Abstract
Abstract We present, for the first time, a Lagrangian multiform for the complete Kadomtsev–Petviashvili hierarchy—a single variational object that generates the whole hierarchy and encapsulates its integrability. By performing a reduction on this Lagrangian multiform, we also obtain Lagrangian multiforms for the Gelfand–Dickey hierarchy of hierarchies, comprising, among others, the Korteweg–de Vries and Boussinesq hierarchies.
Highlights
A feature of integrable systems is the existence of hierarchies of mutually compatible equations
A significant limitation of using traditional Lagrangians for such hierarchies is that they do not capture this compatibility. This limitation was overcome by the Lagrangian multiform [1], which allows compatible Lagrangians (i.e., Lagrangians of compatible equations) to be combined into a single variational object
A Lagrangian multiform for the discrete KP hierarchy was given in [11], whilst a Lagrangian multiform for the first two flows of the continuous KP hierarchy was presented in [7]. This continuous KP Lagrangian multiform was limited in the sense that extending it to contain higher flows of the hierarchy would result in non-local terms in the multiform, and there was no algorithmic method to perform such an extension
Summary
A feature of integrable systems is the existence of hierarchies of mutually compatible equations. A Lagrangian multiform for the discrete KP hierarchy (the first example of a Lagrangian 3-form) was given in [11], whilst a Lagrangian multiform for the first two flows of the continuous KP hierarchy was presented in [7] This continuous KP Lagrangian multiform was limited in the sense that extending it to contain higher flows of the hierarchy would result in non-local terms in the multiform, and there was no algorithmic method to perform such an extension. In this paper we assemble Dickey’s KP Lagrangians, along with a new set of Lagrangians to create Lagrangian multiform for the full KP hierarchy This is the first ever example of a continuous Lagrangian 3-form for a complete integrable hierarchy.
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