Lipschitz metric for the Camassa-Holm equation on the line

TL;DR

This paper introduces a new Lipschitz metric for the Camassa-Holm equation that ensures stability of solutions over time and clarifies its relation to standard norms, extending to a broader class of equations.

Contribution

The paper develops a novel Lipschitz metric for the Camassa-Holm equation that guarantees solution stability and extends to generalized hyperelastic-rod equations.

Findings

Established a Lipschitz metric with exponential stability bounds.

Clarified the relationship between the new metric and standard norms.

Extended the method to generalized hyperelastic-rod equations.

Abstract

We study stability of solutions of the Cauchy problem on the line for the Camassa-Holm equation $u_t-u_{xxt}+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ with initial data $u_0$. In particular, we derive a new Lipschitz metric $d_\D$ with the property that for two solutions $u$ and $v$ of the equation we have $d_\D(u(t),v(t))\le e^{Ct} d_\D(u_0,v_0)$. The relationship between this metric and the usual norms in $H^1$ and $L^\infty$ is clarified. The method extends to the generalized hyperelastic-rod equation $u_t-u_{xxt}+f(u)_x-f(u)_{xxx}+(g(u)+\frac12 f"(u)(u_x)^2)_x=0$ (for $f$ without inflection points).