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

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 $u_0\in H^1(\mathbb{R})$ has a unique solution, globally in time.

References in corpus (2)

• Lipschitz metric for the Camassa-Holm equation on the line
• Lipschitz metric for the periodic Camassa-Holm equation

Cited by in corpus (1)

• Ghostpeakons and Characteristic Curves for the Camassa-Holm, Degasperis-Procesi and Novikov Equations