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

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 μ-versions of the above equations.