The dynamics of conservative peakons in a family of U(1)‐invariant integrable equations of NLS‐Hirota type. Issue 4 (15th October 2018) - Anco, Stephen C.; Chang, Xiangke; Szmigielski, Jacek
Abstract: The tri‐Hamiltonian splitting method applied to the Hirota hierarchy has yielded two U (1)‐invariant nonlinear Partial Differential Equations (PDEs) that admit peakons ( nonsmooth solitons ). In the present paper, these two peakon PDEs are generalized to a family of U (1)‐invariant peakon PDEs parameterized by the real projective line R P 1 . All equations in this family are shown to possess conservative peakon solutions (whose Sobolev H 1 ( R ) norm is time invariant). The Hamiltonian structure for the sector of conservative peakons is identified and the peakon Ordinary Differential Equations (ODEs) are shown to be Hamiltonian with respect to several Poisson structures. As the main result, it is shown that inverse spectral methods allow one to solve explicitly the dynamics of conservative peakons using explicit solutions to a certain interpolation problem. The graphs of multipeakon solutions confirm the existence of multipeakon breathers as well as asymptotic formation of pairs of two peakon bound states in the nonperiodic time domain.