Integration of pair flows of the Camassa-Holm hierarchy

We present the integration of the “pair” flows associated to the Camassa–Holm (CH) hierarchy i.e., an explicit exact formula for the update of the initial velocity profile in terms of initial data when run by the flow associated to a Hamiltonian which (up to a constant factor) is given by the sum of the reciprocals of the squares of any two eigenvalues of the underlying spectral problem. The method stems from the integration of “individual” flows of the CH hierarchy described in [Loubet 2006; McKean 2003], and is seen to be more general in scope in that it may be applied when considering more complex flows (e.g., when the Hamiltonian involves an arbitrary number of eigenvalues of the associated spectral problem) up to when envisaging the full CH flow itself which is nothing but a superposition of commuting individual actions. Indeed, by incorporating piece by piece into the Hamiltonian the distinct eigenvalues describing the spectrum associated to the initial profile, we may recover McKean’s Fredholm determinant formulas [McKean 2003] expressing the evolution of initial data when acted upon by the full CH flow. We also give account of the large-time (and limiting remote past and future) asymptotics and obtain (partial) confirmation of the thesis about soliton genesis and soliton interaction raised in [Loubet 2006].