Symmetry Properties of a Generalized Korteweg-de Vires Equation and some Explicit Solutions

TL;DR

This paper applies symmetry group methods to a generalized Korteweg-de Vries equation, deriving explicit solutions including polynomial, trigonometric, and elliptic functions, and explores reductions to simpler equations under specific constraints.

Contribution

It introduces a novel application of symmetry analysis to a generalized KdV equation, providing explicit solutions and reduction techniques not previously detailed.

Findings

Explicit polynomial, trigonometric, and elliptic solutions derived.

Reduction to a first-order equation under a differential constraint.

Conditions for reciprocal solutions and a first integral identified.

Abstract

The symmetry group method is applied to a generalized Korteweg-de Vries equation and several classes of group invarint solution for it are obtained by means of this technique. Polynomial, trigonometric and elliptic function solutions can be calculated. It is shown that this generalized equation can be reduced to a first-order equation under a particular second-order differential constraint which resembles a Schrodinger equation. For a particular instance in which the constraint is satisfied, the generalized equation is reduced to a quadrature. A condition which ensures that the reciprocal of a solution is also a solution is given, and a first integral to this constraint is found.