= Solution
Treat $\phi$ and $\bar\phi$ as independent fields. The <Lagrangian density> is
$$
\mathcal L=\frac i2(\phi\dot{\bar\phi}-\bar\phi\dot\phi)
+\frac12\phi_x\bar\phi_x-\frac12(\phi\bar\phi)^2
+\epsilon V(\epsilon x)\phi\bar\phi.
$$
The <Euler-Lagrange equation> for $\bar\phi$ is
$$
-i\phi_t-\frac12\phi_{xx}-|\phi|^2\phi
+\epsilon V(\epsilon x)\phi=0,
$$
or
$$
\boxed{i\phi_t=-\frac12\phi_{xx}-|\phi|^2\phi
+\epsilon V(\epsilon x)\phi}.
$$
The equation from variation of $\phi$ is its complex conjugate.
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