\section{Introduction}
\label{sec:introduction}

It has been proposed that the low energy dynamics of D-branes can
be captured by the Dirac-Born-Infeld (DBI) effective action. For
unstable D-branes, there are a or more tachyon fields included in
the DBI action \cite{Garousi:2000tr,Sen:2003bc,Sen:2004nf}. In
type II superstring theory, BPS D-branes are supersymmetric and
stable while non-BPS D-branes are non-supersymmetric and unstable.
For a single non-BPS D-brane, the DBI effective action contains a
real tachyon field. The validity of this tachyon field action has
been justified in string field theory
\cite{Kutasov:2003er,Niarchos:2004rw}, where the tachyon DBI
action is derived based on a general effective action setup by
adopting a time-dependent tachyon profile. This implies that there
should be some relation between tachyon profiles and the DBI
effective action.

The equations of motion from the DBI action of a tachyon are
usually complicated in the spacetime-dependent case and usually
studied by numerical simulations, even in the case in the absence
of other world-volume fields. However, we can not get complete
numerical spacetime-dependent solutions due to singularities near
kinks and the vacuum
\cite{Felder:2002sv,Felder:2004xu,Barnaby:2004dz,Cline:2003vc,
Hindmarsh:2007dz}.

In our previous work \cite{Li:2008ah}, we have shown that the
equation of motion of a general scalar, a tachyon or an ordinary
scalar, from the DBI action with well selected potentials has
exact classical solutions after appropriate field redefinitions.
For a tachyon, the solution is a tachyon profile of a single
momentum mode. However, the tachyon profile solution in the
spacetime-dependent case is complex and hence is invalid in the
DBI theory of a real tachyon field. We can get real tachyon
profile solutions only in the purely time-dependent case or the
purely space-dependent case. A special form of the potential that
leads to the tachyon profile solution is the one derived from
string field theory in \cite{Kutasov:2003er}. For massless and
massive scalars, the spacetime-dependent solutions can be real and
so are valid.

In this paper, we apply the results obtained in \cite{Li:2008ah}
to study the dynamics of a general scalar field on a stable or
unstable D-brane in the presence of world-volume massless fields.
We adopt the potentials and field redefinition relations that can
lead to exact solutions to study the multi-field system in the DBI
effective theory, in which we also find exact solutions for all
fields. It is shown that the solution of a specific field in the
DBI action of multi-fields is exactly the one obtained in the case
with all other fields vanishing, except that there are some
momentum coupling constraints between fields. From these solutions
and momentum coupling conditions, we can study the dynamics of
stable and unstable D-branes in the presence of these massless
fields.

The world-volume massless fields we consider include the massless
scalars in the transverse directions to D-branes and the $U(1)$
gauge fields. As we know, the transverse massless scalars describe
the fluctuation of the D-brane in the transverse directions.
Hence, from the exact solutions obtained in the case of coupling
to the transverse massless scalars we can learn the dynamics of
D-branes that move or fluctuate in the bulk in different modes.
From the solutions obtained in the case of coupling to gauge
fields, we can learn the dynamics of D-branes that carry constant
electromagnetic fields in arbitrary directions or propagating
electromagnetic waves.

This redefinition method provides a useful way to simplify the
discussion and to extract exact information in the DBI effective
theory, in particular in some complicated circumstances.

This paper is organised as follows. In Sec.\ 2, we present the
exact solution of a general scalar in the DBI effective theory in
the absence of other world-volume fields obtained in
\cite{Li:2008ah}. In Sec.\ 3, we apply the results in Sec.\ 2 to
derive exact solutions in the DBI theory of the general scalar
coupled to transverse massless scalars. In Sec.\ 4, we derive
exact solutions in the case that the general scalar couples to
gauge fields. Finally, we summarise the results in Sec.\ 5.

\section{Exact solutions in the DBI effective theory}
\label{sec:excsol}

Generalising the DBI action of tachyon field
\cite{Garousi:2000tr,Bergshoeff:2000dq,Sen:2002an}, we can have
the DBI action of a general scalar field $X$ on a D$p$-brane
\begin{equation}\label{e:DBIA}
 S=-\int d^{p+1}x\mathcal{C}(X)\sqrt{1+\eta^{\mu\nu}
\partial_\mu{X}\partial_\nu{X}},
\end{equation}
where $\mathcal{C}(X)$ is the potential. In the following
discussion, the field $X$ can be a tachyon $T$, a massless scalar
$Y$ or a massive scalar $\Phi$.

The equation of motion from this action is:
\begin{equation}\label{e:Teom}
 (1+\partial{X}\cdot\partial{X})\left(\Box X-\frac
{\mathcal{C}'}{\mathcal{C}}\right)=\partial^\mu{X}
\partial^\nu{X}\partial_\mu\partial_\nu{X},
\end{equation}
where $\Box=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}$ and
$\mathcal{C}'=\partial\mathcal{C}(X)/\partial X$. It is clear that
any solutions satisfying $1+\partial X\cdot\partial X=0$ are always
solutions to this equation as we note that the right hand side of
this equation can be recast into $(1/2)\partial{X}\cdot\partial(1+\partial
X\cdot\partial X)$. For the tachyon field $X=T$, it is proved that the
equation $1+\partial T\cdot\partial T=0$ can be achieved only near the
vacuum at $T\rightarrow\pm\infty$, to be an alternative equation
to the equation of motion (\ref{e:Teom})
\cite{Cline:2003vc,Barnaby:2004dz,Hindmarsh:2007dz}.

Following the previous work \cite{Li:2008ah}, we rewrite the DBI
action (\ref{e:DBIA}) by adopting the following potentials
$\mathcal{C}(X)$ and field redefinition relations respectively for
a tachyon, massless scalar and massive scalar:

(i) For a tachyon field $X=T$, we adopt the potential
\begin{equation}\label{e:hyppot}
 \mathcal{C}(T)=\frac{\mathcal{C}_m}{\cosh(\beta T)}
\end{equation}
and the field redefinition relation
\begin{equation}\label{e:tacmapping}
 T=\frac{1}{\beta}\sinh^{-1}(\beta \phi).
\end{equation}
where $\mathcal{C}_m$ is the maximum value of the potential,
located at $T=0$, and $\beta$ is a constant. This constant
$\mathcal{C}_m$ is related to the tension of the non-BPS D-brane.

(ii) For a massless scalar $X=Y$, the potential
$\mathcal{C}(Y)=\mathcal{C}_m$ is constant and we set $Y=\phi$.

(iii) For a scalar field $X=\Phi$, we adopt
\begin{equation}\label{e:cospot}
 \mathcal{C}(\Phi)=\frac{\mathcal{C}_m}{\cos(\gamma\Phi)},
\textrm{ }\textrm{ }\textrm{ }
-\frac{\pi}{2}\leq\gamma\Phi\leq\frac{\pi}{2},
\end{equation}
and
\begin{equation}\label{e:scamapping}
 \Phi=\frac{1}{\gamma}\sin^{-1}(\gamma\phi),
\textrm{ }\textrm{ }\textrm{ } -1\leq\gamma\phi\leq 1,
\end{equation}
where $\mathcal{C}_m$ is the minimum value of the potential,
located at $\Phi=0$, and $\gamma$ is a constant. The constant
$\mathcal{C}_m$ here is related to the tension of the stable
D-brane.

With the above potentials and field redefinition relations, the
action (\ref{e:DBIA}) can be changed to
\begin{equation}\label{e:phidbi}
 \mathcal{L}=-V(\phi)\sqrt{U(\phi)+\partial_{\mu}{\phi}
\partial^{\mu}{\phi}},
\end{equation}
where $U(\phi)=\mathcal{C}_m/V(\phi)=1+\alpha\phi^2$. For a tachyon
$\phi$, $\alpha=\beta^2$; For a massless scalar $\phi$, $\alpha=0$; For a
massive scalar $\phi$, $\alpha=-\gamma^2$. The equation of motion from
this Lagrangian is:
\begin{equation}\label{e:phieq}
 U(\Box{\phi}+\alpha\phi)+[(\partial{\phi}\cdot
\partial{\phi})\Box{\phi}-\partial_\mu\phi\partial_
\nu\phi\partial^\mu\partial^\nu{\phi}]=0.
\end{equation}

It is clear that this equation has the following
spacetime-dependent solution
\begin{eqnarray}\label{e:phisol1}
 \phi(x^\mu)=\phi_+e^{ik\cdot x}+\phi_-e^{-ik\cdot x},
& k^2=k^\mu k_\mu=\alpha,
\end{eqnarray}
where $\phi_\pm$ are constants and $k_\mu=(k_0,\vec{k})$ are the
momenta of the field $\phi$. Note that this solution is only of a
single momentum mode, while the solution to the equation
$\Box{\phi}+\alpha\phi=0$ can be any combinations of the solution
(\ref{e:phisol1}) of all possible momentum modes $\vec{k}$.

Since we are considering the DBI theory (\ref{e:DBIA}) of a real
scalar field, the solution $\phi$ should be real, which requires
that the coefficients must satisfy $\phi_+^*=\phi_-$ and
$\phi_-^*=\phi_+$. Actually, the real solution (\ref{e:phisol1})
can be more explicitly expressed as
\begin{equation}\label{e:phisol2}
 \phi=\phi_s\sin(k\cdot x)+\phi_c\cos(k\cdot x),
\textrm{ }\textrm{ }\textrm{ } k^2=\alpha,
\end{equation}
where $\phi_{s,c}$ are constants.

For the exact solution $\phi$ (\ref{e:phisol1}) or
(\ref{e:phisol2}), one can prove that
$c=\partial\phi\cdot\partial\phi+\alpha\phi^2$ is a constant:
$c=4\alpha|\phi_+|^2=4\alpha|\phi_-|^2$ for (\ref{e:phisol1}) and
$c=\alpha(\phi_s^2+\phi_c^2)$ for (\ref{e:phisol2}). With the
constant $c$, we can derive the energy-momentum tensor from the
exact solution (\ref{e:phisol1}) or (\ref{e:phisol2}), as will
discussed below. It can be proved that the energy-momentum tensor
of the field $\phi$ is equal to that of $T$, i.e.,
$T_{\mu\nu}(\phi)=T_{\mu\nu}(X)$. The latter is given by
\begin{equation}\label{e:enmo1}
 T_{\mu\nu}(X)=\mathcal{C}(X)\left[\frac{\partial_\mu X
\partial_\nu X}{\sqrt{1+\partial X\cdot\partial X}}-
\eta_{\mu\nu}\sqrt{1+\partial X\cdot\partial X}\right].
\end{equation}

In what follows, we will discuss in detail the above exact
solutions and their energy-momentum tensors respectively for a
tachyon, massless scalar and massive scalar.

\subsection{Tachyon}

For a tachyon $\phi$, there are two situations for the solution
(\ref{e:phisol1}) or (\ref{e:phisol2}) in terms of the condition
$k^2=-k_0^2+\vec{k}^2=\beta^2$. (a) ${\vec{k}}^2\geq\beta^2$: All
components of $k_\mu$ are real. The solution $\phi$ in this case
describes an oscillating scalar field travelling faster than light
with $\vec{k}^2=k_0^2+\beta^2$. (b) ${\vec{k}}^2<\beta^2$: The
space-like components $\vec{k}$ are still real but the time-like
component $k_0$ must be imaginary, i.e., $k_0=i\omega$ with $\omega$
real. Then the solution (\ref{e:phisol1}) can be expressed in the
form:
\begin{eqnarray}\label{e:ctachprof}
 \phi(x^\mu)=\phi_+e^{\omega t}e^{i\vec{k}\cdot\vec{x}}+
\phi_-e^{-\omega t}e^{-i\vec{k}\cdot\vec{x}}, &
\omega^2+{\vec{k}}^2=\beta^2,
\end{eqnarray}
which describes an unstable process, like the decay of unstable
D-branes. The field $\phi$ with this solution contains two modes:
one shrinks to $0$ and the other grows to $\infty$ with time.

However, the solution (\ref{e:ctachprof}) is complex. So it is not
a valid solution in the DBI effective theory of a real tachyon
field. But there exist real solutions in the purely time-dependent
or space-dependent case. In the homogeneous case, Eq.\
(\ref{e:phieq}) for a tachyon $\phi$ reduces to
$\ddot{\phi}=\beta^2\phi$ whose solution is a time-dependent
tachyon profile: $\phi=\phi_+e^{\beta t}+\phi_-e^{-\beta t}$. In the
static and $p=1$ case, Eq.\ (\ref{e:phieq}) for a tachyon reduces
to $\phi''=-\beta^2\phi$ whose solution is a space-dependent
tachyon profile: $\phi=\phi_s\sin(\beta x)+\phi_c\cos(\beta x)$.

Since the solution (\ref{e:ctachprof}) is valid in the
space-dependent and the time-dependent cases, we still give the
energy-momentum tensor from this solution
\begin{equation}\label{e:enmo2}
 T_{\mu\nu}=\frac{\mathcal{C}_m}{\sqrt{1+c}}\left[
\frac{1+c}{\cosh^2(\beta T)}\left(\frac{k_\mu k_\nu}{\beta^2}
-\eta_{\mu\nu}\right)-\frac{k_\mu k_\nu}{\beta^2}\right].
\end{equation}
It is easy to prove that the energy-momentum tensor obeys the
conservation law: $\partial^\mu T_{\mu\nu}=0$. In the homogeneous
case with $k_0^2=-\omega^2=-\beta^2$ and $\vec{k}^2=0$, the components
of the energy-momentum tensor are:
\begin{equation}\label{e:tachvacEnPr}
 T_{00}=\frac{\mathcal{C}_m}{\sqrt{1+c}},
\textrm{ }\textrm{ }\textrm{ } T_{ii}=0.
\end{equation}
It is a pressureless state with finite energy density. It is shown
by numerical simulations \cite{Felder:2002sv,Felder:2004xu} that
the spacetime-dependent tachyon field develops into a nearly
homogeneous field towards the end of condensation. So the
energy-momentum tensor (\ref{e:tachvacEnPr}) should exist as the
tachyon approaches the vacuum. This is consistent with the
previous analysis in conformal field theory
\cite{Sen:2002in,Sen:2002an} and in the DBI effective theory
\cite{Sen:2002an,Felder:2002sv,Felder:2004xu,Hindmarsh:2009hs},
where it was shown that the tachyon field approaching the vacuum
evolved into a pressureless state with finite energy density.

\subsection{Massless scalar}

For a massless scalar $\phi$, $\alpha=0$. The equation of motion
(\ref{e:phieq}) reduces to
\begin{equation}\label{e:masslessscaeom}
 \Box{\phi}+[(\partial{\phi}\cdot\partial{\phi})
\Box{\phi}-\partial_\mu\phi\partial_\nu\phi\partial^\mu
\partial^\nu{\phi}]=0.
\end{equation}
In this case, the field $Y=\phi$ is an appropriate massless scalar
describing the fluctuation of a D-brane in one of its transverse
directions. There are two kinds of solutions to the above massless
scalar field equation.

(a) One possible solution is the one of an oscillating mode
(\ref{e:phisol1}), which can be more explicitly written as
\begin{equation}\label{e:maslesscasol1}
 Y=\phi=\phi_+e^{ik\cdot x}+\phi_-e^{-ik\cdot x},
\textrm{ }\textrm{ }\textrm{ } k_0^2=\vec{k}^2.
\end{equation}
This solution describes the oscillation of a D-brane in one of its
transverse directions.

From this solution of a single oscillating mode, one can derive
the energy-momentum tensor
\begin{equation}
 T_{\mu\nu}=\mathcal{C}_m(\partial_\mu Y\partial_\nu
Y-\eta_{\mu\nu}),
\end{equation}
where the first term comes from the oscillation of $Y$, while the
second term comes from the negative contribution of the D-brane
tension. It is easy to prove that this energy-momentum tensor is
conserved: $\partial^\mu T_{\mu\nu}=0$. A feature indicated by the
energy-momentum tensor is that the pressure of the D-brane of an
oscillating mode is generically negative, whereas in ordinary
field theory of massless scalars the pressure is generically
positive.

(b) Besides the above solution, there is a special solution to the
equation (\ref{e:masslessscaeom})
\begin{equation}\label{e:maslesscasol2}
 \partial_\mu Y=\partial_\mu\phi=a_\mu,
\end{equation}
with $a^2\geq -1$ due to the positiveness of the kinetic part
$1+\partial Y\cdot\partial Y$ in the DBI action. This solution
describes a D-brane that is inclined at an angle decided by the
parameters $a_i$ and moves with the velocity $a_0$. If the
inclined angle is zero, i.e., $a_i=0$, the solution
(\ref{e:maslesscasol2}) is simply $\dot{Y}=a_0$, which is the same
as the zero mode solution of a string.

For the solution (\ref{e:maslesscasol2}) describing the uniform
motion of a D-brane, the energy-momentum tensor is
\begin{equation}
 T_{\mu\nu}=\mathcal{C}_m\left[\frac{a_\mu a_\nu}{
\sqrt{1+a^2}}-\eta_{\mu\nu}\sqrt{1+a^2}\right],
\end{equation}
which is constant and hence obeys the conservation law. For a
D-brane with zero inclined angle or a D-particle, the energy
density is $T_{00}=\mathcal{C}_m/\sqrt{1-a_0^2}$. This expression
is similar to the expression of the effective mass $m$ of an
ordinary particle in relativity: $m=m_0/\sqrt{1-v^2}$, with the
rest mass $m_0=\mathcal{C}_m$ and the velocity $v=a_0$. But, at
the speed $a_0=\pm1$, the pressure of D-particles
$T_{ii}=-\mathcal{C}_m\sqrt{1-a_0^2}$ is zero. For ordinary
particles, the pressure is zero when they are at rest.

\subsection{Massive scalar}

For a massive scalar $\phi$, $\alpha=-\gamma^2$. The massive scalar in
the DBI effective theory may describe confined fluctuation of the
D-brane, in contrast to transverse massless scalars. The real
solution of the massive scalar $\phi$ is (\ref{e:phisol1}) or
(\ref{e:phisol2}). The energy-momentum tensor from this exact
solution is
\begin{equation}
 T_{\mu\nu}=\frac{\mathcal{C}_m}{\sqrt{1+c}}\left[
\frac{k_\mu k_\nu}{\gamma^2}-\frac{1+c}{\cos^2(\gamma\Phi)}
\left(\frac{k_\mu k_\nu}{\gamma^2}+\eta_{\mu\nu}\right) \right].
\end{equation}
It also obeys the conservation law $\partial^\mu T_{\mu\nu}=0$.
Note that in this case $c<0$ and
$1+c\leq\cos^2(\gamma\Phi)=1-\gamma^2\phi^2$ for the real solution
(\ref{e:phisol2}). Thus, the energy density $T_{00}$ is positive.
But, the pressure is also generically negative due to the negative
contribution from the tension of the stable D-brane.

\section{Coupling to transverse massless scalars}
\label{sec:coupmassless}

In the rest of this paper, we shall adopt the method used in the
previous section for deriving exact solutions to explore solutions
in the DBI theory including world-volume massless fields.

As done for the action (\ref{e:DBIA}), we can generalise the DBI
action of a tachyon field in the presence of world-volume massless
fields \cite{Garousi:2000tr,Sen:2003bc,Sen:2004nf} to get the one
of a general scalar field $X$:
\begin{equation}\label{e:simbacdbi}
 S=-\int d^{p+1}x\mathcal{C}(X)\sqrt{-\det(\eta_{\mu\nu}
+\partial_\mu{X}\partial_\nu{X}+\partial_\mu{Y^I}
\partial_\nu{Y^I}+F_{\mu\nu})},
\end{equation}
where $Y^I$ ($I=p+1,\cdots,D-1$) are a set of massless scalar
fields for each direction transverse to the D$p$-brane, with
$D=26$ for bosonic strings and $D=10$ for superstrings, and
$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ is the field
strength of the $U(1)$ gauge field $A_\mu$. $Y^I$ describe
fluctuations of the D-brane in transverse directions.

In this section, we first study the dynamics of the general scalar
$X$ in the presence of only the transverse massless scalars $Y^I$.
Working out the determinant in the DBI action (\ref{e:simbacdbi})
with the gauge fields vanishing, we get the Lagrangian
\begin{equation}\label{e:dbicouscalar2}
 \mathcal{L}=-\mathcal{C}{(X)}\left[1+M+\partial_\mu{X}
\partial^\mu{X}+
\frac{1}{2}H_{\mu\nu}^{IJ}H^{IJ\mu\nu}+\frac{1}{2}
H_{\mu\nu}^I(X)H^{I\mu\nu}(X)\right]^{\frac{1}{2}},
\end{equation}
where $M=\sum_I\partial_\mu{Y^I}\partial^\mu{Y^I}$ and
\begin{equation}
 H_{\mu\nu}^I(X)=\partial_\mu{X}\partial_\nu{Y^I}-
\partial_\nu{X}\partial_\mu{Y^I},
\end{equation}
\begin{equation}
 H_{\mu\nu}^{IJ}=\partial_\mu{Y^I}\partial_\nu{Y^J}-
\partial_\nu{Y^I}\partial_\mu{Y^J}.
\end{equation}

Now we adopt the potentials and field redefinition relations
respectively for the tachyon, massless scalar and massive scalar
given in the previous section to rewrite this Lagrangian as
\begin{equation}\label{e:phidbicoumassless}
 \mathcal{L}=-V(\phi)\left[\left(1+M+\frac{1}{2}
H_{\mu\nu}^{IJ}H^{IJ\mu\nu}\right)U(\phi)+\partial_
\mu{\phi}\partial^\mu{\phi}+\frac{1}{2}H_{\mu\nu}^I
H^{I\mu\nu}\right]^{\frac{1}{2}},
\end{equation}
where $U(\phi)=\mathcal{C}_m/V(\phi)=1+\alpha\phi^2$ and
\begin{equation}
 H_{\mu\nu}^I=\partial_\mu{\phi}\partial_\nu{Y^I}-
\partial_\nu{\phi}\partial_\mu{Y^I},
\end{equation}

The equations of motion of $Y^I$ and $\phi$ from this Lagrangian
are respectively
\begin{equation}\label{e:Yphieom}
 \partial^\mu\left(\frac{\partial_\mu{Y^I}+H_{\mu\nu}^{IJ}
\partial^\nu Y^J-V(\phi)H_{\mu\nu}^I\partial^\nu{\phi}}{y}
\right)=0,
\end{equation}
\begin{equation}\label{e:phiYeom}
 \partial^\mu\left(\frac{\partial_\mu{\phi}+H_{\mu\nu}^I
\partial^\nu{Y^I}}{y}\right)+\left(1+M+\frac{1}{2}H^{IJ}
\cdot H^{IJ}\right)\frac{\alpha\phi}{y}=0,
\end{equation}
where $y$ is the kinetic part of the DBI action
(\ref{e:phidbicoumassless}): $\mathcal{L}=-Vy$.

\subsection{Time evolution}

Let us first consider the simple case, the homogeneous case, in
which the coupling terms $H_{\mu\nu}^I$ and $H_{\mu\nu}^{IJ}$
automatically vanish. Denoting $M_0=\sum_I(\dot{Y}^I)^2$, the
equation of motion of the massless scalar $Y^I$ (\ref{e:Yphieom})
can be expressed as
\begin{equation}
 [(1-M_0)U-\dot{\phi}^2]\ddot{Y}^I+\dot{Y}^I[\dot{\phi}\ddot{\phi}
-\alpha(1-M_0)\phi\dot{\phi}+\frac{1}{2}U\dot{M}_0]=0,
\end{equation}
where the dots denote time derivatives. Multiplying $\dot{Y}^I$ on
both sides of the equation and summing over all components $I$, we
can get
\begin{equation}\label{e:homYphieq}
 \dot{M}_0(U-\dot{\phi}^2)+2M_0\dot{\phi}[\ddot{\phi}
-\alpha(1-M_0)\phi]=0,
\end{equation}
On the other hand, the equation of motion of $\phi$
(\ref{e:phiYeom}) is
\begin{equation}\label{e:homphiYeq}
 2(1-M_0)[\ddot{\phi}-\alpha(1-M_0)\phi]+\dot{\phi}\dot{M}_0=0.
\end{equation}

For the massless scalars $Y^I$, we can have $\ddot{Y}^I=0$, which
imply that $\dot{Y}^I$ are constant: $\dot{Y}^I=a_0^I$, where
$a_0^I$ are real constants. In the special case when $a_0^2=1$,
the tachyon or massive scalar $\phi$ satisfies $\ddot{\phi}=0$,
behaving as a massless scalar. When $a_0^2<1$, denoting
$a_0^2=\sum_I(a_0^I)^2$, we can have the solution of $\phi$
\begin{equation}\label{e:homotacphisol}
 \phi=\phi_+e^{\omega t}+\phi_-e^{-\omega t},
\textrm{ }\textrm{ }\textrm{ } \omega^2=(1-a_0^2)\beta^2
\end{equation}
for a tachyon with $\alpha=\beta^2$, and
\begin{equation}\label{e:homoscaphisol}
 \phi=\phi_s\sin{\left(\omega t\right)}+\phi_c
\cos{\left(\omega t\right)}, \textrm{ }\textrm{ }\textrm{ }
\omega^2=(1-a_0^2)\gamma^2
\end{equation}
for a massive scalar with $\alpha=-\gamma^2$.

Thus, the absolute values of the mass squared $|-k_0k^0|=\omega^2$
for both cases become smaller than the original values due to the
presence of the transverse massless scalars $Y^I$. At the critical
value $a_0=\pm1$, the field $\phi$ becomes a massless scalar.
Moreover, how much the absolute values of the mass squared
decrease depends on the velocity $a_0$ of the whole D-brane but
not necessarily on its components $a_0^I$. To get the same mass
squared $\omega^2$, we only need to keep the velocity $a_0$ of the
whole D-brane the same value, while the values of its components
$a_0^I$ may vary.

For a tachyon $\phi$, the decreasing $\omega^2$ leads to smaller
growth rate of the field, which at late time is
$\dot{T}\simeq\sqrt{1-a_0^2}\leq1$. It means that unstable
D-branes decay slower when they move faster. For a massive scalar
$\phi$, the decreasing $\omega^2$ leads to smaller oscillating
frequency, which is $|\omega|=\gamma\sqrt{1-a_0^2}\leq\gamma$.

Finally, we need to make clear one missing point about the two
field equations (\ref{e:homYphieq}) and (\ref{e:homphiYeq}): we
actually can get the following relation from these two equations
\begin{equation}\label{e:homYphiequiveq}
 1-M_0=\frac{\dot{\phi}^2}{U}=\dot{X}^2.
\end{equation}
One can prove that any solutions of $M_0(t)$ and $\phi(t)$
satisfying this equation can sufficiently satisfy Eqs.\
(\ref{e:homYphieq}) and (\ref{e:homphiYeq}) (note that
$\dot{M}_0=-2\dot{\phi}[\ddot{\phi}-\alpha(1-M_0)\phi]/U$ from Eq.\
(\ref{e:homYphiequiveq})). Actually, this relation
(\ref{e:homYphiequiveq}) leads to the vanishing of the kinetic
parts of the DBI action (\ref{e:dbicouscalar2}) and
(\ref{e:phidbicoumassless}) in the homogeneous case, i.e.,
$1+M+\partial X\cdot\partial X=0$ and $y=0$ respectively. This is similar to
the results for the equation of motion (\ref{e:Teom}): the
solutions satisfying $1+\partial X\cdot\partial X=0$ are always solutions to
the equation of motion (\ref{e:Teom}), as pointed out in the
previous section. We also mentioned that, for the tachyon field
$X=T$ case, the equation $1+\partial T\cdot\partial T=0$ can be achieved
only when the tachyon rolls down to the vacuum. Hence, we can
conjecture that the equation (\ref{e:homYphiequiveq}) for a
tachyon $X=T$ could be achieved only towards the end of
condensation.

\subsection{The spacetime-dependent case}

In the spacetime-dependent case, denoting $M=\sum_J\partial
Y^J\cdot\partial Y^J$ and expanding Eq.\ (\ref{e:Yphieom}) give
\begin{eqnarray}\label{e:Yphieom2}
 [U(1+M+\frac{1}{2}H^{IJ}\cdot H^{IJ})+\partial\phi\cdot
\partial\phi+\frac{1}{2}H^I\cdot H^I][\Box Y^I+\partial^
\mu(H_{\mu\nu}^{IJ})\partial^\nu Y^J
\nonumber \\
-V\partial^\mu(H_{\mu\nu}^I)\partial^\nu\phi] -\frac{1}{4}
(\partial_\mu Y^I+H_{\mu\nu}^{IJ}\partial^\nu Y^J-V
H_{\mu\nu}^I\partial^\nu\phi)[2U\partial^\mu M+
\nonumber \\
U\partial^\mu(H^{IJ}\cdot H^{IJ})+\partial^\mu(H^I\cdot
H^I)+2\alpha\phi\partial^\mu\phi(H^{IJ}\cdot H^{IJ})+
\nonumber \\
4\alpha\phi(1+M+\frac{1}{2}H^{IJ}\cdot
H^{IJ})\partial^\mu\phi+4\partial_\rho\phi
\partial^\mu\partial^\rho\phi]=0.
\end{eqnarray}
Expanding Eq.\ (\ref{e:phiYeom}) gives
\begin{equation}\label{e:phiYeom2}
 I_1+I_3=0,
\end{equation}
where the linear and cubic order terms of $\phi$ are respectively
\begin{eqnarray}
 I_1=(1+M+\frac{1}{2}H^{IJ}\cdot H^{IJ})[\Box\phi+
\alpha\phi(1+M+\frac{1}{2}H^{IJ}\cdot H^{IJ})+
\nonumber\\
\partial^\mu(H_{\mu\nu}^I)\partial^\nu Y^I]
-\frac{1}{2}(\partial_\mu\phi+H_{\mu\nu}^I\partial^\nu
Y^I)\partial^\mu(M+\frac{1}{2}H^{IJ}\cdot H^{IJ}),
\nonumber
\end{eqnarray}
\begin{eqnarray}
 I_3=[\alpha\phi^2(1+M+\frac{1}{2}H^{IJ}\cdot H^{IJ})
+\frac{1}{2}H^I\cdot H^I][\Box\phi+\alpha\phi(1+M+
\nonumber \\
\frac{1}{2}H^{IJ}\cdot H^{IJ})+\partial^\mu(H_{\mu\nu}
^I)\partial^\nu Y^I]+[(\partial\phi\cdot\partial\phi)
\Box\phi-\partial_\mu\phi\partial_\nu\phi\partial^\mu
\partial^\nu\phi]
\nonumber \\
+(\partial\phi\cdot\partial\phi)\partial^\mu(H_{\mu\nu}
^I)\partial^\nu Y^I-H_{\mu\nu}^I\partial^\nu Y^I
[\partial_\rho\phi\partial^\mu\partial^\rho \phi
+\alpha\phi\partial^\mu\phi(1
\nonumber \\
+M+\frac{1}{2}H^{IJ}\cdot H^{IJ})]-\frac{1}{4}
(\partial_\mu\phi+H_{\mu\nu}^I\partial^\nu
Y^I)[2\alpha\phi^2\partial^\mu M
\nonumber \\
+\alpha\phi^2\partial^\mu( H^{IJ}\cdot{H^{IJ}})
+\partial^\mu(H^I\cdot{H^I)}]. \nonumber
\end{eqnarray}

For a tachyon or a massive scalar $\phi$ with $\alpha\neq0$, the
exact solutions of $\phi$ and $Y^I$ satisfying Eqs.\
(\ref{e:Yphieom2}) and (\ref{e:phiYeom2}) are respectively
\begin{equation}\label{e:stYphisol}
 \partial_\mu Y^I(x^\mu)=a_\mu^I, \textrm{ }
\textrm{ }\textrm{ } a^2=\sum_Ia_\mu^Ia^{I\mu}\geq-1,
\end{equation}
\begin{equation}\label{e:stphiYsol}
 \phi(x^\mu)=\phi_+e^{ik\cdot x}+\phi_-e^{-ik\cdot x},
\textrm{ }\textrm{ }\textrm{ } k^2=(1+a^2)\alpha,
\end{equation}
with their momenta satisfying the coupling relations
\begin{equation}\label{e:Yphimomcoup1}
 \frac{a_\mu^I}{a_\nu^I}=\frac{a_\mu^J}{a_\nu^J}
 =\frac{k_\mu}{k_\nu},
\end{equation}
which equivalently give
\begin{equation}\label{e:Yphimomcoup2}
 \frac{a_0^I}{a_0^J}=\frac{a_1^I}{a_1^J}=\cdots\frac{a_p^I}
{a_p^J}=\kappa^{IJ}, \textrm{ }\textrm{ }\textrm{ }
\frac{a_0^I}{k_0}=\frac{a_1^I}{k_1}=\cdots=
\frac{a_p^I}{k_p}=\kappa^{I},
\end{equation}
where $\kappa^{IJ}$ and $\kappa^{I}$ are constants. These momentum
coupling relations give rise to the vanishing of the coupling
terms
\begin{eqnarray}\label{e:coupvan}
 H_{\mu\nu}^I=0, & H_{\mu\nu}^{IJ}=0.
\end{eqnarray}

For the solution of the massless scalars $Y^I$, we choose the
solution (\ref{e:maslesscasol2}) rather than
(\ref{e:maslesscasol1}) because the coupling terms $H_{\mu\nu}^I$
and $H_{\mu\nu}^{IJ}$ cannot vanish if we choose the latter as the
solution of $Y^I$. The momenta of the solution
(\ref{e:maslesscasol1}) satisfy the condition $k^2=0$, while those
of the tachyon or massive scalar field $\phi$ satisfy
$k^2=\alpha\neq0$. Thus, they cannot give the coupling relation
(\ref{e:Yphimomcoup1}) which comes from the vanishing of
$H_{\mu\nu}^I$ and $H_{\mu\nu}^{IJ}$. Since the transverse
massless scalars $Y^I$ describe the fluctuation of the D-brane in
the transverse directions, the solutions of $Y^I$
(\ref{e:stYphisol}) imply that the D-brane is inclined at an angle
related to $a_i^I$ and moves with a uniform speed related to
$a_0^I$ in the bulk, with $a_\mu^I$ satisfying the coupling
relations (\ref{e:Yphimomcoup2}).

From the solution of $\phi$ (\ref{e:stphiYsol}), we can see that
the absolute values of the mass squared $|-k^2|$ also become
smaller than the original values due to the uniform motion of the
transverse massless scalars $Y^I$. We can also learn that the mass
squared $|-k^2|$ is dependent on the value $a^2$ but not
necessarily on its components $a_\mu^I$.

For a tachyon $\phi$, the decreasing $|-k^2|$ implies that
inclined unstable D-branes also decay slower when moving faster.
But, in this case, the spacetime-dependent tachyon solution is
complex when we require $k_0$ to be imaginary. We can only have
real solutions in the homogeneous case and in the static case. The
former case has been discussed in the previous subsection.

For a massive scalar $\phi$, the spacetime-dependent solution
(\ref{e:stphiYsol}) is real and so is valid. In this case,
$-1<a^2<0$ in terms of the coupling relations
(\ref{e:Yphimomcoup2}). It is clear that the mass of the massive
scalar is given by $-k^2=\gamma^2(1+a^2)$, which is smaller than
$\gamma^2$ since $-1<a^2<0$. In terms of the momentum coupling
relation, we can give a relation between the field $\phi$ and
$Y^I$ by neglecting the constant term of $Y^I$ and denoting
$Y=Y^I/\kappa^I$
\begin{eqnarray}
 \phi=\phi_s\sin(Y)+\phi_c\cos(Y).
\end{eqnarray}
From it, we further have the relation between $\Phi$ and $Y$
\begin{eqnarray}
 \Phi=\frac{1}{\gamma}\sin^{-1}[\gamma\phi_s\sin(Y)+\gamma\phi_c\cos(Y)].
\end{eqnarray}
That is, the scalar $\Phi$ oscillates in a mode proportional to
that of the uniformly moving D-brane.

For a massless scalar $\phi$ with $\alpha=0$, the solutions of $Y^I$
can be either (\ref{e:maslesscasol1}) or (\ref{e:maslesscasol2}).
Correspondingly, the solution of $\phi$ is (\ref{e:maslesscasol1})
or (\ref{e:maslesscasol2}) as well.

\section{Coupling to gauge fields}
\label{sec:coupgauge}

The case of coupling to gauge fields is more complicated. We first
work out the determinant in the DBI action including both a
general scalar and the gauge fields and write it in an explicit
form. But these forms of the determinant are different on D-branes
of different dimensions. Another point is that, for varying gauge
fields, we usually need to choose gauge conditions.

The DBI effective action of a general scalar field $X$ coupled to
the $U(1)$ gauge fields is given in the form (\ref{e:simbacdbi})
without the transverse massless scalars $Y^I$
\begin{equation}\label{e:dbicougauge}
 S=-\int d^{p+1}x \mathcal{C}{(X)}\sqrt{-\det(\eta_{\mu\nu}
+\partial_\mu{X}\partial_\nu{X}+F_{\mu\nu})}.
\end{equation}

Aspects of static kink (and vertex) solutions from this action for
a tachyon have been studied in
\cite{Mukhopadhyay:2002en,Sen:2003tm,Sen:2003zf,Kim:2003in,Kim:2003ma,
Brax:2003wf,Afonso:2006ws,Cho:2006pg}, where it was shown that the
electromagnetic fields were constant. Due to the existence of the
constant electromagnetic fields, the separations between kinks and
anti-kinks are reduced. The electromagnetic fields are also
restricted to be constant in the homogeneous case
\cite{Kim:2003he}, in which the tachyon grows slower with time in
the presence of constant electromagnetic fields. In what follows,
we shall use the method of redefinition adopted in Sec.\ 2 to
explore solutions in the coupled system of a general scalar and
gauge fields on D-branes of various dimensions.

\subsection{D-strings}

For D-strings, the action (\ref{e:dbicougauge}) is simply
\begin{equation}\label{e:1dordbi}
 S=-\int dtdx \mathcal{C}{(X)}\sqrt{1-E(t,x)^2
+\partial_\mu{X}\partial^\mu{X}},
\end{equation}
where $E(t,x)=F_{01}$ is the electric field along the $x$
direction on a D-string. Implementing the field redefinitions on
the field $X$ as given in Sec. 2, the action (\ref{e:1dordbi}) can
be rewritten as:
\begin{equation}\label{e:1ddbi}
 S=-\int dtdx V(\phi)\sqrt{\left(1-E^2\right)U(\phi)
+\partial_\mu{\phi}\partial^\mu{\phi}},
\end{equation}
where $U(\phi)=\mathcal{C}_m/V(\phi)=1+\alpha\phi^2$ as before and
$\mu,\nu$ runs over $0,1$.

The equations of motion of the electric field and the field $\phi$
are respectively:
\begin{equation}\label{e:1delee1}
 \partial^\mu{\left(\frac{E}{y}\right)}=0,
\end{equation}
\begin{equation}\label{e:1dphie1}
 \partial^\mu{\left(\frac{\partial_\mu{\phi}}{y}\right)}+
\left(1-E^2\right) \frac{\alpha\phi}{y}=0,
\end{equation}
where $y$ is the kinetic part of the action (\ref{e:1ddbi}):
$\mathcal{L}=-Vy$.

For the pure gauge field case with vanishing $\phi$, only the
equation (\ref{e:1delee1}) is left, which reduces to $\partial^\mu
E=0$. Hence, the electric field $E$ without coupling to any other
fields in the DBI theory can only be constant on D-strings.

With non-trivial $\phi$, the above equations of motion
(\ref{e:1delee1}) and (\ref{e:1dphie1}) give respectively if
we denote $r=E^2$
\begin{equation}\label{e:1delee2}
 \frac{1}{2}(U+\partial\phi\cdot\partial\phi)\partial^\mu
r-r[\frac{1}{2}\partial^\mu(\partial\phi\cdot\partial
\phi)+\alpha(1-r)\phi\partial^\mu\phi]=0,
\end{equation}
\begin{eqnarray}\label{e:1dphie2}
 U(1-r)[\Box\phi+\alpha(1-r)\phi]+\frac{1}{2}U\partial
r\cdot\partial\phi
\nonumber \\
+[(\partial\phi
\cdot\partial\phi)\Box\phi-\partial_\mu\phi\partial_\nu
\phi\partial^\mu\partial^\nu\phi]=0
\end{eqnarray}
It is clear to see that there is an exact solution for $\phi$ when
the electric field is constant:
\begin{equation}\label{e:s1dphiemsol}
 \phi(x^\mu)=\phi_+e^{ik\cdot x}+\phi_-e^{-ik\cdot x},
\textrm{ }\textrm{ } \textrm{ } k^2=(1-r)\alpha.
\end{equation}
In the special case $r=1$, $\phi$ becomes a massless scalar and
has the massless scalar solutions (\ref{e:maslesscasol1}) and
(\ref{e:maslesscasol2}).

In the homogeneous case, Eqs.\ (\ref{e:1delee2}) and
(\ref{e:1dphie2}) become respectively
\begin{equation}\label{e:hom1delee2}
 \dot{r}(U-\dot{\phi}^2)+2r\dot{\phi}[\ddot{\phi}
-\alpha(1-r)\phi]=0,
\end{equation}
\begin{equation}\label{e:hom1dphie2}
 2(1-r)[\ddot{\phi}-\alpha(1-r)\phi]+\dot{r}\dot{\phi}=0.
\end{equation}
These two equations are respectively similar to Eqs.\
(\ref{e:homYphieq}) and (\ref{e:homphiYeq}) for the case of
coupling to transverse massless scalars. Similarly, from them one
can get
\begin{equation}\label{e:1demphiequiveq}
 1-r=\frac{\dot{\phi}^2}{U}=\dot{X}^2,
\end{equation}
which also leads to the vanishing of the kinetic parts of the DBI
actions (\ref{e:1dordbi}) and (\ref{e:1ddbi}). For the tachyon
field $X=T$, this equation is also expected to be an achieved as
the tachyon approaches the vacuum.

When the electric field is constant, the homogeneous solution of
$\phi$ is
\begin{equation}
 \phi=\phi_+e^{\omega t}+\phi_-e^{-\omega t},
\textrm{ }\textrm{ }\textrm{ } \omega^2=\left(1-r\right)\beta^2.
\end{equation}
for a tachyon, or
\begin{equation}
 \phi=\phi_s\sin{(\omega t)}+\phi_c\cos{(\omega t)},
\textrm{ }\textrm{ }\textrm{ } \omega^2=\left(1-r\right)\gamma^2,
\end{equation}
for an ordinary scalar. Thus the tachyon grows slower due to the
presence of non-vanishing constant electric field. At late time,
$|\dot{T}|\simeq\sqrt{1-r}$ and it is less than 1. Hence, the
constant electric field tends to slow down the decay process of
unstable D-strings. At the critical value $r=1$, the tachyon
becomes a massless scalar which implies that the unstable D-string
with a sufficiently strong electric field becomes stable. For a
massive scalar $\phi$, the oscillating frequency of the field
becomes smaller due to the presence of a non-vanishing electric
field.

\subsection{D2-branes}

For D2-branes in the presence of gauge fields, the action
(\ref{e:dbicougauge}) can be rewritten as
\begin{equation}\label{e:2demdbi}
 S=-\int d^3x \mathcal{C}{(X)}\left[1+\partial_\mu{X}
\partial^\mu{X}+\frac{1}{2}F_{\mu\nu}F^{\mu\nu}-G_X^2
\right]^{\frac{1}{2}},
\end{equation}
where
\begin{equation}
 G_X=\widetilde{F}^\mu\partial_\mu{X},
\end{equation}
\begin{equation}
 \widetilde{F}^\mu=\frac{1}{2}\epsilon^{\mu\nu\rho}
F_{\nu\rho}.
\end{equation}
$G_X$ are the coupling terms between the general scalar field $X$
and the dual gauge field strength. $\epsilon_{\mu\nu\rho}$ is a
3-dimensional Levi-Civita tensor with
$\epsilon_{\mu\nu\rho}=-\epsilon^{\mu\nu\rho}$ and
$\epsilon_{012}=1$.

After the field redefinitions given in Sec.\ 2, we reach the
action of $\phi$:
\begin{equation}\label{e:2dphidbi}
 S=-\int d^3xV(\phi)y=-\int d^3xV(\phi)\left[\left(1+
\frac{1}{2}F_{\mu\nu}F^{\mu\nu}\right)U(\phi)+\partial_\mu
{\phi}\partial^\mu{\phi}-G^2\right]^{\frac{1}{2}},
\end{equation}
where $G=\widetilde{F}^\mu\partial_\mu{\phi}$. The equations of
motion for the gauge field can be written:
\begin{equation}
 \partial^\mu\left(\frac{F_{\mu\nu}-V(\phi)G\epsilon_{\mu\nu\rho}
\partial^\rho{\phi}}{y}\right)=0,
\end{equation}
which can be recast into
\begin{equation}\label{e:2demeq}
 \partial^\mu\left(\frac{F_{\mu\nu}}{y}\right)=V\epsilon_
{\mu\nu\rho}\partial^\mu\left(\frac{G}{y}\right)\partial^
\rho\phi.
\end{equation}

The equation of motion for the field $\phi$ is
\begin{equation}\label{e:2dphieq}
 \partial^\mu\left(\frac{\partial_\mu{\phi}-G
\widetilde{F}_\mu}{y}\right)+\left(1+\frac{1}{2}
F_{\mu\nu}F^{\mu\nu}\right)\frac{\alpha\phi}{y}=0.
\end{equation}

\subsubsection{Time evolution}

First, we consider the space-independent solutions from the DBI
action (\ref{e:2dphidbi}) with the field strength elements:
$\phi(t)$, $F_{01}=E_1(t)$, $F_{02}=E_2(t)$ and $F_{12}=B=const$.
In this case, the only non-vanishing coupling term is
$G=-\dot{\phi}B$ and Eq.\ (\ref{e:2demeq}) becomes
\begin{equation}
 \partial^0\left(\frac{E_1(t)}{y}\right)=0, \textrm{ }\textrm{ }
\textrm{ } \partial^0\left(\frac{E_2(t)}{y}\right)=0.
\end{equation}
They lead to
\begin{equation}
 [(1-r)U-\dot{\phi}^2]\dot{E}_j+[\dot{\phi}\ddot{\phi}
-\alpha(1-r)\phi\dot{\phi}+\frac{1}{2}U\dot{r}]E_j=0,
\end{equation}
where the index $j$ runs over $1,2$ and
\begin{equation}
 r=\frac{E_1^2+E_2^2}{1+B^2}.
\end{equation}
Multiplying $E_j$ on both sides of this equation and summating
over $j=1,2$, we get a single equation
\begin{equation}
 \dot{r}(U-\dot{\phi}^2)+2r\dot{\phi}[\ddot{\phi}
-\alpha(1-r)\phi]=0,
\end{equation}
On the other hand, the equation of motion of $\phi$
(\ref{e:2dphieq}) becomes
\begin{equation}
 2(1-r)[\ddot{\phi}-\alpha(1-r)\phi]+\dot{r}\dot{\phi}=0,
\end{equation}
Thus the above two equations of $r$ and $\phi$ are respectively
the same as Eqs.\ (\ref{e:hom1delee2}) and (\ref{e:hom1dphie2}),
except that the definition of $r$ is different here. Therefore, we
can also get the same relation as Eq.\ (\ref{e:1demphiequiveq})
from the two equations.

From the above equations of motion, we know that, when the ratio
$r$ of the total energy densities of electromagnetic fields is
constant, there is a time-dependent solution of $\phi$, which is
\begin{equation}
 \phi=\phi_+e^{\omega t}+\phi_-e^{-\omega t},
\textrm{ }\textrm{ }\textrm{ } \omega^2=\left(1-r\right)\beta^2,
\end{equation}
for a tachyon, and
\begin{equation}
 \phi=\phi_s\sin{(\omega t)}+\phi_c\cos{(\omega t)},
\textrm{ }\textrm{ }\textrm{ } \omega^2=\left(1-r\right)\gamma^2,
\end{equation}
for a massive scalar.

Thus the effective mass squared $\omega^2$ also decreases for a
non-vanishing value $r$. Specially, when $r=1$, the field $\phi$
becomes a massless scalar. Moreover, from the above solutions we
can also learn the feature that if only we keep the ratio $r$
constant we will get the same solution $\phi$ with the same
effective mass $\omega^2$, while the components of the
electromagnetic fields $E_1$, $E_2$ and $B$ are allowed to change.

Another feature we can learn from the expressions of $r$ and
$\omega^2$ is that the electric fields tend to decrease while the
magnetic field tends to increase the absolute values of the mass
squared $\omega^2$.

For a tachyon, this also leads to a slower growth rate of the
field. At late time, the tachyon grows at a rate
$|\dot{T}|\simeq\sqrt{1-r}<1$, which means that unstable D-branes
decay slower in the presence of non-vanishing constant
electromagnetic fields.

For a massive scalar, the field has a lower oscillating frequency
in the presence of non-vanishing constant electromagnetic fields,
indicating that the massive scalar has a smaller effective mass.

\subsubsection{Fixed electromagnetic fields}

Next, we consider the spacetime-dependent case by setting all
components of the electromagnetic fields $F_{01}=E_1$,
$F_{02}=E_2$ and $F_{12}=B$ to be constant. Then one can express
Eq.\ (\ref{e:2dphieq}) as
\begin{eqnarray}
 U(1+\frac{1}{2}F\cdot F)[\Box\phi-\widetilde{F}_\mu
\widetilde{F}_\nu\partial^\mu\partial^\nu\phi+(1+\frac{1}{2}F\cdot
F)\alpha\phi]+
\nonumber \\
\textrm{$[$}(\partial\phi\cdot\partial\phi)\Box\phi
-\partial_\mu\phi\partial_\nu\phi\partial^\mu\partial^\nu\phi]+\widetilde{F}
_\mu\widetilde{F}_\nu[\partial^\nu\phi\partial^\mu(\partial\phi\cdot\partial\phi)
\nonumber \\
-(\partial\phi\cdot\partial\phi)\partial^\mu\partial^\nu
\phi-\partial^\mu\phi\partial^\nu\phi\Box\phi]=0,
\end{eqnarray}
where $\widetilde{F}_\mu=(B,E_2,-E_1)$.

This equation still has the classical solution (\ref{e:phisol1})
or (\ref{e:phisol2}) but with the momenta satisfying
\begin{equation}\label{e:D2fixmomcou}
\begin{array}{ccl}
 k\cdot k & = &(\widetilde{F}\cdot k)^2
+(1+\frac{1}{2}F\cdot F)\alpha \\
 & = &(k_0B+E_1k_2-E_2k_1)^2+(1+B^2)(1-r)\alpha.
\end{array}
\end{equation}
This momentum condition implies that the momenta $k_\mu$ vary on a
surface in the momentum space, which is decided by the three
components of the electromagnetic fields. When $k_1=k_2=0$, we
recover the time-dependent solution obtained in the previous
discussion. Let us discuss the momentum coupling relation in
detail below respectively for a tachyon, massless scalar and
massive scalar.

For the case $\alpha=0$, the solution $\phi$ does not necessarily
describe a massless scalar since its mass squared $k^2\neq0$ when
$\widetilde{F}\cdot k\neq0$ or $G\neq0$. If $G\neq0$, $\phi$
becomes a tachyon with $k^2>0$, propagating faster than light. For
instance, setting $k_1/E_1=k_2/E_2$, we get the momentum condition
$(1+B^2)k_0^2=k_1^2+k_2^2$, which describes a massless scalar in a
transverse direction propagating faster than light. To avoid
superluminal propagation, the allowed momentum mode $k_\mu$ must
satisfy
\begin{equation}
 \widetilde{F}\cdot k=0,
\end{equation}
when $\alpha=0$. In this case, the massless scalar remains as a
massless scalar, but fluctuate only in some specific momentum mode
that is decided by the constant electromagnetic fields on the
D-brane.

For $\alpha<0$, it is also possible that the mass squared $k^2$
becomes positive from the originally negative value $k^2=-\gamma^2$,
making the field $\phi$ become a tachyon from a massive scalar.
Thus, to avoid becoming a tachyon, the field $\phi$ can only be in
some specific modes $k_\mu$ that satisfy
\begin{equation}
 (\widetilde{F}\cdot k)^2\leq(1+B^2)(1-r)\gamma^2.
\end{equation}

For $\alpha>0$, the solution $\phi$ will be always a tachyon with
$k^2>0$ when $r<1$. But, for the tachyon $\phi$, the time-like
component $k_0$ is complex
$k_0=i\omega$. In this case, there are two situations for the above
momentum condition (\ref{e:D2fixmomcou}) in order to keep $\omega$,
$k_1$ and $k_2$ to be real:

(a) When $B=0$, the momenta satisfy
\begin{equation}
 (1-E_2^2)k_1^2+(1-E_1^2)k_2^2+2E_1E_2k_1k_2
=(1-E_1^2-E_2^2)\beta^2-\omega^2.
\end{equation}
For a given $\omega$, $k_1$ and $k_2$ can only vary on the
circumstance of an ellipse that is decided by the values of $E_1$
and $E_2$. The maximum value of $|\omega|$ happens
when $k_1=k_2=0$. So the constraint on $\omega$ is
$|\omega|\leq\beta\sqrt{1-E_1^2-E_2^2}$.

(b) When $B\neq 0$, $k_1$ and $k_2$ must satisfy
$k_1/E_1=k_2/E_2$. Then the relations between the momenta are
\begin{equation}\label{e:momreii}
 k_1^2=\frac{E_1^2}{r}[(1-r)\beta^2-\omega^2], \textrm{ }\textrm{ }
\textrm{ } k_2^2=\frac{E_2^2}{r}[(1-r)\beta^2-\omega^2],
\end{equation}
where $r=(E_1^2+E_2^2)/(1+B^2)$ as before. The constraint on $\omega$
is $|\omega|\leq\beta\sqrt{1-r}$. From Eq.\ (\ref{e:momreii}), we know
that $\partial_1\phi=0$ if $E_1=0$ and $\partial_2\phi=0$ if $E_2=0$ when
$B\neq0$. This is consistent with the requirement of symmetry.

\subsubsection{Electromagnetic wave}

Now we consider the case with a single gauge field component
$A_2(x^\mu)$ along the $x^2$ direction. We assume that $A_2$ is
homogeneous along $x^2$ direction, i.e., $A_2=A_2(x^0,x^1)$, so
that it satisfies the Lorentz gauge $\partial^\mu A_\mu=\partial^2 A_2=0$.
In the following discussion, we denote the indices $\mu,\nu=0,1,2$
and $m,n,a,b=0,1$. The components of the field strength are:
\begin{eqnarray}
 F_{01}=0, & F_{02}=\partial_0 A_2(x^m)=E_2(x^m), &
F_{12}=\partial_1 A_2(x^m)=B(x^m).
\end{eqnarray}
For convenience, we denote $A_2$ by $A$.

As before, let us consider the DBI action for pure gauge field
with vanishing field $\phi=0$. The equation of motion
(\ref{e:2demeq}) of the gauge field becomes:
\begin{equation}\label{e:2dbiemeq}
 \partial^m\partial_m A+[(\partial_mA\partial^mA)\partial^n\partial_nA-\partial_mA\partial_nA\partial^m\partial^nA]=0.
\end{equation}
Obviously, a solution from this equation is
\begin{equation}\label{2demsol}
 A=A_+e^{iq_mx^m}+A_-e^{-iq_mx^m},
\textrm{ }\textrm{ }\textrm{ } q_0^2=q_1^2,
\end{equation}
where $A_\pm$ are constants. It describes a beam of planar
electromagnetic waves propagating along the $x^2$ direction on the
D2-brane. It is the same solution as the one from the Maxwell
equation under the Lorentz gauge $\partial^\mu F_{\mu\nu}=\Box A=0$.

Let us further consider the case with non-trivial $\phi$. Since
there is only a single component of the gauge field $A=A_2$, the
coupling term can be expressed as
$G=\epsilon_{mn2}\partial^m\phi\partial^nA=\epsilon_{mn}\partial^m\phi\partial^nA$.

Eq.\ (\ref{e:2demeq}) contains three equations, with $\nu=0,1,2$.
For the $\nu=0,1$ components, $\partial^\mu F_{\mu\nu}=0$. They
are respectively
\begin{eqnarray}
 \epsilon_{\mu 0\rho}\{\frac{1}{2}G[\partial^\mu(\partial\phi\cdot\partial\phi)
+U\partial^\mu(\partial^mA\partial_mA)]-[U(1+\partial^mA\partial_mA)+\partial\phi\cdot\partial\phi]
\nonumber \\
\partial^\mu G\}\partial^\rho\phi+U\partial_0A[\alpha\phi(1+\partial^mA\partial_mA)
\partial^2\phi+\frac{1}{2}\partial^2(\partial\phi\cdot\partial\phi)-G
\partial^2G]=0,
\end{eqnarray}
and
\begin{eqnarray}
 \epsilon_{\mu 1\rho}\{\frac{1}{2}G[\partial^\mu(\partial\phi\cdot\partial\phi)
+U\partial^\mu(\partial^mA\partial_mA)]-[U(1+\partial^mA\partial_mA)+\partial\phi\cdot\partial\phi]
\nonumber \\
\partial^\mu G\}\partial^\rho\phi+U\partial_1A[\alpha\phi(1+\partial^mA\partial_mA)
\partial^2\phi+\frac{1}{2}\partial^2(\partial\phi\cdot\partial\phi)-G
\partial^2G]=0.
\end{eqnarray}
For the $\nu=2$ component, we use $\partial^\mu
F_{\mu\nu}=\partial_m\partial^mA$,
$\partial_\mu\widetilde{F}^{\mu\nu}=0$ and
$\epsilon_{m2n}=-\epsilon_{mn}$ to get
\begin{equation}
 J_1+J_3=0,
\end{equation}
where the first and the third order terms of $A$ are respectively
\begin{eqnarray}
 J_1=U(U+\partial\phi\cdot\partial\phi)\partial^m\partial_mA+\epsilon_{mn}[
\frac{1}{2}G\partial^m\phi\partial^n(\partial\phi\cdot\partial\phi)
-(\partial\phi\cdot\partial\phi)\partial^m\phi\partial^nG]
\nonumber \\
-U[\epsilon_{mn}\partial^m\phi\partial^nG+\alpha\phi\partial^m\phi
\partial_mA+\frac{1}{2}\partial^mA\partial_m(\partial\phi\cdot\partial\phi)],
\nonumber
\end{eqnarray}
\begin{eqnarray}
 J_3=U(\partial^mA\partial_mA)[U\partial^m\partial_mA-\epsilon_{mn}\partial^m\phi\partial^n
G]-U[\frac{1}{2}U\partial^mA\partial_m(\partial^aA\partial_aA)+
\nonumber \\
\alpha\phi(\partial^aA\partial_aA)\partial^m\phi\partial_mA]
+UG[\partial^mG\partial_mA-G\partial^m\partial_mA+
\nonumber \\
\frac{1}{2}\epsilon_{mn}\partial^m\phi\partial^n(\partial^aA\partial_aA)]
\nonumber
\end{eqnarray}

The equation of motion of $\phi$ is
\begin{equation}
 I_1+I_3=0,
\end{equation}
where the first and the third order terms of $\phi$ are
respectively
\begin{eqnarray}
 I_1=(1+\partial^mA\partial_mA)[\Box\phi+(1+\partial^mA\partial_mA)\alpha\phi-
\epsilon_{mn}\epsilon_{ab}\partial^m(\partial^a\phi\partial^bA)\partial^nA]
\nonumber \\
+\frac{1}{2}[\epsilon_{mn}\epsilon_{ab}\partial^m(\partial^cA\partial_cA)
\partial^nA\partial^a\phi\partial^bA-\partial_m\phi\partial^m(\partial^aA\partial_aA)],
\nonumber
\end{eqnarray}
\begin{eqnarray}
 I_3=\alpha\phi^2(1+\partial^mA\partial_mA)[\Box\phi+(1+\partial^mA\partial_mA)
\alpha\phi-\epsilon_{mn}\epsilon_{ab}\partial^m(\partial^a\phi\partial^bA)
\partial^nA]
\nonumber \\
+[(\partial\phi\cdot\partial\phi)\Box\phi-\partial_\mu\phi\partial_\nu\phi
\partial^\mu\partial^\nu\phi]-\epsilon_{mn}\epsilon_{ab}(\partial\phi
\cdot\partial\phi)\partial^m(\partial^a\phi\partial^bA)\partial^nA
\nonumber \\
+G\partial_\mu\phi\partial^\mu G +G\epsilon_{mn}[\frac{1}{2}\partial^m(\partial
\phi\cdot\partial\phi)\partial^nA-\partial^m\phi\partial^nA\Box\phi]
\nonumber \\
+\frac{1}{2}\alpha\phi^2[\epsilon_{mn}G\partial^m(\partial^aA\partial_aA)
\partial^nA-\partial\phi\cdot\partial(\partial^aA\partial_aA)]. \nonumber
\end{eqnarray}
Not that we denote $\partial\phi\cdot\partial\phi=\partial_\mu\phi\partial^\mu\phi$
and $\Box\phi=\partial_\mu\partial^\mu\phi$ in the above equations.

The solutions of the gauge field and the general scalar field can
be of the form
\begin{equation}\label{e:r2demphisol}
 A=A_+e^{iq_mx^m}+A_-e^{-iq_mx^m},
\textrm{ }\textrm{ }\textrm{ } q_0^2=q_1^2,
\end{equation}
\begin{equation}\label{e:2dphiemsol}
 \phi(x^\mu)=\phi_+e^{ik_\mu x^\mu}+\phi_-e^{-ik_\mu x^\mu},
\textrm{ }\textrm{ } \textrm{ }\frac{k_0}{q_0}=\frac{k_1}{q_1},
\textrm{ }\textrm{ }\textrm{ } k^2=k_2^2=\alpha,
\end{equation}
where $A_\pm$ and $\phi_{1,2}$ are all arbitrary constants. The
momentum coupling relation is equivalent to
\begin{eqnarray}\label{e:2dphiemcourel}
 q_1=\pm q_0, & k_1=\pm k_0, & k_2=\pm\sqrt{\alpha}.
\end{eqnarray}
Therefore, the gauge field solution with non-vanishing $\phi$ is
just the same as that from the DBI action for the pure gauge
field. The above solutions of $A$ and $\phi$ that satisfy the
coupling condition $k_0/q_0=k_1/q_1$ gives rise to
\begin{equation}
 G=0.
\end{equation}

Let us discuss the solutions (\ref{e:r2demphisol}) and
(\ref{e:2dphiemsol}) in detail for different $\alpha$ in what
follows:

(i) For a massless scalar $\phi$ with $\alpha=0$, its mass squared is
$k^2=k_2=0$. In this case, the solutions (\ref{e:r2demphisol}) and
(\ref{e:2dphiemsol}) of the two massless fields $A$ and $\phi$ are
real and so are valid. The solutions indicate that D$2$-branes
carrying a beam of electromagnetic waves are fluctuating in an
oscillating mode, since the field $\phi$ with $\alpha=0$ can be
viewed as a transverse massless scalar.

(ii) For a tachyon $\phi$, $k_2^2=\beta^2$ and so $k_2=\pm\beta$. From
the coupling relation in Eq.\ (\ref{e:2dphiemsol}), we learn that
the momentum components $k_0$ and $k_1$ must be both real or both
imaginary since $q_0$ and $q_1$ are real. When $k_0$ and $k_1$ are
both imaginary, the field $\phi$ is a tachyon with a growing mode.
In this case, the solution (\ref{e:2dphiemsol}) is easily seen to
be complex. If we demand that $\phi$ is real, then $k_0$ and $k_1$
must also be real. In this case, the field $\phi$ oscillates and
does not grow and it describes a scalar field propagating faster
than light.

There is another possibility that we can get a real solution for
the field $\phi$. In terms of the momentum coupling condition in
(\ref{e:2dphiemcourel}), we specially set $k_0=k_1=0$. In this
case, the solution (\ref{e:2dphiemsol}) is a stationary kink
solution: $\phi(x^\mu)=\phi(x^2)=\phi_s\sin(\beta
x^2)+\phi_c\cos(\beta x^2)$.

(iii) For a massive scalar $\phi$, $k_2^2=-\gamma^2$ and so $k_2=\pm
i\gamma$ is imaginary. The momenta $k_0$ and $k_1$ can also be both
real or both imaginary. When they are real, the
spacetime-dependent solution (\ref{e:2dphiemsol}) is complex since
$k_2$ is imaginary. If we demand the solution $\phi$ is real, then
$k_0$ and $k_1$ should be imaginary. In this case, the field
$\phi$ grows.

We can also get a real solution for the field $\phi$ in the case
of $k_0=k_1=0$, which leads the solution (\ref{e:2dphiemsol}) to a
stationary solution: $\phi=\phi_+e^{\beta x^2}+\phi_-e^{-\beta x^2}$.

\subsection{D3-branes}

In the $p=3$ case, the DBI action (\ref{e:dbicougauge}) can be
rewritten as
\begin{equation}\label{e:3ddbi}
 S=-\int d^4x\mathcal{C}{(X)}\left[1+\partial_\mu{X}
\partial^\mu{X}+\frac{1}{2}F_{\mu\nu}F^{\mu\nu}-\frac
{1}{16}(F_{\mu\nu}\widetilde{F}^{\mu\nu})^2-G_X^2
\right]^{\frac{1}{2}},
\end{equation}
where
\begin{equation}
 \widetilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu
\rho\lambda}F_{\rho\lambda},
\end{equation}
\begin{equation}
 G_X^2=(\widetilde{F}^{\mu\nu}\partial_\nu
X)(\widetilde{F}_{\mu\rho}\partial^\rho X).
\end{equation}

After the field redefinitions, the action for a general scalar
$\phi$ is obtained
\begin{equation}\label{e:3dphidbi}
 \mathcal{L}=-Vy=-V\left[\left(1+\frac{1}{2}F\cdot F
-\frac{1}{16}(F\cdot\widetilde{F})^2\right)U+\partial_\mu
{\phi}\partial^\mu{\phi}-G^2\right]^{\frac{1}{2}},
\end{equation}
where $G^2=G^\mu G_\mu=(\widetilde{F}^{\mu\nu}\partial_\nu\phi)
(\widetilde{F}_{\mu\rho}\partial^\rho\phi)$.

The equations of motion of the gauge field and the field $\phi$
are respectively
\begin{equation}\label{e:3demeq}
 \partial^\mu\left(\frac{F_{\mu\nu}-\frac{1}{4}(F\cdot
\widetilde{F})\widetilde{F}_{\mu\nu}}{y}\right)=V
\epsilon_{\mu\nu\rho\lambda}\partial^\mu\left(\frac{G^\rho}
{y}\right)\partial^\lambda\phi,
\end{equation}
and
\begin{equation}\label{e:3dphieq}
 \partial^\mu\left(\frac{\partial_\mu\phi+\widetilde{F}
_{\mu\nu}G^\nu}{y}\right)+\left(1+\frac{1}{2}F\cdot F-
\frac{1}{16}(F\cdot\widetilde{F})^2\right)\frac{\alpha
\phi}{y}=0.
\end{equation}

\subsubsection{Time evolution}

First, we also consider the space-independent case with the
electromagnetic field components:
$\textbf{E}(t)=(E_1(t),E_2(t),E_3(t))$ and
$\textbf{B}=(B_1,B_2,B_3)=const$. So
$G^2=\dot{\phi}^2(B_1^2+B_2^2+B_3^2)$ and Eq.\ (\ref{e:3demeq})
gives
\begin{eqnarray}
 \partial^0\left(\frac{E_1+(\textbf{B}\cdot\textbf{E})B_1}
{y}\right)=0, \nonumber \\
\partial^0\left(\frac{E_2+(\textbf{B}\cdot\textbf{E})B_2}
{y}\right)=0, \\
\partial^0\left(\frac{E_3+(\textbf{B}\cdot\textbf{E})B_3}
{y}\right)=0. \nonumber
\end{eqnarray}
From these three equations, we can get
\begin{equation}\label{e:3dtemeq}
 \partial^0\left(\frac{E_1}{y}\right)=
\partial^0\left(\frac{E_2}{y}\right)=
\partial^0\left(\frac{E_3}{y}\right)=0.
\end{equation}

Following the procedure in the previous discussion, we can get the
solution of $\phi$ when the electric fields are constant:
\begin{equation}
 \phi=\phi_+e^{\omega t}+\phi_-e^{-\omega t}, \textrm{ }
\textrm{ }\textrm{ } \omega^2=\left(1-r\right)\beta^2,
\end{equation}
for a tachyon, and
\begin{equation}
 \phi=\phi_s\sin{(\omega t)}+\phi_c\cos{(\omega t)}, \textrm{ }
\textrm{ }\textrm{ } \omega^2=\left(1-r\right)\gamma^2,
\end{equation}
for a massive scalar, where
\begin{equation}
 r=\frac{\textbf{E}\cdot\textbf{E}}
{1+\textbf{B}\cdot\textbf{B}}.
\end{equation}

The results are similar to those on D-strings and D$2$-brane: for
a tachyon field $\phi$, the growth rate becomes smaller which
means that unstable D$3$-branes decay slower, and, for a massive
scalar $\phi$, the oscillating frequency becomes smaller due to
the presence of non-vanishing constant electromagnetic fields. The
mass squared of $\phi$ also relies on the ratio $r$ but not
necessarily on the components of the electromagnetic fields. We
can also learn from the expressions of $r$ and $\omega^2$ that the
electric fields tend to decrease while the magnetic fields tend to
increase the absolute values of the mass squared $\omega^2$. Finally,
we can still get the same equation as Eq.\
(\ref{e:1demphiequiveq}) but with a different definition of $r$
here.

\subsubsection{Fixed electromagnetic fields}

If we set the electromagnetic fields $F_{\mu\nu}$ to be constant,
Eq.\ (\ref{e:3dphieq}) can be expressed as
\begin{eqnarray}
 U[1+\frac{1}{2}F\cdot F-\frac{1}{16}(F\cdot\widetilde{F})^2]
\{\Box\phi-\widetilde{F}_{\mu\nu}\widetilde{F}^{\mu\rho}\partial^
\nu\partial_\rho\phi+\alpha[1+\frac{1}{2}F\cdot F-
\nonumber \\
\frac{1}{16}(F\cdot\widetilde{F})^2]\phi\}
+[(\partial\phi\cdot\partial\phi)\Box\phi-\partial_\mu\phi\partial_\nu\phi\partial^\mu
\partial^\nu\phi]+\widetilde{F}_{\mu\nu}\widetilde{F}^{\mu\rho}[
\partial^\nu\phi\partial_\rho\phi\Box\phi
\nonumber \\
-\frac{1}{2}\partial_\alpha\phi\partial^\alpha(\partial^\nu\phi\partial_\rho\phi)
+(\partial\phi\cdot\partial\phi)\partial^\nu\partial_\rho\phi-\frac{1}{2}\partial_\rho
\phi\partial^\nu(\partial\phi\cdot\partial\phi)]+
\nonumber \\
\widetilde{F}_{\mu\nu}\widetilde{F}^{\mu\rho}\widetilde{F}_{
\alpha\beta}\widetilde{F}^{\alpha\gamma}[\partial^\beta\phi\partial_\gamma\phi\partial^\nu
\partial_\rho\phi-\frac{1}{2}\partial_\rho\partial^\nu(\partial^\beta\phi\partial_\gamma
\phi)]=0.
\end{eqnarray}

The solution of this equation can be the one (\ref{e:phisol1})
or (\ref{e:phisol2}) with the momenta satisfying
\begin{equation}
 k^2=(\widetilde{F}_{\mu\nu}k^\nu)(\widetilde{F}^{\mu\rho}
k_\rho)+[1+\textbf{B}^2-\textbf{E}^2-(\textbf{E}\cdot
\textbf{B})^2]\alpha,
\end{equation}
which are similar to the results on D$2$-branes. To guarantee the
massless scalar and massive scalar propagating no faster than
light, the momenta $k_\mu$ of the field $\phi$ must be restricted
to be some particular values.

\subsubsection{Electromagnetic waves}

Let us switch on a gauge field background with a single component
$A_3$ along the $x^3$ direction on the D$3$-brane. To satisfy the
Lorentz gauge, we still assume that $A_3$ is homogeneous along the
$x^3$ direction, i.e., $\partial^3A_3=0$. Then the non-vanishing
components of the field strength are:
\begin{eqnarray}
 F_{03}=\partial_0A_3=E_3, &
F_{13}=\partial_1A_3=B_2, & F_{23}=\partial_2A_3=-B_1.
\end{eqnarray}
So $F_{\mu\nu}\widetilde{F}^{\mu\nu}=0$. For convenience, we
denote $A_3$ by $A$ and $m=0,1,2$ in what follows.

Setting $\phi=0$, the DBI action (\ref{e:3ddbi}) and
(\ref{e:3dphidbi}) becomes the DBI action for the gauge field in
$d=4$ dimensions. Then the gauge field equation (\ref{e:3demeq})
becomes the same form as Eq.\ (\ref{e:2dbiemeq}) but for $m=0,1,2$
and correspondingly the solution has the same form as the one
(\ref{2demsol}), describing a beam of planar electromagnetic waves
propagating along the $x^3$ direction. This solution also
satisfies the Maxwell equation $\partial^\mu\partial_\mu A=0$.

We now include the nontrivial field $\phi$ in the discussion.
The components of $G_\mu$ are
\begin{eqnarray}
 G_0=\partial_1A\partial_2\phi-\partial_2A\partial_1\phi, &
G_1=\partial_0A\partial_2\phi-\partial_2A\partial_0\phi, \nonumber \\
G_2=\partial_1A\partial_0\phi-\partial_0A\partial_1\phi, & G_3=0.
\end{eqnarray}

We can expect that the solutions of $A$ and $\phi$ in Eq.\
(\ref{e:3demeq}) should be similar to those on D2-branes except
that we are considering the case with one more dimension. From the
discussion on D2-branes, we have learned that the coupled
solutions of $A$ and $\phi$ would lead to $G^2=0$. So what we need
to do is to check whether the conditions $G_\mu=0$ can give the
electromagnetic wave solution and the tachyon profile solution to
their equations of motion (\ref{e:3demeq}) and (\ref{e:3dphieq}).
Assuming $G_\mu=0$, Eq.\ (\ref{e:3demeq}) and Eq.\
(\ref{e:3dphieq}) can be reduced to
\begin{equation}
 \partial^m\left(\frac{\partial_mA}{y}\right)=0,
\end{equation}
\begin{equation}
 \partial^\mu\left(\frac{\partial_\mu\phi}{y}\right)
+(1+\partial^mA\partial_mA)\frac{\alpha\phi}{y}=0,
\end{equation}
where $y=\sqrt{(1+\partial^mA\partial_mA)U+\partial\phi\cdot\partial\phi}$.
The two equations respectively give
\begin{eqnarray}
 [U(1+\partial_mA\partial^mA)+\partial\phi\cdot\partial\phi]\partial^n\partial_nA-
\partial_nA[\frac{1}{2}U\partial^n(\partial_mA\partial^mA)
\\ \nonumber
+\alpha\phi(1+\partial_mA
\partial^mA)\partial^n\phi+\frac{1}{2}\partial^n(\partial\phi\cdot\partial\phi)]=0,
\end{eqnarray}
\begin{eqnarray}
 U(1+\partial_mA\partial^mA)[\Box\phi+\alpha\phi(1+\partial_mA\partial^mA)]-
\frac{1}{2}U\partial\phi\cdot\partial(\partial_mA\partial^mA)
\nonumber \\
+[(\partial\phi\cdot\partial\phi)\Box\phi-\partial_\mu\phi\partial_\nu\phi
\partial^\mu\partial^\nu\phi]=0
\end{eqnarray}

The solutions of $A$ and $\phi$ from the above equations of
motion can be given respectively:
\begin{equation}\label{e:3demphisol}
 A=A_+e^{iq_mx^m}+A_-e^{-iq_mx^m}, \textrm{ }\textrm{ }
\textrm{ } q_0^2=q_1^2+q_2^2,
\end{equation}
\begin{equation}\label{e:3dphiemsol}
 \phi(x^\mu)=\phi_+e^{ik_\mu x^\mu}+\phi_-e^{-ik_\mu x^\mu},
\textrm{ }\textrm{ } \textrm{ }\frac{k_0}{q_0}=\frac{k_1}{q_1}
=\frac{k_2}{q_2}, \textrm{ }\textrm{ }\textrm{ } k_3^2=\alpha,
\end{equation}
where the momentum coupling relation in Eq.\ (\ref{e:3dphiemsol})
comes from the condition $G_\mu=0$. Thus, the coupling relation is
similar to that on D$2$-brane. (i) For a massless scalar $\phi$
with $\alpha=0$, the solution (\ref{e:3dphiemsol}) can be real and so
is valid. The solution and momentum coupling relation indicate
that the massless scalar $\phi$ propagates together with the gauge
field $A$, which describes a D$3$-brane fluctuating in an
oscillating mode in one of the transverse direction. (ii) For a
tachyon $\phi$ with $\alpha=\beta^2$, the real solution can exist when
$k_0$, $k_1$ and $k_2$ are all real since $k_3=\sqrt{\alpha}$ is
real. But in this case the field $\phi$ describes an oscillating
scalar that propagates faster than light. Another possibility that
there exists a real solution is that we set $k_0=k_1=k_2=0$. Then
we can get a stationery tachyon kink solution. (iii) For a massive
scalar $\alpha=-\gamma^2$, the real solution can be obtained when $k_0$,
$k_1$ and $k_2$ are all imaginary since $k_3$ is imaginary. But
the solution indicates that the field grows with time and so is
unstable. In the case of $k_0=k_1=k_2=0$, we can also get a real
stationary solution.

\section{Conclusions}
\label{sec:conclusionsch7w}

We have shown that the equations of motion from the DBI effective
action for a tachyon or an ordinary scalar field admit exact
classical solutions if we choose appropriate potentials and field
redefinitions. These solutions can also be obtained from the DBI
action including the world-volume massless fields, which makes it
possible to investigate the exact coupling relation between a
general scalar $\phi$ (it can be a tachyon or an ordinary scalar)
and the massless fields on a D-brane. The obtained solution of an
arbitrary field from the DBI action including multiple fields is
simply the one obtained from the DBI action with all other fields
vanishing, except that in the former case there is a momentum
coupling constraint which comes from the vanishing of the coupling
terms in the DBI action.

The main results we obtained from these exact solutions and
momentum coupling relations include:

(i) For a tachyon, the obtained spacetime-dependent solution can
only be complex, which is invalid since we are considering the DBI
effective theory of a real field. But in the purely time-dependent
case or purely space-dependent case, there exist real solutions.
For an ordinary scalar, we find real-valued spacetime-dependent
solutions corresponding to travelling waves with arbitrary
amplitude.

(ii) In the case of coupling to the massless scalar fields $Y^I$
in the transverse directions to the D-brane, we find solutions in
which the tachyon or massive scalar field $\phi$ couples to
uniformly moving $Y^I$, but not oscillating ones, in terms of the
momentum coupling conditions between $\phi$ and $Y^I$. The
solution of $\phi$ implies that the effective mass of the tachyon
or massive scalar $\phi$ appears to decrease due to the uniform
motion of $Y^I$ (or of the whole D-brane since $Y^I$ describe the
fluctuation of the D-brane). The amount of the reduced mass
depends on the value of the velocity of the whole D-brane. The
values of its components are allowed to change, i.e. the direction
of the velocity of the whole D-brane can vary. At a critical value
of the velocity, the tachyon or massive scalar field $\phi$ even
loses all its mass and becomes a massless scalar.

The result of mass loss implies that unstable D-branes decay
slower when moving faster and that the massive scalar oscillates
less frequently on D-branes that move faster.

(iii) In the case of coupling to gauge fields, we considered three
situations. (a) In the space-independent case on D-branes of
arbitrary dimensions or on D-strings, the effective mass of the
general scalar $\phi$ coupled to constant electromagnetic fields
also appears to decrease. The amount of the reduced mass depends
on the ratio $r$ of the energy densities of the electromagnetic
fields. At the critical value $r=1$, the general scalar $\phi$
even loses all its mass. The solution of $\phi$ indicates that the
electric fields tend to reduce while the magnetic fields tend to
increase its effective mass. For the tachyon field case, this
means that the electric fields tend to slow down while the
magnetic fields tend to expedite the decay process of unstable
D-branes. (b) In the spacetime-dependent case on D$p$-branes with
$p\geq2$, the momenta of the general scalar that is coupled to
constant electromagnetic fields can only vary on some restricted
surface in the momentum space that is decided by the components of
the electromagnetic fields. In order to keep the massless scalar
and massive scalar propagating no faster than light, their
momentum must satisfy some extra constraints. (c) On D$p$-branes
with $p\geq2$, the gauge fields can vary as a beam of propagating
electromagnetic waves. When the solution of $\phi$ is real, the
tachyon field $\phi$ is stable, behaving in an oscillating mode,
and the massive scalar field $\phi$ is unstable, behaving in a
growing mode. In the presence of propagating electromagnetic
waves, real stationery kink solutions of $\phi$ can also exist.
For a massless scalar $\phi$, the real solution indicates that it
propagates together with the gauge field $A$. That is, D-branes
carrying a beam of propagating electromagnetic waves are
fluctuating in an oscillating mode that is related to the
oscillating mode of the electromagnetic waves.

\section*{Acknowledgements\markboth{Acknowledgements}{Acknowledgements}}

We would like to thank M. Hindmarsh for many useful discussions.

\newpage
\bibliographystyle{JHEP}


\section{Introduction}
As quantum computing and deep learning have recently begun to draw attentions, notable research achievements have been pouring over past decades. In the field of deep learning, the problems which were considered as their inherent limitations like gradient vanishing, local minimum, learning inefficiencies in large-scale parameter training are gradually being conquered~\cite{pieee202105park}. On the one hand, innovative new deep learning algorithms such as quantum neural network (QNN), convolutional neural network (CNN), and recurrent neural network (RNN) are completely changing the way various kinds of data are processed. Meanwhile, the field of quantum computing has also undergone rapid developments in recent years. Quantum computing, which has been recognized only for its potential for a long time, has opened up a new era of enormous potentials with the recent advances of variational quantum circuits (VQC). The surprising potentials of the variational quantum algorithms were made clear by solving various combinatorial optimization problems and the intrinsic energy problems of molecules, which were difficult to solve using conventional methods, and further extensions are considered to design machine learning algorithms using quantum computing. Among them, quantum deep learning fields are growing rapidly, inheriting the achievements of existing deep learning research. Accordingly, numerous notable achievements related to quantum deep learning have been published, and active follow-up studies are being conducted at this time. In this paper, we first briefly introduce the background knowledge, basic principles of quantum deep learning, and look at the current research directions. We then discuss the various directions and challenges of future research in quantum deep learning.

\subsection{Quantum Computing}
Quantum computers use qubits as the basic units of computation, which represent a superposition state between $|0\rangle$ and $|1\rangle$~\cite{app20choi,ictc19choi,icoin20choi,ictc20oh}. A single qubit state can be represented as a normalized two-dimensional complex vector, i.e.,
\begin{equation}
    \label{eq:qubit}
|\psi\rangle = \alpha|0\rangle + \beta|1\rangle ,\|\alpha\|^2 + \|\beta\|^2 = 1
\end{equation}
and $\|\alpha\|^2$ and $\|\beta\|^2$ are the probabilities of observing $|0\rangle$ and $|1\rangle$ from the qubit, respectively. This can be also geometrically represented using polar coordinates $\theta$ and $\phi$,
\begin{equation}\label{eq:bloch}
|\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle,
\end{equation}
where $0\leq\theta\leq\pi$ and $0\leq\phi\leq\pi$.
This representation maps a single qubit state into the surface of 3-dimensional unit sphere, which is called Bloch sphere.
A multi qubit system can be represented as the tensor product of $n$ single qubits, which exists as a superposition of $2^n$ basis states from $|00...00\rangle$ to $|11...11\rangle$. Quantum entanglement appears as a correlation between different qubits in this system. For example, in a 2-qubit system $\frac{1}{\sqrt{2}}|00\rangle + \frac{1}{\sqrt{2}}|11\rangle$, the observation of the first qubit directly determines that of the second qubit. Those systems are controlled by quantum gates in a quantum circuit to perform a quantum computation on its purpose~\cite{icoin21choi,icoin21oh}.

Quantum gates are unitary operators mapping a qubit system into another one, and as classical computing, it is known that every quantum gate can be factorized into the combination of several basic operators like rotation operator gates and CX gate~\cite{electronics20choi}.
Rotation operator gates $R_x(\theta), R_y(\theta), R_z(\theta)$ rotates a qubit state in Bloch sphere around corresponding axis by $\theta$ and CX gate entangles two qubits by flipping a qubit state if the other is $|1\rangle$. Those quantum gates utilizes quantum superposition and entanglement to take an advantage over classical computing, and it is well known that quantum algorithms can obtain an exponential computational gain over existing algorithms in certain tasks such as prime factorization~\cite{shor1999polynomial}.
\section{Quantum Deep Learning}
\subsection{Variational Quantum Circuits (VQC)}
A variational quantum circuit (VQC) is a quantum circuit using rotation operator gates with free parameters to perform various numerical tasks, such as approximation, optimization, classification. An algorithm using a variational quantum circuit is called variational quantum algorithm (VQA), which is a classical-quantum hybrid algorithm because its parameter optimization is often performed by a classical computer. Since its universal function approximating property~\cite{biamonte2021universal}, many algorithms using VQC~\cite{cerezo2020variational} are designed to solve various numerical problems~\cite{farhi2014quantum,kandala2017hardware,app20choi,electronics20choi,apwcs21kim}.
This flow led to many applications of VQA in machine learning and is also for replacing the artificial neural network of the existing model with VQC~\cite{schuld2019quantum,cong2019qcnn,bausch2020recurrent,dong2008quantum}.
VQC is similar to artificial neural networks in that it approximates functions through parameter learning, but has differences due to the several characteristics of quantum computing. Since all quantum gate operations are reversible linear operations, quantum circuits use entanglement layers instead of activation functions to have multilayer structures. These VQCs are called quantum neural networks, and this paper will look at them through classification according to their structure and characteristics.

\subsection{Quantum Neural Networks}
\begin{figure}[htp] \centering \includegraphics[width=1\columnwidth]{QNN.PNG} \caption{Illustration of QNN with the input $|\psi\rangle$, the parameter $\theta$ and linear entanglement structure.} \label{fig:QNN} \end{figure}

In this section, we try to demonstrate how a basic quantum neural network(QNN) works with a simple example described in the Fig.~\ref{fig:QNN}. The way a QNN processes data is as follows. First, the input data is encoded into the corresponding qubit state of an appropriate number of qubits. Then, the qubit state is transformed through the parameterized rotation gates and entangling gates for a given number of layers. The transformed qubit state is then measured by obtaining expected value of a hamiltonian operator, such as Pauli gates. These measurements are decoded back into the form of appropriate output data. The parameters are then updated by an optimizer like Adam optimizer. A neural network constructed in the form of VQC can perform various roles in various forms, which will be explored as quantum neural networks.

\subsubsection{Quantum Convolutional Neural Networks}
\begin{figure}[htp] \centering \includegraphics[width=1\columnwidth]{QCNN.PNG} \caption{Illustration of QCNN with the input $|\psi\rangle$, the parameter $\theta$ with single convolution and pooling layer.} \label{fig:QCNN} \end{figure}
Quantum convolutional neural network (QCNN) was proposed in \cite{cong2019qcnn}, implementing the convolution layer and pooling layer on the quantum circuits. According to the previous research results in \cite{ictc20oh,garg2020advances}, the QCNN circuit computation proceeds as follows. The first step is same as any other QNN models, encoding input data into a qubit state with rotation operator gates. Then the convolution layer with quasi-local unitary gates filters the input data into a feature map. The pooling layer with controlled rotation operators then downsizes the feature map. By repeating this process sufficiently, the fully connected layer acts on the qubit state as classical CNN models. Finally, the measurement of the qubit state is decoded into an output data with desired sizes. The circuit parameters are updated with gradient descent based optimizer after each measurements.

Unfortunaltely, in the current quantum computing environment~\cite{preskill2018quantum}, QCNN is difficult to perform better than the existing classical CNN. However, it is expected that the QCNN will be able to obtain sufficient computational gains over the classical ones in future quantum computing environment where larger-size quantum calculations are possible~\cite{cong2019qcnn,ictc20oh}.

\section{Future Work Directions and Challenges}
\subsection{Applications of Quantum Deep Learning to Reinforcement Learning}
There are many research results applying deep learning to reinforcement learning to derive optimal actions from a complex state space~\cite{mnih2013playing,twc201912choi,tvt201905shin,tvt202106jung}. However, reinforcement learning research using quantum deep learning~\cite{dong2008quantum,chen2020variational,jerbi2021variational} is still in its infancy. The current approach step is to replace the policy training network with a quantum neural network from the existing deep neural network, but there remains the possibility of many algorithms applying various ideas of classical deep reinforcement learning researches. In particular, if it is proved that quantum computational gains can be obtained through QNN in a situation of high computational complexity due to the complex Markov decision process environment, quantum reinforcement learning will open a new horizon for reinforcement learning research.

\subsection{Applications of Quantum Deep Learning to Communication Networks}
The QNN and quantum reinforcement learning algorithms can be used in various research fields, and this paper considers the applications in terms of communications and networks.
In terms of the acceleration of computation in fully distributed platforms, e.g., blockchain~\cite{isj202003saad,isj2021boo}, QNN can be used.

In addition, various advanced communication technologies such as Internet of Things (IoT)~\cite{jsac201811dao,isj2021dao}, millimeter-wave networks~\cite{tvt2021jung,jcn201410kim}, caching networks~\cite{tmc202106malik,twc201910choi,twc202012choi,twc202104choi}, and video streaming/scheduling~\cite{ton201608kim,jsac201806choi,tmc201907koo,tmc2021yi} are good applications of QNN and quantum reinforcement learning algorithms.

\subsection{Challenges}
\subsubsection{Gradient Vanishing}
Vanishing gradient is a crucial problem in quantum deep learning as of classical deep learning. The problem of gradient disappearance while backpropagating many hidden layers has been considered a chronic problem in deep neural network computation. Since quantum neural networks also use gradient descent method training their parameters as classical ones, they have to solve the same problem. Classical deep learning models solve this problem by utilizing an appropriate activation function, but quantum deep learning does not use an activation function, thus eventually, a different solution is needed. A former research~\cite{mcclean2018barren} called this quantum gradient vanishing pheonomena as barren plateaus, while proving that when the number of qubits increases, the probability of occurring barren plateaus increases exponentially. This can be avoided by setting good initial parameters in small-scale QNN, but it is unavoidable to deal with this problem when designing large-scale QNN. This is an open problem for which a solution is not yet clear.

\subsubsection{Near-Term Device Compatibility}
Noisy intermediate scale quantum (NISQ)~\cite{preskill2018quantum}, which means fewer qubits and a lot of computational error of near-term quantum devices, has already become a familiar term to quantum researchers. Many algorithms designed to implement quantum computational gains do not work at all in this NISQ environment, and are expected to be implemented at least several decades later. For example, a practical implementation of the Shor's algorithm requires at least thousand of qubits even without an error correction processes, current quantum devices have only a few tens of qubits with non-negligible computational error rate of several percent. However, due to the relatively small circuit depth and qubit requirements, VQA and QNN based on them are tolerant to these environmental constraints. Nevertheless, in order to increase the data processing capability of quantum neural network, it is necessary to consider near-term device compatibility. For example, using many multi-qubit controlling gates for quantum entanglement is theoretically thought to increase the performance of QNN, but it entails a large error rate and a complicated error correction process. Therefore, it is essential to design an algorithm regarding these tradeoffs in quantum deep learning research.

\subsubsection{The Quantum Advantage}
The term quantum supremacy may lead to the illusion that quantum algorithms are always better than classical algorithms performing the same function. However, given the inherent limitations of quantum computing, quantum computing benefits can only be realized through well-thought-out algorithms under certain circumstances. In fact, especially among variational quantum-based algorithms, only a few of them have proven their quantum advantage in a limited situation.

Due to the universal approximation property of QNN, it is known that quantum deep learning can perform most of the computations performed in classical deep learning~\cite{biamonte2021universal}. Nevertheless, if one approaches simply based on this fact without the consideration of quantum gain, the result may be much inefficient compared to the existing classical algorithm. Therefore, in designing a new QNN-based deep learning algorithm, it is necessary to justify it by articulating its advantages over the corresponding classical models.

\section{Conclusion}\label{sec:sec5}
This paper introduces the basic concepts of quantum neural networks and their applications scenarios in various fields. Furthermore, this paper presents research challenges and potential solutions of quantum neural network computation.

\section*{Acknowledgment}
This work was supported by the National Research Foundation of Korea (2019M3E4A1080391).
Joongheon Kim is a corresponding author of this paper.
\bibliographystyle{IEEEtran}


\section{\label{sec:Intro} Introduction}
Phonons are the quasiparticles of lattice vibrations and therefore one of the fundamental excitations in a solid body. Of special interest are surface acoustic waves (SAWs), which are bound to the surface of a body. These waves find various applications in industry and research \cite{Delsing2019,Edmonson2004,Mauricio2005}. Since the velocity of acoustic waves is about 100.000 times slower than the speed of light, they are used in miniaturized devices at GHz frequencies. Furthermore, SAWs can be excited in a controlled and efficient manner by so-called interdigital transducers (IDT) on piezoelectric substrates. Therefore, devices based on SAWs are essential building blocks for RF technology and widely used in current wireless devices for telecommunication (LTE and 5G), sensors \cite{Polewczyk2020,Kuszewski2018} or radar systems. Beside these technological applications, SAWs are also subject of fundamental research. Due to the advances in nano-lithography, higher frequencies and smaller wavelengths can be achieved. This opens the possibility to efficiently couple SAWs to other systems (e.g., qubits\cite{Satzinger2018, Manenti2017}, spin waves \cite{Verba2018,Dreher2012} and other hybrid systems \cite{Whiteley2019}). In this context, since SAW can carry phonon angular momentum, the conversion of angular momentum between the phonon and spin system has recently attracted significant attention \cite{Sasaki,Long2018}.\\
Techniques based on IDTs working as sender and receiver offer a very sensitive way to probe the transmitted power. However, this technique provides only limited information about the spacial distribution of the acoustic field in between the IDTs. This field, however, is an important basis for many phenomena. It can be mapped by various methods like atomic force microscopy \cite{Hesjedal2001}, scanning electron microscopy \cite{Eberharter1980} and x-ray measurements \cite{Sauer1999}. In this paper, we want to emphasize micro-focussed Brillouin light scattering spectroscopy ($\upmu$BLS) as an additional method for mapping SAWs in the GHz regime. This method is already well established in the field of magnonics for probing the spacial distribution of spin waves on the micrometer scale \cite{Sebastian2015}.

\section{\label{sec:Methods}Methods}
Brillouin light scattering is the inelastic scattering of photons by quasiparticles such as phonons. During the BLS process, momentum and energy of the scattered photons are changed. Wavevector-resolved BLS spectroscopy is a standard tool to study thermally\cite{Wittkowski2000} and externally excited\cite{Vincent2005,Kruger2004} phonons. By varying the angle of incidence, the probed wavevector can be defined, whereby the dispersion relation can be mapped. However, the spatial resolution is limited by the laser spot size, which is usually some tens of $\upmu$m. Hence, this technique is not suited to investigate phonons on the micro- and nanoscale.\\
For this reason, Carlotti et \textit{al}. \cite{Carlotti2018} emphasized to use micro-focussed BLS, a technique developed in the field of magnonics \cite{Chumak2015}. A scheme of the setup is shown in Fig.\ref{fig:fig1} (a). While the physical principle of the scattering process stays the same, the light of a solid state laser ($\lambda=\unit[532]{nm}$) is strongly focussed onto the sample by a microscope objective. In this study, the objective has a numerical aperture of $NA=0.75$ and a magnification of 100x. This allows a spatial resolution down to $\unit[250]{nm}$ \cite{Sebastian2015}. Due to the focussing, the angle of incidence is no longer well-defined. Therefore, the BLS signal averages over a large wave-vector regime. The shift in frequency of the reflected photons can be measured by a Tandem Fabry-P\'erot interferometer (TFPI). The intensity of the scattered light is proportional to the power of the acoustic wave. Furthermore, the polarization of the backscattered light can be analysed. The polarization entering the interferometer can be chosen by a $\lambda$/2-plate in front of it. Since the interferometer itself is optimized for only one polarization axis, this allows to investigate the polarization of the scattered light. For investigations at the microscale, it is essential to control the position of the laser spot on the sample. Hence, an automatized positioning system combined with microscopic imaging is used. Both the data acquisition by the TFPI and the positioning system were controlled by central programming interfaces developed by THATec Innovation GmbH.\\
The investigated samples consists of single SAW resonators, which have been fabricated on commercial GaN/Si wafer (produced by NTT-AT Japan). Undoped GaN ($\unit[1]{\upmu m}$) is grown on a Si substrate with a $\unit[0.3]{\upmu m}$ buffer layer. This is a so-called "fast on slow" structure, where confined modes, like Sezawa waves, can exist beside the Rayleigh mode. Three different IDT structures have been produced by electron beam lithography and conventional lift-off techniques. Their finger/interdigit widths are $w=\unit[170]{nm},~\unit[200]{nm}~\mathrm{and}~\unit[250]{nm}$. The IDTs are contacted by a PicoProbe connected to an RF generator.

\begin{figure}[h]
	\begin{center}
		\scalebox{1}{\includegraphics[width=8.0 cm, clip]{Fig1neu1}}
	\end{center}
	\caption{\label{fig:fig1} a) schematic BLS setup. The laser light is focussed by a microscope objective on the sample. The polarization of the backscattered light can be rotated by a $\lambda$/2-plate and is analysed by the polarization-sensitive Tandem Fabry-P\'erot interferometer. b) Normalized BLS intensity (logarithmic scale) for different excitation frequencies $f_{\mathrm{RF}}$ applied to the IDT. The square of the calculated excitation efficiency for the Rayleigh mode (blue) and Sezawa mode (red) is shown. c) Two-dimensional time-averaged BLS intensity map of a SAW emitted from an IDT with $f_{\mathrm{RF}}=\unit[6.25]{GHz}$. The SAW form a confined beam with a periodic interference pattern.}
\end{figure}

\section{\label{sec:Experiment}Experimental Results}
An RF voltage applied to the IDT generates an alternating strain in the piezoelectric layer, and SAWs can be generated, which propagate away from the IDT. Fig.\ref{fig:fig1} (b) shows the time-averaged BLS intensity close to the IDT with a finger/interdigit width $w=\unit[170]{nm}$ as a function of the RF frequency $f_{\mathrm{RF}}$. The different maxima can be identified as the Rayleigh mode (R) at $\unit[6.25]{GHz}$ and the first Sezawa mode (S) at $\unit[7.0]{GHz}$. Furthermore, a pseudo-bulk mode (PB) has been found at higher frequencies. These three modes can also be observed for the other two devices with different finger/interdigit widths and therefore different wavevectors. The extracted frequencies for all three devices are shown in Fig.\ref{fig:fig3} (a). They correspond well to the dispersion relation that was obtained based on data presented by Mueller et \textit{al}. \cite{Muller2015} for a GaN layer on Si. Moreover, the square of excitation efficiency has been calculated based on the Fourier transform of the IDTs geometry \cite{Royer2000} as a function of the wavevector $k$. Taking into account the dispersion relation, the excitation efficiency can be calculated as a function of the frequency for a given mode. This is shown in blue for the Rayleigh mode and in red for the first Sezawa mode. Both frequencies of the intensity maxima correspond well to the measured data. The noise level of the measurement has been added to the calculated curves. Please note that a absolute comparison of the calculated excitations efficiencies of Rayleigh and Sezawa mode is not possible, since the model does not take into account the strain profile of the IDT, the amplitude profile of the SAW (over the film thickness, respectively) and the surface sensitivity of the BLS. However, for the further discussion, it is important to notice that the excitation efficiency of the Sezawa mode at the Rayleigh mode resonance is not vanishing and vice versa.\\
An additional interesting parameter, which has to be addressed is the polarization of the scattered light. It has been analyzed by a rotation of the $\lambda$/2-plate in combination with the polarization-sensitive TFPI. Rayleigh waves have been excited at $f=\unit[6.25]{GHz}$ and Sezawa waves at $f=\unit[6.9]{GHz}$. As a reference polarization, elastically scattered light from a laser mode has been used. Figure \ref{fig:fig0} shows that the measured BLS intensity follows a $\sin^2$ dependence of the $\lambda$/2-plate angle, indicating that the scattered light is linearly polarized and not changed in its polarization direction. This is in agreement to studies carried out with wavevector-resolved BLS\cite{Carlotti2018}.

\begin{figure}[h]
	\begin{center}
		\scalebox{1}{\includegraphics[width=8.0 cm, clip]{PolRS}}
	\end{center}
	\caption{\label{fig:fig0} BLS intensity of Rayleigh waves a) and Sezawa waves b) as a function of the angle of the $\lambda$/2-plate in Fig.1. Zero angle corresponds to the reference polarization of elastically scattered light. The polarization of the backscattered light is not changed during the scattering process with Rayleigh waves or Sezawa waves.}
\end{figure}

As mentioned above, $\upmu$BLS is a scanning method, which allows to measure the acoustic field at the surface of the sample. As an example, Fig.\ref{fig:fig1} (c) shows the time-averaged BLS intensity of emitted SAWs with a frequency of $\unit[6.25]{GHz}$. Since the aperture of the IDT ($a=\unit[50]{\upmu m}$) is large compared to the wavelength of the SAW ($\lambda=\unit[680]{nm}$), the waves are emitted in a confined beam with no visible diffraction. Hence, the wave can be treated as a plain wave in the following explanation.

\begin{figure}[h]
	\begin{center}
		\scalebox{1}{\includegraphics[width=8.0 cm, clip]{Fig2_final}}
	\end{center}
	\caption{\label{fig:fig2} a) Normalized time-averaged BLS intensity for an excitation frequency $f=\unit[6.25]{GHz}$ along the propagation direction extracted from Fig.\ref{fig:fig1}c). b) Normalized FFT of the BLS intensity. The wave-vector difference $\Delta k$ can be extracted from the position of the maximum.}
\end{figure}

It is important to note that the time-averaged BLS intensity oscillates along the propagation direction of the SAW with a periodicity $p=\unit[4.59]{\upmu m}$. For better visibility of this phenomenon Fig.\ref{fig:fig2} (a) shows the BLS intensity summed over the width of the measured area. This periodic oscillation in intensity can be explained by the interference of co-propagating Rayleigh and Sezawa waves with the same frequency, but different wavevectors. This is possible due to the finite length of the IDT and the associated linewidth in k-space (see Fig~\ref{fig:fig1}~(b)).\\
We assume two plane waves $\Psi_i$:
\begin{equation}
\Psi_i = A_i \cos(k_{i}x-\omega t).
\label{eqn:eqn1}
\end{equation}
The BLS intensity is proportional to the SAW intensity averaged over one oscillation period, which is expressed by:
\begin{equation}
\label{eqn:eqn3}
\begin{split}
I(x) & =\vert{\int{\Psi_R(x,t)+\Psi_S(x,t)\ dt }}\vert^2\\
& =A_R^2+A_S^2+{2{A_R A_S}}\cos[\Delta k x]
\end{split}
\end{equation}
with
\begin{equation}
\Delta k = k_R-k_S = {{2\pi}\over p}.
\label{eqn:eqn4}
\end{equation}
From the amplitude of the observed modulation the ratio between the amplitude of Rayleigh and Sezawa wave can be estimated to $A_S/A_R \approx 0.5$. This value is larger than one might estimate from the simple model shown in Fig. 1. However, as already explained before, the different excitation efficiencies together with the different BLS scattering cross sections for the two modes prevents a direct comparison. \\

\begin{figure}[h]
	\begin{center}
		\scalebox{1}{\includegraphics[width=8.0 cm, clip]{Fig3avr}}
	\end{center}
	\caption{\label{fig:fig3} a) Dispersion relation for SAWs in a $\unit[1.3]{\upmu m}$ thick GaN layer on Si substrate. From the BLS measurements for different excitation frequencies the maxima are evaluated and the positions are shown (black dots) for all tested devices (see Fig. \ref{fig:fig2} (b)). b) The wave-vector difference $\Delta k$ for a given frequency between Rayleigh and Sezawa mode has been determined by BLS measurements along the propagation direction. The predicted wave-vector differences based on the dispersion relation are represented in green color.}
\end{figure}

The wave-vector difference $\Delta k$ has been extracted from the Fourier transform of the BLS intensity along the propagation direction, as shown in Fig.\ref{fig:fig2} (b). To confirm the interpretation of the two co-propagating and interfering waves, additional measurements at different excitation frequencies $f_{\mathrm{RF}}$ have been performed. Therefore, the BLS intensity was mapped on a line along the propagation direction and the beating pattern was analyzed like shown above. All measurements showed similar oscillations. The resulting wave-vector differences are shown in Fig.\ref{fig:fig3} (b). In the investigated frequency range the wave-vector difference $\Delta k$ is nearly constant, which is in a good agreement with the prediction based on the dispersion relation of the GaN-Si system. While for small wavevectors the slope for the Rayleigh and the Sezawa mode varies it is comparable to waves with frequencies above $\unit[2]{GHz}$. Deviations can be attributed to the fact that the value of the wave-vector difference is very sensitive to small changes of dispersion.\\

\section{\label{sec:Conclusion}Conclusion}
In this article we have used micro-focussed Brillouin light scattering spectroscopy to investigate the excitation and propagation of SAWs in a GaN layer on a Si substrate. In the first part, the excitation efficiency of the IDT and dispersion relation has been verified. We have found that Rayleigh and Sezawa waves are excited at the same frequency by an IDT structure. These waves interfere coherently while they propagate and their time-averaged intensity is spatially varying with a periodicity determined by their wavevector difference at the excitation frequency. This pattern offers the opportunity to have a time-independent, space-modulated acoustic field with well defined wave vector direction.

\begin{acknowledgments}
Financial support by the EU Horizon 2020 research and innovation program within the CHIRON project (contract no. 801055) is gratefully acknowledged.
\end{acknowledgments}

\bibliographystyle{apsrev4-1}