Singular solutions of a modified two-component Camassa-Holm equation

TL;DR

This paper introduces a modified two-component Camassa-Holm system that admits new singular solutions and exhibits unique peakon interaction behaviors, expanding understanding of shallow water wave models.

Contribution

The paper proposes a modified CH2 system allowing peakon solutions in velocity and average density, revealing new collision dynamics and divergence phenomena.

Findings

Modified system admits peakons in velocity and average density

Numerical simulations show little short-term effect of average density modification

Peakons can exhibit diverging phase shifts in certain interactions

Abstract

The Camassa-Holm equation (CH) is a well known integrable equation describing the velocity dynamics of shallow water waves. This equation exhibits spontaneous emergence of singular solutions (peakons) from smooth initial conditions. The CH equation has been recently extended to a two-component integrable system (CH2), which includes both velocity and density variables in the dynamics. Although possessing peakon solutions in the velocity, the CH2 equation does not admit singular solutions in the density profile. We modify the CH2 system to allow dependence on average density as well as pointwise density. The modified CH2 system (MCH2) does admit peakon solutions in velocity and average density. We analytically identify the steepening mechanism that allows the singular solutions to emerge from smooth spatially-confined initial data. Numerical results for MCH2 are given and compared with the pure CH2 case. These numerics show that the modification in MCH2 to introduce average density has little short-time effect on the emergent dynamical properties. However, an analytical and numerical study of pairwise peakon interactions for MCH2 shows a new asymptotic feature. Namely, besides the expected soliton scattering behavior seen in overtaking and head-on peakon collisions, MCH2 also allows the phase shift of the peakon collision to diverge in certain parameter regimes.