  The appearance of a fundamental long-time asymptotic regime in the two space
one time dimensional hyperbolic nonlinear Schr\"odinger (HNLS) equation is
discussed. Based on analytical and extensive numerical simulations an
approximate self-similar solution is found for a wide range of initial
conditions -- essentially for initial lumps of small to moderate energy. Even
relatively large initial amplitudes, which imply strong nonlinear effects,
eventually lead to local structures resembling those of the self-similar
solution, with appropriate small modifications. These modifications are
important in order to properly capture the behavior of the phase of the
solution. This solution has aspects that suggest it is a universal attractor
emanating from wide ranges of initial data.
