\exp\left\{
-i\frac{\tau}{\xi} \left( t - x\frac{ (v_1^{-1} + v_2^{-1})}{2} \right) +
i  \frac{x\, \mu \sqrt{1+\tau^2}}{2 c_1 c_2 k_{\rm sh}}
\right\}
\frac{d\tau}{\sqrt{1 + \tau^2}}
,
\]
where $\sigma$ is the contour encircling the cut $[-i,i]$ in the positive direction.
This integral can be expressed in terms of Bessel function $J_0$:
\begin{equation}
\label{Jasympt}
I_j = -\frac{\pi^2}{c_1^2c_2^2 k_{\rm sh}^2
(v_2^{-1} - v_1^{-1}) }
A_j(\omega_{\rm sh},k_{\rm sh}) \exp\{ik_{\rm sh}x -i \omega_{\rm sh} t \}
J_0(b),
\end{equation}
\[
b = \frac{ \mu \sqrt{(t - x / v_1)(x / v_2 - t)}}{c_1 c_2 k_{\rm sh} (v_2^{-1} - v_1^{-1})}
\]

{\bf Q}: This asymptotics is valid when the DOIs of the saddle points
$\omega_{\ast 1}$, $\omega_{\ast 2}$, and $\omega_{\ast 3}$
overlap (or when the DOIs
of $\omega_{\ast 1}$, $\omega_{\ast 3}$, and $\omega_{\ast 4}$ overlap). Although we don't present an explicit expression for the Q-asymptotics here, we describe a way to obtain it.

In \figurename~\ref{cont_Q} we plot deformed contours of integration for
the lower domain with the Q-asymptotics
when the DOIs of $\omega_{\ast 1}$, $\omega_{\ast 3}$, and $\omega_{\ast 4}$ overlap,
and $\omega_{\rm ast 2}$ is isolated.
The bold green contour on Sheet~2 going along the cut corresponds to the Q-asymptotics.

\begin{figure}[!h]
\centering
\includegraphics[scale = 1]{cont_Q-eps-converted-to}
\caption{The deformed contour of integration for the points
$(t, V)$ with the asymptotics Q}
\label{cont_Q}
\end{figure}

Some detailed consideration shows that
an
 approximation of the field corresponding to the Q-asymptotics
  can be expressed through a new special function
\begin{equation}
\label{asymptQ}
Q(\beta,z) =  \int \limits_{\Gamma} \frac{1}{\sqrt{1 + \tau^2}}
\exp\left\{i (\sqrt{1 + \tau^2} + z\tau + \beta \tau^2)\right\} d \tau,
\end{equation}
where $\beta$, and $z$ are some constants, which depend on the waveguide parameters,
contour $\Gamma$ is shown in \figurename~\ref{cont_gamma_Q}.
\begin{figure}[!h]
\centering
\includegraphics[scale = 0.9]{cont_gamma_Q-eps-converted-to}
\caption{Contour $\Gamma$ for defenition of $Q(\beta,z)$}
\label{cont_gamma_Q}
\end{figure}

{\bf B}: In this case the DOIs of all saddle points overlap. This zone is the parent for all, and
no simplification of the integral
\eqref{Ser_int_repr} can be made.

\section{Conclusion}

The saddle point method allows one to obtain asymptotic
estimations of the field in the far zone.
This zone is the set of points $(t, x)$,
such  that the
domains of influence of different saddle points do not overlap.

If the domains of influences of different saddle points overlap, one has to build new asymptotics. A parent / child relation between the new and old asymptotics
can be established.

To visualize all the obtained asymptotics, we propose a zone diagram.
This is a branched diagram in the $(t, V)$-plane.

In this work we built such set of asymptotics for a system described by \eqref{MKGE}.
A formula for the J-asymptotics is derived.
A sketch of obtaining a Q-asymptotics is given.

\section{Acknowledgements}
The work is supported by the RFBR grant 19-29-06048.