Global Solution to a nonlinear wave equation of liquid crystal in the constant electric field

We construct a global conservative weak solution to the Cauchy problem for the non-linear variational wave equation $v_{tt} - c(v)(c(v)v_x)_x + \frac{1}{2}(v+v^3)= 0$ where $c(\cdot)$ is any smooth function with uniformly positive bounded value. This wave equation is derived from a wave system modelling nematic liquid crystals in a constant electric field.
Comment: 30 pages, 6 figures