Title: Symmetries and multipeakon solutions for the modified two-component Camassa-Holm system,
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.