Variational lambda-symmetries and exact solutions to Euler-Lagrange equations lacking standard symmetries

Abstract

Variational lambda-symmetries are used to find exact solutions to second- and fourth-order Euler-Lagrange equations associated to variational problems for which standard procedures fail. A one-parameter family of exact solutions in terms of Bessel functions is obtained for a first-order variational problem whose Euler-Lagrange equation does not admit Lie symmetries. A family of second- order equations, involving arbitrary functions and parameters, is first written in variational form. The variational lambda-symmetry method successes in finding one-parameter families of exact solutions, despite the lack of Lie point and variational symmetries. A three-parameter family of exact solutions for a fourth-order equation with absence of Lie point symmetries is also deduced.