Pseudo-peakons and Cauchy analysis for an integrable fifth-order equation of Camassa-Holm type

In this paper we introduce a hierarchy of integrable higher order equations of Camassa-Holm (CH) type, that is, we present infinitely many nonlinear equations depending on inertia operators which generalize the standard momentum operator A2=∂xx−1 appearing in the Camassa-Holm equation mt=−mxu−2mux, m=A2(u). Our higher order CH-type equations are integrable in the sense that they possess an infinite number of local conservation laws, quadratic pseudo-potentials, and zero curvature formulations. We focus mainly on the fifth order CH-type equation and we show that it admits {\em pseudo-peakons}, this is, bounded solutions with differentiable first derivative and continuous and bounded second derivative, but whose higher order derivatives blow up. Furthermore, we investigate the Cauchy problem of our fifth order equation on the real line and prove local well-posedness for initial conditions u0∈Hs(R), s\u3e7/2. In addition, we discuss conditions for global well-posedness in H4(R) as well as conditions causing local solutions to blow up in a finite time. We finish our paper with some comments on the geometric content of our equations of CH-type

Pseudo-peakons and Cauchy analysis for an integrable fifth-order equation of Camassa-Holm type

In this paper we discuss integrable higher order equations {\em of Camassa-Holm (CH) type}. Our higher order CH-type equations are "geometrically integrable", that is, they describe one-parametric families of pseudo-spherical surfaces, in a sense explained in Section 1, and they are integrable in the sense of zero curvature formulation ($\simeq$ Lax pair) with infinitely many local conservation laws. The major focus of the present paper is on a specific fifth order CH-type equation admitting {\em pseudo-peakons} solutions, that is, weak bounded solutions with differentiable first derivative and continuous and bounded second derivative, but such that any higher order derivative blows up. Furthermore, we investigate the Cauchy problem of this fifth order CH-type equation on the real line and prove local well-posedness under the initial conditions $u_0 \in H^s(\mathbb{R})$, $s > 7/2$. In addition, we study conditions for global well-posedness in $H^4(\mathbb{R})$ as well as conditions causing local solutions to blow up in a finite time. We conclude our paper with some comments on the geometric content of the high order CH-type equations.Comment: 6 figures; 32 page

On (t2,t3)−(t_2,t_3)-Zakharov-Shabat equations of generalized Kadomtsev-Petviashvili hierarchies

We review the integration of the KP hierarchy in several non-standard contexts. Specifically, we consider KP in the following associative differential algebras: an algebra equipped with a nilpotent derivation; an algebra of functions equipped with a derivation that generalizes the gradient operator; an algebra of quaternion-valued functions; a differential Lie algebra; an algebra of polynomials equipped with the Pincherle differential; a Moyal algebra. In all these cases we can formulate and solve the Cauchy problem of the KP hierarchy. Also, in each of these cases we derive different Zakharov-Shabat $(t_2,t_3)$-equations -- that is, different Kadomtsev-Petviashvili equations -- and we prove existence of solutions arising from solutions to the corresponding KP hierarchy