Nonautonomous analysis of steady Korteweg–de Vries waves under nonlocalised forcing

• Obtains analytical approximations for near-uniform and near-solitary wave profiles.
• Genuinely nonautonomous approach associates with hyperbolic and homoclinic solutions.
• Computed using normal and tangential deformations of stable and unstable manifolds.
• Small forcing need not have compact support, decay at infinity, or differentiability.
• Provides a tool for finding the number of solitary waves for a given small forcing.

Recently developed nonautonomous dynamical systems theory is applied to quantify the effect of bottom topography variation on steady surface waves governed by the Korteweg–de Vries (KdV) equation. Arbitrary (but small) nonlocalised bottom topographies are amenable to this method. Two classes of free surface solutions–hyperbolic and homoclinic solutions of the associated augmented dynamical system–are characterised. The first of these corresponds to near-uniform free-surface flows for which explicit formulæ are developed for a range of topographies. The second corresponds to solitary waves on the free surface, and a method for determining their number is developed. Formulæ for the shape of these solitary waves are also obtained. Theoretical free-surface profiles are verified using numerical KdV solutions, and excellent agreement is obtained.