Integrability, existence of global solutions, and wave breaking criteria for a generalization of the Camassa–Holm equation

Abstract

Recent generalizations of the Camassa-Holm equation are studied from the point of view of existence of global solutions, criteria for wave breaking phenomena and integrability. We provide conditions, based on lower bounds for the first spatial derivative of local solutions, for global well-posedness in Sobolev spaces for the family under consideration. Moreover, we prove that wave breaking phenomena occurs under certain mild hypothesis. Based on the machinery developed by Dubrovin [Commun. Math. Phys. 267, 117-139 (2006)] regarding bi-Hamiltonian deformations, we introduce the notion of quasi-integrability and prove that there exists a unique bi-Hamiltonian structure for the equation only when it is reduced to the Dullin-Gotwald-Holm equation. Our results suggest that a recent shallow water model incorporating Coriollis effects is integrable only in specific situations. Finally, to finish the scheme of geometric integrability of the family of equations initiated in a previous work, we prove that the Dullin-Gotwald-Holm equation describes pseudo-spherical surfaces.

Wave breaking for shallow water models with time decaying solutions

“…As we shall show in section Section 3, it implies that the global solutions of (1.2), as well as their corresponding functional of energy, vanish as t → ∞. Moreover, for λ sufficiently small, the conditions for wave breaking of the solutions (1.2) are unaltered when compared to (1.3), as one can observe comparing our results with those proved in [16] regarding this matter.…”

“…The presence of the cubic and quartic nonlinearities in (1.2) brings some challenges in the qualitative analysis of the solutions of (1.2) when compared with similar works dealing with (1.3). This is somewhat expected, since the presence of these higher order nonlinearities introduces substantial modifications on the behaviour of the solutions of (1.2) in comparison with (1.3), as one can infer by comparing 1 the results in [2,[7][8][9][10]14,20,29,31,38,40,41] with those in [6,15,16,23,24,37]. In particular, equation (1.3) does not seem to be integrable unless β = γ = 0, see [15,16].…”

“…which was studied in [15,16], and also in [6,23,24,37] for specific choices of the parameters α, β, γ and Γ, as already mentioned.…”

“…If λ = 0, equation (1.1) includes a shallow water model with Coriolis effects proposed in [6,23,24,37], see also [15,16].…”

A Novikov equation describing pseudo‐spherical surfaces, its pseudo‐potentials, and local isometric immersions

“…Since the seminal work, 8 many equations such as () have been studied from this perspective, see Refs. 9, 10, 12, 22–25. More recently, Reyes formulated the notion of geometric integrability , 21 see also Refs.…”