  In the present paper, we consider the discontinuous Galerkin (DG) methods for
solving short pulse (SP) type equations. The short pulse equation has been
shown to be completely integrable, which admits the loop-soliton,
cuspon-soliton solutions as well as smooth-soliton solutions. Through hodograph
transformations, these nonclassical solutions can be profiled as the smooth
solutions of the coupled dispersionless (CD) system or the sine-Gordon
equation. Thus, DG methods can be developed for the CD system or the
sine-Gordon equation to simulate the loop-soliton or cuspon-soliton solutions
of the SP equation. The conservativeness or dissipation of the Hamiltonian or
momentum for the semi-discrete DG schemes can be proved. Also we modify the
above DG schemes and obtain an integration DG scheme. Theoretically the
a-priori error estimates have been provided for the momentum conserved DG
scheme and the integration DG scheme. We also propose the DG scheme and the
integration DG scheme for the sine-Gordon equation, in case the SP equation can
not be transformed to the CD system. All these DG schemes can be adopted to the
generalized or modified SP type equations. Numerical experiments are provided
to illustrate the optimal order of accuracy and capability of these DG schemes.
