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Singular solutions of a modified two-component shallow-water equation

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The Camassa-Holm equation (CH) is a well known integrable Hamiltonian equation describing the velocity dynamics of shallow water waves. In its dispersionless limit, 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 system does not admit singular solutions in its density profile.

We modify the CH2 system to allow dependence on average density as well as on pointwise density. The modified CH2 system (MCH2) now admits peakon solutions in both velocity and average density, although it may no longer be integrable. We analytically identify the steepening mechanism that summons the emergent singular solutions 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 both overtaking and head-on peakon collisions, MCH2 also allows the phase shift of the peakon collision to diverge in certain parameter regimes.

This talk is part of the Applied and Computational Analysis series.

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