On the Cauchy problem for the integrable Camassa-Holm type equation with cubic nonlinearity

Abstract

Considered in this paper is the modified Camassa-Holm equation with cubic nonlinearity, which is integrable and admits the single peaked solitons and multi-peakon solutions. The short-wave limit of this equation is known as the short-pulse equation. The main investigation is the Cauchy problem of the modified Camassa-Holm equation with qualitative properties of its solutions. It is firstly shown that the equation is locally well-posed in a range of the Besov spaces. The blow-up scenario and the lower bound of the maximal time of existence are then determined. A blow-up mechanism for solutions with certain initial profiles is described in detail and nonexistence of the smooth traveling wave solutions is also demonstrated. In addition, the persistence properties of the strong solutions for the equation are obtained.

30 pages

References in corpus (2)

• An integrable shallow water equation with peaked solitons
• Ultra-short pulses in linear and nonlinear media

Cited by in corpus (6)

• On the Cauchy problem for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity
• The Cauchy problem and wave-breaking phenomenon for a generalized sine-type FORQ/mCH equation
• The Cauchy problem for fractional Camassa-Holm equation in Besov space
• Existence of peakons for a cubic generalization of the Camassa-Holm equation
• Stability of peakons for the generalized modified Camassa-Holm equation
• Algebraic Properties for Certain Form of the Members of Sequence on Generalized Modified Camassa-Holm Equation