# Symmetry reduction and soliton-like solutions for the generalized Korteweg-de Vries equation

## TL;DR

This paper investigates the symmetry properties of the generalized Korteweg-de Vries (gKdV) equation, identifies conditions for soliton-like solutions, and derives explicit plane wave solutions using symmetry reduction techniques.

## Contribution

It provides a comprehensive analysis of symmetry reductions for gKdV with arbitrary functions and derives explicit soliton-like and plane wave solutions.

## Key findings

- gKdV's symmetry algebra is generally 2-dimensional, leading to plane wave solutions
- Exceptional symmetries occur for specific functions f(u), enabling additional reductions
- Explicit hyperbolic secant type plane wave solutions are derived

## Abstract

We analyze the gKdV equation, a generalized version of Korteweg-de Vries with an arbitrary function $f(u)$. In general, for a function $f(u)$ the Lie algebra of symmetries of gKdV is the $2$-dimensional Lie algebra of translations of the plane $xt$. This implies the existence of plane wave solutions. Indeed, for some specific values of $f(u)$ the equation gKdV admits a Lie algebra of symmetries of dimension grater than $2$. We compute the similarity reductions corresponding to these exceptional symmetries. We prove that the gKdV equation has soliton-like solutions under some general assumptions, and we find a closed formula for the plane wave solutions, that are of hyperbolic secant type.