Wave‐Breaking and Persistence Property for a Two‐Component Shallow‐Water Wave System With Higher‐Order Nonlinearities

ABSTRACT This paper investigates wave‐breaking and persistence properties for a two‐component shallow‐water wave system with higher‐order nonlinearities, derived from the rotational Green‐Naghdi equations under the Camassa‐Holm regime. We first establish a new wave‐breaking criterion that directly links the initial slope of the velocity to the system's conserved quantities and a bound on the component . Next, utilizing the framework of moderate weight functions, we demonstrate the persistence of strong solutions in weighted spaces for a general class of weights. As a consequence of this persistence, we derive the precise spatial asymptotic profiles of solutions that exhibit exponential decay. These results extend classical findings for the Camassa‐Holm equation to a more complex system with higher‐order nonlinearities.