2012•Studies in Applied MathematicsRequires access

A Super Camassa–Holm Equation with N‐Peakon Solutions

Xianguo Geng, Bo Xue, Lihua Wu

Abstract

A super Camassa–Holm equation with peakon solutions is proposed, which is associated with a 3 × 3 matrix spectral problem with two potentials. With the aid of the zero‐curvature equation, we derive a hierarchy of super Harry Dym type equations and establish their Hamiltonian structures. It is shown that the super Camassa–Holm equation is exactly a negative flow in the hierarchy and admits exact solutions with N peakons. As an example, exact 1‐peakon solutions of the super Camassa–Holm equation are given. Infinitely many conserved quantities of the super Camassa–Holm equation and the super Harry Dym type equation are, respectively, obtained.

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Key concepts: Peakon, Camassa–Holm equation, Mathematics, Hierarchy, Curvature, Matrix (chemical analysis), Lax pair, Mathematical physics