  This work develops an energy-based discontinuous Galerkin (EDG) method for
the nonlinear Schr\"odinger equation with the wave operator. The focus of the
study is on the energy-conserving or energy-dissipating behavior of the method
with some simple mesh-independent numerical fluxes we designed. We establish
error estimates in the energy norm that require careful selection of a test
function for the auxiliary equation involving the time derivative of the
displacement variable. A critical part of the convergence analysis is to
establish the L2 error bounds for the time derivative of the approximation
error in the displacement variable by using the equation that determines its
mean value. Using a specially chosen test function, we show that one can create
a linear system for the time evolution of the unknowns even when dealing with
nonlinear properties in the original problem. Extensive numerical experiments
are provided to demonstrate the optimal convergence of the scheme in the L2
norm with our choices of the numerical flux.
