Uniqueness of Conservative Solutions to the Camassa-Holm Equation via Characteristics

2014/01/01 by Alberto Bressan, Geng Chen, Bressan, Alberto +3
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1401.0312

openalex publication_date 2014/01/01 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

The paper provides a direct proof the uniqueness of solutions to the Camassa-Holm equation, based on characteristics. Given a conservative solution u=u(t,x), an equation is introduced which singles out a unique characteristic curve through each initial point. By studying the evolution of the quantities u and v= 2\arctan u_x along each characteristic, it is proved that the Cauchy problem with general initial data u₀∈ H¹(ℝ) has a unique solution, globally in time.

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