  The initial-boundary value problem in a bounded domain with moving boundaries
and nonhomogeneous boundary conditions for a higher order nonlinear
Schr\"odinger (HNLS) equation is considered. Existence and uniqueness of global
weak solutions are proved as well as the stability of the solution.
Additionally, a conservative numerical method of finite differences is
introduced that also verifies stability properties with respect to the
$L^2$-norm, and along with proving its convergence, some interesting numerical
examples are shown that illustrate the behavior of the solution.
