  We prove global well-posedness of the initial value problem for a class of
variational quasilinear wave equations, in one spatial dimension, with initial
data that is not-necessarily small. Key to our argument is a form of
quasilinear null condition (a "nilpotent structure") that persists for our
class of equations even in the large data setting. This in particular allows us
to prove global well-posedness for $C^2$ initial data of moderate decrease,
provided the data is sufficiently close to that which generates a simple
traveling wave.
  We take here a geometric approach inspired by works in mathematical
relativity and recent works on shock formation for fluid systems. First we
recast the equations of motion in terms of a dynamical double-null coordinate
system; we show that this formulation semilinearizes our system and decouples
the wave variables from the null structure equations. After solving for the
wave variables in the double-null coordinate system, we next analyze the null
structure equations, using the wave variables as input, to show that the
dynamical coordinates are $C^1$ regular and covers the entire space-time.
