Nonlinear initial-boundary value problems using Taylor expansion of parameter

Nonlinear initial-boundary value problems using Taylor expansion of parameter

Let $u(x,t; \epsilon)$ satisfy the nonlinear initial boundary value
problem $$ u_{tt} = (u_{x} + u_{x}^3)_{x} + u_{xxt}, \space 0 \lt x \lt 1
$$ $$ u(0,t) = 0 \\ u(1,t) = 0 \\ u(x,0) = \epsilon f(x) \\ u_{t}(x,0) =
\epsilon g(x) $$ where $f(0) = 0 = f(1)$ and $g(0) = 0 = g(1)$. How do I
go about finding the equation satisfied by $u$ if I assume that the
solution depends smoothly on $\epsilon$ in the neighborhood of $0$, so
that $u$ has the beginnings of a Taylor expansion in $\epsilon:$ $\\\\
u(x,t; \epsilon) = 0 + u_{1}(x,t)\epsilon +
\frac{1}{2}u_{2}(x,t)\epsilon^2 + \frac{1}{3}u_{3}(x,t)\epsilon^3 +
\space...\space + \frac{1}{n}u_{n}(x,t)\epsilon^n $
And is it possible to prove that such a solution is unique? Also,
regarding notation, does $(u_{x} + u_{x}^3)_{x}$ signify the partial
derivative of $(u_{x} + u_{x}^3)$ with respect to x? Is this equivalent to
$u_{xx} + (u_{x}^3)_{x}$?