The positive and negative Camassa-Holm-γ hierarchies, zero curvature representations, bi-Hamiltonian structures, and algebro-geometric solutions

In this paper, we extend the Camassa-Holm-gamma (CH-gamma) equation to two different hierarchies of nonlinear evolution equations by resorting to two Lenard recursion sequences. One is of positive order CH-gamma hierarchy including a Harry-Dym-type equation, while the other is of negative order CH-gamma hierarchy of integrodifferential equations with nonzero integration constants including the CH-gamma equation. We provide the zero curvature representations for both hierarchies by solving a key matrix equation. Moreover, the Lenard sequences are used to construct a Lax matrix that satisfies a stationary zero curvature equation, which enables us to establish bi-Hamiltonian structures for both hierarchies by applying trace identity. In addition, the Lenard sequence and Lax pair are used to systematically derive the Its-Matveev trace formula and the Dubrovin-type equations associated with the CH-gamma equation. The hyperelliptic curve and Abel-Jacobi coordinates are then introduced to linearize the associated flow, from which the algebro-geometric solutions to the CH-gamma equation are constructed by using standard Jacobi inversion technique.