Multi-symplectic method for peakon-antipeakon collision of quasi-Degasperis-Procesi equation

Focusing on the local geometric properties of the shockpeakon for the Degasperis-Procesi equation, a multi-symplectic method for the quasi-Degasperis-Procesi equation is proposed to reveal the jump discontinuity of the shockpeakon for the Degasperis-Procesi equation numerically in this paper. The main contribution of this paper lies in the following: (1) the uniform multi-symplectic structure of the b-family equation is constructed; (2) the stable jump discontinuity of the shockpeakon for the Degasperis-Procesi equation is reproduced by simulating the peakon-antipeakon collision process of the quasi-Degasperis-Procesi equation. First, the multi-symplectic structure and several local conservation laws are presented for the b-family equation with two exceptions (b=3 and b=4). And then, the Preissman Box multi-symplectic scheme for the multi-symplectic structure is constructed and the mathematical proofs for the discrete local conservation laws of the multi-symplectic structure are given. Finally, the numerical experiments on the peakon-antipeakon collision of the quasi-Degasperis-Procesi equation are reported to investigate the jump discontinuity of shockpeakon of the Degasperis-Procesi equation. From the numerical results, it can be concluded that the peakon-antipeakon collision of the quasi-Degasperis-Procesi equation can be simulated well by the multi-symplectic method and the simulation results can reveal the jump discontinuity of shockpeakon of the Degasperis-Procesi equation approximately.