2006/12/28 by Alberto Bressan, ALBERTO BRESSAN, Adrian Constantin +1 · 532 citations
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Bounded function #Bounded variation #Breaking wave #Camassa–Holm equation #Cauchy problem #Continuation #Dissipative system #Gravitational singularity #Initial value problem #Integrable system #Mathematical analysis #Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Semigroup #Space (punctuation) #Term (time) #Transformation (genetics) #Wave equation #Wave propagation
paper · doi:10.1142/s0219530507000857
published in Analysis and Applications 05(01), 1-27 (World Scientific)
openalex publication_date 2006/12/28 · crossref created 2006/12/28 · crossref issued 2007/01/01 · crossref published 2007/01/01 · crossref published-print 2007/01/01 · crossref published-online 2011/11/20 · crossref deposited 2024/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05 · crossref indexed 2026/08/07
This paper is devoted to the continuation of solutions to the Camassa–Holm equation after wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear hyperbolic system in an L ∞ space, containing a non-local source term which is discontinuous but has bounded directional variation. For a given initial condition, the Cauchy problem has a unique solution obtained as fixed point of a contractive integral transformation. Returning to the original variables, we obtain a semigroup of global dissipative solutions, defined for every initial data [Formula: see text], and continuously depending on the initial data. The new variables resolve all singularities due to possible wave breaking and ensure that energy loss occurs only through wave breaking.