Uniqueness of global conservative weak solutions for two-component higher-order shallow water system

We investigate the uniqueness of global-in-time conservative weak solutions for the generalized two-component higher-order shallow water system on R. The proof strategy relies on characteristics. Given a conservative weak solution, we introduce the characteristic equation to uniquely determine a characteristic curve through each initial point, transforming the Eulerian coordinate into a Lagrangian coordinate. We demonstrate that the Cauchy problem of the generalized two-component higher-order Camassa-Holm system, equipped with initial data z₀ = (u₀, ρ₀)∈ (H²( R)∩ W2, p( R))×(L²( R)∩ L^∞( R)), has a unique global conservative weak solution.