The Lagrangian multiform approach to integrable systems

A Lagrangian multiform enables the multi-dimensional consistency of a set of PDEs to be captured at the variational level. We offer a new perspective on the multiform Euler-Lagrange equations in terms of the variational derivative of the exterior derivative of a Lagrangian multiform and present for the first time in their full generality the multiform Euler-Lagrange equations for discrete Lagrangian multiforms. Then, by considering the closure property of a Lagrangian multiform as a conservation law, we use Noether's theorem to show that every variational symmetry of a Lagrangian leads to a Lagrangian multiform. In doing so, we provide a systematic method for constructing Lagrangian multiforms for which the closure property and the multiform Euler-Lagrange both hold. We present three examples, including what was at the time the first known example of a continuous Lagrangian 3-form: a Lagrangian multiform for the Kadomtsev-Petviashvili equation. We show that the Zakharov-Mikhailov Lagrangian structure for integrable nonlinear equations derived from a general class of Lax pairs possesses a Lagrangian multiform structure. We show that, as a consequence of this multiform structure, we can formulate a variational principle for the Lax pair itself, a problem that to our knowledge was never previously considered. As an example, we present an integrable NxN matrix system that contains the AKNS hierarchy. Finally, we present, for the first time, a Lagrangian multiform for the complete Kadomtsev-Petviashvili (KP) hierarchy: a single variational object that generates the whole hierarchy and encapsulates its integrability. By performing a reduction on this Lagrangian multiform, we are able to obtain Lagrangian multiforms for the Gelfand-Dickey hierarchy of hierarchies comprising, amongst others, the Korteweg-de Vries and Boussinesq hierarchies.