  We study the Strang splitting scheme for quasilinear Schr\"odinger equations.
We establish the convergence of the scheme for solutions with small initial
data. We analyze the linear instability of the numerical scheme, which explains
the numerical blow-up of large data solutions and connects to analytical
breakdown of regularity of solutions to quasilinear Schr\"odinger equations.
Numerical tests are performed for a modified version of the superfluid thin
film equation.
