\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 40, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/40\hfil Weak separation limit]
{Weak separation limit of a two-component Bose-Einstein condensate}

\author[C. Sourdis \hfil EJDE-2018/40\hfilneg]
{Christos Sourdis}

\address{Christos Sourdis \newline
Department of Mathematics,
University of Ioannina, 
Ioannina, Greece}
\email{sourdis@uoc.gr}


\dedicatory{Communicated by Peter Bates}

\thanks{Submitted October 10, 2017. Published January 31, 2018.}
\subjclass[2010]{34C37, 34C45, 34C14, 35J61}
\keywords{Geometric singular perturbation theory; heteroclinic connection; 
\hfill\break\indent hamiltonian system;
Bose-Einstein condensate; phase separation}

\begin{abstract}
 This article studies of the behaviour of the wave functions of a
 two-component Bose-Einstein condensate in the case of weak segregation.
 This amounts to the study of the asymptotic behaviour of a heteroclinic
 connection in a conservative Hamiltonian system of two coupled second order
 ODE's, as the strength of the coupling tends to its infimum.
 For this purpose, we apply geometric singular perturbation theory.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction} \label{secIntro}

We consider the  \emph{heteroclinic connection problem}
\begin{gather}\label{eqEqGen}
\begin{gathered}
 \lambda^2 \ddot{u}= u^3-u+\Lambda v^2 u, \\
 \ddot{v}=v^3-v+\Lambda u^2 v;
 \end{gathered} \\
\label{equvPos}
u,v>0; \\
\label{eqBdryGen}
 (u,v )\to (0,1) \text{ as } z\to -\infty,\quad (u,v)\to (1,0)\ \text{ as }
 z\to +\infty,
 \end{gather}
for values of the parameter $\Lambda>1$, where for the constant $\lambda$
we may assume without loss of generality that $\lambda \geq 1$.

This problem arises in the study of two-component Bose-Einstein condensates in
 the case of segregation, see \cite{aftalionSourdis} and the references therein,
but also in the study of certain amplitude equations
(see \cite{pismen2006patterns,van2000domain}).

 The heteroclinic connection problem
\eqref{eqEqGen}-\eqref{equvPos}-\eqref{eqBdryGen} always admits a solution
which minimizes the associated enegy in Proposition \ref{proEnergy}
 below (see \cite{alamaARMA15,van2000domain}).
This type of heteroclinics enjoy the following monotonicity property:
\begin{equation}\label{eqmonotS}
\dot{u}>0,\quad \dot{v}<0,
\end{equation}
(actually this is an implication of their stability, see \cite{alamaARMA15});
in the special case where $\lambda=1$, it also holds that the function
\begin{equation}\label{eqARCT}
\arctan(v/u)\text{ is decreasing}
\end{equation}
and $u(z+z_0)\equiv v(z_0-z)$ for some $z_0 \in \mathbb{R}$
(see \cite{van2000domain}). Moreover, any solution of
\eqref{eqEqGen}, \eqref{eqBdryGen} satisfies $u^2+v^2<1$
(see \cite{alamaARMA15}) and the hamiltonian identity
\begin{equation}\label{eqHam}
\lambda^2\frac{(\dot{u})^2}{2}+\frac{(\dot{v})^2}{2}-\frac{(1-u^2-v^2)^2}{4}
-\frac{\Lambda-1}{2}u^2v^2\equiv 0.
\end{equation}
Remarkably, if there were more general constant coefficients in \eqref{eqEqGen},
then they could be absorbed in $\lambda, \Lambda$ by a rescaling, as they
would have to satisfy a balancing condition in order for the
corresponding heteroclinic solutions to exist.

It was shown recently in \cite{aftalionSourdis} that solutions of
\eqref{eqEqGen}-\eqref{equvPos}-\eqref{eqBdryGen} satisfying the monotonicity
property \eqref{eqmonotS} are unique up to translations; interestingly enough,
it was also shown that the monotonicity of just one of the components is
enough to reach the same conclusion. Even more recently, and after the first
 version of the current paper was completed, it was shown in
 \cite{farina2017monotonicity}
that solutions of \eqref{eqEqGen}-\eqref{equvPos}-\eqref{eqBdryGen}
are indeed monotone in the sense of \eqref{eqmonotS}, and thus there is
uniqueness modulo translations without the need of imposing a-priori
a monotonicity assumption.

There are two singular limits associated with
\eqref{eqEqGen}-\eqref{equvPos}-\eqref{eqBdryGen}:
$\Lambda \to +\infty$ and $\Lambda \to 1^+$
which are called the strong and the weak separation limit, respectively.
Both limits were studied formally in \cite{barankov} (see also \cite{vaninterface}
and \cite{malomed1990domain} for more formal arguments in the strong and weak
separation limits, respectively). In particular, it was predicted therein that
the components of an energy minimizing solution satisfy $uv\to 0$ and
$u^2+v^2\to 1^-$, at least pointwise, as $\Lambda \to +\infty$ and
$\Lambda \to 1^+$, respectively.
The strong separation limit was studied rigorously and in great detail
recently in \cite{aftalionSourdis}. The scope of the current article is to
study rigorously the weak separation limit, i.e., $\Lambda \to 1^+$.
 To the best of our knowledge, the only rigorous result in this direction
is contained in the recent paper \cite{goldman2015phase}, where the authors
employed $\Gamma$-convergence techniques to obtain a first order
asymptotic expansion of the minimal energy.

It turns out that, in contrast to the strong separation limit, here we can
apply by now standard arguments from geometric singular perturbation theory
(see \cite{kuhen} and the references therein). To this end, we first have
to put system \eqref{eqEqGen} in the appropriate slow-fast form.
At this point we will rely on the intuition of the physicists in the
aforementioned papers. In this regard, a main observation is that
from \eqref{eqEqGenEpsilon}, by letting $\varepsilon =0$,
we find that $u^2 + v^2 =1$ is a slow manifold (or critical manifold).
This motivates the introduction of the polar coordinates in \eqref{eqPolar}.
This task will be carried out in Section \ref{secSF}. We will analyze
the resulting slow-fast system using geometric singular perturbation
theory in Section \ref{secGeom}. Armed with this analysis, we will prove
our main result in Section \ref{secMain} which provides fine estimates
for a heteroclinic solution of \eqref{eqEqGen}-\eqref{eqBdryGen},
as $\Lambda \to 1^+$, expressed in terms of suitable polar coordinates.
One can then directly go back and estimate the original $u,v$ via
 \eqref{eqEpsilon}, \eqref{eqxz}, \eqref{eqPolar} and \eqref{eqR}.
 Lastly, in Section \ref{secEnergy} we will show that this solution coincides
with the unique (up to translations) minimizing heteroclic connection of
\eqref{eqEqGen}-\eqref{eqBdryGen}, and provide an asymptotic expression
 for its energy.

 \section{Slow-fast system}\label{secSF}
We let
\begin{equation}\label{eqEpsilon}
\varepsilon=\sqrt{\Lambda-1},
\end{equation}
and consider the slow variable
\begin{equation}\label{eqxz}
x=\varepsilon z.
\end{equation}
In the rest of the paper, unless specified otherwise, we will assume that
$\varepsilon>0$.
Then, system \eqref{eqEqGen} is equivalent to
\begin{equation}\label{eqEqGenEpsilon}
 \begin{gathered}
 \lambda^2 \varepsilon^2 u''=u^3-u+ v^2 u+ \varepsilon^2 v^2 u, \\
 \varepsilon^2v''= v^3-v+ u^2 v+ \varepsilon^2 u^2 v,
 \end{gathered}
 \end{equation}
where $'=d/{dx}$ (the relations \eqref{equvPos} and \eqref{eqBdryGen}
remain the same).
Next, motivated from \cite{barankov,malomed1990domain}, we express $(u,v)$
in polar coordinates as
\begin{equation}\label{eqPolar}
u=R\cos \varphi,\quad v=R\sin \varphi,
\end{equation}
and write \eqref{eqEqGenEpsilon}-\eqref{equvPos}-\eqref{eqBdryGen} equivalently as
\begin{gather*}
\varepsilon^2 \left[R''- R (\varphi')^2\right]
=(R^3-R)\big[1+\big(\frac{1}{\lambda^2}-1 \big)\cos^2 \varphi \big]
 +\varepsilon^2R^3\big(\frac{1}{\lambda^2}+1 \big)\sin^2\varphi\cos^2\varphi, \\
\begin{aligned}
\varepsilon^2 \left(R \varphi''+2 R'\varphi'\right)
&= -\big(\frac{1}{\lambda^2}-1 \big)(R^3-R) \sin \varphi \cos \varphi \\
&\quad +\varepsilon^2 R^3\big(\sin \varphi \cos^3 \varphi-\frac{1}{\lambda^2}
\cos \varphi \sin^3 \varphi\big), 
\end{aligned}\\
R>0,\quad 0<\varphi<\frac{\pi}{2},\\
R\to 1 \text{ as } x\to \pm \infty, \quad
\varphi \to \frac{\pi}{2}\text{ as } x\to -\infty,\quad
\varphi \to 0 \text{ as } x\to +\infty.
\end{gather*}
Subsequently, we blow-up the neighborhood near $R=1$ by setting
\begin{equation}\label{eqR}
R=1-\varepsilon^2 w,
\end{equation}
and get the equivalent problem:
\begin{gather*}
\begin{aligned}
-\varepsilon^2w''- (1-\varepsilon^2 w) (\varphi')^2
&= (1-\varepsilon^2 w)(\varepsilon^2w^2-2w)
 \big[1+\big(\frac{1}{\lambda^2}-1 \big)\cos^2 \varphi \big]\\
&\quad + (1-\varepsilon^2 w)^3\big(\frac{1}{\lambda^2}
 +1 \big)\sin^2\varphi\cos^2\varphi,
\end{aligned} \\
\begin{aligned}
(1-\varepsilon^2 w) \varphi''-2 \varepsilon^2 w'\varphi'
&=\big(1-\frac{1}{\lambda^2} \big)(1-\varepsilon^2 w)(\varepsilon^2w^2-2w)
 \sin \varphi \cos \varphi \\
&\quad +(1-\varepsilon^2 w)^3\big(\sin \varphi \cos^3 \varphi-\frac{1}{\lambda^2}
\cos \varphi \sin^3 \varphi\big),
\end{aligned} \\
0<\varphi<\frac{\pi}{2}, \\
w\to 0 \text{ as } x\to \pm \infty, \quad
\varphi \to \frac{\pi}{2} \text{ as } x\to -\infty,\quad
\varphi \to 0 \text{ as } x\to +\infty.
\end{gather*}
Now we can define
\begin{equation}\label{eqTransFi}
w_1=w,\quad w_2=\varepsilon w_1',\quad \varphi_1= \varphi,\quad
 \varphi_2=\varphi_1',
\end{equation}
and write the problem equivalently in the following slow-fast form,
with $(w_1,w_2)$ being the fast variables and $(\varphi_1,\varphi_2)$
the slow ones:
\begin{gather}\label{eqSF}
 \varepsilon w_1' = w_2, \\
 \begin{aligned}
 \varepsilon w_2' & =  - (1-\varepsilon^2 w_1) \varphi_2^2 
-(1-\varepsilon^2 w_1)(\varepsilon^2w_1^2-2w_1)
 \big[1+\big(\frac{1}{\lambda^2}-1\big)\cos^2 \varphi_1 \big] \\
 &\quad - (1-\varepsilon^2 w_1)^3\big(\frac{1}{\lambda^2}+1\big)
 \sin^2\varphi_1\cos^2\varphi_1,
\end{aligned} \\
 \varphi_1'  =  \varphi_2, \\
\label{e2.10}
\begin{aligned}
 \varphi_2' & =  \frac{2 \varepsilon w_2\varphi_2}{1-\varepsilon^2 w_1}
 +\big(1-\frac{1}{\lambda^2}\big)(\varepsilon^2w_1^2-2w_1) 
 \sin \varphi_1 \cos \varphi_1 \\
 &\quad +(1-\varepsilon^2 w_1)^2\Big(\sin \varphi_1 \cos^3 \varphi_1
-\frac{1}{\lambda^2} \cos \varphi_1 \sin^3 \varphi_1\Big),
 \end{aligned}\\
\label{eqPhi1pos}
0<\varphi_1<\frac{\pi}{2}, \\
\label{eqSFbdryu}
\begin{gathered}
 w_1, w_2\to 0 \quad \text{as } x\to \pm \infty, \\
 \varphi_1 \to \frac{\pi}{2} \text{ as } x\to -\infty,\quad
 \varphi_1 \to 0 \text{ as } x\to +\infty,\quad
 \varphi_2\to 0  \text{ as } x\to \pm \infty.
\end{gathered}
\end{gather}

\subsection{Analysis at equilibria}\label{subsubLinearization}
It is easy to check that the eigenvalues of the linearization of 
\eqref{eqSF}-\eqref{e2.10}
at the equilibria $(0,0,\frac{\pi}{2},0)$ and $(0,0,0,0)$ that we wish to 
connect are
\begin{equation}\label{eqEVs}
\pm \frac{\sqrt{2}}{\varepsilon}, \quad \pm \frac{1}{\lambda}, \quad
\pm \frac{\sqrt{2}}{\lambda \varepsilon}, \quad \pm 1,
\end{equation}
respectively. Moreover, as associated eigenfunctions we can choose the following:
\begin{equation}\label{eqEigen}
\big(\pm \frac{1}{\sqrt{2}},1,0,0 \big),\quad
(0,0,\pm \lambda,1), \quad \big(\pm \frac{\lambda}{\sqrt{2}},1,0,0 \big),\quad
(0,0,\pm 1,1),
\end{equation}
respectively.

\section{Geometric singular perturbation theoretic analysis}\label{secGeom}

Having put the problem in the standard slow-fast form, we can now start 
analyzing it using geometric singular perturbation theory.

\subsection{The $\varepsilon=0$ limit slow system}\label{subsecSFslow}
The slow-fast system \eqref{eqSF}-\eqref{e2.10} is in the so called slow form. 
Switching back to the variable $z$ (recall \eqref{eqxz}) gives us the corresponding 
fast form. They are equivalent as long as $\varepsilon$ is positive, but they 
provide different information when we formally set $\varepsilon=0$. 
For the problem at hand, we will only need the information that comes from 
the slow $\varepsilon=0$ limit problem, which is the following:
\begin{equation}\label{eqSFlimit}
\begin{gathered}
 0  =  w_2, \\
 0  =  - \varphi_2^2 +2w_1[1+\big(\frac{1}{\lambda^2}-1 \big)
\cos^2 \varphi_1 ] 
- (\frac{1}{\lambda^2}+1)\sin^2\varphi_1\cos^2\varphi_1, \\
 \varphi_1' =  \varphi_2, \\
 \varphi_2'  =  -2\big(1-\frac{1}{\lambda^2} \big)w_1 \sin \varphi_1 \cos \varphi_1 
+\sin \varphi_1 \cos^3 \varphi_1-\frac{1}{\lambda^2} 
\cos \varphi_1 \sin^3 \varphi_1.
 \end{gathered}
\end{equation}

\subsubsection*{Critical manifold $\mathcal{M}_0$} 
The first two equations of \eqref{eqSFlimit} define the \emph{critical manifold}, 
which is
\begin{equation}\label{eqCritiqMan}
\mathcal{M}_0=\big\{w_1=\frac{\varphi_2^2 + (\frac{1}{\lambda^2}
+1 )\sin^2\varphi_1\cos^2\varphi_1}
{2[1+(\frac{1}{\lambda^2}-1 )\cos^2 \varphi_1 ]},\;
 w_2=0,\; (\varphi_1,\varphi_2)\in \mathbb{R}^2 \big\}.
\end{equation}

\subsubsection*{Reduced problem}\label{subsecRedux}
The last two equations of \eqref{eqSFlimit} define a flow on the critical
 manifold $\mathcal{M}_0$, which is given by the lifting on $\mathcal{M}_0$ 
of the trajectories of the following two-dimensional \emph{reduced system}:
\begin{equation}\label{eqRedux}
\begin{gathered}
 \varphi_1'  =  \varphi_2, \\
\begin{aligned}
 \varphi_2' & =  -\big(1-\frac{1}{\lambda^2}\big)
 \big[\frac{\varphi_2^2 + \big(\frac{1}{\lambda^2}+1\big)
 \sin^2\varphi_1\cos^2\varphi_1}{1+\big(\frac{1}{\lambda^2}-1\big)
 \cos^2 \varphi_1 }\big] \sin \varphi_1 \cos \varphi_1 \\ 
 &\quad +\sin \varphi_1 \cos^3 \varphi_1-\frac{1}{\lambda^2} 
 \cos \varphi_1 \sin^3 \varphi_1.
 \end{aligned}
\end{gathered}
\end{equation}
The form of the above system may be discouraging at first sight,
 but a closer look reveals that it can be written in the following simple 
form for $\varphi_1$:
\begin{equation}\label{eqHamReducia}
\frac{d}{dx}\big\{\big[1+\big(\frac{1}{\lambda^2}-1\big)\cos^2 \varphi_1 \big]
(\varphi_1')^2 \big\}
=\frac{1}{4\lambda^2}\frac{d}{dx}\left\{\sin^2(2\varphi_1) \right\}.
\end{equation}
Then, in view of the asymptotic behaviour \eqref{eqSFbdryu}, 
the \emph{reduced problem} becomes
\begin{equation}\label{eqRB}
\begin{gathered}
 \varphi_1' =-\frac{1}{2\lambda}\sin(2\varphi_1)
\Big[1+\big(\frac{1}{\lambda^2}-1\big)\cos^2 \varphi_1 \Big]^{-1/2}, \\
 \varphi_1 \to \frac{\pi}{2} \text{ as } x\to -\infty,\quad
 \varphi_1 \to 0 \text{ as } x\to +\infty.
 \end{gathered}
\end{equation}
Clearly, the above problem admits a unique solution $\varphi_{1,0}$ such 
that $\varphi_{1,0}(0)=\frac{\pi}{4}$. Moreover, it holds 
$\varphi_{2,0}=\varphi_{1,0}'<0$. We note that this limit problem also arose 
in the $\Gamma$-convergence argument of \cite{goldman2015phase}. 
The lifting of the orbit $(\varphi_{1,0},\varphi_{2,0})$ on the critical 
manifold $\mathcal{M}_0$ is called \emph{singular heteroclinic orbit or connection}.
 We note that $(\frac{\pi}{2},0)$ and $(0,0)$ are saddle equilibria 
for \eqref{eqRedux} with corresponding eigenvalues $\pm \frac{1}{\lambda}$ 
and $\pm 1$, respectively; the associated eigenvectors are 
$\left(\pm \lambda,1 \right)$ and $\left(\pm 1,1 \right)$, respectively.
 It is useful to compare with Subsection \ref{subsubLinearization}.

\subsection{Locally invariant manifold $\mathcal{M}_\varepsilon$}
\subsubsection{Normal hyperbolicity of $\mathcal{M}_0$} 
The critical manifold $\mathcal{M}_0$ corresponds to a two-dimensional
 manifold of equilibria for the $\varepsilon=0$ limit fast system 
(recall the discussion in the beginning of Subsection \ref{subsecSFslow}). 
The associated linearization at such an equilibrium point is
\[
\begin{pmatrix}
 0 & 1 & 0 & 0 \\
 2+2\big(\frac{1}{\lambda^2}-1\big)\cos^2 \varphi_1 & 0 & 0 & 0 \\
 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & 0
 \end{pmatrix}.
\]
The eigenvalues of this matrix are 
$\pm \sqrt{2+2\big(\frac{1}{\lambda^2}-1\big)\cos^2 \varphi_1}$ and zero (double). 
Therefore, as there are no other eigenvalues on the imaginary axis besides of 
zero whose multiplicity is equal to the dimension of $\mathcal{M}_0$, we 
infer that the critical manifold $\mathcal{M}_0$ is normally hyperbolic.

\subsubsection{Persistence of $\mathcal{M}_0$ for $0<\varepsilon \ll 1$}
\label{subsubPersist}
Since $\mathcal{M}_0$ is normally hyperbolic and a $C^\infty$ graph over the 
$(\varphi_1,\varphi_2)$ plane, as a particular consequence of Fenichel's first 
theorem (see \cite{fenichelJDE}, \cite{jones1995geometric} or \cite[Ch. 3]{kuhen}), 
we deduce that, given an integer $m\geq 1$ and a compact subset $\mathcal{K}$ 
of the $(\varphi_1,\varphi_2)$ plane, there are functions 
$h_i(\varphi_1,\varphi_2,\varepsilon)\in C^m\left(\mathcal{K}\times 
[0,\infty) \right)$, $i=1,2$, and an $\varepsilon_0>0$ so that for 
$\varepsilon \in (0,\varepsilon_0)$ the graph $\mathcal{M}_\varepsilon$ over
 $\mathcal{K}$ described by 
\begin{equation}\label{eqh}
w_1=\frac{\varphi_2^2 + \big(\frac{1}{\lambda^2}+1\big)
\sin^2\varphi_1\cos^2\varphi_1}{2\left[1+\big(\frac{1}{\lambda^2}-1\big)\cos^2 
\varphi_1 \right]}+\varepsilon h_1(\varphi_1,\varphi_2,\varepsilon),\quad
 w_2=\varepsilon h_2(\varphi_1,\varphi_2,\varepsilon),
\end{equation}
is locally invariant under \eqref{eqSF}-\eqref{e2.10}. In passing, we note that this property 
also follows by appending the equation $\dot{\varepsilon}=0$ to the equivalent 
fast form of \eqref{eqSF}-\eqref{e2.10}, applying the usual center manifold theorem at each 
equilibrium on $\mathcal{M}_0\times \{0\}$, and then taking slices for 
$\varepsilon$ fixed (see \cite[Ch. 2]{berglund2006noise}). 
As a center-like manifold, $\mathcal{M}_\varepsilon$ is generally not unique. 
We choose the compact set $\mathcal{K}$ to be the closure of a smooth domain 
that contains the heteroclinic connection $(\varphi_{1,0}, \varphi_{2,0})$ 
of the reduced system \eqref{eqRedux}. The equilibria $(0,0,\frac{\pi}{2},0)$ 
and $(0,0,0,0)$ of \eqref{eqSF}-\eqref{e2.10} lie on $\mathcal{M}_\varepsilon$, that is
\begin{equation}\label{eqEquiv0}
h_i\big(\frac{\pi}{2},0,\varepsilon\big)=0,\quad
h_i\left(0,0,\varepsilon\right)=0,\quad i=1,2,\quad
 \varepsilon \in [0,\varepsilon_0).
\end{equation} 
This is because every invariant set of \eqref{eqSF}-\eqref{e2.10} in a 
sufficiently small
 $\varepsilon$-indepen\-dent neighborhood of $\mathcal{M}_0$ must be on 
$\mathcal{M}_\varepsilon$.

\subsubsection{Equivariant aspects of $\mathcal{M}_\varepsilon$}
In this subsection, we will discuss some symmetry properties of 
$\mathcal{M}_\varepsilon$ that are inherited from \eqref{eqSF}-\eqref{e2.10}. 
We point out that these properties will only be used in order to get 
precise exponents in the exponential decay rates in \eqref{eqturn}.
 More precisely, we will just use that $\mathcal{M}_\varepsilon$ may be
 assumed to be tangential to $\mathcal{M}_0$ at either one of the equilibria 
that we wish to connect (see \eqref{eqEquiv1} below). 
Therefore, depending on the reader's preference, this subsection may be 
skipped at first reading.

We observe that if $(w_1,w_2,\varphi_1,\varphi_2)$ solves \eqref{eqSF}-\eqref{e2.10}, 
then so do 
\begin{equation}\label{eqSymas}
(w_1,w_2,-\varphi_1,-\varphi_2)\quad \text{and}\quad 
(w_1,w_2,\pi-\varphi_1,-\varphi_2).
\end{equation} 
Then, by further assuming that $\mathcal{K}$ is symmetric with respect to 
the lines $\varphi_1=0$, $\varphi_1=\frac{\pi}{2}$ and $\varphi_2=0$,
 the invariant manifold $\mathcal{M}_\varepsilon$ can be constructed so that 
the flow on it preserves at least one of these two properties. 
More precisely, we may assume that one of the following identities holds:
\begin{equation}\label{eqEquiv1}
h_i\left(-\varphi_1,-\varphi_2,\varepsilon\right)
=h_i\left(\varphi_1,\varphi_2,\varepsilon\right)\quad \text{or} \quad
 h_i\left(\pi-\varphi_1,-\varphi_2,\varepsilon\right)
=h_i\left(\varphi_1,\varphi_2,\varepsilon\right),
\end{equation}
for $i=1,2$ and $\varepsilon \in [0,\varepsilon_0)$. In any case, we can always
 assume $h_i(\cdot,\cdot,\varepsilon)$, $i=1,2$, to be even with respect to 
$\varphi_2$.

This follows from the way that $\mathcal{M}_\varepsilon$ is constructed 
(see \cite{jones1995geometric}), which we briefly recall. Firstly, one appropriately 
modifies the last two equations of \eqref{eqSF}-\eqref{e2.10} outside of $\mathcal{K}$ and 
constructs a unique, three-dimensional, positively invariant center-stable 
manifold for that modified system (note that the last relation on 
page 67 of the aforementioned reference should be with the opposite sign). 
Similarly, one constructs a unique, three-dimensional, negatively invariant, 
center-unstable manifold for an analogous extension of \eqref{eqSF}-\eqref{e2.10}.
It is easy to see that these two modifications can be performed
while preserving one of the symmetries in \eqref{eqSymas}.
In turn, as a consequence of their uniqueness, the corresponding 
center-stable and center-unstable manifolds inherit the chosen symmetry. 
In particular, so does their intersection over $\mathcal{K}$, namely 
$\mathcal{M}_\varepsilon$. For related arguments, we refer the interested 
reader to \cite[Sec. 5.7]{chossat2000methods} and \cite[Ap. B]{haragus2010local}.

Let us henceforth assume that the locally invariant manifold 
$\mathcal{M}_\varepsilon$ enjoys the first symmetry in \eqref{eqSymas}, 
that is the first relation in \eqref{eqEquiv1} holds. However, as we will see,
 the second relation in \eqref{eqEquiv1} will be a-posteriori satisfied along 
the heteroclinic orbit on $\mathcal{M}_\varepsilon$ that we will construct 
in Theorem \ref{thmMan} below.

\section{Main result}\label{secMain}

We are now all set for our main result.
\begin{theorem}\label{thmMan}
For each $\varepsilon>0$ sufficiently small, there is a heteroclinic orbit 
$(w_{1,\varepsilon},w_{2,\varepsilon},\varphi_{1,\varepsilon},
\varphi_{2,\varepsilon})$ of \eqref{eqSF}-\eqref{e2.10} connecting the equilibria 
$(0,0,\pi/2,0)$ and \break
$(0,0,0,0)$ which lies on $\mathcal{M}_\varepsilon$.
 More precisely, the following estimates hold:
\begin{equation}\label{eqturn}
\begin{gathered}
 w_{1,\varepsilon}=\frac{\varphi_{2,\varepsilon}^2 
+ \big(\frac{1}{\lambda^2}+1\big)\sin^2\varphi_{1,\varepsilon}
 \cos^2\varphi_{1,\varepsilon}}{2[1+\big(\frac{1}{\lambda^2}-1\big)\cos^2 
\varphi_{1,\varepsilon} ]}
+\mathcal{O}(\varepsilon)\min\{e^{\frac{2x}{\lambda}},e^{-2x}\}, \\
w_{2,\varepsilon}=\mathcal{O}(\varepsilon)
\min\{e^{\frac{2x}{\lambda}},e^{-2x}\}, \\
\varphi_{i,\varepsilon}=\varphi_{i,0}+\mathcal{O}(\varepsilon)
 \min\{e^{\frac{x}{\lambda}},e^{-x}\},\quad i=1,2,
\end{gathered}
\end{equation}
uniformly in $\mathbb{R}$, as $\varepsilon \to 0$. Moreover, it holds 
\begin{equation}\label{eqThmMonot}
\varphi_{2,\varepsilon}<0.
\end{equation}
\end{theorem}

\begin{proof}
In light of the analysis in Subsection \ref{subsubLinearization}, each of the 
two equilibria has a two-dimensional (global) stable and unstable manifold,
 which is tangent at that point to the corresponding two-dimensional eigenspace 
in \eqref{eqEigen}. Let us call them $W_\varepsilon^s(0,0,\frac{\pi}{2},0)$, 
$W_\varepsilon^u(0,0,\frac{\pi}{2},0)$ and
 $W_\varepsilon^s(0,0,0,0)$, $W_\varepsilon^u(0,0,0,0)$.
The first two eigenvalues in each relation of \eqref{eqEVs} correspond to motion 
normal to $\mathcal{M_\varepsilon}$, while the latter two correspond to motion 
on $\mathcal{M_\varepsilon}$. The dynamical system within $\mathcal{M_\varepsilon}$ 
therefore has a saddle point at each of these equilibria, with one-dimensional 
stable and unstable manifolds given by 
$W_\varepsilon^s(0,0,\frac{\pi}{2},0)\cap \mathcal{M_\varepsilon}$, 
$W_\varepsilon^u(0,0,\frac{\pi}{2},0)\cap \mathcal{M_\varepsilon}$ and
 $W_\varepsilon^s(0,0,0,0)\cap \mathcal{M_\varepsilon}$,
 $W_\varepsilon^u(0,0,0,0)\cap \mathcal{M_\varepsilon}$. Our goal is to show 
that $W_\varepsilon^u(0,0,\frac{\pi}{2},0)\cap \mathcal{M_\varepsilon}$ and 
$W_\varepsilon^s(0,0,0,0)\cap \mathcal{M_\varepsilon}$ meet. Thus, since 
they are one-dimensional, they have to coincide.

 We begin by deriving the equations on $\mathcal{M}_\varepsilon$. 
By  \eqref{eqh}, the flow of \eqref{eqSF}-\eqref{e2.10} on $\mathcal{M}_\varepsilon$ is 
determined by a smooth, for $\varepsilon \in [0,\varepsilon_0)$, 
$\mathcal{O}(\varepsilon)$-regular perturbation of the reduced system 
\eqref{eqRedux}. We will refer to this as the \emph{$\varepsilon$-reduced system}.
 Thanks to \eqref{eqEquiv0}, the points $(\frac{\pi}{2},0)$ and $(0,0)$ are saddles 
for the $\varepsilon$-reduced system with associated linearized eigenvalues 
and eigenfunctions given by smooth $\mathcal{O}(\varepsilon)$-regular perturbations, 
for $\varepsilon \in [0,\varepsilon_0)$, of the corresponding ones at the end 
of Subsection \ref{subsecRedux}. Actually, as we have assumed the validity 
of the first condition in \eqref{eqEquiv1}, the corresponding linearization 
at $(0,0)$ is independent of $\varepsilon \in [0,\varepsilon_0)$. 
Our interest will be in the unstable manifold $W^u_\varepsilon(\frac{\pi}{2},0)$ 
of $(\frac{\pi}{2},0)$ and in the stable manifold $W^s_\varepsilon (0,0)$ of 
$(0,0)$. In fact, these are the projections to the $(\varphi_1,\varphi_2)$ plane 
of $W_\varepsilon^u(0,0,\frac{\pi}{2},0)\cap \mathcal{M_\varepsilon}$ and 
$W_\varepsilon^s(0,0,0,0)\cap \mathcal{M_\varepsilon}$, respectively.

 The manifolds $W^u_\varepsilon(\frac{\pi}{2},0)$ and $W^s_\varepsilon (0,0)$ 
depend smoothly on $\varepsilon \in [0,\varepsilon_0)$ (see for instance 
\cite[Ch. 9]{teschl2012ordinary}). From now on, with this notation, we will 
only refer to the parts of these invariant manifolds that shadow the 
heteroclinic orbit $(\varphi_{1,0}, \varphi_{2,0})$. Then,
 $W^u_\varepsilon(\frac{\pi}{2},0)$ and $W^s_\varepsilon (0,0)$ intersect the 
line $\phi_1=\frac{\pi}{4}$ at the points $(\frac{\pi}{4},\phi_{2,\varepsilon}^-)$
 and $(\frac{\pi}{4},\phi_{2,\varepsilon}^+)$, respectively, such that
 \begin{equation}\label{eqContradict0}
\phi_{2,\varepsilon}^\pm-\varphi_{2,0}(0)=\mathcal{O}(\varepsilon)\quad \text{as }
 \varepsilon \to 0,
 \end{equation}
(recall Subsection \ref{subsecRedux}).
 Let $\left(w_{1,\varepsilon}^-, w_{2,\varepsilon}^-,
\frac{\pi}{4},\phi_{2,\varepsilon}^-\right)$ and 
$\left(w_{1,\varepsilon}^+, w_{2,\varepsilon}^+,\frac{\pi}{4},
\phi_{2,\varepsilon}^+\right)$, respectively, be their lifting to 
$\mathcal{M}_\varepsilon$ for $\varepsilon \in [0,\varepsilon_0)$. 
The values $w_{i,\varepsilon}^\pm$, $i=1,2$, depend smoothly on 
$\varepsilon \in [0,\varepsilon_0)$; in particular, it holds
 \begin{equation}\label{eqContra2}
 w_{i,\varepsilon}^\pm-w_{i,0}=\mathcal{O}(\varepsilon),\quad
 i=1,2,\quad \text{as } \varepsilon \to 0,
 \end{equation}
 where $\left(w_{1,0},w_{2,0} \right)$ is the image of 
$\left(\frac{\pi}{4}, \varphi_{2,0}(0) \right)$ on the graph of $\mathcal{M}_0$.
 We will show that 
\begin{equation}\label{eqMeet}
w_{i,\varepsilon}^-=w_{i,\varepsilon}^+,\quad i=1,2,\quad \text{and}\quad
\phi_{2,\varepsilon}^-=\phi_{2,\varepsilon}^+ ,
\end{equation}
 provided that $\varepsilon>0$ is sufficiently small.

 Notice that we want to determine uniquely three variables, although \eqref{eqh} 
furnishes only two equations. The third equation will be provided by the 
hamiltonian identity \eqref{eqHam} (see also \cite{alikakos2007singular} 
for a related argument in a simpler problem).
Taking into account \eqref{eqEpsilon}, \eqref{eqxz}, \eqref{eqPolar}, \eqref{eqR}, 
and dividing by $\varepsilon^2/2$, we find that the identity \eqref{eqHam} becomes
 \begin{equation}\label{eqRHS}
\begin{split}
0&= \lambda^2\left[\varepsilon^2 w_2^2 \cos ^2 \varphi_1+(1-\varepsilon^2 w_1)^2
 \varphi_2^2 \sin^2\varphi_1+\varepsilon w_2(1-\varepsilon^2 w_1)
 \sin 2 \varphi_1 \right]
 \\
&\quad +\varepsilon^2 w_2^2 \sin ^2 \varphi_1+(1-\varepsilon^2 w_1)^2\varphi_2^2 
 \cos^2\varphi_1-\varepsilon w_2 (1-\varepsilon^2 w_1) \sin2 \varphi_1 \\
&\quad -\frac{\varepsilon^2}{2}(2w_1-\varepsilon^2 w_1^2)^2
 -\frac{1}{4}(1-\varepsilon^2 w_1)^4 \sin^22 \varphi_1,
\end{split}
\end{equation}
which is valid along trajectories of \eqref{eqSF}-\eqref{e2.10} on either one of 
$W_\varepsilon^{s/u}\left(0,0,\frac{\pi}{2},0\right)$ or \\
$W_\varepsilon^{s/u}\left(0,0,0,0\right)$, for $\varepsilon>0$.
Moreover, it will be important in the sequel to observe that, thanks to 
\eqref{eqHamReducia}, the above identity continues to hold for $\varepsilon=0$, 
i.e., along $(\varphi_{1,0},\varphi_{2,0})$.

We consider the smooth map 
$F: \mathbb{R}^2 \times \mathcal{K}\times [0,\infty)\to \mathbb{R}^3$ defined by
\[
F\begin{pmatrix}
 w_1 \\
 w_2 \\
 \varphi_1 \\
 \varphi_2 \\
 \varepsilon
 \end{pmatrix}
=\begin{pmatrix}
 w_1-\frac{\varphi_2^2 + (\frac{1}{\lambda^2}+1)
\sin^2\varphi_1\cos^2\varphi_1}{2[1+(\frac{1}{\lambda^2}-1)
\cos^2 \varphi_1]}-\varepsilon h_1(\varphi_1,\varphi_2,\varepsilon) 
 \\
 w_2-\varepsilon h_2(\varphi_1,\varphi_2,\varepsilon) 
 \\
 H(w_1,w_2,\varphi_1,\varphi_2, \varepsilon)
 \end{pmatrix},
\]
where $H$ is the function defined by the righthand side of \eqref{eqRHS}.
We observe that
\begin{equation}\label{eqContraFinale}
F\big(w_{1,\varepsilon}^\pm,w_{2,\varepsilon}^\pm,\frac{\pi}{4},
\phi_{2,\varepsilon}^\pm,\varepsilon \big)=(0,0,0),\quad \varepsilon 
\in (0,\varepsilon_0).
\end{equation}
Furthermore, it holds 
\begin{equation}\label{eqContra3}
F\big(w_{1,0},w_{2,0},\frac{\pi}{4},\phi_{2,0}(0),0 \big)=(0,0,0).
\end{equation}
Moreover, it follows readily that
\begin{equation}\label{eqital}
\partial_{w_1,w_2,\varphi_2}F\begin{pmatrix}
 w_1 \\
 w_2 \\
 \varphi_1 \\
 \varphi_2 \\
 0
 \end{pmatrix}
=\begin{pmatrix}
 1 & 0 & -\frac{\varphi_2}{1+(\frac{1}{\lambda^2}-1)\cos^2 \varphi_1} \\
 0 & 1 & 0 \\
 0 & 0 & \lambda^2 \varphi_2 \sin^2\varphi_1 +\varphi_2 \cos^2\varphi_1
 \end{pmatrix}.
\end{equation}
In particular, this matrix is invertible at the point 
$\left(w_{1,0},w_{2,0},\frac{\pi}{4},\varphi_{2,0}(0),0 \right)$.
Thus, recalling \eqref{eqContra3}, we deduce by the implicit function theorem 
that there exists $\delta>0$ such that, for 
$\varphi_1 \in \left(\frac{\pi}{4}-\delta,\frac{\pi}{4}+\delta \right)$ 
and $\varepsilon \in [0, \delta)$, the equation
\[
F\left(w_1,w_2,\varphi_1,\varphi_2,\varepsilon \right)=(0,0,0)
\]
has at most one solution $(w_1,w_2,\varphi_2)$ such that $|w_i-w_{i,0}|<\delta$, 
for $i=1,2$, and $|\varphi_2-\varphi_{2,0}(0)|<\delta$.
 Hence, applying this property for $\varphi_1=\frac{\pi}{4}$, we infer 
from \eqref{eqContradict0}, \eqref{eqContra2} and \eqref{eqContraFinale} 
that the desired relation  \eqref{eqMeet} is true, provided that 
$\varepsilon>0$ is sufficiently small.


Let $\left(w_{1,\varepsilon},w_{2,\varepsilon},\varphi_{1,\varepsilon},
\varphi_{2,\varepsilon} \right)$
denote the heteroclinic connection of \eqref{eqSF}, \eqref{eqSFbdryu} on 
$\mathcal{M}_\varepsilon$ which passes through the point
$\left(w_{1,\varepsilon}^+,w_{2,\varepsilon}^+,\frac{\pi}{4},
\phi_{2,\varepsilon}^+ \right)$ at $x=0$. We will first establish the 
validity of properties \eqref{eqPhi1pos} and \eqref{eqThmMonot}. 
For this purpose, we recall that the trajectory curve of 
$(\varphi_{1,\varepsilon},\varphi_{2,\varepsilon})$ on the 
$(\varphi_{1},\varphi_{2})$ phase plane is given by 
$W_\varepsilon^u\left(\frac{\pi}{2},0\right)\cap W^s_\varepsilon(0,0)$, 
and varies smoothly for $\varepsilon \geq 0$ small. The asserted properties 
now follow at once from the fact that the limiting curve 
$W_0^u\left(\frac{\pi}{2},0\right)\cap W^s_0(0,0)$ is
contained in the half-strip $\mathcal{S}=\left\{ 0\leq \varphi_1 
\leq \frac{\pi}{2},\; \varphi_2\leq 0 \right\}$, 
and touches the boundary of $\mathcal{S}$ only at $\left(0,0 \right)$ and 
$\left(\frac{\pi}{2},0 \right)$ in a non-tangential manner
(keep in mind the linearized analysis from the end of Subsection \ref{subsecRedux}).

We next turn our attention to the last relation in \eqref{eqturn}. 
We will first show it for $x\geq 0$. To this end, we will need the preliminary 
estimates
\begin{equation}\label{eqPrelim}
\varphi_{i,\varepsilon}(x) =(-1)^{i-1}a_+ \left(1+o(1)\right) e^{-x},\quad
 i=1,2, \quad \text{as } x\to +\infty,
\end{equation}
where the constant $a_+> 0$ is independent of small $\varepsilon >0$, 
and these limits hold uniformly with respect to $\varepsilon$. 
The above relation follows directly from the refined version of the stable 
manifold theorem in \cite[Thm. 4.3, Ch. 13]{coddington1955theory}; recall that 
the linearization of the $\varepsilon$-reduced system at $(0,0)$ 
has eigenvalues $\pm 1$ for $\varepsilon \geq 0$ small. 
The latter property about the linearized problem implies that the pair 
$\Psi_\varepsilon=(\psi_{1,\varepsilon},\psi_{2,\varepsilon})$, where
\[
\psi_{i,\varepsilon}=\frac{\varphi_{i,\varepsilon}-\varphi_{i,0}}{\varepsilon},\quad
 i=1,2,
\]
satisfies 
\begin{gather*}
 \Psi_\varepsilon'=A\Psi_\varepsilon
+\mathcal{O}\left(\varepsilon|\Psi_\varepsilon|^2\right)
+\mathcal{O}\left(\varphi_{1,\varepsilon}^2+\varphi_{2,\varepsilon}^2 \right),\quad
 x\geq 0; \\
\Psi_\varepsilon(0)=\mathcal{O}(1),\quad \Psi_\varepsilon(\infty)=0,
 \end{gather*}
with the obvious notation, uniformly as $\varepsilon \to 0$, where $A$ 
is the aforementioned linearized matrix (recall also \eqref{eqContradict0}).
Then, using \eqref{eqPrelim} to estimate the last term in the righthand 
side and working as in the previously mentioned stable manifold theorem 
in \cite{coddington1955theory}, we obtain that
\[
|\Psi_\varepsilon(x)|\leq Ce^{-x},\quad x\geq 0,
\]
for some constant $C>0$ independent of small $\varepsilon>0$, which implies 
the validity of the last relation of \eqref{eqturn} for $x\geq 0$. 
In turn, the corresponding estimates in the first two relations of \eqref{eqturn} 
follow at once via the second identity in \eqref{eqEquiv0} and the first 
one in \eqref{eqEquiv1}.

 The sole obstruction in showing the corresponding estimates for $x\leq 0$ 
is that the linearization of the $\varepsilon$-reduced system at 
$(\pi/2,0 )$ may not be independent of $\varepsilon$ 
(recall that we could only choose one of the symmetries in \eqref{eqSymas}). 
Nevertheless, this can be surpassed easily by noting that the 
constructed heteroclinic connection of \eqref{eqSF}-\eqref{e2.10} on 
$\mathcal{M}_\varepsilon$ 
should also be on an analogous invariant manifold $\tilde{\mathcal{M}}_\varepsilon$ 
which enjoys the second symmetry in \eqref{eqSymas} 
(recall the concluding remark in Subsection \ref{subsubPersist}), provided 
that $\varepsilon>0$ is sufficiently small. Then, the arguments 
for $x\leq 0$ go through as before. In passing, we note that the graphs of 
$\mathcal{M}_\varepsilon$ and $\tilde{\mathcal{M}}_\varepsilon$ over 
$\mathcal{K}$ have the same expansion in powers of $\varepsilon$ up to any 
order (see \cite[Ch. 3]{kuhen} for more details).
The proof is complete.
\end{proof}

\begin{remark} \label{rmk4.1} \rm
We suspect that the calculation in \eqref{eqital} provides the required 
nondegeneracy condition in
\cite[Sec. 5]{mackay2004slow}
which allows to choose $\mathcal{M}_\varepsilon$ so that the corresponding 
$\varepsilon$-reduced system is hamiltonian (in $p=\cos \varphi_1$, 
$q=\sin \varphi_1$).
\end{remark}


\begin{remark} \label{rmk} \rm 
From the invariance of $\mathcal{M}_\varepsilon$ and the equation 
$w_2=\varepsilon w_1'$, via the second equation of \eqref{eqh}, we obtain that
\begin{align*}
 \frac{w_{2,\varepsilon}}{\varepsilon} 
& =  -2\big(1-\frac{1}{\lambda^2}\big)\varphi_{2,\varepsilon}
 \sin \varphi_{1,\varepsilon} \cos \varphi_{1,\varepsilon}\frac{\varphi_2^2 
 + \big(\frac{1}{\lambda^2}+1\big)\sin^2\varphi_{1,\varepsilon}
 \cos^2\varphi_{1,\varepsilon}}{[1+\big(\frac{1}{\lambda^2}-1\big)
 \cos^2 \varphi_{1,\varepsilon} ]^2} \\
&\quad +\frac{\varphi_{2,\varepsilon}}{[1+\big(\frac{1}{\lambda^2}-1\big)
 \cos^2 \varphi_{1,\varepsilon} ]}
 \Big(\sin \varphi_1 \cos^3 \varphi_1-\frac{1}{\lambda^2} 
 \cos \varphi_1 \sin^3 \varphi_1\Big) \\
&\quad +\frac{1}{2}\big(1+\frac{1}{\lambda^2} \big)
 \varphi_{2,\varepsilon}\frac{\sin 2\varphi_{1,\varepsilon}
 -4\cos\varphi_{1,\varepsilon}\sin^3\varphi_{1,\varepsilon}}
 {1+\big(\frac{1}{\lambda^2}-1\big)\cos^2 \varphi_{1,\varepsilon} } \\
&\quad  +\mathcal{O(\varepsilon)}\min\{e^{\frac{2x}{\lambda}},e^{-2x} \},
 \end{align*}
uniformly in $\mathbb{R}$ as $\varepsilon \to 0$. Analogously, we can refine
 the $w_1$ component of the constructed heteroclinic. Then, plugging these 
refinements in the $\varepsilon$-reduced system, we can refine the 
$\varphi_1,\varphi_2$ components too (by the solution of a linear inhomogeneous
 problem), and so on. We note, however, that formally the correct spatial 
decay in the above relation should be 
$\min\{e^{\frac{3x}{\lambda}},e^{-3x} \}$. This observation points in the 
direction that $\mathcal{M}_\varepsilon$ should be close beyond all orders 
of $\varepsilon$ to $\mathcal{M}_0$ at the two equilibria (recall the proofs 
of the corresponding decay estimates in \eqref{eqturn} and the concluding 
remark in the proof of Theorem \ref{thmMan}).
\end{remark}

\section{Further properties of the constructed heteroclinic connection}
\label{secEnergy}
\subsection{Variational characterization}
In view of \eqref{eqThmMonot}
and the comments leading to \eqref{eqARCT}, we expect that the corresponding 
solution to \eqref{eqEqGen}-\eqref{eqBdryGen}, provided by Theorem \ref{thmMan} 
via the transformations \eqref{eqEpsilon}, \eqref{eqxz}, \eqref{eqPolar}, 
\eqref{eqR} and \eqref{eqTransFi}, minimizes the associated energy.
By the uniqueness result of \cite{aftalionSourdis} that we mentioned in the 
introduction, to verify this, it suffices to show that one of its components 
satisfies the corresponding monotonicity property in \eqref{eqmonotS}.
For this purpose, we note that
\[
u'=-\varepsilon w_2 \cos\varphi_1- (1-\varepsilon^2 w_1)\varphi_2\sin\varphi_1.
\]
Hence, by virtue of \eqref{eqturn} and \eqref{eqThmMonot}, given any fixed interval 
$I$, it holds $u'>0$ in $I$ for sufficiently small $\varepsilon>0$.
 We infer that $u'>0$ outside of $I$ by means of \eqref{eqPrelim} 
(and the analogous relation for $x\leq 0$).
 Alternatively, similarly to \cite{aftalionSourdis}, we just have to fix a 
sufficiently large $I$ so that we can apply the maximum principle componentwise 
in the linear elliptic system for $u',v'$ in $\mathbb{R}\setminus I$ 
(note that such an interval can be chosen to be independent of $\varepsilon$).

\subsection{Energy expansion}
By exploiting the above observation and making mild use of the estimates in 
Theorem \ref{thmMan}, we are in position to give an asymptotic expression 
for the minimal energy of the heteroclinic connection problem 
\eqref{eqEqGen}-\eqref{eqBdryGen} as $\Lambda \to 1^+$. The limiting value 
of the minimal energy, appropriately renormalized (so that it does not 
converge to zero), was identified rigorously very recently in 
\cite{goldman2015phase}, using the variational technique of $\Gamma$-convergence. 
We recover their result but also provide a rate of convergence to this minimal value.

\begin{proposition}\label{proEnergy}
Let
$\sigma_\Lambda=\inf_\mathcal{X} E_\Lambda(u,v)$,
where
\begin{gather*}
E_\Lambda(u,v)=\int_{-\infty}^{\infty}\big[\lambda^2\frac{(\dot{u})^2}{2}
+\frac{(\dot{v})^2}{2}+\frac{(1-u^2-v^2)^2}{4}+\frac{\Lambda-1}{2}u^2v^2 \big]dz,\\
\mathcal{X}=\big\{(u, v)\in W^{1,2}_{loc}(\mathbb{R})\times W^{1,2}_{loc}(\mathbb{R})
  \text{ satisfying } \eqref{eqBdryGen} \big\}.
\end{gather*}
It holds
\[
\sigma_\Lambda= \frac{1}{3} \frac{1-\lambda^3}{1-\lambda^2} 
(\Lambda-1)^{1/2} +\mathcal{O}\left(\Lambda-1 \right)\quad
 \text{as } \Lambda \to 1^+,
\]
with the obvious meaning for $\lambda=1$.
\end{proposition}

\begin{proof}
It follows from \eqref{eqRHS}, paying attention to the comment leading to it, that
\[
\sigma_\Lambda= \frac{1}{4} \Big(\int_{-\infty}^{\infty}\sin^2(2\varphi_{1,0} 
)dx\Big) (\Lambda-1)^{1/2} +\mathcal{O}\left(\Lambda-1 \right)\quad
 \text{as }\Lambda \to 1^+,
\]
where $\varphi_{1,0}$ is the prescribed solution of \eqref{eqRB}.
It therefore remains to compute the above integral. Using \eqref{eqHamReducia}, 
we find that
\begin{align*}
\int_{-\infty}^{\infty}\sin^2\left(2\varphi_{1,0}\right) dx
=& -2\lambda\int_{-\infty}^{\infty}\sin\left(2\varphi_{1,0}\right)
\big[1+\big(\frac{1}{\lambda^2}-1\big)\cos^2\varphi_{1,0} \big]^{1/2}
 \varphi_{1,0}'dx \\
=&2\lambda \int_{0}^{1}\big[1+\big(\frac{1}{\lambda^2}-1\big)t \big]^{1/2}dt \\
=& \frac{4}{3}\frac{1-\lambda^3}{1-\lambda^2},
 \end{align*}
which implies the assertion of the proposition.
\end{proof}

\subsection*{Acknowledgments} 
I would like to thank Prof. Aftalion for bringing this problem to my attention 
and for useful discussions. Moreover, I would like to thank Prof. Scheel 
for bringing the paper \cite{van2000domain} to my attention, from the references 
therein I also found out about \cite{malomed1990domain} and \cite{van1997domain}. 
Lastly, we would like to thank the anonymous referee for some suggestions. 
This project has received funding from the European Union's Horizon 2020 
research and innovation programme under the Marie Sk\l{}odowska-Curie grant 
agreement No 609402-2020 researchers: Train to Move (T2M).

\begin{thebibliography}{00}

\bibitem{aftalionSourdis}
Amandine Aftalion, Christos Sourdis; \emph{Interface layer of a
  two-component bose--einstein condensate}, Communications in Contemporary
  Mathematics \textbf{19} (2017), no.~05, 1650052.

\bibitem{alamaARMA15}
Stan Alama, Lia Bronsard, Andres Contreras, Dmitry~E. Pelinovsky;
  \emph{Domain walls in the coupled {G}ross--{P}itaevskii equations}, Archive
  for Rational Mechanics and Analysis \textbf{215} (2015), no.~2, 579--610.

\bibitem{alikakos2007singular}
ND~Alikakos, PC~Fife, G~Fusco,  C~Sourdis;
 \emph{Singular perturbation
  problems arising from the anisotropy of crystalline grain boundaries},
  Journal of Dynamics and Differential Equations \textbf{19} (2007), no.~4,
  935--949.

\bibitem{barankov}
RA~Barankov; 
\emph{Boundary of two mixed {B}ose-{E}instein condensates},
  Physical Review A \textbf{66} (2002), no.~1, 013612.

\bibitem{berglund2006noise}
Nils Berglund, Barbara Gentz;
 \emph{Noise-induced phenomena in slow-fast
  dynamical systems: a sample-paths approach}, Springer Science \& Business
  Media, 2006.

\bibitem{chossat2000methods}
Pascal Chossat, Reiner Lauterbach;
 \emph{Methods in equivariant bifurcations
  and dynamical systems}, World Scientific Publishing Co., Inc., 2000.

\bibitem{coddington1955theory}
Earl~A Coddington and Norman Levinson;
 \emph{Theory of ordinary differential
  equations}, Tata McGraw-Hill Education, 1955.

\bibitem{farina2017monotonicity}
Alberto Farina, Berardino Sciunzi, Nicola Soave;
\emph{Monotonicity and
  rigidity of solutions to some elliptic systems with uniform limits}, arXiv
  preprint arXiv:1704.06430 (2017).

\bibitem{fenichelJDE}
N~Fenichel;
 \emph{Geometric singular perturbation theory for ordinary
  differential equations}, J. Differential Equations \textbf{31} (1979),
  53--98.

\bibitem{goldman2015phase}
Michael Goldman and Beno{\i}t Merlet;
 \emph{Phase segregation for binary
  mixtures of {B}ose--{E}instein condensates}, SIAM Journal on Mathematical
  Analysis \textbf{49} (2017), no.~3, 1947--1981.

\bibitem{haragus2010local}
Mariana Haragus and G{\'e}rard Iooss;
 \emph{Local bifurcations, center
  manifolds, and normal forms in infinite-dimensional dynamical systems},
  Springer Science \& Business Media, 2010.

\bibitem{jones1995geometric}
Christopher~KRT Jones;
 \emph{Geometric singular perturbation theory}, Dynamical
  systems, {Lecture Notes in Mathematics 1609}, Springer, 1995, pp.~44--118.

\bibitem{kuhen}
Christian Kuehn; \emph{{M}ultiple {T}ime {S}cale {D}ynamics}, {A}pplied
  {M}athematical {S}ciences, vol. 191, Springer, 2015.

\bibitem{mackay2004slow}
RS~MacKay;
 \emph{Slow manifolds}, Energy localisation and Transfer, vol.~22,
  World Scientific, 2004, pp.~149--192.

\bibitem{malomed1990domain}
Boris~A Malomed, Alexander~A Nepomnyashchy,  Michael~I Tribelsky;
  \emph{Domain boundaries in convection patterns}, Physical Review A
  \textbf{42} (1990), no.~12, 7244.

\bibitem{pismen2006patterns}
Len~M Pismen;
 \emph{Patterns and interfaces in dissipative dynamics}, Springer
  Science \& Business Media, 2006.

\bibitem{teschl2012ordinary}
Gerald Teschl;
 \emph{Ordinary differential equations and dynamical systems},
  vol. 140, American Mathematical Society Providence, RI, 2012.

\bibitem{van2000domain}
GJB Van~den Berg and RCAM Van~der Vorst;
 \emph{A domain-wall between
  single-mode and bimodal states}, Differential and Integral Equations
  \textbf{13} (2000), no.~1-3, 369--400.

\bibitem{van1997domain}
M~Van~Hecke and BA~Malomed;
 \emph{A domain wall between single-mode and bimodal
  states and its transition to dynamical behavior in inhomogeneous systems},
  Physica D: Nonlinear Phenomena \textbf{101} (1997), no.~1, 131--156.

\bibitem{vaninterface}
Bert Van~Schaeybroeck;
 \emph{Interface tension of {B}ose-{E}instein
  condensates}, Physical Review A, \textbf{78} (2008), no.~2, 023624.

\end{thebibliography}

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%