Lipschitz metric for the modified coupled Camassa–Holm system

Abstract

In this paper, we study the Lipschitz continuity of the conservative solutions for the modified coupled Camassa–Holm system on the real line. By introducing a new characteristic, the original Camassa–Holm system can be reformulated to an equivalent semilinear system in Lagrangian coordinates, and the solutions on the equivalence classes (established from the relabeling functions in Lagrangian coordinates) can construct a semigroup . There exists a bijection between the conservative solutions in original coordinates and the equivalence classes in Lagrangian coordinates such that we can construct a semigroup of conservative solutions in Eulerian coordinates. Moreover, the Lipschitz continuity of the semigroup ensures that the semigroup with a new metric is Lipschitz continuous.