A note on the convergence of the solution of the Novikov equation

Abstract

We consider the Novikov and Camass-Holm equations, which contain nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solution of the dispersive equation converges to the unique entropy solution of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the L p setting.

A view of the peakon world through the lens of approximation theory

“…The literature surrounding the Novikov equation is not yet quite as overwhelming as for the CH and DP equations, but there is no shortage of articles about PDE-analytic questions, in addition to the one just mentioned [253,290,305,176,147,314,306,135,150,195,315,196,307,53,304,143,34,149,328,302,63,228,278,207]. Stability of peakons has been considered by several researchers [215,299,257,258,54,55], and likewise solitons [237,206,260,259,301,229,326] and integrability aspects [281,30,186,29,273].…”