2005/12/01 by Enrique Loubet · 6 citations
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Nonlinear Photonic Systems #Algebraic structures and combinatorial models #Superposition principle #Mathematics #Hamiltonian (control theory) #Hierarchy #Eigenvalues and eigenvectors #Soliton #Flow (mathematics) #Mathematical analysis #Statistical physics #Physics #Quantum mechanics #Geometry #Nonlinear system
paper · doi:10.1002/cpa.20109
published in Communications on Pure and Applied Mathematics 59(3), 408-465 (Wiley)
openalex publication_date 2005/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08
Abstract The present work offers a detailed account of the large‐time development of the velocity profile run by a single “individual” Hamiltonian flow of the Camassa‐Holm (CH) hierarchy, the Hamiltonian employed being the reciprocal of any eigenvalue of the underlying spectral problem. In this simpler scenario, I prove some of the conjectures raised by McKean [27]. Notably, I confirm the ultimate shaping into solitons of the cusps that appear, near blowup sites, of any velocity profile emanating from an initial disposition for which breakdown of the wave in finite time is sure to happen. The careful large‐time asymptotic analysis is carried from exact expressions describing the velocity in terms of initial data, the integration involving a “Lagrangian” scale and three “theta functions,” the rates at which the latter reach their common values at each end of the line characterizing the region where soliton genesis is expected. In fact, the present method also suggests how solitons may arise from initial conditions not leading to breakdown. The full CH flow is nothing but a superposition of such commuting “individual” actions. Therein lies the hope that the present account will pave the way to elucidate soliton formation for more complex flows, in particular for the CH flow itself. © 2005 Wiley Periodicals, Inc.