Lagrangian formulation of the Darboux system

TL;DR

This paper formulates the Darboux system as a Lagrangian PDE, links it to the KP hierarchy, and constructs scalar Lagrangians for various discrete and continuous versions, revealing integrable structures.

Contribution

It provides the first explicit Lagrangian formulation of the Darboux system and its discrete variants, connecting them to the KP hierarchy and integrable Lagrangians.

Findings

Lagrangian form of the Darboux PDE is derived

Scalar Lagrangians for discrete and continuous cases are constructed

Dispersionless limits yield a classification of 3D integrable Lagrangians

Abstract

The classical Darboux system governing rotation coefficients of three-dimensional metrics of diagonal curvature possesses an equivalent formulation as a sixth-order PDE for a scalar potential (related to the corresponding $\tau$-function). We demonstrate that this PDE is Lagrangian and can be viewed as an explicit scalar form of the `generating PDE of the KP hierarchy' as discussed recently in Nijhoff [arXiv:2406.13423] in the Lagrangian multiform approach to the Darboux and KP hierarchies. Scalar Lagrangian formulations for differential-difference and fully discrete versions of the Darboux system are also constructed. In the first three cases (continuous and differential-difference with one and two discrete variables), the corresponding Lagrangians are expressible via elementary functions (logarithms), whereas the fully discrete case requires special functions (dilogarithms). Remarkably, dispersionless limits of the above Lagrangians provide a complete list of 3D second-order integrable Lagrangians of the form $\int f(u_{xy}, u_{xt}, u_{yt})\, dxdydt$.