A small dispersion limit to the Camassa–Holm equation: A numerical study

Abstract

In this paper we take up the question of a small dispersion limit for the Camassa–Holm equation. The particular limit we study involves a modification of the Camassa–Holm equation, seen in the recent theoretical developments by Himonas and Misiołek, as well as the first author, where well-posedness is proved in weak Sobolev spaces. This work led naturally to the question of how solutions actually behave in these modified equations as time evolves. While the dispersive limit studied here is inspired by the work of Lax and Levermore on the zero dispersion limit of the Korteweg–de Vries equation to the inviscid Burgers’ equation, here there is no known Inverse Scattering theory. Consequently, we resort to a sophisticated numerical simulation to study two representative (one smooth and one peakon), but by no means exhaustive, initial conditions in the modified Camassa–Holm equation. In both cases there appears to be a strong limit of the modified Camassa–Holm equation to the Camassa–Holm equation as the dispersive parameter tends to zero, provided that solutions have not evolved for too long (time sufficiently small). For the smooth initial condition considered, this time must be chosen before the solution approaches steepening; beyond this time the computed solution becomes increasingly complicated as the dispersive term tends towards zero, and there does not appear to be a limit. By contrast, for the peakon initial condition this limit does appear to exist for all times considered. While in many cases the computations required few discretization points, there were some very challenging cases (particularly for the small dispersion computations) where an enormous number of unknowns were required to properly resolve the solution.