2007/08/01 by Luca Zampogni · 1 citation
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Algebraic structures and combinatorial models #Advanced Differential Equations and Dynamical Systems #Mathematics #Subvariety #Parallelogram #Type (biology) #Pure mathematics #Mathematical analysis #Hierarchy #Zero (linguistics) #Space (punctuation) #Inverse #Variety (cybernetics) #Geometry
paper · pdf · doi:10.1515/ans-2007-0303
openalex publication_date 2007/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/05/21
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 I 0 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, t 0 ) 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).