Hamiltonian formulation for the gradient-power family

Construct a standard Hamiltonian formulation for the gradient-power dispersive equation when the gradient exponent satisfies $r\ne1$.

Background

The equation preserves mass and a nonlinear $L^{p+1}$ quantity, but the paper does not identify a Hamiltonian structure for general $r\ne1$. The authors distinguish this family from the Cooper–Shepard–Sodano compacton models, whose Hamiltonian or Lagrangian framework does not directly apply here.

A Hamiltonian formulation could provide variational methods and stability criteria that are currently unavailable for the evolution generator. The unresolved status is stated explicitly by the authors.