The paper by Adrian Constantin and David Lannes explores the hydrodynamic relevance of the Camassa-Holm and Degasperis-Procesi equations, which are nonlinear dispersive partial differential equations that arise in the modeling of shallow water waves over a flat bed. These equations capture stronger nonlinear effects compared to classical models like the Benjamin-Bona-Mahoney and Korteweg-de Vries equations, particularly in the context of wave breaking phenomena. The authors provide a rigorous mathematical foundation for earlier asymptotic procedures by Johnson, extending them to a firm and mathematically rigorous basis. They investigate the unidirectional asymptotics for water waves, leading to the Green-Naghdi equations, and derive asymptotic equations for the velocity and surface elevation under the Camassa-Holm scaling. The paper focuses on the Camassa-Holm and Degasperis-Procesi equations, which are bi-Hamiltonian systems and thus completely integrable. These equations admit wave breaking, a fundamental phenomenon in water wave dynamics. The authors prove the well-posedness of the unidirectional equations and rigorously justify their approximation to the Green-Naghdi equations, showing that they provide a good approximation with the same accuracy. A key result is the precise blow-up pattern for the free surface elevation, demonstrating that wave breaking occurs in the form of surging breakers, not plunging breakers, as expected for the velocity component. This is illustrated through numerical computations and detailed mathematical analysis. Overall, the paper provides a comprehensive study of the hydrodynamic relevance of the Camassa-Holm and Degasperis-Procesi equations, highlighting their ability to accurately model wave breaking phenomena in shallow water waves.The paper by Adrian Constantin and David Lannes explores the hydrodynamic relevance of the Camassa-Holm and Degasperis-Procesi equations, which are nonlinear dispersive partial differential equations that arise in the modeling of shallow water waves over a flat bed. These equations capture stronger nonlinear effects compared to classical models like the Benjamin-Bona-Mahoney and Korteweg-de Vries equations, particularly in the context of wave breaking phenomena. The authors provide a rigorous mathematical foundation for earlier asymptotic procedures by Johnson, extending them to a firm and mathematically rigorous basis. They investigate the unidirectional asymptotics for water waves, leading to the Green-Naghdi equations, and derive asymptotic equations for the velocity and surface elevation under the Camassa-Holm scaling. The paper focuses on the Camassa-Holm and Degasperis-Procesi equations, which are bi-Hamiltonian systems and thus completely integrable. These equations admit wave breaking, a fundamental phenomenon in water wave dynamics. The authors prove the well-posedness of the unidirectional equations and rigorously justify their approximation to the Green-Naghdi equations, showing that they provide a good approximation with the same accuracy. A key result is the precise blow-up pattern for the free surface elevation, demonstrating that wave breaking occurs in the form of surging breakers, not plunging breakers, as expected for the velocity component. This is illustrated through numerical computations and detailed mathematical analysis. Overall, the paper provides a comprehensive study of the hydrodynamic relevance of the Camassa-Holm and Degasperis-Procesi equations, highlighting their ability to accurately model wave breaking phenomena in shallow water waves.