On the higher Poisson structures of the Camassa-Holm hierarchy
The article under review is to generate higher-order Poisson structures related to the Camassa-Holm equation. The key point is to adopt the squared solutions to a linear second-order differential equation associated with a linear combination of the Hamiltonian pair. Then the authors represent the generating series of higher-order Poisson structures as a compact formula using the squared solutions and the inverse operator \(\partial ^{-1}\). The article is clearly written and only the domain and range space for the inverse operator \(\partial ^{-1}\) is not sufficiently treated. In particular, the condition \(\lim _{x\to \infty}\xi (x) =-\int _{x\to -\infty} \xi (x) \in \mathbb{R} \) for functions in the space requires \(\int_{-\infty}^\infty \xi (y) \, dy =0\) to guaranntee the convergence of the two integrals \(\int_{-\infty}^x \xi (y) \, dy \) and \(\int_{-\infty}^x \xi (y) \, dy \). This makes that the inverse operator \(\partial ^{-1}\) goes out into another space.
scientific article; zbMATH DE number 5319014
scientific article