The paper discusses the hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations in modeling shallow water wave propagation. These equations, derived from the Green-Naghdi equations under specific scaling assumptions, capture stronger nonlinear effects than classical equations like the Korteweg-de Vries (KdV) and Benjamin-Bona-Mahoney (BBM) equations. They are capable of modeling wave breaking phenomena, a key feature in water wave dynamics. The Camassa-Holm (CH) and Degasperis-Procesi (DP) equations are shown to be integrable and possess bi-Hamiltonian structures, which are essential for their complete integrability and the existence of soliton solutions. The CH equation is derived from the Green-Naghdi equations under the Camassa-Holm scaling, where the nonlinear effects are stronger, allowing for wave breaking. Similarly, the DP equation is derived under a different scaling and also exhibits wave breaking behavior. The paper provides a rigorous justification for these equations as approximations to the governing equations of water waves. It shows that both equations can be derived from the Green-Naghdi equations through asymptotic analysis and that they accurately model the behavior of water waves, including wave breaking. The CH equation is shown to describe plunging breakers, while the DP equation describes surging breakers, highlighting the different behaviors captured by these equations. The mathematical analysis confirms the well-posedness of these equations and their ability to approximate the exact solutions of the Green-Naghdi equations. The study also demonstrates that the CH and DP equations can model the evolution of the free surface elevation and the associated velocity, providing a more accurate description of wave dynamics, including the formation of singularities (wave breaking) in finite time. The results validate the use of these equations in modeling shallow water waves and their relevance in understanding complex wave phenomena.The paper discusses the hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations in modeling shallow water wave propagation. These equations, derived from the Green-Naghdi equations under specific scaling assumptions, capture stronger nonlinear effects than classical equations like the Korteweg-de Vries (KdV) and Benjamin-Bona-Mahoney (BBM) equations. They are capable of modeling wave breaking phenomena, a key feature in water wave dynamics. The Camassa-Holm (CH) and Degasperis-Procesi (DP) equations are shown to be integrable and possess bi-Hamiltonian structures, which are essential for their complete integrability and the existence of soliton solutions. The CH equation is derived from the Green-Naghdi equations under the Camassa-Holm scaling, where the nonlinear effects are stronger, allowing for wave breaking. Similarly, the DP equation is derived under a different scaling and also exhibits wave breaking behavior. The paper provides a rigorous justification for these equations as approximations to the governing equations of water waves. It shows that both equations can be derived from the Green-Naghdi equations through asymptotic analysis and that they accurately model the behavior of water waves, including wave breaking. The CH equation is shown to describe plunging breakers, while the DP equation describes surging breakers, highlighting the different behaviors captured by these equations. The mathematical analysis confirms the well-posedness of these equations and their ability to approximate the exact solutions of the Green-Naghdi equations. The study also demonstrates that the CH and DP equations can model the evolution of the free surface elevation and the associated velocity, providing a more accurate description of wave dynamics, including the formation of singularities (wave breaking) in finite time. The results validate the use of these equations in modeling shallow water waves and their relevance in understanding complex wave phenomena.