the 3-4 plane, as well as the translation symmetries along $t$, $x_3$ and $x_4$. This leads to the following ansatz, $$\label{E:inhomogeneousAnsatz}
\begin{split}
&\tilde A_0 = f(r)\;,\qquad \tilde A_1=g_1(r)\;, \qquad \tilde A_2=g_2(r)\\
&\tilde A:=\tilde A_3 +i \tilde A_4 = h(r) e^{-i k x_2}\;.
\end{split}$$ Note that $g_1(r)$ can be set to vanish by a gauge choice, and $g_2(r)$ has to vanish by the Maxwell equation $\frac{\delta S}{\delta \tilde A_2}=0$. The non-trivial equations of motion are $$\begin{split}
-\partial_r (r^3 f'(r))-4 k h(r) h'(r) &=0\\
\partial_r( H(r) r h'(r) ) - \frac 1 r k^2 h(r) + 4 k h(r) f'(r)&=0\;.
\end{split}$$ The first equation can be integrated and becomes $$r^3 f'(r) + 2 k h(r)^2 = \tilde E\;.$$ Eliminating $f'(r)$ in the second equation by using the above relation, we obtain $$\label{E:EOMinRNAdSBH}
\partial_r( H(r) r h'(r) ) - \frac 1 r k^2 h(r) + 4 k \frac{h(r)}{r^3} (\tilde E-2 k h(r)^2)=0\;.$$

Second Order Phase Transition
-----------------------------

We have solved the differential equation numerically. For each initial condition at the horizon, the equation is integrated numerically toward the $AdS_5$ boundary. In general, we find a linear combination of normalizable and non-normalizable modes near the boundary. Since the new phase of the system should 