Discontinuous Galerkin methods for short pulse type equations via hodograph transformations

Abstract

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. Therefore, 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 applied 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.