# Exact analytical solution of viscous Korteweg-deVries equation for water waves

## TL;DR

This paper derives an exact analytical solution for the viscous Korteweg-de Vries equation modeling water waves, revealing complex soliton dynamics including Peregrine soliton formation and decay.

## Contribution

It provides the first exact analytical solution to the viscous KdV equation for water waves, including soliton behavior and bifurcations, extending previous weakly nonlinear analyses.

## Key findings

- Analytical solution shows Peregrine soliton formation and decay.
- Weak nonlinearity leads to specific soliton bifurcations.
- Viscous effects significantly influence soliton dynamics.

## Abstract

The evolution of a solitary wave with very weak nonlinearity which was originally investigated by Miles [4] is revisited. The solution for a one-dimensional gravity wave in a water of uniform depth is considered. This leads to finding the solution to a Korteweg-de Vries (KdV) equation in which the nonlinear term is small. Also considered is the asymptotic solution of the linearized KdV equation both analytically and numerically. As in Miles [4], the asymptotic solution of the KdV equation for both linear and weakly nonlinear case is found using the method of inversescattering theory. Additionally investigated is the analytical solution of viscous-KdV equation which reveals the formation of the Peregrine soliton that decays to the initial sech^2(\xi) soliton and eventually growing back to a narrower and higher amplitude bifurcated Peregrine-type soliton.