Extensions of the General Solution Families for the Inverse Problem of the Calculus of Variations for Sixth- and Eighth-order Ordinary Differential Equations

TL;DR

This paper develops new higher-order Lagrangian hierarchies for sixth- and eighth-order ODEs, expanding the class of variational formulations with greater flexibility and applications to nonlinear wave models.

Contribution

It introduces generalized Lagrangians with four arbitrary functions satisfying geometric criteria for inverse variational problems of high-order ODEs.

Findings

Derived new variational hierarchies for sixth- and eighth-order ODEs.

Found families of solitary and embedded solitons in the generalized models.

Demonstrated applications to nonlinear wave equations and physical models.

Abstract

New third- and fourth-order Lagrangian hierarchies are derived in this paper. The free coefficients in the leading terms satisfy the most general differential geometric criteria currently known for the existence of a variational formulation, as derived by solution of the full inverse problem of the Calculus of Variations for scalar sixth- and eighth-order ordinary differential equations (ODEs). The Lagrangians obtained here have greater freedom since they require conditions only on individual coefficients. In particular, they contain four arbitrary functions, so that some investigations based on the existing general criteria for a variational representation are particular cases of our families of models. The variational equations resulting from our generalized Lagrangians may also represent traveling waves of various nonlinear evolution equations, some of which recover known physical models. For a typical member of our generalized variational ODEs, families of regular and embedded solitary waves are also derived in appropriate parameter regimes. As usual, the embedded solitons are found to occur only on isolated curves in the part of parameter space where they exist.