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

Holm, DD, O Náraigh, L and Tronci, C (2009) Singular solutions of a modified two-component Camassa-Holm equation. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), 79 (1), 016601.

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The Camassa-Holm (CH) equation 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 a dependence on the average density as well as the pointwise density. The modified CH2 system (MCH2) does admit peakon solutions in the 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 the MCH2 system are given and compared with the pure CH2 case. These numerics show that the modification in the MCH2 system to introduce the average density has little short-time effect on the emergent dynamical properties. However, an analytical and numerical study of pairwise peakon interactions for the MCH2 system shows a different asymptotic feature. Namely, besides the expected soliton scattering behavior seen in overtaking and head-on peakon collisions, the MCH2 system also allows the phase shift of the peakon collision to diverge in certain parameter regimes.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
Holm, DD
O Náraigh, L
Date : January 2009
DOI : 10.1103/PhysRevE.79.016601
Depositing User : Symplectic Elements
Date Deposited : 17 May 2017 12:20
Last Modified : 10 Jun 2019 13:08

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