The Cauchy problem and multi-peakons for the mCH-Novikov-CH equation with quadratic and cubic nonlinearities

TL;DR

This paper studies the well-posedness, blow-up phenomena, and peakon solutions of a generalized Camassa-Holm equation with quadratic and cubic nonlinearities, extending previous models and providing new insights into wave-breaking and soliton solutions.

Contribution

It establishes local well-posedness, blow-up criteria, and constructs peakon and multi-peakon solutions for the mCH-Novikov-CH equation with quadratic and cubic nonlinearities.

Findings

Proved local well-posedness in Besov spaces.

Derived blow-up criteria and wave-breaking conditions.

Constructed explicit peakon and multi-peakon solutions.

Abstract

This paper investigates the Cauchy problem of a generalized Camassa-Holm equation with quadratic and cubic nonlinearities (alias the mCH-Novikov-CH equation), which is a generalization of some special equations such as the Camassa-Holm (CH) equation, the modified CH (mCH) equation ((alias the Fokas-Olver-Rosenau-Qiao equation), the Novikov equation, the CH-mCH equation, the mCH-Novikov equation, and the CH-Novikov equation. We first show the local well-posedness for the strong solutions of the mCH-Novikov-CH equation in Besov spaces by means of the Littlewood-Paley theory and the transport equations theory. Then, the Holder continuity of the data-to-solution map to this equation are exhibited in some Sobolev spaces. After providing the blow-up criterion and the precise blow-up quantity in light of the Moser-type estimate in the Sobolev spaces, we then trace a portion and the whole of the precise blow-up quantity, respectively, along the characteristics associated with this equation, and obtain two kinds of sufficient conditions on the gradient of the initial data to guarantee the occurance of the wave-breaking phenomenon. Finally, the non-periodic and periodic peakon and multi-peakon solutions for this equation are also explored.