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

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 µ-versions of the above equations.2010 Mathematics Subject Classification. 35Q35,35C05,35C99.

Orbital stability and dynamical behaviors of solitary waves for the Camassa–Holm equation with quartic nonlinearity

“…The subtle balance between the dispersion and the convection gives rise to solitary waves which can cause qualitative changes in the nature of genuinely nonlinear phenomena [1][2][3]. The dispersion term stretches the solitary wave, gradually elongating it until the solitary wave vanishes.…”

Global existence and propagation speed for a Degasperis-Procesi equation with both dissipation and dispersion

“…After the Degasperis-Procesi equation was derived, many papers were devoted to its study, cf. [3,8,12,14,21] and the citations therein.…”

On the optimal control problem for the Novikov equation with strong viscosity

“…In [21], Yan, Li and Zhang considered the weakly dissipative Novikov equation y t + u 2 y x + 3uu x y + λy = 0, y = u − u xx , (1.5) and obtained the global existence and blow-up phenomenon for this equation. However, it has been pointed out in [22] that these weakly dissipative On the other hand, the optimal control of viscous partial differential equation(PDE) are widely investigated. Let us mention some papers concerning this issue.…”

“…The optimal control of viscous peakon PDEs with square nonlinearity, such as viscous Camassa-Holm equation [30], viscous Degasperis-Procesi equation [31] and viscous Dullin-Gottwalld-Holm equation [32,41] up to a trivial change of variables [22], we intend to investigate an optimal control problem governed by the strongly viscous Novikov equation…”