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A nonlinear second-order differential equation with a small parameter at the derivatives is considered in the case where the limit algebraic equation has a multiple root. The matching method is applied to construct an asymptotic expansion of the solution of the boundary value problem. Two boundary layer variables with different scales are used to describe the asymptotic solution near the boundary.

For the model of surface waves, we perform an asymptotic analysis with respect to a small parameter e 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 ≈ e−2, where the Korteweg-de Vries approximation becomes inapplicable.