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. 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.

References in corpus (14)

• The short pulse equation is integrable
• Solitary wave solutions of the short pulse equation
• Ultra-short pulses in linear and nonlinear media
• The Bi-Hamiltonian Structure of the Short Pulse Equation
• Complex short pulse and coupled complex short pulse equations
• A novel multi-component generalization of the short pulse equation and its multisoliton solutions
• A defocusing complex short pulse equation and its multi-dark soliton solution by Darboux transformation
• Primal-dual algorithm for quasi-static contact problem with Coulomb's friction
• Transformation and integrability of a generalized short pulse equation
• Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions
• Integrable multi-component generalization of a modified short pulse equation
• Discontinuous Galerkin methods for the Ostrovsky-Vakhnenko equation
• Integrable discretizations and self-adaptive moving mesh method for a coupled short pulse equation
• A focusing and defocusing semi-discrete complex short pulse equation and its varioius soliton solutions

Cited by in corpus (1)

• Dissipative and conservative local discontinuous Galerkin methods for the Fornberg-Whitham type equations