Euler-Poincaré formalism of peakon equations with cubic nonlinearity (Q887234)

The paper addresses a recently introduced integrable equation with the cubic nonlinearity: \[ u_t - u_{xxt} + 4u^2u_x = 3uu_xu_{xx} + u^2u_{xxx}. \] This equation admits exact peakon solutions, in the form of localized pulses with a jump of the first derivative at the center. Integrable equations which give rise to peakon solutions were known before, but with quadratic nonlinear terms, such as the Camassa-Holm equation. The objective of the work is to analyze the Hamiltonian structure of the equation. It is demonstrated that the structure can be made equivalent to the Euler-Poincaré formulation.