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 $\varepsilon$ 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 $t\approx\varepsilon^{-2}$, where the Korteweg–de Vries approximation becomes inapplicable.