Formal recursion operators of integrable nonevolutionary equations and Lagrangian systems

TL;DR

This paper develops a framework for understanding the structure of formal recursion operators in non-evolutionary equations, enabling classification of integrable Lagrangian systems with higher order Lagrangians.

Contribution

It introduces a method based on homogeneity of determining equations to classify integrable Lagrangian systems and extends results to more general equations.

Findings

Derived the structure of formal recursion operators for non-evolutionary equations

Classified integrable Lagrangian systems with higher order Lagrangians

Extended results to general equations with multiple derivatives

Abstract

We derive the general structure of the space of formal recursion operators of nonevolutionary equations~$q_{tt}=f(q,q_{x},q_t,q_{xx},q_{xt},q_{xxx},q_{xxxx})$. This allows us to classify integrable Lagrangian systems with a higher order Lagrangian of the form~$\mathscr{L}=\frac12 L_2(q_{xx}, q_x, q)\,q_t^2 + L_1(q_{xx}, q_x, q)\, q_{t} + L_0(q_{xx}, q_x, q)$. The key technique relays on exploiting a homogeneity of the determining equations of formal recursion operators. This technique allows us to extend the main results to more general equations~$q_{tt}=f(q,q_{x},\ldots,q_{n};q_{t},q_{xt},\ldots,q_{mt})$.