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gradient flow, Ginzburg-Landau, magnetic vortex
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\documentclass[11pt]{article}
\usepackage{amsmath,amsfonts,amssymb}
\begin{document}
\newtheorem{theorem}{Theorem}
\newtheorem{proposition}{Proposition}
\newtheorem{lemma}{Lemma}
\tolerance=1000
\title{\bf Effective Dynamics of a Magnetic Vortex in a Local potential}
\author{Y. Strauss${}^{\, a)\,c)}$ and I.M. Sigal${}^{\, a)\,b)}$}
\date{}
\maketitle
\begin{itemize}
\item[{\small a)}] {\footnotesize Department of Mathematics, University of Toronto,\\
Toronto, ON, Canada, M5S 3G3. URL: www.math.toronto.edu/sigal}
\item[{\small b)}] {\footnotesize Department of Mathematics, University of Notre Dame,\\
Notre Dame, IN, USA, 46556-4618.}
\item[{\small c)}] {\footnotesize Einstein Institute of Mathematics, The Hebrew University
of Jerusalem,\\ Jerusalem, Israel 91904. E-mail: ystrauss@math.huji.ac.il}
\end{itemize}
\maketitle
\begin{abstract}
We study simplified mean field model of superconductor dynamics in the
presence of impurities or for variable superconductor depth. This model is
given by the gradient-flow version of the Ginzburg-Landau equations
(Grokov-Eliashberg equations) with an addition of a potential term. We find
a dynamical law of motion of the vortex center, involving the potential, such
that for datum close to a (static) magnetic vortex the solution
is close, for all times, to a magnetic vortex whose center obeys this law.
\end{abstract}
%
\section{Introduction}
\label{intro}
\numberwithin{equation}{section}
\subsection{Preliminary discussion and statement of the problem}
\label{prob}
\par In this work we consider the gradient flow version of the Ginzburg-Landau
equations in the presence of a local potential. Specifically, we study certain
solutions of the system of equations
%
\begin{equation}
 \begin{split}
   \partial_t\psi&=\Delta_A\psi+\frac{\lambda}{2}
   (1-\vert \psi\vert^2)\psi-\epsilon W\psi\\
   \partial_t A&= -\nabla\times\nabla\times
   A+Im(\overline \psi\nabla_A\psi)\\
 \end{split}
\tag{1.1}
\end{equation}
%
where $(\psi(t), A(t)): \mathbb R^2\mapsto \mathbb C\times \mathbb R^2$
for each $t\geq 0$, $\nabla_A=\nabla-iA$ and
$\Delta_A=\nabla^2_A$. The function $W: \mathbb R^2\mapsto \mathbb R$ is a localized
potential and is assumed to be smooth and sufficiently fast decaying at
infinity. The parameter $\epsilon$ measures the strength of the potential. In
the study of superconductivity
the function $\psi$ is the order parameter with $\vert\psi\vert$ measuring the
density of superconducting Cooper pairs and the vector field $A$ is the vector
potential for the magnetic field. The potential $W(x)$ arises with impurities,
variations of superconductor geometry or other inhomogeneities in the matter.
\par Setting $\epsilon=0$ in Eq. (1.1), one obtains the following
set of coupled equations for $\psi$ and $A$
%
\begin{equation}
 \begin{split}
   \partial_t\psi&=\Delta_A\psi
   +\frac{\lambda}{2}(1-\vert\psi\vert^2)\psi\\
   \partial_t A&=-\nabla\times\nabla\times A+{\rm Im}\,(\overline\psi
   \nabla_A\psi)\,.\\
 \end{split}
\tag{1.2}
\end{equation}
%
This system of equations is used to describe the dynamics of superconductors
in the mean field approximation and near the normal-superconducting state
transition point. They are
called the {\it Gorkov-Eliashberg} equations or {\it time-dependent Ginzburg-Landau}
(TDGL) equations \cite{GE,T}.
\par The global well--posedness of the TDGL equations was shown by Demoulini
and Stuart [DS]. Their proof can easily be extended to Eq. (1.1). This will be
outlined elsewhere.
\par The stationary version of the TDGL equations is the Ginzburg-Landau (GL)
equations
%
\begin{equation}
 \begin{split}
   \Delta_A\psi+\frac{\lambda}{2}(1-\vert\psi\vert^2)\psi&=0\\
   -\nabla\times\nabla\times A+Im(\overline\psi\nabla_A\psi)&=0\,.\\
 \end{split}
 \tag{1.3}
\end{equation}
%
Let $u=(\psi,A)$. Denote by ${\cal E}_{GL}(u)$ the Ginzburg-Landau energy functional
%
$$ {\cal E}_{GL}(u)=\frac{1}{2}\int_{\mathbb R^2}\left[ \vert\nabla_A\psi\vert^2
   +\vert\nabla \times A\vert^2+\frac{\lambda}{4}(\vert\psi\vert^2-1)^2\right]
   \,.\eqno(1.4)$$
%
The Ginzburg-Landau equations, Eq. (1.3) are the Euler-Lagrange equations
derived from ${\cal E}_{GL}$.
\par Superconductors are characterized by two length scales called
{\it penetration depth} and {\it coherence depth}. In the scaling units
corresponding to the form of ${\cal E}_{GL}$ given in Eq. (1.4) the penetration
depth, measuring the scale of variations in the magnetic field, is equal to
1. The coherence length, measuring the scale of variations in the order
parameter $\psi$ is $1/m_\lambda$ where $m_\lambda=\min(\sqrt{\lambda},2)$.
For $\lambda=1$ the two length scales are equal and we distinguish between
type I superconductors, for which $\lambda<1$ and type II superconductors
for which $\lambda>1$. Experimentally, type I and type II materials differ
in their magnetic behavior. In type I superconductors magnetic fields are
excluded from the bulk of the material except for a very thin 
layer near the surface. In type II superconductors magnetic fields penetrate
the material in vortex structures. In general, Type II superconductors can
sustain magnetic fields much higher than type I superconductors without
losing their superconducting state.
The existence of magnetic vortices and of type II superconductors was
predicted in 1957 by Abrikosov \cite{A}(in addition Abrikosov predicted the
existence of large arrays of magnetic vortices, called Abrikosov lattices, in
type II materials).
\par Configurations $(\psi,A)$ with finite energy are classified by
the Brouwer degree of $\psi$, i.e., the topological degree
%
$$ deg(\psi):=deg\left(\frac{\psi}{\vert\psi\vert}
   {\bigg\vert}_{\vert x\vert=R}\right),\qquad deg(\psi)\in \mathbb Z$$
%
where $R$ is large enough. For the magnetic field $B=\nabla\times A$ the
degree, $deg(\psi)$, is related to the magnetic flux as
%
$$ \int_{\mathbb R^2} B=2\pi\deg(\psi)\in \mathbb Z$$
%
(quantization of magnetic flux).
\par Magnetic (Abrikosov) vortices are equivariant solutions of Eq. (1.3) of
the form
%
$$ \psi^{(n)}(x)=f_n(r)e^{i n\theta},\qquad
   A^{(n)}(x)=a_n(r)\nabla(n\theta)
   \eqno(1.5)$$
%
where $(r,\theta)$ are the polar coordinates of $x\in\mathbb R^2$ and $n\in\mathbb Z$.
Calculation of the degree of $\psi^{(n)}$ for a magnetic vortex solution given
by Eq. (1.5) yields $deg(\psi^{(n)})=n$. The pair $(\psi^{(n)},A^{(n)})$ is called
an $n$-vortex. The existence of $n$-vortices was proved by Plohr \cite{P} and
by Berger and Chen \cite{BC} by variational methods. Stability of $n$-vortices
was proved by Gustafson and Sigal \cite{GS1} and Gustafson \cite{G2}. It is
shown in \cite{JT} that $0<f_n<1$, $0<a_n<1$ and that the asymptotic behavior
for $a_n(r)$ and $f_n(r)$ as $r\to\infty$ is given by
%
$$ f_n(r)=1+O(e^{-m_\lambda r}),\qquad a_n(r)=1+O(e^{-r})\,.$$
%
It is also known that $f_n(r)$ vanishes like $r^n$ and $a_n(r)$ vanishes like
$r^2$ as $r$ goes to the origin.
\par For an $n$-vortex $v^{(n)}=(\psi^{(n)}, A^{(n)})$
the GL energy functional ${\cal E}_{GL}$ is a smooth functional on the affine
space given by
%
$$ X^{(n)}\equiv\left\{ u: \mathbb R^2\to \mathbb C\times \mathbb R^2
   \mid u-v^{(n)}\in H^1(\mathbb R^2;\mathbb C\times \mathbb R^2)\right\}\,.
   \eqno(1.6)$$
\par As mentioned above, type II superconductors can sustain very large
magnetic fields (over $10^5$ Gauss). However, a major obstacle in the attempt
to produce large magnetic fields is the
dissipation of energy due to the {\it creeping} or {\it flow} of vortices
\cite{T}. One way to overcome this problem is to {\it pin down} the vortices
to particular locations in the material.
The pinning down of vortices is achieved by the presence of
point defects, impurities, inhomogeneities or by a variation in the thickness
of the sample of superconducting material \cite{DG}. Rubinstein \cite{Rub1}
considered a model of the effect of pinning in which the quartic term in
${\cal E}_{GL}$ is replaced by the modified term
%
$$ \frac{\lambda}{2}(\vert\psi\vert^2-f^2(x))^2\,.$$
%
The function $f(x)$ can be thought of as measuring the quality of the
superconductor (consider for example the large $\lambda$ limit).
\par In \cite{BBH} Bethuel, Brezis and H\'elein study the large $\lambda$ limit of
a simplified version of the GL functional without magnetic field: 
%
$$ {\cal E}_\epsilon(\psi)=\int_\Omega \left[\frac{1}{2}\vert\nabla\psi\vert^2
   +\frac{1}{4\epsilon^2}(\vert\psi\vert^2-1)^2\right]\, ,$$
%
where $\Omega$ is a simply connected bounded domain in $\mathbb R^2$. Since there is
no magnetic field in the problem one needs to introduce a mechanism for the
confinment of vortices to $\Omega$. The minimization problem for
${\cal E}_\epsilon(\psi)$ is considered, therefore, over all $\psi$ satisfying
the Dirichlet boundary condition
%
$$ \psi(x)=g(x),\qquad x\in\partial\Omega$$
%
where $g:\partial\Omega\to S^1$ satisfies $deg\,g=d\not=0$. In the general
context of the Bethuel-Brezis-H\'elein theory Andre and
Shafrir \cite{AS} analyzed a model for the pinning of vortices in superconductors
with small variable thickness which was proposed by Du and Gunzburger \cite{DG}.
This model for the pinning effect involves a weighted GL functional of the
form
%
$$ {\cal E}_\epsilon(\psi)=\int_\Omega p(x)\left[\frac{1}{2}
   \vert\nabla\psi\vert^2
   +\frac{1}{4\epsilon^2}(\vert\psi\vert^2-1)^2\right]\,.$$
%
For this model Andre and Shafrir proved the Bethuel-Brezis-H\'elein type result
that the set of points $a_1,\ldots,a_d$, on which the limiting $\psi_\epsilon$
is concentrated, consists of points (possibly repeating) in which $p$ achieves
its global minimum.
\par More recently the effect of pinning was studied by Aftalion, Sandier and
Serfaty \cite{ASaSe}, Andre, Bauman and Philips \cite{ABP} and I.M. Sigal and
F. Ting \cite{ST1}. In addition, Sigal and Ting investigated the stability of
pinned vortices \cite{ST2}.
\par Our goal in the present paper is to provide a description of the
dynamics of the pinning effect, i.e., we aim to derive rigorously the law
governing the dynamics of a vortex in the presence of a potential.
Formally speaking, this constitutes
a reduction of the dynamics of the various fields in the problem to an
effective dynamics of a smaller number of degrees of freedom, i.e, the vortex
center and the gauge function. The methods utilized for the proof of the main
theorem in the current work are close to those used by Gustafson
and Sigal in their study of the dynamics of magnetic vortices \cite{GS2}. We
refer to \cite{GS2} for a short survey of various
rigorous and non-rigorous results concerning the dynamics of vortices in
various time dependent versions of the GL equations. In particular, we mention
here the non-rigorous results for the superconducting model of Eq. (1.2)
obtained by Perez and Rubinstein \cite{PeRu}
and W. E \cite{E} and the rigorous results obtained, for the same equations,
by Gustafson and Sigal \cite{GS2}. A simplified version of the TDGL
equations, called the {\it nonlinear heat flow} (NLHF) equation, in which
$A\equiv 0$, was studied by Neu \cite{N}, Lin \cite{L}, Jerard and Soner
\cite{JS} and Rubinstein and Sternberg \cite{RS} (see also \cite{Rub2} and
references therein).
\par We conclude this subsection by noting that reviews of the
Ginzburg-Landau theory of superconductivity can be found, for example, in
\cite{B, BFGLV, G1, JT, Rit, Riv, Rub2}.
%
\subsection{Symmetries of the Ginzburg-Landau equations}
\label{symmetry}
\par The Ginzburg-Landau energy functional ${\cal E}_{GL}$ is invariant under
the following symmetries:
%
\begin{itemize}
\item[(i)] Translation symmetry $x\to x'=x+z$
%
$$ \psi(x)\to\psi_z(x)=\psi(x-z),\qquad A(x)\to A_z(x)=A(x-z)\,;$$
%
\item[(ii)] Gauge symmetry 
%
$$ \psi(x)\to \psi_\gamma(x)=e^{i\gamma(x)}\psi(x),
   \qquad A(x)\to A_\gamma(x)=A(x)+\nabla\gamma(x)\,;$$
%
\item[(iii)] Rotation Symmetry $x\to x'=Rx$, $R\in SO(2)$
%
$$ \psi(x)\to \psi_R(x)=\psi(R^{-1}x),\qquad
   A(x)\to A_R(x)=RA(R^{-1}x)\,.$$
%
\end{itemize}
%
\par\noindent We note that for an $n$-vortex solution the application of the
rotation symmetry group correspond to a subgroup of the gauge symmetry.
Therefore, for $n$-vortices the relevant symmetry operations are the
translation and gauge groups. Accordingly, given an $n$-vortex 
$v^{(n)}=(\psi^{(n)},A^{(n)})$, we can apply translation symmetry with
translation parameters ${\bf z}=(z_1,z_2)\in\mathbb R^2$ and gauge symmetry with a
gauge function $\gamma$ and find another solution $v_{{\bf z}\gamma}$ of
Eq. (1.3) where
%
$$ v_{{\bf z}\gamma}=(\psi_{{\bf z}\gamma},\,A_{{\bf z}\gamma})
   =(e^{i\gamma}\psi^{(n)}(\cdot-{\bf z}),
   A^{(n)}(\cdot-{\bf z})+\nabla\gamma)
   \eqno(1.7)$$
%
with $\psi_{{\bf z}\gamma}\equiv e^{i\gamma}\psi^{(n)}(\cdot-{\bf z})$ and
$A_{{\bf z}\gamma}\equiv A^{(n)}(\cdot-{\bf z})+\nabla\gamma$. The latter
functions can be also written as $\psi_{{\bf z}\gamma}=e^{i\gamma}\psi_{{\bf z} 0}$
and $A_{{\bf z}\gamma}= A_{{\bf z}0}+\nabla\gamma$, where 
$\psi_{{\bf z} 0}\equiv \psi^{(n)}(\cdot-{\bf z})$ and
$A_{{\bf z} 0}\equiv A^{(n)}(\cdot-{\bf z})$. For reasons that will become
clear below we consider gauge transforms of the form
%
$$ \gamma=\sum_{i=1,2} z_i A^{(n)}_i(\cdot-{\bf z})+\tilde\gamma
   \eqno(1.8)\,.$$
%
Then, according to Eq. (1.6) and Eq. (1.7) we require that 
$\tilde\gamma\in H^2(\mathbb R^2; \mathbb R)$. In this case $v_{{\bf z}\gamma}$ in
Eq. (1.7) will always be in $X^{(n)}$. Applying symmetry transformations
in the form of Eq. (1.7) for all possible values of ${\bf z}$ and $\gamma$ we
obtain the symmetry manifold of the $n$-vortex
%
$$ M_{sym}^{(n)}=\left\{ v_{{\bf z}\gamma}\mid {\bf z}\in\mathbb R^2,
   \tilde\gamma\in H^2(\mathbb R^2;\mathbb R)\right\}
   \eqno(1.9)$$
%
with $\tilde\gamma$ given in Eq. (1.8). It is obvious that all points in
$M_{sym}^{(n)}$ are solutions of the Ginzburg-Landau equations, Eq. (1.3).
\par In order to obtain Eq. (1.1) we add to ${\cal E}_{GL}$ an interaction
term of the form
%
$$ {\cal E}_{int}(u)=\int d^2x\, W(x)\vert\psi(x)\vert^2
   \eqno(1.10)$$
%
where $u\in X^{(n)}$. Define a modified energy functional
%
$$ {\cal E}_\epsilon(u)={\cal E}_{GL}(u)+\epsilon\,{\cal E}_{int}(u)\,.
   \eqno(1.11)$$
%
With this definition Eq. (1.1) can be written
%
$$ \partial_t u=-{\cal E}'_\epsilon(u)
   \eqno(1.12)$$
%
where ${\cal E}'_\epsilon$ is the Fr\'echet derivative or, more precisely,
the $L^2$--gradient, of ${\cal E}_\epsilon$.
We observe that the interaction term in Eq. (1.10) does not break the gauge
symmetry. However, ${\cal E}_{int}$ does break the translation symmetry and
if we translate $\psi$ the interaction energy depends on the translation
parameter ${\bf z}$. In particular, applying translations to the $n$-vortex
solution $v^{(n)}$ we define the interaction energy
%
$$ W^{(n)}_{int}({\bf z}) \equiv\epsilon
   \int d^2x\, W({\bf x})\vert\psi^{(n)}({\bf x}-{\bf z})\vert^2\,.
   \eqno(1.13)$$
%
\subsection{Summary of results}
\label{summary}
\par The main result in this work states that the force derived from the
interaction energy $W^{(n)}_{int}({\bf z})$ is governing
the effective dynamics of the magnetic vortex. More specifically, given an
initial data $u_0=(\psi_0,A_0)$ close to
a point $v_{{\bf z}_0\gamma_0}\in M_{sym}^{(n)}$, we show
that there exists a curve in $M_{sym}^{(n)}$, parametrized by the time $t$,
such that for all $t\geq 0$ the solution $u(t)=(\psi(t),A(t))$ satisfies
%
$$ u(t)-v_{{\bf z}(t)\gamma(t)}=O(\epsilon)$$
where the equation for the effective dynamics of the vortex center is given
by
%
$$ a_n\dot{{\bf z}}+\nabla_{{\bf z}}W^{(n)}_{int}({\bf z})=O(\epsilon^2)$$
and
%
$$ a_n=\frac{1}{2}\Vert\nabla_{A^{(n)}}\psi^{(n)}\Vert_2^2
   +\Vert\nabla\times A^{(n)}\Vert_2^2\,.
   \eqno(1.14)$$
%
\subsection{Effective dynamics of vortices - main theorem}
\label{dynamics}
\par We formulate our main result giving the effective dynamics
of the gradient flow Ginzburg-Landau equations with a
local potential. Recall that $v^{(n)}=(\psi^{(n)},A^{(n)})$ denotes an
$n$-vortex solution of the static GL equations, Eq. (1.3), and $M_{sym}^{(n)}$
denotes the symmetry manifold for $v^{(n)}$ defined in Eq. (1.9). We have:
%
\begin{theorem}[effective dynamics]
\label{theorem_A}
Let $n$ be an arbitrary integer if $0<\lambda<1$ and
$n=\pm 1$\  if $\lambda>1$.
There exist constants $\epsilon_0,c,c_0>0$, with $c,c_0$
depending only on $\epsilon_0$, such that for $0<\epsilon<\epsilon_0$ and for
a datum $u_0$ satisfying the condition
%
$$ \Vert u_0-v_{{\bf z}_0\gamma_0}\Vert_{X^{(n)}}
   \leq c_0\epsilon\ ,$$
for some $v_{{\bf z}_0\gamma_0}\in M_{sym}^{(n)}$, the solution $u(t)$ of
Eq. (1.1) with datum $u_0$ is of the form
%
$$ u(t)=v_{{\bf z}(t)\gamma(t)}+\zeta(t)$$
with $v_{{\bf z}(t)\gamma(t)}\in M_{sym}^{(n)}$, the functions $z(t)$, $\gamma(t)$
satisfying the differential equations
%
\begin{equation*}
 \begin{split}
   a_n\frac{d{\bf z}}{dt}&=-\nabla_{{\bf z}}W^{(n)}_{int}({\bf z})
   +O(\epsilon^2)\\
   \frac{d\gamma}{dt}&=\sum_i \dot z_iA^{(n)}_i(\cdot-{\bf z})
   +O_{H^{1-s}}(\epsilon^2),\qquad s>0\\
 \end{split}
\tag{1.15}
\end{equation*}
%
where $a_n$ is given by Eq. (1.14) 
and the function $\zeta(t)$ in $X^{(n)}$ and bounded as
%
$$ \Vert \zeta(t)\Vert_{X^{(n)}}\leq c\epsilon\,.
   \eqno(1.16)$$
\end{theorem}
\begin{flushright}$\square$\end{flushright}
%
\par Remark: The notation $O_X(\epsilon)$ stands for a function in $X$ whose
$X$-norm is bounded by $C\epsilon$.
\par Eqns. (1.15) will be called the {\it effective vortex dynamics equations}.
We see that the solution to Eq. (1.1) which is initially close to an $n$-vortex
stays close to a vortex for all times provided its center and gauge evolve
according to Eqns. (1.15).
\par The proof of Theorem \ref{theorem_A} is given in
Section 2 with several technical statements proved in Sections 3 and 4.
Finally, Appendix A contains explicit expressions for the Taylor expansions
of the energy functional and its Fr\'echet derivative (r.h.s. of the equations
of motion).
%
\section{Proof of Theorem \ref{theorem_A}}
%
\label{proof_A}
\par We begin with some comments concerning notation. Throughout the
discussion below the symbol $H^s$ always means the Sobolev space
$H^s(\mathbb R^2;\mathbb C\times \mathbb R^2)$. Let
$\zeta=(\xi,F)\in L^2(\mathbb R^2;\mathbb C\times \mathbb R^2)$ and
$\eta=(\lambda,G)\in L^2(\mathbb R^2;\mathbb C\times \mathbb R^2)$. The $L^2$
inner product of $\zeta$ and $\eta$ is denoted by $\langle\zeta,\eta\rangle$
and is given by
%
$$ \langle\zeta,\eta\rangle
   \equiv\int_{\mathbb R^2}(Re(\overline\xi\lambda)+F\cdot G)\,.$$
In order to make the notation less cubersome we shall denote below the 
parameters $\{{\bf z},\gamma\}$ by $\sigma$. For example, according to this 
convention $v_\sigma$ stands for $v_{{\bf z}\gamma}$. We restore the full
notation whenever this is necessary or if it adds to the clarity of argument.
Otherwise, the shorter notation is kept throughout the course of the proof.
In several steps that do not require reference to the parametrization of
$M_{as}$ we omit it altogether.
%
\subsection{The manifold of approximate solutions}
%
\par Let
%
$$ \Sigma=\{({\bf z},\gamma)\mid {\bf z}\in \mathbb R^2, 
   \gamma-\sum_i z_iA^{(n)}_i(\cdot-{\bf z})\in H^2(\mathbb R^2;\mathbb R)\}\,.
   \eqno(2.1)$$
\par Denote the elements of the translation group by $T_{{\bf z}}$
(${\bf z}\in \mathbb R^2$) and the elements of the gauge group by $G_\gamma$
($\gamma\in H^2(\mathbb R^2;\mathbb C)$).
Applying the translation and gauge transformations to a vortex
solution $v^{(n)}$ of Eq. (1.3) we obtain a manifold $M_{as}$ of solutions of
this equation
%
$$ M_{as}=\{ G_\gamma T_{{\bf z}}\,v^{(n)}\mid ({\bf z},\gamma)\in\Sigma\}
   =\{v_{{\bf z}\gamma}\mid ({\bf z},\gamma)\in\Sigma\}
   =M_{sym}^{(n)}\subset X^{(n)}
   \eqno(2.2)$$
The manifold $M_{as}$ is, therefore, parametrized by ${\bf z}$ and $\gamma$.
A point in $M_{as}$ corresponding to a given value $\{{\bf z},\gamma\}$ of the
parameters (for $\gamma$ this means a definite gauge function) is denoted by 
$v_{{\bf z}\gamma}$. 
\par The manifold $M_{as}$ is called {\it manifold of approximate
solutions}. We argue for the introduction of $M_{as}$ and the terminology used
as follows: Inserting $v_{{\bf z}\gamma}\in M_{as}$ as a trial solution into 
Eq. (1.12) we obtain in the r.h.s 
%
$$ {\cal E}'_\epsilon(v_{{\bf z}\gamma})={\cal E}'_{GL}(v_{{\bf z}\gamma})+
   \epsilon\,{\cal E}'_{int}(v_{{\bf z}\gamma})
   =\epsilon\,{\cal E}'_{int}(v_{{\bf z}\gamma})
   \eqno(2.3)$$
hence the r.h.s of Eq. (1.12) (or, equivalently, of Eq. (1.1)) is of order 
$\epsilon$ and $v_{{\bf z}\gamma}$ is seen to be an approximate static 
solution of Eq. (1.1)  for $\epsilon$ small.
\par Denote by $T_{{\bf z}\gamma}\Sigma$ the tangent space to $\Sigma$ at the 
point
$({\bf z},\gamma)$ and by $T_{v_{{\bf z}\gamma}}M_{as}$ the tangent space to 
$M_{as}$ at
the point $v_{{\bf z}\gamma}$. A basis for $T_{{\bf z}\gamma}\Sigma$ is given by
%
$$ \{\partial_{z_i}+\langle {\bf z}\cdot\partial_i A,\partial_{\tilde\gamma}
   \rangle, \partial_{\tilde\gamma}\}
   \eqno(2.4)$$
where $\tilde\gamma$ is given by Eq. (1.8) and for a function $g$ we set
$\langle g, \partial_{\tilde\gamma}\rangle=\int d^2x\,g(x)
\partial_{\tilde\gamma(x)}$. Using the natural parametrization
map $\beta: \Sigma\to M_{as}$ defined by
$\beta({\bf z},\gamma)=v_{{\bf z}\gamma}$ we can push forward the basis given in
Eq. (2.4) in order to obtain a basis for $T_{v_{{\bf z}\gamma}}M_{as}$. This 
later basis, is obtained via the mapping of the basis in Eq. (2.4) by the 
Fr\'echet derivative 
$D_{{\bf z}\gamma}(\beta({\bf z},\gamma))=D_{{\bf z}\gamma}v_{{\bf z}\gamma}$.
Denoting $\partial_{z_i}^A=\partial_{z_i}+\langle A_i,\partial_\gamma\rangle$,
we get in $T_{v_{{\bf z}\gamma}}M_{as}$ the basis vectors
%
$$ T^{{\bf z}\gamma}_i=\partial^A_{z_i}v_{{\bf z}\gamma},\ i=1,2\qquad
   G^{{\bf z}\gamma}_{\delta(x)}=\partial_{\gamma(x)} v_{{\bf z}\gamma}
   \eqno(2.5)$$
where the (covariant) translation vectors $T^{{\bf z}\gamma}_i$ are given by
%
$$ T^{{\bf z}\gamma}_i=\left((\nabla_{A_{{\bf z}\gamma}}
   \psi_{{\bf z}\gamma})_i\ , 
   (\nabla\times A_{{\bf z}\gamma})\hat e_i \right)\,.
   \eqno(2.6)$$
Here $\hat e_1=(0,1)$ and $\hat e_2=(-1,0)$. For a definite 
gauge function $\chi$ the gauge basis vectors $G^{{\bf z}\gamma}_{\delta(x)}$
in Eq. (2.5) are defined by the relation
%
$$ \int d^2x\, G^{{\bf z}\gamma}_{\delta(x)}\chi(x)=G^{{\bf z}\gamma}_{\chi}
   \eqno(2.7)$$
where
%
$$ G^{{\bf z}\gamma}_\chi=(i\chi\psi_{{\bf z}\gamma}, \nabla\chi)\,.
   \eqno(2.8)$$
The vectors in Eq. (2.5) form an orthogonal basis for 
$T_{v_{{\bf z}\gamma}}M_{as}$. We have 
$\langle G^{{\bf z}\gamma}_{\delta(x)},T^{{\bf z}\gamma}_i\rangle=0$ and 
$\langle T^{{\bf z}\gamma}_i,T^{{\bf z}\gamma}_j\rangle=\delta_{ij}a_n$ with
%
$$ a_n=\Vert T^{{\bf z}\gamma}_1\Vert_2^2=\Vert T^{{\bf z}\gamma}_2\Vert_2^2
   =\frac{1}{2}\Vert\nabla_{A^{(n)}}\psi^{(n)}\Vert_2^2
   +\Vert\nabla\times A^{(n)}\Vert_2^2
   \eqno(2.9)$$
and, in addition
%
$$ \langle G^{{\bf z}\gamma}_{\delta(x)},G^{{\bf z}\gamma}_{\delta(y)}\rangle
   =K_{{\bf z}\gamma}(x,y)
   \eqno(2.10)$$
where $K_{{\bf z}\gamma}(x,y)$ is the integral kernel for the operator
%
$$ K_{{\bf z}\gamma}= -\Delta+\vert\psi_{{\bf z}\gamma}\vert^2\,.
   \eqno(2.11)$$ 
We note that $K_{{\bf z}\gamma}$ is self-adjoint and that $K_{{\bf z}\gamma}>0$.
\par We will see below that one of the main ingredients that enter into the 
proof of Theorem \ref{theorem_A} is the projection of Eq. (1.1), for each time $t$, on the 
tangent space $T_{v_{{\bf z}\gamma}}M_{as}$ (where ${\bf z}$ and $\gamma$
depend on $t$
in a suitable way). Denote $P_v: T_vX^{(n)}\to T_vM_{as}$ the projection on 
the tangent space to $M_{as}$ at the point $v_{{\bf z}\gamma}$. 
We are able to use
the orthogonal basis of Eq. (2.5) to obtain an explicit expression for
$P_{v_{{\bf z}\gamma}}$. For any $\zeta\in T_{v_{{\bf z}\gamma}}X^{(n)}$ we 
have
%
$$ P_{v_{{\bf z}\gamma}}\zeta=\sum_{i=1,2} a_n^{-1}T_i^{{\bf z}\gamma}
   \langle T_i^{{\bf z}\gamma},\zeta\rangle
   +\int d^2x\int d^2y\, G^{{\bf z}\gamma}_{\delta(x)}
   K_{{\bf z}\gamma}^{-1}(x,y)
   \langle G^{{\bf z}\gamma}_{\delta(y)},\zeta\rangle\,.
   \eqno(2.12)$$ 
In particular, if $v(t)=v_{{\bf z}(t)\gamma(t)}$ is a path in $M_{as}$,
depending on the time $t$, then we have
%
$$ \dot v_{{\bf z}\gamma}=P_v\dot v_{{\bf z}\gamma}
   =\sum_i(-\dot z_i)T^{{\bf z}\gamma}_i
   +G^{{\bf z}\gamma}_{\dot{\tilde\gamma}}
   \eqno(2.13)$$
with
%
$$ \dot{\tilde\gamma}=\dot\gamma-\sum_i \dot z_iA^{(n)}_i(\cdot-{\bf z})\,.
   \eqno(2.14)$$
In Eq. (2.13) and Eq. (2.14) we use the notation
$\dot v_{{\bf z}\gamma}=\partial_t v_{{\bf z}\gamma}$,
$\dot\gamma=\partial_t\gamma$, $\dot{\tilde\gamma}=\partial_t{\tilde\gamma}$
and $\dot{{\bf z}}=\partial_t{\bf z}$.
%
\subsection{Local Sobolev spaces on the manifold of approximate solutions}
%
\label{local_sobolev}
\par In this subsection we define the notion of local Sobolev spaces on the
manifold of approximate solutions $M_{as}$. The introduction of this concept 
helps to facilitate many of the estimates in this work and is based on the 
observation that certain estimates are not uniform on $M_{as}$ but depend on 
the point $v\in M_{as}$. More specifically, at the point $v_{{\bf z}\gamma}$ 
the constants appearing in the estimates depend on the gauge function 
$\gamma$.
\par As an example for the motivation for the definition of local Sobolev
spaces on $M_{as}$ we give an estimate which is used below in the proof of 
Lemma \ref{lemma_B}.
Consider vectors $(\xi,F)\in T_{v_{{\bf z}\gamma}}X^{(n)}$ with the 
component $\xi$ 
transforming covariantly under gauge transformations and $F$ gauge invariant.
Let $\zeta=(\xi,F)$ be such a vector and let $G_\gamma\zeta$ be the action of
(an appropriate unitary representation of) an element $G_\gamma$ of the gauge
group on $\zeta$. We have $G_{\gamma}\zeta=\zeta'$ where 
$\zeta'=(\xi',F')=(e^{i\gamma}\xi,F)$.
Construct a vector $\zeta_v=(0,Im(\overline\xi\nabla_A\xi))$ where $\xi$ is
the first component of $\zeta$. Then $\zeta_v\in T_{v_{{\bf z}\gamma}}X^{(n)}$
has the desired transformation properties under the action of the gauge group.
Suppose that we want to estimate the $H^{-s}$ ($s>0$) norm of $\zeta_v$. we 
have
%
$$ \Vert\zeta_v\Vert_{H^{-s}}=\Vert\overline\xi\nabla_A\xi\Vert_{H^{-s}}=
   \Vert\overline\xi\nabla\xi-iA\vert\xi\vert^2\Vert_{H^{-s}}
   \leq \Vert\overline\xi\nabla\xi\Vert_{H^{-s}}+
   \Vert A\vert\xi\vert^2\Vert_{H^{-s}}\,.
   \eqno(2.15)$$
For the first term on the r.h.s. of Eq. (2.15) we have the estimate
%
\begin{multline}
   \Vert\overline\xi\nabla\xi\Vert_{H^{-s}}
   =\sup_{\Vert\eta\Vert_{H^s}\leq 1}
   \vert\langle\eta,\overline\xi\nabla\xi\rangle\vert
   \leq \sup_{\Vert\eta\Vert_{H^s}\leq 1}
   \Vert\eta\xi\Vert_2\Vert\nabla\xi\Vert_2\\
   \leq\sup_{\Vert\eta\Vert_{H^s}\leq 1}
   \Vert\eta\Vert_p\Vert\xi\Vert_q\Vert\xi\Vert_{H^1}
   \leq C\Vert\xi\Vert_{H^1}^2\,.\\
\tag{2.16}
\end{multline}
%
where $1/2=1/p+1/q$ and $q$ is large enough so that $H^s\subset L^p$.
In order to estimate the second term on the r.h.s. of Eq. (2.15) we recall 
that at $v_{{\bf z}\gamma}$ we have 
$A_{{\bf z}\gamma}=A^{(n)}(\cdot-{\bf z})+\nabla\gamma$ and so
%
$$ \Vert A\vert\xi\vert^2\Vert_{H^{-s}}=
   \Vert (A^{(n)}(\cdot-{\bf z})+\nabla\gamma)\vert\xi\vert^2\Vert_{H^{-s}}
   \leq \Vert A^{(n)}\Vert_\infty\Vert\xi\Vert^2_2
   +\Vert\nabla\gamma\vert\xi\vert^2\Vert_{H^{-s}}\,.
   \eqno(2.17)$$ 
For the second term on the r.h.s. of Eq. (2.17) we have the estimate
%
\begin{multline}
   \Vert\nabla\gamma\vert\xi\vert^2\Vert_{H^{-s}}
   =\sup_{\Vert\eta\Vert_{H^s}\leq 1}
   \vert\langle\eta,(\nabla\gamma)\vert\xi\vert^2\rangle\vert
   \leq\sup_{\Vert\eta\Vert_{H^s}\leq 1}
   \Vert\eta\vert\xi\vert^2\Vert_2\Vert\nabla\gamma\Vert_2\\
   \leq\sup_{\Vert\eta\Vert_{H^s}\leq 1}
   \Vert\eta\Vert_p\Vert\xi^2\Vert_q\Vert\gamma\Vert_{H^1}
   \leq C\Vert\xi\Vert_{2q}^2\Vert\gamma\Vert_{H^1}
   \leq C\Vert\xi\Vert_{H^1}^2\Vert\gamma\Vert_{H^1}\\
\tag{2.18}
\end{multline}
%
where again $1/2=1/q+1/p$ and $q$ is large enough so that $H^s\subset L^p$.
Using Eq. (2.16)-(2.18) we obtain
%
$$ \Vert\zeta_v\Vert_{H^{-s}}
   \leq C_\gamma\Vert\zeta_v\Vert_{H^1}^2
   \eqno(2.19)$$
where
%
$$ C_\gamma=C(1+\Vert\gamma\Vert_{H^1})
   \eqno(2.20)$$
we see that this estimate is not uniform on $M_{as}$ but depends on the 
gauge function $\gamma$. 
\par In order to simplify estimates of the type considered here we define the 
notion of {\it local Sobolev spaces} on $M_{as}$. With a point 
$v_{{\bf z}\gamma}\in M_{as}$, and for any real $s$, we associate a local 
Sobolev space $H^s_{v_{{\bf z}\gamma}}$ such that, given 
$\zeta=(\xi,F)\in T_{v_{{\bf z}\gamma}}X^{(n)}$ with the gauge transformation 
properties described above, its $H^s_{v_{{\bf z}\gamma}}$ norm is defined to 
be
%
$$ \Vert\zeta\Vert_{H^s_{v_{{\bf z}\gamma}}}\equiv 
   \Vert G_{-\gamma}\zeta\Vert_{H^s}\,.
   \eqno(2.21)$$
The definition of $H^s_{v_{{\bf z}\gamma}}$ preserves all of the properties of
Sobolev spaces. In particular, the Sobolev embedding theorems are all valid
for the local Sobolev spaces. Note also that $L^p_{v_{{\bf z}\gamma}}=L^p$ for
all $p$. 
\par Suppose we want to obtain an estimate for the vector 
$\zeta_v=(0,Im(\overline\xi\nabla_A\xi))$ in the local 
$H^{-s}_{v_{{\bf z}\gamma}}$ ($s>0$) norm. We have
%
$$ \Vert\zeta_v\Vert_{H^{-s}_{v_{{\bf z}\gamma}}}
   =\Vert \overline\xi\nabla_A\xi\Vert_{H^{-s}}
   =\Vert(\overline{e^{-i\gamma}\xi})(e^{-i\gamma}\nabla_A\xi)\Vert_{H^{-s}}
   =\Vert\overline{\xi'}\nabla_{A^{(n)}}\xi'\Vert_{H^{-s}}
   \eqno(2.22)$$
where $\xi'=e^{-i\gamma}\xi$. Repeating the sequence of estimates in 
Eq. (2.16)-(2.18) for the r.h.s. of Eq. (2.22) we find that
%
$$ \Vert\overline{\xi'}\nabla_{A^{(n)}}\xi'\Vert_{H^{-s}}
   \leq C\Vert\xi'\Vert_{H^1}^2=C\Vert e^{-i\gamma}\xi\Vert_{H^1}^2
   =C\Vert\xi\Vert_{H^1_{v_{{\bf z}\gamma}}}^2
   \eqno(2.23)$$
with $C=1+\Vert A^{(n)}\Vert_\infty$, a constant independent of $\gamma$. 
Thus we arrive at the (local) estimate
%
$$ \Vert\zeta_v\Vert_{H^{-s}_{v_{{\bf z}\gamma}}}
   \leq C\Vert\zeta_v\Vert_{H^1_{v_{{\bf z}\gamma}}}^2
   \eqno(2.24)$$
which has the same form as Eq. (2.19). The emphasis is on the fact 
that the constant $C$ here does not depend on $\gamma$.
\par In accord with the discussion here most of the estimates
involved in the proof of the main theorem of this work are performed using the
local Sobolev spaces $H^s_{v_{{\bf z}\gamma}}$.
%
\subsection{Proof of Theorem \ref{theorem_A}}
%
\par The strategy for the proof of Theorem \ref{theorem_A} is as follows: Given
$\epsilon>0$ small enough we show that there exists a neighborhood of
$M_{as}$ such that, if the initial data $u_0$ for Eq. (1.1) is
in this neighborhood, then the distance of the solution $u(t)$
from $M_{as}$  can be controlled for all times $t\geq 0$.
For such a solution we define in a suitable way a projection on $M_{as}$
so as to obtain, for each $t\geq 0$ a unique point
$v(t)=v(u(t))\in M_{as}$ corresponding to $u(t)$. We think
of $u(t)$ as inducing a (uniquely defined) trajectory on $M_{as}$,
traced by $v(t)$. We then obtain effective equations of motion for the point
$v(t)$ in terms of the parametrization of $M_{as}$ by the translation
parameters ${\bf z}$ and the gauge function $\gamma$, i.e, we obtain
equations of motion for these parameters and prove their accuracy to order
$\epsilon^2$ for all $t\geq 0$. The process is considered as a reduction of
the problem of the analysis of the dynamics generated by Eq. (1.1) to a
dynamical problem consisting of a small number (in a sense) of degrees of
freedom, providing an effective overall behavior of the original system.
\par As mentioned above, the proof of Theorem \ref{theorem_A} consists of several steps:
\bigskip
\par{\it Step 1- Decomposition}
\smallskip
\par The first step in the proof is to establish the validity of an appropriate
decomposition for any $u\in X^{(n)}$ which is close enough to $M_{as}$:
%
\begin{proposition}[decomposition]
\label{proposition_A}
Define a neighborhood $U_\delta\subset X^{(n)}$ of $M_{as}$ in $X^{(n)}$ as
follows
%
$$ U_\delta=\{ u\mid u\in X^{(n)}\ {\rm and}\ \exists({\bf z},\gamma)
   \in \Sigma,\, \Vert u-v_{{\bf z}\gamma}\Vert_{X^{(n)}}<\delta\}\,.
   \eqno(2.25)$$
Then there exist $\delta>0$ and a $C^1$ map $M_{dec}: U_\delta
\mapsto \Sigma$ such that, if we denote $v(u)=\beta(M_{dec}(u))$ for 
$u\in U_\delta$ (where $\beta({\bf z},\gamma)=v_{{\bf z}\gamma}$; the
parametrization mapping for $M_{as}$), we have
$u-v(u)\in H^1(\mathbb R^2;\mathbb C \times\mathbb R^2)$ and
%
$$ P_{v(u)}(u-v(u))=0\,.
   \eqno(2.26)$$
\end{proposition}
\begin{flushright}$\square$\end{flushright}
%
\par{\bf Proof:}
\smallskip
\par Given the expression in Eq. (2.12) for the projection operator
$P_{v_{{\bf z}\gamma}}$ we see that condition (2.26) is equivalent to the
following conditions
%
$$ \langle T^{{\bf z}\gamma}_i,(u-v_{{\bf z}\gamma})\rangle=0,\ i=1,2\qquad
   \langle G^{{\bf z}\gamma}_{\delta_{(x)}},(u-v_{{\bf z},\gamma})\rangle=0\,.
   \eqno(2.27)$$
\par Define a function $g: U_\delta\times\Sigma\to
\mathbb R^2\times L^2(\mathbb R^2)$ by
%
$$ g(u;{\bf z},\gamma)=(\langle T^{{\bf z}\gamma}_i,(u-v_{{\bf 
z}\gamma})\rangle,
   \langle G^{{\bf z}\gamma}_{\delta_{(x)}},(u-v_{{\bf z}\gamma})\rangle)\,.
   \eqno(2.28)$$
It is obvious that $g(v_{\sigma}; \sigma)=0$. Taking the
Fr\'echet derivative we obtain a map
$D_{\sigma}g(v_{\sigma};\sigma):
T_{\sigma}\Sigma\mapsto \mathbb R^2\times L^2(\mathbb R^2)$
%
$$ D_{\sigma}g(v_{\sigma};\sigma)
   = -(\langle T^{\sigma}_i,D_{\sigma}v_{\sigma}\rangle,
   \langle G^{\sigma}_{\delta_{(x)}}, D_{\sigma}v_{\sigma}\,\rangle)\,.
   \eqno(2.29)$$
Using the basis of $T_{\sigma}\Sigma$ given in Eq. (2.4) we can express
Eq. (2.28) as a transformation on the coordinate vector in that basis. We
then obtain a matrix representation
$[D_{\sigma}g(v_{\sigma};\sigma)]_R$
of $D_{\sigma}g(v_{\sigma};\sigma)$ with
% 
$$ [D_{\sigma}g(v_{\sigma};\sigma)]_R: \mathbb R^2\times H^2(\mathbb R^2;\mathbb R)
   \mapsto \mathbb R^2\times L^2(\mathbb R^2)$$
and
%
$$ [D_{\sigma}g(v_{\sigma};\sigma)]_R=diag\{-a_n, -a_n, K_{\sigma}\}\,.
   \eqno(2.30)$$
Since $K_{\sigma}$ is invertible
we find that $[D_{\sigma}g(v_{\sigma};\sigma)]_R$ is
invertible (hence also $D_{\sigma}g(v_{\sigma};\sigma)$). The implicit
function theorem then implies that for any
$v_{\sigma}\in M_{as}$ there
is a unique $C^1$ map $M_{dec}$ from $B_{X^{(n)}}(v_{\sigma};\delta)$, a 
ball in $X^{(n)}$ of size $\delta$ centered
at $v_{\sigma}$, to $\Sigma$ such that, for $\sigma=M_{dec}(u)$,
Eq. (2.27) and hence Eq. (2.26) are satisfied. We note that
$\Vert K_{\sigma}^{-1}\Vert_{L^2\to H^2}$ is uniformly bounded on
$M_{as}$ and also that $D_{\sigma}g(u;\sigma)$ and
$D^2_{\sigma}g(u;\sigma)$ contain only covariant derivatives
of $T^{\sigma}_i$ and $G^{\sigma}_{\delta_{(x)}}$. We therefore find
that the coordinate transformation representations
%
$$ [D_{\sigma}g(u;\sigma)]_R:
   X^{(n)}\times(\mathbb R^2\times H^2(\mathbb R^2;\mathbb R))
   \mapsto \mathbb R^2\times L^2(\mathbb R^2)$$
and
%
$$ [D^2_{\sigma}g(u;\sigma)]_R:
   X^{(n)}\times(\mathbb R^2\times H^2(\mathbb R^2;\mathbb R))
   \times(\mathbb R^2\times H^2(\mathbb R^2;\mathbb R))
   \mapsto \mathbb R^2\times L^2(\mathbb R^2)$$
are uniformly bounded in $\sigma$ for all balls
$B_{X^{(n)}}(v_{\sigma};\delta_0)$, where $\delta_0$ is independent of
$\sigma$. Thus, we can choose $\delta$ independent of the point
$v_{\sigma}$ and the existence of an appropriate neighborhood
$U_\delta$ is established.\hfill$\blacksquare$
\bigskip
\par Consider a solution $u(t)$ of Eq. (1.1) which, for a time 
interval $t\in[0,T_\delta]$ satisfies $u(t)\in U_\delta$.
Proposition \ref{proposition_A} then implies that for such a solution it is
possible to find, for each $t\in[0,T_\delta]$ a unique point $v(u(t))\in M_{as}$
such that the condition in Eq. (2.26) holds. If we denote
$\zeta_v(t)=u(t)-v(u(t))$ then
$\zeta(t)\in X^{(n)}$ and we obtain a unique decomposition
%
$$ u(t)=v(u(t))+\zeta_v(t)
   \eqno(2.31)$$
with
%
$$ P_{v(u(t))}\zeta_v(t)=0\,.
   \eqno(2.32)$$
For the sake of bravity we omit in the sequel from our notation the time
$t$ and all indications for the procedure of obtaining the point 
$v(u(t))$ when given a solution $u(t)$. Thus we write,
unless a temporary need arises to restore the full notation, $u(t)$,
$v(u(t))$ and $\zeta_v(t)$ as $u$, $v$ and $\zeta_v$ 
respectively. Eq. (2.31) will then be written in short as
$u=v+\zeta_v$, Eq. (2.32) will be written $P_v\zeta_v=0$, etc.
\bigskip
\par{\it Step 2- Effective dynamics equations}
\smallskip
\par Once the existence of the decomposition, Eq. (2.31)-(2.32), is 
established, we can project Eq. (1.1) on the tangent space to $M_{as}$ in 
order to obtain the effective equations of motion. Applying the projection 
$P_v$ to Eq. (1.12) we get  
%
$$ P_v\dot u= -P_v{\cal E}'_\epsilon(u)\,.
   \eqno(2.33)$$
Making use of the decomposition in Eq. (2.31) we obtain
%
$$ P_v\dot v= -P_v{\cal E}'_\epsilon(u)
   -P_v\dot\zeta_v\,.
   \eqno(2.34)$$
We expand in a Taylor series the Fr\'echet derivative of the energy functional
%
$$ {\cal E}'_\epsilon(u)={\cal E}'_\epsilon(v)
   +L_{\epsilon;v}\zeta_v+N_v(\zeta_v)
   \eqno(2.35)$$
where $L_{\epsilon;v}\equiv {\cal E}^{\prime\prime}_\epsilon(v)$
and $N_v(\zeta_v)$ is the non-linear term defined by 
$N_v(\zeta_v)={\cal E}'_\epsilon(u)-{\cal E}'_\epsilon(v)
-L_{\epsilon;v}\zeta_v$. Inserting Eq. (2.35) into Eq. (2.34) we get
%
$$ P_v\dot v= -P_v({\cal E}'_\epsilon(v)+L_{\epsilon;v}\zeta_v+N_v(\zeta_v))
   -P_v\dot\zeta_v\,.
   \eqno(2.36)$$
Note further that $P_v\dot v=\dot v$ (since $\dot v\in T_vM_{as}$) and 
hence
%
$$ \dot v+P_v{\cal E}'_\epsilon(v)=-P_vL_{\epsilon;v}\zeta_v-P_vN_v(\zeta_v)
   -P_v\dot\zeta_v\,.
   \eqno(2.37)$$
The last term on the r.h.s. of Eq. (2.37) can be written in a more convenient
form by using Eq. (2.32). We have
%
$$ 0=\partial_t(P_v\zeta_v)=\dot P_v\zeta_v+P_v\dot\zeta_v$$
and so
%
$$ P_v\dot\zeta_v= -\dot P_v\zeta_v\,.
   \eqno(2.38)$$
Inserting Eq. (2.38) into Eq. (2.37) we obtain
%
$$ \dot v+P_v{\cal E}'_\epsilon(v)=-P_vL_{\epsilon;v}\zeta_v-P_vN_v(\zeta_v)
   +\dot P_v\zeta_v\,.
   \eqno(2.39)$$
Eq. (2.39) is the starting point for the derivation of the effective dynamics
on the manifold $M_{as}$ induced by the gradient flow equations.
\par We begin by estimating Eq. (2.39):
%
\begin{proposition}
\label{proposition_B}
Suppose that a solution $u$ of Eq. (1.1)
satisfy the condition that $u(t)\in U_\delta$ for $t\in[0,T_\delta]$,
where $U_\delta\subset X^{(n)}$ is the neighborhood of $M_{as}$ given by 
Proposition \ref{proposition_A}. For such $u$ the decomposition $u=v+\zeta_v$
is valid and we have the following estimate for $t\in[0,T_\delta]$
%
\begin{multline}
   \Vert\dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)
   \Vert_{H^{-s}_{v_\sigma}}
   \leq C[\epsilon\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}
   +\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}^2\\
   +\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}^3
   +(\vert\dot{\bf z}\vert+\Vert\dot{\tilde\gamma}\Vert_2)
   \Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}]
\tag{2.40}
\end{multline}
%
for some $C>0$.\hfill$\square$
\end{proposition}
%
\bigskip
\par{\bf Proof:}
\par A first estimate of Eq. (2.39) gives
%
\begin{multline}
   \Vert \dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)
   \Vert_{H^{-s}_{v_\sigma}}\leq
   \Vert P_{v_\sigma}L_{\epsilon;v_\sigma}\zeta_{v_\sigma}
   \Vert_{H^{-s}_{v_\sigma}}
   +\Vert P_{v_\sigma} N_{v_\sigma}(\zeta_{v_\sigma})
   \Vert_{H^{-s}_{v_\sigma}}\\
   +\Vert\dot P_{v_\sigma}\zeta_{v_\sigma}
   \Vert_{H^{-s}_{v_\sigma}}\,.
\tag{2.41}
\end{multline}
%
The proposition is a result of Lemma \ref{lemma_A}, Lemma \ref{lemma_B} and
Lemma \ref{lemma_C} below, which provide estimates on the terms on the r.h.s.
of Eq. (2.41):
%
\begin{lemma}[approximate zero modes property]
\label{lemma_A}
There exists a constant $C>0$ such that, for all $v\in M_{as}$ and any vector
$\zeta\in T_vX^{(n)}$, we have
%
$$ \Vert P_vL_{v;\epsilon}\zeta\Vert_2
   \leq \epsilon C\Vert\zeta\Vert_2
   \eqno(2.42)$$
where $\epsilon$ is the potential strength parameter appearing in Eq. (1.1).
\hfill$\square$
\end{lemma}
%
\bigskip
\par Lemma \ref{lemma_A} is proved in Section 4. This lemma provides a bound on the
first term on the r.h.s. of Eq. (2.41) (since $L^2=L^2_{v_{{\bf z}\gamma}}$).
A straightforward technical calculation provides an estimate for the middle
term on the r.h.s of Eq. (2.41). This is also done in Section 4 where
we obtain the following result:
%
\begin{lemma}
\label{lemma_B}
For $\zeta\in T_vX^{(n)}$ and $s>0$ we have
%
$$ \Vert P_{v_\sigma}N_{v_\sigma}(\zeta)\Vert_{H^{-s}_{v_\sigma}}
   \leq C(\,\Vert\zeta\Vert_{H^1_{v_{\sigma}}}^2
   +\Vert\zeta\Vert_{H^1_{v_\sigma}}^3)\,.
   \eqno(2.43)$$
   \end{lemma}
\hfill$\square$
%
\par An estimate of the last term on the r.h.s. of Eq. (2.41) is given in
Lemma \ref{lemma_C}, proved in Section 4:
%
\begin{lemma}
\label{lemma_C}
For $\zeta_{v_\sigma}\in T_{v_\sigma}X^{(n)}$
satisfying $P_{v_\sigma}\zeta_{v_\sigma}=0$ we have
%
$$ \Vert\dot P_{v_\sigma}\zeta_{v_\sigma}\Vert_2
   \leq C(\vert\dot{{\bf z}}\vert+\Vert\dot{\tilde\gamma}\Vert_2)
   \Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}
   \eqno(2.44)$$
where $\dot{\tilde\gamma}$ is given in Eq. (2.14).\hfill$\square$
\end{lemma}
%
insertion of the inequalities, Eq. (2.42)-(2.44) into Eq. (2.41) completes the
proof of Proposition \ref{proposition_B}.
\bigskip
\par{\it Step 3- A bound on the error term}
\smallskip
\par Given the decomposition $u=v+\zeta_v$, for a solution $u$
satisfying the conditions of Proposition \ref{proposition_A}, we show that the remainder term 
$\zeta_v$
is of order $\epsilon$ for all times $t>0$, where $\epsilon$ is the potential
strength parameter (this result also justifies the assumption that the
decomposition implied by Proposition \ref{proposition_A} exists for all times if the initial 
data $u(0)$ satisfies certain requirments which are specified
below). The following proposition is the main step in the proof of Theorem
\ref{theorem_A}:
%
\begin{proposition}
\label{proposition_C}
Let $u(t)$ be a solution of Eq. (1.1)
with initial data $u(0)=u_0=(\psi_0,A_0)$.
There exist constants $\epsilon_0, c_0,c>0$, with $c_0$ and
$c$ depending only on $\epsilon_0$, such that, for any $0<\epsilon<\epsilon_0$,
if $u_0$ is chosen in
such a way that there exists a point $v_{{\bf z}_0\gamma_0}\in M_{as}$ with
$\Vert u_0-v_{{\bf z}_0\gamma_0}\Vert_{X^{(n)}}\leq c_0\epsilon$,
then Proposition \ref{proposition_A} holds for $u(t)$ for all $t\geq 0$ and the
decomposition $u(t)=v_\sigma(t)+\zeta_{v_\sigma}(t)$ satisfies
%
$$ \Vert\zeta_{v_\sigma}(t)\Vert_{X^{(n)}}\leq c\,\epsilon,\qquad t\geq 0
   \eqno(2.45)$$
and
%
$$ \Vert\dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)
   \Vert_{H^{-s}_{v_\sigma}}\leq c\,\epsilon^2,\qquad t\geq 0\,.
   \eqno(2.46)$$
\hfill$\square$
\end{proposition}
%
\par{\bf Proof:} The central result underlying the proof of Theorem \ref{theorem_A}
is the {\it Linear Stability Theorem} for the GL equations proved by 
S. Gustafson and I.M. Sigal \cite{GS1}. This linear stability property corresponds
to the {\it coercivity of the Hessian}. We start the proof of Proposition
\ref{proposition_C} by stating this theorem in a form convenient for our purposes.
\par Suppose that the manifold $M_{as}$ corresponds to a vortex with index 
$n$, i.e., $M_{as}=M_{sym}^{(n)}$. An $n$-vortex is called stable if there 
exists some constant $\nu>0$ such that 
%
$$ L_{0;v}\mid_{(T_vM_{as})^\perp}\geq\nu\,.$$
An $n$-vortex is called unstable if $L_{0;v}$ has a negative eigenvalue. We
have
\bigskip
\par{\bf Linear Stability Theorem (S. Gustafson and I.M. Sigal):}
{\it
%
\begin{itemize}
\item[(i)] For all $\lambda>0$ the vortex with $n=\pm 1$ is stable.
\item[(ii)] For $\lambda<1$ a vortex with $\vert n\vert\geq 2$ is stable.
\item[(ii)] For $\lambda>1$ a vortex with $\vert n\vert\geq 2$ is unstable.
\end{itemize}
\hfill$\square$}
%
\par In our work here we consider only stable vortices. Then 
$L_{0;v}\mid_{(T_v M_{as})^\perp}$ is a positive operator. In this case, for 
any vector $\zeta_v\in T_vX^{(n)}$ such that $P_v\zeta_v=0$, the linear 
stability theorem implies that
%
$$ \langle\zeta_v,L_{0;v}\zeta_v\rangle\geq\nu \Vert\zeta_v\Vert_{H^1}^2
   \eqno(2.47)$$
for some constant $\nu>0$.
\smallskip
Introducing the potential $W$ along with the potential strength parameter
$\epsilon$ we have the following corollary to the linear stability theorem:
\bigskip
\par{\bf Corollary A:} {\it For a vector $\zeta_v\in T_v X^{(n)}$ with
$P_v\zeta_v=0$ and for $\epsilon>0$ we have
%
$$ \langle\zeta_v,L_{\epsilon;v}\zeta_v\rangle\geq\nu
   \Vert\zeta_v\Vert_{H^1}^2\,.
   \eqno(2.48)$$}
\hfill$\square$
\bigskip
\par{\bf Proof:} We calculate
%
\begin{equation*}
 \begin{split}
   \langle\zeta_v,L_{\epsilon;v}\zeta_v\rangle=
   \langle\zeta_v,(L_{0;v}+\epsilon W)\zeta_v\rangle
   =\langle\zeta_v,L_{0;v}\zeta_v\rangle
   &+\epsilon\langle\zeta_v,W\zeta_v\rangle\\
   &\geq\langle\zeta_v,L_{0;v}\zeta_v\rangle
   \geq \nu\Vert\zeta_v\Vert_{H^1}^2\\
 \end{split}
\end{equation*}
%
since the term containing the potential $W$ is positive.\hfill$\blacksquare$
\bigskip
\par The following result, which we use below, is also an immediate corollary 
of the linear stability theorem
\bigskip
\par{\bf Corollary B:} {\it For a vector $\zeta_{v_\sigma}\in 
T_{v_\sigma}X^{(n)}$ 
satisfying $P_{v_\sigma}\zeta_{v_\sigma}=0$ and for $\epsilon>0$ we have
%
$$ \langle\zeta_{v_\sigma},L_{\epsilon;v_\sigma}
   \zeta_{v_\sigma}\rangle\geq\nu
   \Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}^2\,.
   \eqno(2.49)$$}
\hfill$\square$
\bigskip
\par{\bf Proof:} It is easy to check that the application of the gauge 
transformation $G_{-\gamma}$ to the projection $P_{v_{{\bf z}\gamma}}$ yields
$G_{-\gamma}P_{v_{{\bf z}\gamma}}G^*_{-\gamma}=G_{-\gamma}P_{v_{{\bf z}\gamma}}
G_{\gamma}=P_{v_{{\bf z} 0}}$. This implies that
%
$$ 0=G_{-\gamma}P_{v_\sigma}\zeta_{v_\sigma}
   =G_{-\gamma}P_{v_\sigma}G^*_{-\gamma}G_{-\gamma}\zeta_{v_\sigma}
   =P_{v_{{\bf z} 0}}\zeta_{v_{{\bf z} 0}}
   \eqno(2.50)$$
where $\zeta_{v_{{\bf z} 0}}=G_{-\gamma}\zeta_{v_\sigma}$. In addition we have
$G_{-\gamma}L_{\epsilon;v_\sigma}=L_{\epsilon;v_{{\bf z} 0}}G_{-\gamma}$.
Therefore, we obtain 
%
\begin{multline}
   \langle\zeta_{v_\sigma},L_{\epsilon;v_\sigma}
   \zeta_{v_\sigma}\rangle
   =\langle G_{-\gamma}\zeta_{v_\sigma},
   G_{-\gamma}L_{\epsilon;v_\sigma}\zeta_{v_\sigma}\rangle\\
   =\langle \zeta_{v_{{\bf z} 0}},
   L_{\epsilon;v_{{\bf z} 0}}\zeta_{v_{{\bf z} 0}}\rangle
   \geq\nu\Vert\zeta_{v_{{\bf z} 0}}\Vert_{H^1}^2
   =\nu\Vert G_{-\gamma}\zeta_{v_\sigma}\Vert_{H^1}^2
   =\nu\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}^2\,.
\tag{2.51}\\
\end{multline}
\hfill$\blacksquare$
%
\par In the proof of Proposition \ref{proposition_C} we make use of the dominating linear 
stability properties mentioned above by analyzing the time evolution and 
providing an upper bound on the quantity $\langle\zeta_v,L_{\epsilon;v}
\zeta_v\rangle$.
Corollary A of the linear stability theorem then ensures that this yields
a bound on the error term $\zeta_v$.
\par In Section 3 we prove the following proposition:
%
\begin{proposition}
\label{proposition_D}
Let $u$ be a solution of Eq. (1.1)
satisfying the assumption of Proposition B above and let 
$u=v+\zeta_v$ be the decomposition implied by Proposition \ref{proposition_A}.
Then, for $t\in[0,T_\delta]$, there exist constants $\nu',\,C_1,C_2,C_3>0$ such
that
%
\begin{equation}
 \begin{split}
   &\frac{1}{2}\partial_t\langle\zeta_{v_\sigma},
   L_{\epsilon;v_{\sigma}}\zeta_{v_\sigma}\rangle
   \leq -\frac{1}{2}\nu'\langle\zeta_{v_\sigma},
   L_{\epsilon;v_{\sigma}}\zeta_{v_\sigma}\rangle
   +\big\{-C_1\cr
   &+C_2\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}
   \big[(2+\Vert\zeta_{v_{\sigma}}\Vert_{H^1_{v_\sigma}})
   (1+\epsilon+\Vert\dot v_\sigma+P_{v_\sigma}
   {\cal E}'_\epsilon(v_\sigma)\Vert_{H^{-s}_{v_\sigma}})-1\big]
   \big\}\Vert\zeta_{v_\sigma}\Vert_{H^2_{v_\sigma}}^2\cr
   &+\epsilon C_3(1+\Vert\dot v_\sigma+P_{v_\sigma}
   {\cal E}'_\epsilon(v_\sigma)\Vert_{H^{-s}_{v_\sigma}}
   +\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}
   +\Vert\zeta_{v_\sigma}\Vert^2_{H^1_{v_\sigma}})
   \Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}\,.\cr
 \end{split}
 \tag{2.52}
\end{equation}
\hfill$\square$
\end{proposition}
%
\bigskip
Assume that there exists a maximum time $T_1$ such that
%
$$ \Vert\dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)
   \Vert_{H^{-s}_{v_\sigma}}\leq \epsilon,\qquad t\in [0,T_1]\,.
   \eqno(2.53)$$
By Eq. (2.52) and Eq. (2.53) we have in this time interval
%
\begin{equation}
 \begin{split}
   \frac{1}{2}\partial_t\langle\zeta_{v_\sigma},
   &L_{\epsilon;v_\sigma}\zeta_{v_\sigma}\rangle\leq\,
   -\frac{1}{2}\nu'\,\langle\zeta_{v_\sigma},
   L_{\epsilon;v_\sigma}\zeta_{v_\sigma}\rangle\cr
   &+\big\{-C_1+C_2\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}
   \big[(2+\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}})
   (1+2\epsilon)-1\big]\big\}
   \Vert\zeta_{v_\sigma}\Vert_{H^2_{v_\sigma}}^2\cr
   &+\epsilon C_3(1+\epsilon
   +\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}
   +\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}^2)
   \Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}\,.\cr
 \end{split}
\tag{2.54}
\end{equation}
%
Let $\delta>0$ be such that the decomposition $u=v_\sigma(u)+\zeta_{v_\sigma}$
implied by Proposition \ref{proposition_A} holds for any $u\in U_\delta\subset X^{(n)}$ (see
Eq. (2.25) for the definition of $U_\delta$). The existence of such a
$\delta$ is guaranteed by Proposition \ref{proposition_A}. Suppose that there exists a
maximum time $T_2$ such that
%
$$ \Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}
   \leq \min\left\{1,\frac{C_1}{10C_2},\delta \right\}\ ,
   \qquad t\in[0,T_2]\,.
   \eqno(2.55)$$
Set $\tau=min\{T_1,T_2\}$. Then for $t\in [0,\tau]$ we have,
for some $\tilde C, C>0$,
%
$$ \frac{1}{2}\partial_t\langle\zeta_{v_\sigma},
   L_{\epsilon;v_\sigma}\zeta_{v_\sigma}\rangle
   \leq -\frac{1}{2}\nu'\langle\zeta_{v_\sigma},
   L_{\epsilon;v_\sigma}\zeta_{v_\sigma}\rangle
   -\frac{C_1}{2}\Vert\zeta_{v_\sigma}\Vert_{H^2_{v_\sigma}}^2
   +\epsilon \frac{C}{2}\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}\,.
   \eqno(2.56)$$
Dropping the negative definite second term on the r.h.s. of Eq. (2.56) and
using Corollary B of the linear stability theorem we get
%
$$ \frac{1}{2}\partial_t\langle\zeta_{v_\sigma},
   L_{\epsilon;v_\sigma}\zeta_{v_\sigma}\rangle
   \leq -\frac{1}{2}\nu'\langle\zeta_{v_\sigma},
   L_{\epsilon;v_\sigma}\zeta_{v_\sigma}\rangle
   +\epsilon\frac{\alpha}{2}\langle\zeta_{v_\sigma},
   L_{\epsilon;v_\sigma}\zeta_{v_\sigma}\rangle^{1/2}
   \eqno(2.57)$$
where $\alpha>0$. Set $\langle\zeta_{v_\sigma}(t),L_{\epsilon;v_\sigma(t)}
\zeta_{v_\sigma}(t)\rangle=f^2(t)$, then Eq. (2.57) reads
%
$$ \frac{d}{dt}f\leq -\nu' f+\epsilon\alpha\,.
   \eqno(2.58)$$
Let $g(t)$ be a solution of Eq. (2.58) with an equality sign and initial
condition $g(0)=f(0)=\langle\zeta_{v_\sigma}(0),
L_{\epsilon;v_\sigma(0)}\zeta_{v_\sigma}(0)\rangle^{1/2}$ i.e.,
%
$$ \frac{d}{dt}g=-\nu' g+\epsilon\alpha\,.$$
We get
%
$$ g(t)=f(0)e^{-\nu' t}+\frac{\epsilon\alpha}{\nu'}(1-e^{-\nu' t})
   \leq f(0)+\frac{\epsilon\alpha}{\nu'}\,.$$
With the help of Lemma \ref{lemma_F}, stated in Section 3, we observe that 
%
\begin{multline*}
   f^2(0)=\langle\zeta_{v_\sigma}(0),
   L_{\epsilon;v_\sigma(0)}\zeta_{v_\sigma}(0)\rangle
   =\langle\zeta_{v_\sigma}(0),
   \overline P_{v_\sigma}(0)L_{\epsilon;v_\sigma(0)}
   \zeta_{v_\sigma}(0)\rangle\\
   \leq \Vert\zeta_{v_\sigma}(0)\Vert_{H^1_{v_\sigma(0)}}
   \Vert\overline P_{v_\sigma(0)}L_{\epsilon;v_\sigma(0)}
   \zeta_{v_\sigma}(0)\Vert_{H^{-1}_{v_\sigma(0)}}
   \leq C\Vert\zeta_{v_\sigma}(0)\Vert_{H^1_{v_\sigma(0)}}^2\\
   \leq C'\Vert\zeta_{v_\sigma}(0)\Vert_{H^1}^2\,.\\
\end{multline*}
%
The validity of the last inequality stems from the fact that $v_\sigma(0)$
is some fixed point on $M_{as}$ and we can estimate the gauge function
there.
\par Now, Proposition \ref{proposition_A} guarantees the existence of a neighborhood $U_\delta$
of $M_{as}$ in which the decomposition property holds. Take $u(0)$ to
be an initial data for Eq. (1.1) with $u(0)\in U_\delta$. Then we 
have $\zeta_{v_\sigma}(0)=u(0)-v(u(0))$ with 
$P_{v(u(0))}\zeta_{v_\sigma}(0)=0$. Assume furthermore that 
$u(0)$ is chosen to satisfy the condition that there exists a point 
$v_{{\bf z}_0\gamma_0}=v_{\sigma_0}\in M_{as}$ such that 
$\Vert u(0)-v_{\sigma_0}\Vert_{X^{(n)}}
=\Vert u(0)-v_{\sigma_0}\Vert_{H^1}\leq c_0\epsilon$ for some constant
$c_0>0$. Under these conditions we have
%
$$ \Vert\zeta_{v_\sigma}(0)\Vert_{H^1}
   \leq\Vert u(0)-v_{\sigma_0}\Vert_{H^1}
   +\Vert v_{\sigma_0}-v(u(0))\Vert_{H^1}
   \leq c_0\epsilon+\Vert v_{\sigma_0}-v(u(0))\Vert_{H^1}\,.$$
Denoting $u'=v_{\sigma_0}$, the solution $v(u')$
satisfying the orthogonality condition, Eq. (2.26), is simply 
$v(u')=u'$. We need to show that for $u''$ in the vicinity of
$u'=v_{\sigma_0}$ we have
%
$$ \Vert v(u')-v(u'')\Vert_{H^1}\leq c'\Vert u'-u''\Vert_{H^1}
   \eqno(2.59)$$
for some constant $c'>0$ which may depend only on $\epsilon$,
since then we would have $\Vert v_{\sigma_0}-v(u(0))\Vert_{H^1}
\leq c'\Vert v_{\sigma_0}-u(0)\Vert_{H^1}
\leq c' c_0\epsilon$. In order to see that Eq. (2.59) holds near 
$u'=v_{\sigma_0}$ we first observe that by the implicit function
argument in the proof of Proposition \ref{proposition_A} we have
%
$$ 0=D_u[g(u;M_{dec}(u))]=D_u g(u;M_{dec}(u))
   +D_uM_{dec}(u)\, D_\sigma g(u;M_{dec}(u))$$
and hence
%
$$ D_uM_{dec}(u)=-[D_\sigma g(u;M_{dec}(u))]^{-1}
   D_u g(u;M_{dec}(u))\,.$$
It follows that $\Vert D_uM_{dec}(v_{\sigma_0})\Vert_{X^{(n)}\to \mathbb R^2\times H^2}\leq C$;
since Eq. (2.30) (recall that $K_\sigma$ is uniformly invertible on $M_{as}$)
and the definition of $g(u;\sigma)$ in Eq. (2.28) imply, respectively, that
the two norms $\Vert\,[D_\sigma g(v_{\sigma_0};\sigma_0)]^{-1}
\Vert_{\mathbb R^2\times L^2\to \mathbb R^2\times H^2}$ and
$\Vert D_u g(v_{\sigma_0};\sigma_0)\Vert_{X^{(n)}\to \mathbb R^2\times L^2}$
are uniformly bounded on $M_{as}$. This observation leads to the following
bound
%
\begin{multline*}
   \Vert D_u v(v_{\sigma_0})\Vert_{X^{(n)}\to X^{(n)}}
   =\Vert D_u[\beta(M_{dec}(v_{\sigma_0}))]\ \Vert_{X^{(n)}\to X^{(n)}}\\
   \leq \Vert\ (D_{\sigma}v_{\sigma})_{\sigma_0}
   \Vert_{\mathbb R^2\times H^2\to X^{(n)}}\,
   \Vert D_u M_{dec}(v_{\sigma_0}))\Vert_{X^{(n)}\to \mathbb R^2\times H^2}
   \leq C\\
\end{multline*}
%
and we conclude that, for $c_0=c_0(\epsilon)$ small enough, if 
$\Vert u(0)-v_{\sigma_0}\Vert_{H^1}\leq c_0\epsilon$
then $f(0)\leq c\epsilon$ for some constant $c=c(\epsilon)>0$. Hence, in the
time interval $t\in[0,\tau]$, we have
%
$$ \Vert\zeta_v\Vert_{H^1}
   \leq \nu^{-1}\langle\zeta_v,L_{\epsilon;v}\zeta_v\rangle^{1/2}
   =\nu^{-1}f(t)\leq \nu^{-1}g(t)\leq C\epsilon\,.
   \eqno(2.60)$$
In order to close the proof of Proposition \ref{proposition_C} we need an estimate
on the time derivatives $\dot{{\bf z}}$ and $\dot{\tilde\gamma}$ appearing in
the last term on  the r.h.s of Eq. (2.40). The following lemma is proved in
Section 4:
%
\begin{lemma}
\label{lemma_D}
For $s>0$ we have the following estimate for the
parametric equations of motion
%
$$ \vert a_n\dot{{\bf z}}+\partial^A_{{\bf z}}{\cal E}_\epsilon(v_\sigma)\vert
   +\Vert\dot{\tilde\gamma}\Vert_{H^{1-s}}
   \leq C\Vert\dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)
   \Vert_{H^{-s}_{v_\sigma}}
   \eqno(2.61)$$
where $\dot{\tilde\gamma}$ is defined in Eq. (2.14).\hfill$\square$
\end{lemma}
%
\par Since $\partial^A_{z_i}{\cal E}_\epsilon(v_\sigma)=o(\epsilon)$
(see Eq. (2.67) below) we get that for $0<s<1$
%
\begin{equation}
 \begin{split}
   \vert\dot{{\bf z}}\vert+\Vert\dot{\tilde\gamma}\Vert_2
   &\leq a_n^{-1}(\vert a_n\dot{{\bf z}}
   +\partial^A_{{\bf z}}{\cal E}_\epsilon(v_\sigma)\vert
   +\vert\partial^A_{{\bf z}}{\cal E}_\epsilon(v_\sigma)\vert)
   +\Vert\dot{\tilde\gamma}\Vert_{H^{1-s}}\\
   &\leq C(\Vert\dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)
   \Vert_{H^{-s}_{v_\sigma}}+\epsilon)\\
 \end{split}
 \tag{2.62}
\end{equation}
%
then, under the assumption made in Eq. (2.53), we have
%
$$ \vert\dot{{\bf z}}\vert+\Vert\dot{\tilde\gamma}\Vert_2
   \leq\epsilon C,\qquad t\in[0,\tau]\,.
   \eqno(2.63)$$
Eq. (2.40), Eq. (2.60) and Eq. (2.63) then imply that for $t\in[0,\tau]$
the following inequality is valid
%
\begin{equation}
 \begin{split}
   \Vert \dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)
   \Vert_{H^{-s}_{v_\sigma}}
   \leq C[\epsilon&\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}
   +\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}^2\\
   +&\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}^3
   +(\vert\dot{{\bf z}}\vert+\Vert\dot{\tilde\gamma}\Vert_2)
   \Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}]\leq C\epsilon^2\,.\\
 \end{split}
 \tag{2.64}
\end{equation}
%
The definition of $\tau$ implies that at $t=\tau$ at least one of the
conditions, Eq. (2.53) or Eq. (2.55), is no longer valid. However,
we can choose $\epsilon_0$ in such a way that for $0<\epsilon<\epsilon_0$
this conclusion will stand in contradiction with the results, Eq. (2.60) and
Eq. (2.64). Hence $\tau=\infty$ and the proof of Proposition \ref{proposition_C}
is complete.\hfill$\blacksquare$
\bigskip
\par{\it Step 4 - Effective equations for the parameters}
\smallskip
\par We derive the appropriate equations for the variables ${\bf z}$ and
$\gamma$, parametrizing the manifold $M_{as}$. Thus, solutions of Eq. (1.1)
with an initial condition close enough to $M_{as}$ induce, up to a
small, controllable error, an effective dynamics on $M_{as}$ in terms of
equations of motion for the vortex center and gauge function in the presence
of the potential $W$.
\par The effective equations of motion for the parameters ${\bf z}$ and 
$\tilde\gamma$ are derived using Eq. (2.61) and Proposition \ref{proposition_C}. 
For $\tilde\gamma$ we have simply
%
$$ \Vert\dot{\tilde\gamma}\Vert_{H^{1-s}}\leq C\epsilon^2
   \eqno(2.65)$$ 
and for ${\bf z}$ we get 
%
$$ \vert a_n\dot{{\bf z}}+\partial^A_{{\bf z}}{\cal E}_\epsilon(v_\sigma)\vert
   \leq C\epsilon^2
   \eqno(2.66)$$
Since ${\cal E}_\epsilon(v_\sigma)={\cal E}_{int}(v_\sigma)
=W^{(n)}_{int}({\bf z})$, where
%
$$ W^{(n)}_{int}({\bf z})\equiv
   \epsilon\int d^2x\, W({\bf x})\vert\psi^{(n)}({\bf x}-{\bf z})\vert^2
   \eqno(2.67)$$
does not depend on the vector potential $A$, we get finally
%
$$ \vert a_n\dot{{\bf z}}+\nabla_{{\bf z}} W_{int}({\bf z})\vert\leq C\epsilon^2
   \eqno(2.68)$$
This derivation of the parametric equations of motion completes the proof of
Theorem \ref{theorem_A}.
%
\section{Proof of Proposition \ref{proposition_D}}
\label{prop_D}
\par We calculate
%
\begin{multline}
   \frac{1}{2}\partial_t \langle\zeta_v,L_{\epsilon,v}\zeta_v\rangle=
   \partial_t({\cal E}_\epsilon(u)-{\cal E}_\epsilon(v)
   -\langle {\cal E}'_\epsilon(v),\zeta_v\rangle-R_v(\zeta_v))=\\
   =\langle {\cal E}'_\epsilon(u), \dot u\rangle-
   \langle {\cal E}'_\epsilon(v), \dot v\rangle
   -\langle {\cal E}^{\prime\prime}_\epsilon(v)\dot v,\zeta_v\rangle
   -\langle {\cal E}'_\epsilon(v),\dot\zeta_v\rangle-
   \partial_t[R_v(\zeta_v)]=\\
   =\langle {\cal E}'_\epsilon(u)-{\cal E}'_\epsilon(v),
   \dot u\rangle-\langle L_{\epsilon,v}\dot v,\zeta_v\rangle
   -\partial_t[R_v(\zeta_v)]=\\
   =-\langle {\cal E}'_\epsilon(u)-{\cal E}'_\epsilon(v),
   {\cal E}'_\epsilon(u)\rangle-\langle L_{\epsilon,v} \dot v,\zeta_v\rangle
   -\partial_t[R_v(\zeta_v)]\\
   =-\langle {\cal E}'_\epsilon(u)-{\cal E}'_\epsilon(v),
   \overline P_v{\cal E}'_\epsilon(u)\rangle
   +\langle {\cal E}'_\epsilon(v)-{\cal E}'_\epsilon(u),
   P_v{\cal E}'_\epsilon(u)\rangle
   -\langle L_{\epsilon,v} \dot v,\zeta_v\rangle
   -\partial_t[R_v(\zeta_v)]\\
   \tag{3,1}
\end{multline}
%
where the energy remainder term $R_v(\zeta)$ is given in Appendix A,
Eq. (A.5). Next we define the quantity $\dot R_v(\zeta_v)$ by
%
$$ \partial_t[R_v(\zeta_v)]=\dot R_v(\zeta_v)
   +\langle N_v(\zeta_v),\dot\zeta_v\rangle\,.
   \eqno(3.2)$$
Consider the equation of motion for the error term $\zeta_v$. This equation 
is obtained by projecting Eq. (1.12) on $(T_vM_{as})^\perp$. We have
%
$$ \dot\zeta_v= -\overline P_v{\cal E}'_\epsilon(u)-\dot P_v\zeta_v\,.
   \eqno(3.3)$$
From Eq. (3.2) and Eq. (3.3) we get
%
$$ \partial_t[R_v(\zeta_v)]=\dot R_v(\zeta_v)
   -\langle N_v(\zeta_v),\overline P_v{\cal E}'_\epsilon(u)\rangle
   -\langle N_v(\zeta_v),\dot P_v\zeta_v\rangle\,.
   \eqno(3.4)$$
Inserting Eq. (3.4) into Eq. (3.1) we obtain the following inequality 
%
\begin{multline}
   \frac{1}{2}\partial_t \langle\zeta_v,L_{\epsilon,v}\zeta_v\rangle
   =-\langle L_{\epsilon,v}\zeta_v,\overline P_v{\cal E}'_\epsilon(u)\rangle
   -\langle {\cal E}'_\epsilon(u)-{\cal E}'_\epsilon(v),
   P_v{\cal E}'_\epsilon(u)\rangle\\
   -\langle L_{\epsilon,v} \dot v,\zeta_v\rangle
   -\dot R_v(\zeta_v)+\langle N_v(\zeta_v),\dot P_v\zeta_v\rangle=\cr
   \leq -\langle L_{\epsilon,v}\zeta_v,\overline P_v{\cal E}'_\epsilon(u)\rangle
   -\langle {\cal E}'_\epsilon(u)-{\cal E}'_\epsilon(v),
   P_v{\cal E}'_\epsilon(v)\rangle\\
   -\langle L_{\epsilon,v} \dot v,\zeta_v\rangle
   -\dot R_v(\zeta_v)+\langle N_v(\zeta_v),\dot P_v\zeta_v\rangle=\\
   =-\langle \overline P_v L_{\epsilon,v}\zeta_v,\overline P_v 
   L_{\epsilon,v}\zeta_v\rangle
   +\langle L_{\epsilon,v}\zeta_v,\overline P_v({\cal E}'_\epsilon(v)
   +N_v(\zeta_v))\rangle\\
   -\langle L_{\epsilon;v}\zeta_v,\dot v+P_v{\cal E}'_\epsilon(v)\rangle
   -\langle N_v(\zeta_v),P_v{\cal E}'_\epsilon(v)\rangle
   -\dot R_v(\zeta_v)+\langle N_v(\zeta_v),\dot P_v\zeta_v\rangle\\
   \tag{3.5}
\end{multline}
%
and so
%
\begin{equation}
 \begin{split}
   &\frac{1}{2}\partial_t\langle\zeta_v, L_{\epsilon;v}\zeta_v
   \rangle\leq -\langle \overline P_v L_{\epsilon,v}\zeta_v,\overline P_v 
   L_{\epsilon,v}\zeta_v\rangle
   +\vert\langle L_{\epsilon,v}\zeta_v,\overline P_v({\cal E}'_\epsilon(v)
   +N_v(\zeta_v))\rangle\vert\cr
   &+\vert\langle L_{\epsilon;v}\zeta_v,
   \dot v+P_v{\cal E}'_\epsilon(v)\rangle\vert
   +\vert\langle N_v(\zeta_v),P_v{\cal E}'_\epsilon(v)\rangle\vert
   +\vert\dot R_v(\zeta_v)\vert
   +\vert\langle N_v(\zeta_v),\dot P_v\zeta_v\rangle\vert\cr
 \end{split}
\tag{3.6}
\end{equation}
%
In Section 4 we establish, via straightforward calculations, the
validity of the following estimates
%
\begin{lemma}
\label{lemma_E}
For $\zeta\in T_vX^{(n)}$ we have
% 
$$ \Vert N_{v_\sigma}(\zeta)\Vert_2\leq C\Vert\zeta\Vert_{H^2_{v_\sigma}}
   (\Vert\zeta\Vert_{H^1_{v_\sigma}}+\Vert\zeta\Vert^2_{H^1_{v_\sigma}})
   \eqno(3.7)$$
\hfill$\square$
\end{lemma}
%
and
%
\begin{lemma}
\label{lemma_F}
For $\zeta\in T_vX^{(n)}$ the following inequalities
hold
%
$$ \Vert \overline P_{v_\sigma}
   L_{\epsilon;v_\sigma}\zeta\Vert_{H^{-1}_{v_\sigma}}
   \leq C\Vert\zeta\Vert_{H^1_{v_\sigma}},
   \qquad\Vert\overline P_{v_\sigma}L_{\epsilon;v_\sigma}\zeta\Vert_2
   \leq C\Vert\zeta\Vert_{H^2_{v_\sigma}}\,.
   \eqno(3.8)$$
\hfill$\square$
\end{lemma}
%
Eq. (3.7), Eq. (3.8) and the easily obtainable inequality
%
$$ \Vert{\cal E}'_\epsilon(v_\sigma)\Vert_{H^1_{v_\sigma}}=
   \epsilon\Vert W\psi_{{\bf z} 0}\Vert_{H^1}\leq \epsilon C
   \eqno(3.9)$$
result in the following estimates:
%
$$ \vert\langle \overline P_{v_\sigma}
   L_{\epsilon;v_\sigma}\zeta_{v_\sigma},{\cal E}'_\epsilon(v_\sigma)\rangle
   \vert\leq \Vert\overline P_{v_\sigma}
   L_{\epsilon,v_\sigma}\zeta_{v_\sigma}\Vert_{H^{-1}_{v_\sigma}}
   \Vert{\cal E}'_\epsilon(v_\sigma)\Vert_{H^1_{v_\sigma}}
   \leq \epsilon\,C\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}
   \eqno(3.10)$$
and
\begin{equation}
 \begin{split}
   \vert\langle \overline P_{v_\sigma}
   L_{\epsilon;v_\sigma}\zeta_{v_\sigma},N_{v_\sigma}(\zeta_{v_\sigma})
   \rangle\vert
   &\leq \Vert\overline P_{v_\sigma}
   L_{\epsilon;v_\sigma}\zeta_{v_\sigma}\Vert_2
   \Vert N_{v_\sigma}(\zeta_{v_\sigma})\Vert_2\cr
   &\leq C\Vert\zeta_{v_\sigma}\Vert_{H^2_{v_\sigma}}^2
   (\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}
   +\Vert\zeta_{v_\sigma}\Vert^2_{H^1_{v_\sigma}})\,.\cr
 \end{split}
\tag{3.11}
\end{equation}
%
Furthermore, we have the two estimates 
%
\begin{multline}
   \vert\langle L_{\epsilon;v_\sigma}\zeta_{v_\sigma},
   \dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)\rangle\vert
   =\vert\langle P_{v_\sigma}L_{\epsilon;v_\sigma}\zeta_{v_\sigma},
   \dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)\rangle\vert\cr
   \leq\Vert P_{v_\sigma}L_{\epsilon;v_\sigma}\zeta_{v_\sigma}
   \Vert_{H^1_{v_\sigma}}
   \Vert\dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)
   \Vert_{H^{-1}_{v_\sigma}}
   \leq \epsilon C\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}
   \Vert\dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)
   \Vert_{H^{-s}_{v_\sigma}}\cr
   \tag{3.12}
\end{multline}
%
(see the proof of Lemma \ref{lemma_A}) and
%
\begin{multline}
   \vert\langle N_{v_\sigma}(\zeta_{v_\sigma}),
   P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)\rangle\vert
   \leq\Vert P_{v_\sigma}N_{v_\sigma}(\zeta_{v_\sigma})
   \Vert_{H^{-1}_{v_\sigma}}\Vert{\cal E}'_\epsilon(v_\sigma)
   \Vert_{H^1_{v_\sigma}}\\
   \leq \epsilon C(\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}^2
   +\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}^3)\\
   \tag{3.13}
\end{multline}
%
where in Eq. (3.13) use has been made of Eq. (3.9) and Lemma \ref{lemma_B}.
\par In order to obtain a bound on the term $\dot R_v(\zeta_v)$ we note that
Eq. (3.2) implies
%
$$ \dot R_v(\zeta)=
   \big(\partial_t R_v(\zeta)\big)_{\zeta=const.}=\langle\dot v, 
   \partial_v R_v(\zeta)\rangle
   \eqno(3.14)$$
hence, we have
%
\begin{equation}
 \begin{split}
   \vert\dot R_{v_\sigma}(\zeta_{v_\sigma})\vert
   &=\vert\langle\dot v_\sigma,
   \partial_{v_\sigma} R_{v_\sigma}(\zeta_{v_\sigma})\rangle\vert\cr
   &\leq (\Vert\dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)
   \Vert_{H^{-s}_{v_\sigma}}
   +\Vert P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)\Vert_{H^{-s}_{v_\sigma}})
   \Vert\partial_{v_\sigma} R_{v_\sigma}(\zeta_{v_\sigma})
   \Vert_{H^s_{v_\sigma}}\,.\cr
 \end{split}
 \tag{3.15}
\end{equation}
%
The following estimate is proved in Section 4:
%
\begin{lemma}
\label{lemma_G}
For any $\zeta\in T_{v_\sigma} X$ we have
%
$$ \Vert\partial_{v_\sigma} R_{v_\sigma}(\zeta)\Vert_{H^1_{v_\sigma}}
   \leq C \Vert\zeta\Vert_{H^2_{v_\sigma}}^2\Vert\zeta\Vert_{H^1_{v_\sigma}}\,.
   \eqno(3.16)$$
\hfill$\square$
\end{lemma}
%
Lemma \ref{lemma_G} and Eq. (3.9) then imply the following bound for $0<s<1$
%
$$ \vert\dot R_{v_\sigma}(\zeta_{v_\sigma})\vert
   \leq C(\Vert\dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)
   \Vert_{H^{-s}_{v_\sigma}}
   +\epsilon)\Vert\zeta_{v_\sigma}\Vert_{H^2_{v_\sigma}}^2
   \Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}\,.
   \eqno(3.17)$$
With Eq. (3.10), (3.11), (3.12), (3.13) and (3.17) providing appropriate
bounds on the corresponding terms in Eq. (3.6) we get
%
\begin{multline}
   \frac{1}{2}\partial_t\langle\zeta_{v_\sigma},
   L_{\epsilon;v_\sigma}\zeta_{v_\sigma}\rangle\leq
   -\langle \overline P_{v_\sigma}L_{\epsilon;v_\sigma}\zeta_{v_\sigma},
   \overline P_{v_\sigma}L_{\epsilon;v_\sigma}\zeta_{v_\sigma}\rangle\cr
   +C(1+\epsilon+\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}
   +\Vert\dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)
   \Vert_{H^{-s}_{v_\sigma}})
   \Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}
   \Vert\zeta_{v_\sigma}\Vert^2_{H^2_{v_\sigma}}\cr
   +\epsilon C(1+\Vert\dot v_\sigma+P_{v_\sigma}
   {\cal E}'_\epsilon(v_\sigma)\Vert_{H^{-s}_{v_\sigma}}
   +\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}
   +\Vert\zeta_{v_\sigma}\Vert^2_{H^1_{v_\sigma}})
   \Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}\cr
   +\vert\langle N_{v_\sigma}(\zeta_{v_\sigma}),
   \dot P_{v_\sigma}\zeta_{v_\sigma}\rangle\vert\,.\cr
   \tag{3.18}
\end{multline}
%
We still need to estimate the last term on the r.h.s. of Eq. (3.18). Using
Lemma \ref{lemma_C} and Eq. (2.62) we obtain
%  
$$ \Vert\dot P_{v_\sigma}\zeta_{v_\sigma}\Vert_2\leq C
   (\Vert\dot v_\sigma+P_{v_\sigma}{\cal E'}_\epsilon(v_\sigma)
   \Vert_{H^{-s}_{v_{\sigma}}}+\epsilon)
   \Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}\,.
   \eqno(3.19)$$
With the help of Lemma \ref{lemma_E} and Eq. (3.19) we can provide the
appropriate bound in the form
%
\begin{multline}
   \vert\langle N_{v_\sigma}(\zeta_{v_\sigma}),
   \dot P_{v_\sigma}\zeta_{v_\sigma}\rangle\vert
   \leq\Vert N_{v_\sigma}(\zeta_{v_\sigma})\Vert_2
   \Vert\dot P_{v_\sigma}\zeta_{v_\sigma}\Vert_2\cr
   \leq C(\Vert\dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)
   \Vert_{H^{-s}_{v_\sigma}}+\epsilon)
   (1+\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}})
   \Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}
   \Vert\zeta_{v_\sigma}\Vert^2_{H^2_{v_\sigma}}\,.\cr
   \tag{3.20}
\end{multline}
%
Inserting the bound obtained in Eq. (3.20) into the r.h.s. of Eq. (3.18) we
get
%
\begin{multline}
   \frac{1}{2}\partial_t\langle\zeta_{v_\sigma},
   L_{\epsilon;v_\sigma}\zeta_{v_\sigma}\rangle\leq
   -\langle \overline P_{v_\sigma}L_{\epsilon;v_\sigma}\zeta_{v_\sigma},
   \overline P_{v_\sigma}L_{\epsilon;v_\sigma}\zeta_{v_\sigma}\rangle\cr
   +C_2\big[(2+\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}})(1+\epsilon+
   \Vert\dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)
   \Vert_{H^{-s}_{v_\sigma}})-1\big]
   \Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}
   \Vert\zeta_{v_\sigma}\Vert^2_{H^2_{v_\sigma}}\cr
   +\epsilon C_3(1+\Vert\dot v_\sigma+P_{v_\sigma}
   {\cal E}'_\epsilon(v_\sigma)\Vert_{H^{-s}_{v_\sigma}}
   +\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}
   +\Vert\zeta_{v_\sigma}\Vert^2_{H^1_{v_\sigma}})
   \Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}\,.\cr
   \tag{3.21}
\end{multline}
%
The first term on the r.h.s. of Eq. (3.21) is handled by utilizing the
linear stability properties of the Ginzburg-Landau equations. The Linear
Stability Theorem is used in Section 4 in the proof of the following lemma:
%
\begin{lemma}
\label{lemma_H}
For $\zeta_v\in T_vX^{(n)}$ orthogonal to $T_vM_{as}$,
that is for $P_v\zeta_v=0$, we have
%
$$ \langle\overline P_{v_\sigma}L_{\epsilon;v_\sigma}\zeta_{v_\sigma},
   \overline P_{v_\sigma}L_{\epsilon;v_\sigma}\zeta_{v_\sigma}\rangle
   \geq 2C_1\Vert\zeta_{v_\sigma}\Vert_{H^2_{v_\sigma}}^2
   \eqno(3.22)$$
(the choice of the form of the constant here is purely for reasons of
convenience).\hfill$\square$
\end{lemma}
%
\par We will also need an additional lemma which is also proved in Section 4:
%
\begin{lemma}
\label{lemma_I}
For $\zeta\in T_vX^{(n)}$ with
$\zeta_v\in (T_vM_{as})^\perp$ we have
%
$$ \langle \overline P_vL_{\epsilon;v}\zeta_v,
   \overline P_vL_{\epsilon;v}\zeta_v\rangle
   \geq\nu'\langle\zeta_v,L_{\epsilon;v}\zeta_v\rangle\,.
   \eqno(3.23)$$
\hfill$\square$
\end{lemma}
%
\par Eq. (3.21), Lemma \ref{lemma_H} and Lemma \ref{lemma_I} imply that
%
\begin{multline}
   \frac{1}{2}\partial_t\langle\zeta_{v_\sigma},
   L_{\epsilon;v_\sigma}\zeta_{v_\sigma}\rangle
   \leq -\frac{1}{2}\nu'\langle\zeta_{v_\sigma},
   L_{\epsilon;v_\sigma}\zeta_{v_\sigma}\rangle+
   \Vert\zeta_{v_\sigma}\Vert_{H^2_{v_\sigma}}^2\times\cr
   \left\{-C_1+C_2\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}
   \big[(2+\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}})
   (1+\epsilon+\Vert\dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)
   \Vert_{H^{-s}_{v_\sigma}})-1\big]\right\}\cr
   +\epsilon C_3\left(1+\Vert\dot v_\sigma+P_{v_\sigma}
   {\cal E}'_\epsilon(v_\sigma)\Vert_{H^{-s}_{v_\sigma}}
   +\Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}
   +\Vert\zeta_{v_\sigma}\Vert^2_{H^1_{v_\sigma}}\right)
   \Vert\zeta_{v_\sigma}\Vert_{H^1_{v_\sigma}}\,.\cr
   \tag{3.24}
\end{multline}
%
Eq. (3.24) is the statement of proposition \ref{proposition_D}.
\hfill$\blacksquare$
%
\section{Proofs of Lemmas \ref{lemma_A}-\ref{lemma_I}}
\label{A_I}
\smallskip
\par{\bf Proof of Lemma \ref{lemma_C}:}
\smallskip
\par The starting point to the proof of Lemma \ref{lemma_C} is the explicit
expression
for the projection $P_{v_\sigma}$ given in Eq. (2.12). Taking the time 
derivative of $P_{v_\sigma}$ and applying the resulting operator to a fixed
vector $\zeta\in T_{v_\sigma}X^{(n)}$ satisfying the condition 
$P_{v_\sigma}\zeta=0$ we get
%
$$ \dot P_{v_\sigma}\zeta=a_n^{-2}\sum_{i=1,2} T_i^\sigma 
   \partial_t\langle T_i^\sigma,\zeta\rangle
   +\int d^2x\, G^\sigma_{\delta(x)}
   K_\sigma^{-1}(x,y)
   \partial_t\langle G^\sigma_{\delta(y)},\zeta\rangle\,.
   \eqno(4.1)$$
Eq. (4.1) enable us to obtain the $L^2$ norm of $\dot P_v\zeta$
\begin{equation}
 \begin{split}
   \Vert\dot P_{v_\sigma}\zeta\Vert_2^2&=a_n^{-2}\sum_{i=1,2}
   \vert\langle\partial_t T_i^\sigma,\zeta\rangle\vert^2\cr
   &+\int d^2x\int d^2y\ 
   \overline{\partial_t\langle G^\sigma_{\delta(x)},\zeta\rangle}
   K_\sigma^{-1}(x,y)
   \partial_t\langle G^\sigma_{\delta(y)},\zeta\rangle\,.\cr
 \end{split}
 \tag{4.2}
\end{equation}
%
\par Consider the first term on the r.h.s. of Eq. (4.2). The time derivative
of $T^\sigma_i$ is given by
%
\begin{equation}
   \partial_t T^\sigma_i=i\dot{\tilde\gamma}
   \begin{pmatrix}
   (\nabla_{A_\sigma}\psi_\sigma)_i\\ 0 \\
   \end{pmatrix}
   +\Sigma_j(-\dot z_j)(\nabla_{A_\sigma} T^\sigma_i)_j
 \tag{4.3}
\end{equation}
%
where
%
\begin{equation}
   (\nabla_{A_\sigma} T^\sigma_i)_j=
   \begin{pmatrix}
   (\nabla_{A_\sigma}(\nabla_{A_\sigma}\psi_\sigma)_i)_j\\
   \partial_j (\nabla\times A_\sigma)\hat e_i\\
   \end{pmatrix}\,.
   \tag{4.4}
\end{equation}
%
Thus we arrive at the following estimate
%
\begin{equation}
 \begin{split}
   \Vert \partial_t T^\sigma_i\Vert_2&\leq
   \Vert \dot{\tilde\gamma}(\nabla_{A_\sigma}\psi_{\sigma})_i\Vert_2
   +\Sigma_j\vert\dot z_j\vert\ \Vert(\nabla_{A_\sigma}T^\sigma_i)_j
   \Vert_2\cr
   &\leq \Vert \dot{\tilde\gamma}\Vert_2\Vert(\nabla_{A_\sigma}
   \psi_\sigma)_i\Vert_\infty+\Sigma_j\vert\dot z_j\vert\ 
   \Vert(\nabla_{A_\sigma}T^\sigma_i)_j\Vert_2\,.\cr
 \end{split}
 \tag{4.5}
\end{equation}
%
The norms of the two vectors appearing on the r.h.s. of Eq. (4.5) are gauge
and translation invariant, hence they are constant at all points of $M_{as}$.
We conclude that the following inequality is valid
%
$$  \Vert \partial_t T^\sigma_i\Vert_2
    \leq C (\Vert \dot{\tilde\gamma}\Vert_2+\Sigma_i\vert\dot z_i\vert)
    \eqno(4.6)$$
and so
%
\begin{equation}
 \begin{split}
   \vert\langle\partial_t T^\sigma_i,\zeta\rangle\vert
   &\leq\Vert\partial_t T^\sigma_i\Vert_2\Vert\zeta\Vert_2
   \leq C(\Vert\dot{\tilde\gamma}\Vert_2+\sum_i\vert\dot z_i\vert\,)
   \Vert G_{-\gamma}\zeta\Vert_2\cr
   &\leq C(\Vert\dot{\tilde\gamma}\Vert_2+\sum_i\vert\dot z_i\vert\,)
   \Vert G_{-\gamma}\zeta\Vert_{H^1}
   =C(\Vert\dot{\tilde\gamma}\Vert_2+\sum_i\vert\dot z_i\vert\,)
   \Vert\zeta\Vert_{H^1_{v_\sigma}}\,.\cr
 \end{split}
 \tag{4.7}
\end{equation}
%
\par Consider now the second term on the r.h.s. of Eq. (4.2). Let
$\zeta=(\xi, F)\in T_vX^{(n)}$, the time derivative
$\partial_t\langle G_\chi^\sigma,\zeta\rangle$ is given by (note that
$\zeta$ is a fixed vector)
%
\begin{equation}
 \begin{split}
   \partial_t\langle G_{\chi}^\sigma,\zeta\rangle&=
   \partial_t\langle \int d^2x\, \chi(x)G_{\delta(x)}^\sigma,\zeta\rangle
   =\int d^2x\,\chi(x)\partial_t
   \langle G_{\delta(x)}^\sigma,\zeta\rangle=\cr
   &=\partial_t\int d^2x\, (Im[\chi{\overline\psi}_\sigma\xi]-\nabla\chi F)
   =\int d^2x\, \chi(x)Im[\partial_t{\overline\psi}_\sigma\xi](x)\cr
 \end{split}
 \tag{4.8}
\end{equation}
%
hence
%
$$ \partial_t\langle G_{\delta(x)}^\sigma,\zeta\rangle=
   Im[\partial_t{\overline\psi_\sigma(x)}\xi(x)]\,.
   \eqno(4.9)$$
Now, $K^{-1}_\sigma$ are uniformly bounded on $M_{as}$ as operators from
$H^{-1}$ to $H^1$ (see \cite{ST1}) and we can estimate
%
\begin{multline}
   \bigg\vert\int d^2x\int d^2y\ 
   \overline{\partial_t\langle G^\sigma_{\delta(x)},\zeta\rangle}
   K_\sigma^{-1}(x,y)
   \partial_t\langle G^\sigma_{\delta(y)},\zeta\rangle\bigg\vert=\\
   \big\vert(\partial_t\langle G^\sigma_{\delta(x)},\zeta\rangle,
   K_\sigma^{-1}
   \,\partial_t\langle G^\sigma_{\delta(x)},\zeta\rangle)\big\vert
   \leq\Vert\partial_t\langle G^\sigma_{\delta(x)}
   ,\zeta\rangle\Vert_{H^{-1}}\Vert K_\sigma^{-1}
   \,\partial_t\langle G^\sigma_{\delta(x)},\zeta\rangle)\Vert_{H^1}\\
   \leq \Vert K^{-1}_\sigma\Vert_{H^{-1}\to H^1}
   \Vert\partial_t\langle G^\sigma_{\delta(x)}
   ,\zeta\rangle\Vert_{H^{-1}}^2
   =C\Vert Im[\partial_t\overline\psi_\sigma\xi]\Vert_{H^{-1}}^2
   \leq C\Vert\partial_t\psi_\sigma\xi\Vert_{H^{-1}}^2\\
   \tag{4.10}
\end{multline}
%
were here again $\zeta=(\xi, F)$.
We want to estimate the r.h.s. of this inequality. We have
%
\begin{equation}
 \begin{split}
   \Vert\partial_t\overline\psi_\sigma\xi\Vert_{H^{-1}}&=
   \sup_{\Vert\alpha\Vert_{H^1}\leq 1}
   \langle\alpha,\partial_t\overline\psi_\sigma\xi\rangle
   \leq \sup_{\Vert\alpha\Vert_{H^1}\leq 1}
   \Vert\alpha\xi\Vert_2\Vert\partial_t\psi_\sigma\Vert_2\cr
   &\leq\sup_{\Vert\alpha\Vert_{H^1}\leq 1}\Vert\alpha\Vert_4
   \Vert\partial_t\psi_\sigma\Vert_2\Vert\xi\Vert_4
   =C\Vert\partial_t\psi_\sigma
   \Vert_2\Vert\xi\Vert_{H^1_{v_\sigma}}\,.\cr
 \end{split}
 \tag{4.11}
\end{equation}
%
Furthermore, if we use Eq. (2.13) to write explicitly the time derivative
%
$$ \partial_t \psi_\sigma
   =i\dot{\tilde\gamma}\psi_\sigma+\sum_i(-\dot z_i)
   (\nabla_{A_\sigma}\psi_\sigma)_i
   \eqno(4.12)$$
we obtain the following bound on $\Vert\partial_t\psi_\sigma\Vert_2$
%
$$ \Vert\partial_t\psi_\sigma\Vert_2
   \leq \Vert\dot{\tilde\gamma}\Vert_2+\sum_i \vert\dot z_i\vert
   \,\Vert(\nabla_{A_\sigma}\psi_\sigma)_i\Vert_2
   \leq C(\Vert\dot{\tilde\gamma}\Vert_2+\sum_i \vert\dot z_i\vert)
   \eqno(4.13)$$
where again we use the gauge invariance of 
$\Vert(\nabla_{A_\sigma}\psi_\sigma)_i\Vert_2$.
\par Thus we arrive at the result
%
$$ \Vert\dot P_{v_\sigma}\zeta\Vert_2\leq C(\Vert\dot{\tilde\gamma}\Vert_2
   +\sum_i\vert\dot z_i\vert\,)\Vert\zeta\Vert_{H^1_{v_\sigma}}\,.
   \eqno(4.14)$$
This completes the proof of Lemma \ref{lemma_C}.\hfill$\blacksquare$
\bigskip
\par{\bf Proof of Lemma \ref{lemma_D}:}
\smallskip
\par Our goal is to prove the inequality
%
$$ \vert a_n\dot{\bf z}+\partial^A_{\bf z}{\cal E}'_\epsilon(v_\sigma)\vert
   +\Vert\dot{\tilde\gamma}\Vert_{H^{1-s}}
   \leq C\Vert\dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)
   \Vert_{H^{-s}_{v_\sigma}}\,.
   \eqno(4.15)$$
Starting with $\dot{\tilde\gamma}$ we have
%
\begin{multline}
   \langle G^\sigma_{\delta(x)},
   \dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)\rangle=
   \langle G^\sigma_{\delta(x)},\dot v_\sigma\rangle
   +\langle G^\sigma_{\delta(x)},P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)
   \rangle=
   \langle G^\sigma_{\delta(x)},\dot v_\sigma\rangle\cr
   =(-\Delta+\vert\psi_\sigma(x)\vert^2)\dot{\tilde\gamma}(x)
   =(K_\sigma\dot{\tilde\gamma})(x)\cr
   \tag{4.16}
\end{multline}
%
since $\langle G^{{\bf z}\gamma}_{\delta(x)},P_v{\cal E}'_\epsilon(v)\rangle
=Im[\overline\psi(\epsilon W(x)\psi)]=0$.
We find the following expression for $\dot{\tilde\gamma}$
%
$$ \dot{\tilde\gamma}(x)=\int d^2y\, K^{-1}_\sigma(x,y)
   \langle G^\sigma_{\delta(y)},
   \dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)\rangle
   \eqno(4.17)$$
where,
\begin{multline}
   \langle G^\sigma_{\delta(y)},\dot v_\sigma
   +P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)\rangle=\\
   =\left\{Im[\overline\psi_\sigma(\dot v_\sigma
   +P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma))_1]
   -\nabla\cdot(\dot v_\sigma
   +P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma))_2\right\}(y)\,.\\
   \tag{4.18}
\end{multline}
%
Here$(\dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma))_1$ is the first,
complex scalar, component and
$(\dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma))_2$ is the second,
real vector component. Since $K^{-1}_\sigma$ is a bounded operator from
$H^{-1-s}$ to $H^{1-s}$ (uniformly in $\sigma$) we have
%
\begin{multline}
   \Vert\dot{\tilde\gamma}\Vert_{H^{1-s}}=\Vert K^{-1}_\sigma
   \langle G^\sigma_{\delta(y)},\dot v_\sigma
   +P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)\rangle
   \Vert_{H^{1-s}}\\
   \leq C \Vert\langle G^\sigma_{\delta(y)},
   \dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)
   \rangle\Vert_{H^{-1-s}}\,.\\
   \tag{4.19}
\end{multline}
%
However, using Eq. (4.18) we obtain
%
\begin{multline}
   \Vert\langle G^\sigma_{\delta(y)},
   \dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)\rangle
   \Vert_{H^{-1-s}}=\cr
   =\Vert Im[\overline\psi_{{\bf z} 0}e^{-i\gamma}(\dot v_\sigma
   +P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma))_1]
   -\nabla\cdot(\dot v_\sigma
   +P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma))_2\Vert_{H^{-1-s}}\cr
   \leq C \Vert\dot v_\sigma
   +P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)\Vert_{H^{-s}_{v_\sigma}}\cr
   \tag{4.20}
\end{multline}
%
which is the desired bound. As for $\dot z_i$, we have
%
$$ \vert a_n\dot z_i+\partial^A_{z_i}{\cal E}_\epsilon(v_\sigma)\vert
   =\vert\langle T^\sigma_i,\dot v_\sigma+{\cal E}'_\epsilon(v_\sigma)\rangle
   \vert\leq \Vert T^\sigma_i\Vert_{H^s_{v_\sigma}}
   \Vert \dot v_\sigma+P_{v_\sigma}{\cal E}'_\epsilon(v_\sigma)
   \Vert_{H^{-s}_{v_\sigma}}\,.
   \eqno(4.21)$$
Obviously $\Vert T^\sigma_i\Vert_{H^s_{v_\sigma}}=
\Vert T^{{\bf z} 0}_i\Vert_{H^s}<C$. From the last estimate, together with
Eq. (4.19) and Eq. (4.20), follows Eq. (4.15).\hfill$\blacksquare$
\bigskip
\par{\bf Proof of Lemma \ref{lemma_A}:}
\smallskip
\par For any $v\in M_{as}$, the tangent space $T_vM_{as}$ is 
spanned by zero eigenvectors of $L_{0;v}$. Hence, since $P_v$ is a projection
on $T_vM_{as}$ we have
%
$$ P_vL_{0;v}=L_{0;v}P_v=0\,.
   \eqno(4.22)$$
Furthermore, by assumption the potential $W$ is bounded
%
$$ \sup_{x\in R^2}\vert W(x)\vert=C\,.
   \eqno(4.23)$$
Take any vector $\zeta=(\xi,F)\in T_vX^{(n)}$. The above observation
leads to the following estimate
%
$$ \Vert P_vL_{\epsilon;v}\zeta\Vert_2=
   \Vert P_v(L_{0;v}+\epsilon W)\zeta\Vert_2
   \leq\epsilon\Vert W\zeta\Vert_2
   \leq \epsilon C\Vert\zeta\Vert_2\,.
   \eqno(4.24)$$
\hfill$\blacksquare$
\bigskip
\par {\bf Proof of Lemma \ref{lemma_B}}: 
\par We set out to prove the following inequality
%
$$ \Vert P_v N_v(\zeta)\Vert_{H^{-s}_{v_\sigma}}
   \leq C[\Vert\zeta\Vert_{H^1_{v_\sigma}}^2
   +\Vert\zeta\Vert_{H^1_{v_\sigma}}^3]\,.
   \eqno(4.25)$$
\par The expression for $N_v(\zeta)$ is given in Appendix A, Eq. (A.3), where
$\zeta=(\xi, F)$. The estimates of the terms in $N_v(\zeta)$ which do not
include derivatives are straightforward. For example, denoting
$\tilde N_1=(0,2Re(\overline\psi F\xi))$ we obtain
%
\begin{multline}
   \Vert\tilde N_1\Vert_{H^{-s}_{v_\sigma}}=2
   \Vert Re(\overline\psi F\xi)\Vert_{H^{-s}_{v_\sigma}}
   \leq 4\Vert Re(\overline\psi F\xi)\Vert_{L^2_{v_\sigma}}
   =4\Vert Re(\overline\psi F\xi)\Vert_2\\
   \leq C\Vert\zeta\Vert_4^2
   \leq C\Vert\zeta\Vert_{H^1_{v_\sigma}}^2\,.\\
   \tag{4.26}
\end{multline}
%
The vector $\tilde N_2=(0,Im(\overline\xi\nabla_A\xi))$ is an example
for a term which is more difficult to estimate. An estimate of this term is
given in Section 4. The other problematic terms in $N_v(\zeta)$ are
estimated in the same manner.\hfill$\blacksquare$
\bigskip
\par{\bf Proof of Lemma \ref{lemma_E}:}
\smallskip
\par We want to prove the inequality
%
$$ \Vert N_v(\zeta)\Vert_2\leq C\Vert\zeta\Vert_{H^2_{v_\sigma}}
   (\Vert\zeta\Vert_{H^1_{v_\sigma}}+\Vert\zeta\Vert_{H^1_{v_\sigma}}^2)
   \eqno(4.27)$$
\smallskip
\par Again the terms not containing any derivatives do not present any 
problems (see for example Eq. (4.26)). For the vector
$\tilde N_2=(0,Im(\overline\xi\nabla_{A_\sigma}\xi))$ we have
%
\begin{multline}
   \Vert\tilde N_2\Vert_2
   \leq\Vert\overline\xi\nabla_{A_\sigma}\xi\Vert_2
   \leq\Vert\overline{\xi'}\nabla_{A^{(n)}}\xi'\Vert_2
   \leq\Vert\xi'\Vert_\infty\Vert\nabla_{A^{(n)}}\xi'\Vert_2\cr
   \leq C\Vert\xi'\Vert_{H^2}\Vert\xi'\Vert_{H^1}
   =C\Vert\xi\Vert_{H^2_{v_\sigma}}\Vert\xi\Vert_{H^1_{v_\sigma}}\cr
   \tag{4.28}
\end{multline}
%
where $\xi'=e^{-i\gamma}\xi$. Other terms containing derivatives are
estimated in a similar way.\hfill$\blacksquare$
\bigskip
\par{\bf Proof of Lemma \ref{lemma_F}:}
\smallskip
\par We want to prove the inequalities
%
$$ \Vert\overline P_{v_\sigma}L_{\epsilon;v_\sigma}\zeta\Vert_2
   \leq C\Vert\zeta\Vert_{H^2_{v_\sigma}}
   \eqno(4.29)$$
and
%
$$ \Vert \overline P_{v_\sigma}L_{\epsilon;v_\sigma}
   \zeta\Vert_{H^{-1}_{v_\sigma}}
   \leq C\Vert\zeta\Vert_{H^1_{v_\sigma}}\,.
   \eqno(4.30)$$
\par Since $L_{\epsilon;v_\sigma}$ is a second order linear operator no
difficulties arise and the two inequalities are proved by strightforward
calculations. We just note that
%
$$ \Vert\overline P_{v_\sigma}L_{\epsilon;v_\sigma}\zeta\Vert_2
   =\Vert G_{-\gamma}\overline P_{v_\sigma}L_{\epsilon;v_\sigma}\zeta\Vert_2
   =\Vert\overline P_{v_{{\bf z} 0}}L_{\epsilon;v_{{\bf z} 0}}\zeta'\Vert_2
   \eqno(4.31)$$
and 
%
$$ \Vert\overline P_{v_\sigma}L_{\epsilon;v_\sigma}
   \zeta\Vert_{H^{-1}_{v_\sigma}}
   =\Vert G_{-\gamma}\overline P_{v_\sigma}L_{\epsilon;v_\sigma}
   \zeta\Vert_{H^{-1}}
   =\Vert\overline P_{v_{{\bf z} 0}}
   L_{\epsilon;v_{{\bf z} 0}}\zeta'\Vert_{H^{-1}}
   \eqno(4.32)$$
where $\zeta'=e^{-i\gamma}\zeta$. Since the gauge function $\gamma$ does not 
appear in the operators on the r.h.s. of Eq. (4.31) and 
Eq. (4.32) one obtains by a straightforward calculation
%
$$ \Vert\overline P_{v_{{\bf z} 0}}L_{\epsilon;v_{{\bf z} 0}}\zeta'\Vert_2
   \leq C\Vert\zeta'\Vert_{H^2}=\Vert\zeta\Vert_{H^2_{v_\sigma}}$$
and
%
$$ \Vert\overline P_{v_{{\bf z} 0}}L_{\epsilon;v_{{\bf z} 0}}\zeta'
   \Vert_{H^{-1}}
   \leq C\Vert\zeta'\Vert_{H^1}=C\Vert\zeta'\Vert_{H^1_{v_\sigma}}\,.$$
\hfill$\blacksquare$
\bigskip
\par{\bf Proof of Lemma \ref{lemma_G}:}
\smallskip
We want to prove the inequality
%
$$ \Vert \partial_{v_\sigma}R_{v_\sigma}(\zeta)\Vert_{H^1_{v_\sigma}}
   \leq \Vert\zeta\Vert_{H^2_{v_\sigma}}^2
   \Vert\zeta\Vert_{H^1_{v_\sigma}}\,.$$
\par By derivation of the expression for $R_v(\zeta)$ (Eq. (A.5)) we obtain, 
for $\zeta=(\xi,F)$
%
$$ \partial_v R_v(\zeta)
   =\left( (F^2+\frac{\lambda}{2}\vert\xi\vert^2)\xi\ ,
   F\vert\xi\vert^2\right)\,.
   \eqno(4.33)$$
We want to estimate the norm 
$\Vert\partial_{v_\sigma} R_{v_\sigma}(\zeta)\Vert_{H^1_{v_\sigma}}$. Note
first that
%
$$ G_{-\gamma}\partial_{v_\sigma} R_{v_\sigma}(\zeta)
   =\left((F^2+\frac{\lambda}{2}\vert\xi'\vert^2)\xi'\ ,
   F\vert\xi'\vert^2\right)
   =\partial_{v_\sigma} R_{v_\sigma}(\zeta')$$
where $\xi'=e^{-i\gamma}\xi$, $\zeta'=G_{-\gamma}\zeta$. The estimate of the 
$L^2$ norm (or $L^2_{v_\sigma}$ norm which is the same) is 
straightforward and we get
%
$$ \Vert\partial_{v_\sigma} R_{v_\sigma}(\zeta)\Vert_2^2
   =\Vert(F^2+\frac{\lambda}{2}\vert\xi'\vert^2)\xi'\Vert_2^2
   +\Vert F\vert\xi'\vert^2\Vert_2^2
   \leq C\Vert\zeta'\Vert_{H^1}^6=C\Vert\zeta\Vert_{H^1_{v_\sigma}}^6\,.$$
Let $\tilde F=F_1+iF_2$ be the complexification of $F$. We have
%
\begin{multline}
   \Vert\nabla(\partial_v R_v(\zeta'))\Vert_2^2
   =\Vert\nabla((\vert\tilde F\vert^2+\frac{\lambda}{2}\vert\xi'\vert^2)\xi')
   \Vert_2^2
   +\Vert\nabla(\tilde F\vert\xi'\vert^2)\Vert_2^2=\cr
   =\Vert 2Re[(\nabla\tilde F)\overline{\tilde F}]\xi'
   +\vert\tilde F\vert^2(\nabla\xi')
   +{\lambda}(\nabla\xi')\vert\xi'\vert^2
   +\frac{\lambda}{2}(\nabla\overline{\xi'}){\xi'}^2\Vert_2^2\cr
   +\Vert(\nabla\tilde F)\vert\xi'\vert^2+ 
   2\tilde F\, Re[\overline{\xi'}\nabla\xi']\Vert_2^2\,.\cr
   \tag{4.34}
\end{multline}
%
All the terms on the r.h.s. of Eq. (4.34) are handled in the same manner.
For example:
%
\begin{equation*}
 \begin{split}
   \Vert(\nabla\tilde F)\overline{\tilde F}\,\xi')\Vert_2
   &\leq \Vert\tilde F\Vert_\infty\Vert\xi'\Vert_\infty
   \Vert\nabla\tilde F\Vert_2\cr
   &\leq\Vert\tilde F\Vert_{H^2}\Vert\xi'\Vert_{H^2}\Vert\tilde F\Vert_{H^1}
   \leq \Vert\zeta'\Vert_{H^2}^2\Vert\zeta'\Vert_{H^1}
   =\Vert\zeta\Vert_{H^2_{v_\sigma}}^2\Vert\zeta\Vert_{H^1_{v_\sigma}}\,.\cr
 \end{split}
\end{equation*}
\hfill$\blacksquare$
\bigskip
\par{\bf Proof of Lemma \ref{lemma_H}:}
\smallskip
\par First we use Eq. (4.22) and the linear stability theorem in order to
arrive at the result that for any $\zeta_v\in T_vX^{(n)}$ we have
%
\begin{multline}
   \Vert L_{0;v}\zeta_v\Vert_2^2
   =\langle L_{0;v}\zeta_v,L_{0;v}\zeta_v\rangle
   =\langle L_{0;v}^{1/2}\zeta_v,L_{0;v}\,L_{0;v}^{1/2}\zeta_v\rangle=\cr
   =\langle L_{0;v}^{1/2}\zeta_v,\overline P_vL_{0;v}\overline P_v
   \,L_{0;v}^{1/2}\zeta_v\rangle
   =\langle \overline P_vL_{0;v}^{1/2}\zeta_v,L_{0;v}\,\overline P_v
   L_{0;v}^{1/2}\zeta_v\rangle
   \geq\nu\Vert\overline P_vL_{0;v}^{1/2}\zeta_v\Vert_2^2\,.\cr
   \tag{4.35}
\end{multline}
%
In addition, using Eq. (4.22) again, we have
%
$$ \Vert P_vL_{0;v}^{1/2}\zeta_v\Vert_2^2
   =\langle P_vL_{0;v}L_{0;v}^{-1/2}\zeta_v, P_vL_{0;v}^{1/2}\zeta_v\rangle=0\,.
   \eqno(4.36)$$
From Eq. (4.35), Eq. (4.36) and the linear stability theorem we deduce that
%
$$ \langle L_{0;v}\zeta_v,L_{0;v}\zeta_v\rangle
   \geq\nu\langle \zeta_v,L_{0;v}\zeta_v\rangle
   \geq\nu^2\Vert\zeta_v\Vert_{H^1}^2\,.
   \eqno(4.37)$$
We note that $\Vert (L_{0;v}+\Delta)\zeta_v\Vert_2\leq C_1\Vert\zeta_v
\Vert_{H^1}+\Vert\nabla\nabla F_v\Vert_2$, where $\zeta_v=(\xi_v,F_v)$. 
Furthermore, since $\zeta_v\in (T_vM_{as})^\perp$, it satisfies the gauge 
condition ${\rm Im}\,(\overline\psi_v\xi_v)-\nabla F_v$=0. We conclude that
$L_{0,v}+\Delta$ is a bounded operator from $H^1$ to $L^2$, i.e. 
$\Vert L_{0,v}+\Delta\Vert_{H^1\to L^2}\leq C_1$. we have
%
\begin{multline*}
   \Vert\Delta\zeta\Vert_2^2=\Vert (L_{0;v}+\Delta)\zeta-L_{0;v}\zeta\Vert_2^2
   \leq 2(\Vert (L_{0;v}+\Delta)\zeta\Vert_2^2+
   \Vert L_{0;v}\zeta\Vert_2^2)\\
   \leq 2(C_1\Vert\zeta\Vert_{H^1}^2+
   \Vert L_{0;v}\zeta\Vert_2^2)\\
\end{multline*}
%
and so
%
$$ \frac{1}{2}\Vert\Delta\zeta\Vert_2^2-C_1\Vert\zeta\Vert_{H^1}^2
   \leq\langle L_{0;v}\zeta,L_{0;v}\zeta\rangle\,.$$
Then, for any $0<\delta<1$ we have
%
\begin{equation}
 \begin{split}
   \langle L_{0,v}\zeta_v,L_{0,v}\zeta_v\rangle&=
   \delta\langle L_{0,v}\zeta_v,L_{0,v}\zeta_v\rangle
   +(1-\delta)\langle L_{0,v}\zeta_v,L_{0,v}\zeta_v\rangle\cr
   &\geq \frac{\delta}{2}\Vert\Delta\zeta_v\Vert_2^2
   +[(1-\delta)C_2-\delta C_1]\Vert\zeta_v\Vert_{H^1}^2\,.\cr
 \end{split}
 \tag{4.38}
\end{equation}
%
Choosing $\delta$ small enough the second term on the r.h.s. of Eq. (4.38)
is positive and we get
%
$$ \langle L_{0,v}\zeta_v,L_{0,v}\zeta_v\rangle 
   \geq C\Vert\zeta_v\Vert_{H^2}^2\,.
   \eqno(4.39)$$
In addition to this basic result we have 
%
\begin{equation}
 \begin{split}
   \Vert L_{0;v}\zeta_v\Vert_2^2=\Vert\overline P_v L_{0;v}\zeta_v\Vert_2^2
   &=\Vert\overline P_v(L_{\epsilon;v}\zeta_v-\epsilon W\zeta_v)\Vert_2^2\\
   &\leq 2(\Vert\overline P_vL_{\epsilon;v}\zeta_v\Vert_2^2
   +\epsilon^2\Vert W\Vert_\infty^2\Vert\zeta_v\Vert_2^2)\\
 \end{split}
 \tag{4.40}
\end{equation}
%
which leads to
%
$$ \langle \overline P_vL_{\epsilon;v}\zeta_v,
   \overline P_vL_{\epsilon;v}\zeta_v\rangle
   \geq \left(\frac{1}{2}-\epsilon^2\Vert W\Vert_\infty^2\nu^2\right)
   \langle L_{0;v}\zeta,L_{0;v}\zeta_v\rangle\,.
   \eqno(4.41)$$
The proof of Lemma \ref{lemma_H} is completed by the following simple observation
%
\begin{equation}
 \begin{split}
   \langle \overline P_vL_{\epsilon;v}\zeta_v,
   \overline P_vL_{\epsilon;v}\zeta_v\rangle
   =\langle G_{-\gamma}\overline P_vL_{\epsilon;v}\zeta_v&,
   G_{-\gamma}\overline P_vL_{\epsilon;v}\zeta_v\rangle=\\
   &=\langle \overline P_{v_{{\bf z} 0}}L_{\epsilon;v_{{\bf z} 0}}\zeta'_v,
   \overline P_{v_{{\bf z} 0}}L_{\epsilon;v_{{\bf z} 0}}\zeta'_v\rangle\\
 \end{split}
 \tag{4.42}
\end{equation}
%
where $\zeta'_v=G_{-\gamma}\zeta_v$ and we have $P_{v_{{\bf z} 0}}\zeta'_v=0$.
Eq. (4.39), Eq. (4.41) and Eq. (4.42) imply the desired result.
\hfill$\blacksquare$
\bigskip
\par{\bf Proof of Lemma \ref{lemma_I}:}
\smallskip
\par The inequality in Lemma \ref{lemma_I}, Eq. (3.23), follows from Eq. (4.37),
Eq. (4.41) and the following simple estimate
%
\begin{equation}
 \begin{split}
   \langle\zeta_v, L_{\epsilon;v}\zeta_v\rangle=
   \langle\zeta_v, L_{0;v}\zeta_v\rangle
   +\epsilon\langle\zeta_v,W\zeta_v\rangle
   &\leq\langle\zeta_v, L_{0;v}\zeta_v\rangle+\epsilon\Vert W\Vert_\infty
   \Vert\zeta_v\Vert_2^2\cr
   &\leq (1+\epsilon\Vert W\Vert_\infty\nu)
   \langle\zeta_v, L_{0;v}\zeta_v\rangle\,.\cr
 \end{split}
 \tag{4.43}
\end{equation}
\hfill$\blacksquare$
%
\bigskip
\par{\bf Appendix A - Taylor expansions of ${\cal E}_\epsilon(u)$ and
${\cal E}'_\epsilon(u)$:}
\bigskip
\par In the proof of Theorem \ref{theorem_A} above we frequently refer to the Taylor 
expansion of ${\cal E}'_\epsilon(u)$  
%
$$ {\cal E}'_\epsilon(u)={\cal E}'_\epsilon(v)+L_{\epsilon;v}\zeta
   +N_{\epsilon;v}(\zeta)\eqno(A.1)$$
A strightforward calculation provides us with explicit expressions for the 
linear and nonlinear terms in $\zeta$. Denoting $\zeta=(\xi,F)$, we have
%
$$ L_{\epsilon;u}\zeta=
   \begin{pmatrix}
   [-\Delta_A+\frac{\lambda}{2}(2\vert\psi\vert^2-1)]\xi
   +\frac{\lambda}{2}\psi^2\overline\xi+i[2\nabla_A\psi+\psi\nabla]F
   +\epsilon W\xi\\
   \\
   Im([\overline{\nabla_A}\psi-\overline\psi\nabla_A]\xi) 
   +(-\Delta+\nabla\nabla+\vert\psi\vert^2)F\\
   \end{pmatrix}
   \eqno(A.2)$$
and
%
$$ N_{\psi,A}(\zeta)=
   \begin{pmatrix}
   i(2\nabla_A\xi+\xi\nabla)F+F^2\psi+F^2\xi+\frac{\lambda}{2}
   (\overline\psi\xi^2+2\vert\xi\vert^2\psi+\vert\xi\vert^2\xi)\\ \\
   2Re(\overline\psi F\xi)-Im(\overline\xi\nabla_A\xi)+F\vert\xi\vert^2\\
   \end{pmatrix}\,.
   \eqno(A.3)$$
In addition we refer at several points to the Taylor expansion of the energy
functional ${\cal E}'_\epsilon(u)$. Letting $u=v+\zeta$ we have
%
$$ {\cal E}_\epsilon(u)={\cal E}_\epsilon(v)
   +\langle\zeta,{\cal E}'_\epsilon(v)\rangle
   +\frac{1}{2}\langle\zeta,L_{\epsilon;v}\zeta\rangle+R_v(\zeta)\eqno(A.4)$$
where $L_{\epsilon;v}$ is given in Eq. (A.2) and the remainder $R_v(\zeta)$
(containing all terms which are higher then quadratic in $\zeta$) is given by
%
$$ R_v(\zeta)=\int\left\{ (F^2+\frac{\lambda}{2}\vert\xi\vert^2){\rm Re}\,
   [\overline\xi\psi]-F\cdot{\rm Im}\,[\overline\xi\nabla_A\xi]
   +\frac{1}{2}(F^2\vert\xi\vert^2+\frac{\lambda}{4}\vert\xi\vert^4)
   \right\}d^2x
   \eqno(A.5)$$
\bigskip
\par\centerline{\bf Acknowledgments}
\smallskip
\par The work of Y. Strauss was supported by NSERC Grant N7109, 
THE ISRAEL SCIENCE FOUNDATION
(Grant no. 188/02) and the Edmund Landau Center for Research in Mathematical
Analysis and Related areas, sponsored by the Minerva Foundation (Germany).
The work of I.M. Sigal was supported by NSF Grant DMS-040026.
%
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\end{document}

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