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Conservation laws of the Camassa-Holm equation. (English) Zbl 1076.35100

Summary: We use the bi-Hamiltonian structure of the Camassa–Holm equation to show that its conservation laws \(H_n[m]\) are homogeneous with respect to the scaling \(m \mapsto \lambda_m\). Moreover, a direct argument is presented proving that \(H_{-1}, H_{-2}, \dots\), are of local character. Finally, simple representations of the conservation laws in terms of their variational derivatives are derived and used to obtain a constructive scheme for computation of the \(H_ns\).

MSC:

35Q35 PDEs in connection with fluid mechanics
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
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