On the weakly dissipative Camassa-Holm, Degasperis-Procesi, and Novikov equations

TL;DR

This paper demonstrates that weakly dissipative versions of certain integrable equations can be transformed into their non-dissipative forms through an exponential time scaling, establishing their equivalence.

Contribution

It shows that weak dissipation in these equations can be removed via a simple exponential time transformation, unifying dissipative and non-dissipative models.

Findings

Weakly dissipative equations are equivalent to non-dissipative ones after a change of variables.

The results extend to b-family and two-component versions of these equations.

Dissipative effects can be absorbed into a time-dependent scaling, simplifying analysis.

Abstract

We show that the weakly dissipative Camassa-Holm, Degasperis-Procesi, Hunter-Saxton, and Novikov equations can be reduced to their non-dissipative versions by means of an exponentially time-dependent scaling. Hence, up to a simple change of variables, the non-dissipative and dissipative versions of these equations are equivalent. Similar results hold also for the equations in the so-called b-family of equations as well as for the two-component and \mu-versions of the above equations.