An asymptotic property of the Camassa–Holm equation

Asymptotic densities are used to study an asymptotic property of the dispersionless Camassa–Holm equation. An asymptotic density of a global solution is a weak limit of its scaled momentum density along a sequence of time increasing to infinity. For a global solution with non-negative compactly supported initial momentum density, we show that if the asymptotic density is unique, then it is a positive combination of Dirac measures supported in a bounded interval in the non-negative axis with zero as the only possible accumulation point. In other words, if the scaled momentum density does not oscillate as time goes to infinity, the solution behaves as a combination of peakons moving to the right at different speeds. In contrast to many investigations on the topic, our approach is not spectral theoretic and hence is independent of the structure of the isospectral problem associated with the equation.