On Algebro-Geometric Solutions of the Camassa-Holm Hierarchy

Abstract

We find global solutions of algebro geometric type for all the equations of a new commuting hierarchy containing the Camassa-Holm equation. This hierarchy is built in analogy to the classical K-dV and AKNS hierarchies. We use a zero curvature method to give recursion formulas. The time evolution of the solutions is completely determined, and the motion on a nonlinear subvariety Υ of a generalized Jacobian variety is obtained by solving an inverse problem for the Sturm-Liouville equation L(φ) = −φ″ + φ = λyφ. This is the natural setting for the expression of the solutions which depend linearly with respect to t and x, with coordinates on a curvilinear parallelogram contained in such a subvariety φ. φ is obtained as the restriction of the generalized Abel map I₀ to the space Symm^g(R) of unordered g-tuples of points on R, and the nonlinear parallelogram is the image through the restricted generalized Abel map of the moving poles P_i(x, t) (i = 1, . . . , g) of the Weyl m-functions m_±(x, t, λ) of the dynamical Sturm-Liouville family of equations L_x(φ) = −φ″ + φ = λτ_x(y)φ, where τ is the translation flow. It turns out that the choice of a particular stationary initial condition u_g(x, t₀) completely determines the solution u(x, t) of all the equations in the hierarchy, as functions of the poles P_i(x, t) of the Weyl m-functions corresponding to the family L_x of Sturm-Liouville operators, with density function y(x, t) = u_xx(x, t)/2 − 2u(x, t). For every t∊ℝ, the maps x ↦ y(x, t) lie in an isospectral class of the associated family of Sturm-Liouville equations L_x(φ), and are completely determined by assigning spectral parameters and initial conditions for the poles P_i(x, t).