Wave-Breaking and Peakons for a Modified Camassa–Holm Equation

Abstract

ABSTRACT. In this paper, we investigate the formation of singularities and the existence of peaked traveling-wave solutions for a modified Camassa-Holm equation with cubic nonlinearity. The equation is known to be integrable, and is shown to admit a single peaked soliton and multi-peakon solutions, of a different character than those of the Camassa-Holm equation. Singularities of the solutions can occur only in the form of wave-breaking, and a new wave-breaking mechanism for solutions with certain initial profiles is described in detail.

On the blow-up of solutions to the integrable modified Camassa–Holm equation

“…This observation allows us to establish sharper results than were found in [15]; see Remarks 3.4 and 3.6 below for comparisons. Let us now state the main results in the present paper.…”

“…Finally, if we drop the assumption that u 0 ∈ L 1 (R), then, provided the initial momentum density is of one sign, we can establish the following alternative blowup criterion, which is significantly stronger than that found in [15]; see Remark 3.6 for details.…”

“…The geometric formulation, integrability, local well-posedness, blow-up criteria and wave breaking, existence of peaked solitons (peakons), and the stability of single peakons and periodic peakons to the modified CH equation (1.1) were studied recently in [12,15,23,24]. It is shown that even when the initial momentum density m 0 (x) does not change sign, the solutions to the Cauchy problem (1.1) may blow up in finite time, in contrast to the CH equation, the Degasperis-Procesi (DP) equation [10,18], or the Novikov equation (1.5).…”

“…The goal of the present paper is to derive sharp sufficient conditions on the initial data that guarantees the formulation of singularities in the resulting solution in finite time. We take a different approach from that used in [15]. The first step is to rewrite the mCH equation (1.1) as a transport equation for the momentum density (1.2):…”

“…remains bounded, while the solution blows up in finite time when the slope (1.7) is unbounded from below [12,15]. The fact that the momentum density satisfies a transport equation also implies that, provided it initially does not change sign, then m(t, x) and, in view of the Green's function formula -see (3.4) below -the solution u(t, x) will maintain the same sign where defined.…”