  For the short pulse initial data with a first order outgoing constraint
condition and optimal orders of smallness, we establish the global existence of
smooth solutions to 2D quasilinear wave equations with higher order null
conditions. Such kinds of wave equations include 2D relativistic membrane
equations, 2D membrane equations, and some 2D quasilinear equations which come
from the nonlinear Maxwell equations in electromagnetic theory or from the
corresponding Lagrangian functionals as perturbations of the Lagrangian
densities of linear wave operators. The main ingredients of the analysis here
include looking for a new good unknown, finding some key identities based on
the higher order null conditions and the resulting null frames, as well as
overcoming the difficulties due to the slow decay of solutions to the 2-D wave
equation, so that the solutions can be estimated precisely.
