  In this paper, we present and study discontinuous Galerkin (DG) methods for
one-dimensional multi-symplectic Hamiltonian partial differential equations. We
particularly focus on semi-discrete schemes with spatial discretization only,
and show that the proposed DG methods can simultaneously preserve the
multi-symplectic structure and energy conservation with a general class of
numerical fluxes, which includes the well-known central and alternating fluxes.
Applications to the wave equation, the Benjamin-Bona-Mahony equation, the
Camassa-Holm equation, the Korteweg-de Vries equation and the nonlinear
Schr\"odinger equation are discussed. Some numerical results are provided to
demonstrate the accuracy and long time behavior of the proposed methods.
Numerically, we observe that certain choices of numerical fluxes in the
discussed class may help achieve better accuracy compared with the commonly
used ones including the central fluxes.
