Lipschitz metric for the Camassa-Holm equation on the line (Q379461)

The Cauchy problem for the Camassa-Holm equation NEWLINE\[NEWLINE u_t - u_{xxt} + \kappa u_x + 3uu_{x} -2u_xu_{xx} - uu_{xxx} = 0, \quad \kappa\in\mathbb{R}, NEWLINE\]NEWLINE has attracted a considerable attention due to the fact that it serves as a model for shallow water waves and its rich mathematical structure.NEWLINENEWLINEThe main focus in the paper under review is on the construction of the Lipschitz metric for the semigroup of conservative solutions on the real line. In the periodic case, this problem was studied recently in [\textit{K. Grunert} et al., J. Differ. Equations 250, No. 3, 1460--1492 (2011; Zbl 1205.35252)]. The authors show how to modify the approach used there in the non-periodic setting. Note that there are some subtle and nontrivial differences between the two cases and hence the constructions of the corresponding metrics are distinct.NEWLINENEWLINEFinally, let us mention that the authors consider the case without dispersion, \(\kappa=0\), however, the approach presented in the paper under review can also handle the generalized hyperelastic-rod equation, which includes \(\kappa\neq 0\) as a particular case.