# Symmetries and multipeakon solutions for the modified two-component Camassa-Holm system

## TL;DR

This paper derives a new pointwise invariant for the modified two-component Camassa-Holm system, linking symmetries to multipeakon solutions, and reports a novel periodic solution influenced by an elastic potential.

## Contribution

It introduces a new invariant derived from system symmetries, establishing a direct connection between symmetries and multipeakon solutions in the modified system.

## Key findings

- Derived a new pointwise invariant using Noether's theorem.
- Established the link between symmetries and multipeakon solutions.
- Discovered a previously unreported periodic solution.

## Abstract

Compared with the two-component Camassa-Holm system, the modified two-component Camassa-Holm system introduces a regularized density which makes possible the existence of solutions of lower regularity, and in particular of multipeakon solutions. In this paper, we derive a new pointwise invariant for the modified two-component Camassa-Holm system. The derivation of the invariant uses directly the symmetry of the system, following the classical argument of Noether's theorem. The existence of the multipeakon solutions can be directly inferred from this pointwise invariant. This derivation shows the strong connection between symmetries and the existence of special solutions. The observation also holds for the scalar Camassa-Holm equation and, for comparison, we have also included the corresponding derivation. Finally, we compute explicitly the solutions obtained for the peakon-antipeakon case. We observe the existence of a periodic solution which has not been reported in the literature previously. This case shows the attractive effect that the introduction of an elastic potential can have on the solutions.