Korteweg-de Vries hierarchy as an asymptotic limit of the Boussinesq system

For the model of surface waves, we perform an asymptotic analysis with respect to a small parameter ε for large times where corrections to the approximation described by the Korteweg-de Vries equation must be taken into account. We reveal the appearance of the Korteweg-de Vries hierarchy, which ensures the construction of an asymptotic representation up to the times t ≈ ε−2, where the Korteweg-de Vries approximation becomes inapplicable.