Mathematica NDSolve does not solve with non-linear coefficients. The parsing error seems to be related to Abs use only
Remove the Abs and you'll see the real issue
NDSolve[
I D[u[x, t], x] + 1/2 D[u[x, t], {t, 2}] + u[x, t]^3 == 0, u, {x, 0, 10}, {t, 0, 10}]
NDSolve::femnonlinear: Nonlinear coefficients are not supported in this version of NDSolve. >>
ps. you also missing initial/boundary conditions. This error was seen here also
