Convergence of the dispersion Camassa-Holm N-Soliton

TL;DR

This paper demonstrates that smooth N-soliton solutions of the dispersion Camassa-Holm equation converge to peakon solutions of the dispersionless version as the dispersion parameter approaches zero, using asymptotic analysis.

Contribution

It establishes a rigorous connection between smooth and peakon solutions in the dispersion Camassa-Holm equation through convergence analysis.

Findings

Smooth N-soliton solutions converge uniformly to peakons as dispersion tends to zero.

Asymptotic analysis and determinant identities are key tools used.

Provides a mathematical foundation for understanding peakon emergence from smooth solutions.

Abstract

In this paper, we show that the peakon (peaked soliton) solutions can be recovered from the smooth soliton solutions, in the sense that there exists a sequence of smooth N-soliton solutions of the dispersion Camassa-Holm equation converging to the N-peakon of the dispersionless Camassa-Holm equation uniformly with respect to the spatial variable x when the dispersion parameter tends to zero. The main tools are asymptotic analysis and determinant identities.