The dynamics of conservative peakons in the NLS hierarchy

TL;DR

This paper extends the study of peakon solutions in the nonlinear Schrödinger hierarchy by introducing a family of $U(1)$-invariant PDEs, analyzing their Hamiltonian structures, and explicitly solving their dynamics using inverse spectral methods.

Contribution

It generalizes previous peakon PDEs to a broader family parametrized by $\mathbf{RP}^1$, identifies their Hamiltonian structures, and provides explicit solutions for their conservative peakon dynamics.

Findings

All equations admit conservative peakon solutions with invariant Sobolev norm.

Hamiltonian structures are identified for the peakon sector, with peakon ODEs being Hamiltonian.

Explicit solutions demonstrate multipeakon breathers and bound states.

Abstract

Using the tri-hamiltonian splitting method, the authors of [Anco and Mobasheramini, Physica D, 355:1--23, 2017] derived two $U(1)$-invariant nonlinear PDEs that arise from the hierarchy of the nonlinear Schr\"{o}dinger equation and admit peakons ($non-smooth\ solitons$). In the present paper, these two peakon PDEs are generalized to a family of $U(1)$-invariant peakon PDEs parametrized by the real projective line $ \mathbf{RP} ^1$. All equations in this family are shown to posses $conservative\ peakon \ solutions$ (whose Sobolev $H^1(\mathbf{R})$ norm is time invariant). The Hamiltonian structure for the sector of conservative peakons is identified and the peakon ODEs are shown to be Hamiltonian with respect to several Poisson structures. It is shown that the resulting Hamilonian peakon flows in the case of the two peakon equations derived in [Anco and Mobasheramini, Physica D, 355:1--23, 2017] form orthogonal families, while in general the Hamiltonian peakon flows for two different equations in the general family intersect at a fixed angle equal to the angle between two lines in $ \mathbf{RP} ^1$ parametrizing those two equations. Moreover, 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 non-periodic time domain.