On the Green-Naghdi equations with surface tension in the Camassa-Holm scale

The aim of this paper is to analyze the water waves problem for uneven bottom under the influence of surface tension. For that, we consider an asymptotic model of the 1D Green-Naghdi equations in Camassa-Holm scale and derive in a formal way by using the Whitham technique the Camassa-Holm equation under the influence of surface tension. After that, the well-posdeness of the obtained Camassa-Holm equation is proved by using the Picard iterative scheme which proves that there is no loss of regularity of the solution relative to the initial condition. Also, the $$H^s$$ -consistency, the stability and the convergency of the solution with the 1D Green-Naghdi model in Camassa-Holm scale are showed. Finally, the aspect of breaking wave for the Camassa-Holm equation is discussed in the presence of surface tension.

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