# Soliton Interaction with External Forcing within the Korteweg-de Vries Equation

## TL;DR

This paper investigates the interaction of solitary waves with external forces in the Korteweg-de Vries equation, deriving approximate solutions and validating them through numerical simulations for arbitrary forcing and soliton widths.

## Contribution

It introduces an asymptotic method to analyze forced KdV solutions with arbitrary forcing and soliton widths, extending previous limiting case studies.

## Key findings

- Asymptotic solutions agree well with numerical results.
- Exact solutions are obtainable for specific forcings.
- The method applies to small amplitude external forces.

## Abstract

We revise the solutions of the forced Korteweg-de Vries equation describing a resonant interaction of a solitary wave with external pulse-type perturbations. In contrast to previous works where only the limiting cases of a very narrow forcing in comparison with the initial soliton or a very narrow soliton in comparison with the width of external perturbation were studied, we consider here an arbitrary relationship between the widths of soliton and external perturbation of a relatively small amplitude. In many particular cases, exact solutions of the forced Korteweg-de Vries equation can be obtained for the specific forcings of arbitrary amplitude. We use the earlier developed asymptotic method to derive an approximate set of equations up to the second-order on a small parameter characterising the amplitude of external force. The analysis of exact solutions of the derived equations is presented and illustrated graphically. It is shown that the theoretical outcomes obtained by asymptotic method are in a good agreement with the results of direct numerical modelling within the framework of forced Korteweg-de Vries equation.