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Lagrangian formulation of the Darboux system

Lingling Xue, E. V. Ferapontov, M. V. Pavlov

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 $τ$-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 $\int f(u_{xy}, u_{xt}, u_{yt})\, dxdydt$.

Lagrangian formulation of the Darboux system

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 -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 .
Paper Structure (9 sections, 4 theorems, 109 equations)

This paper contains 9 sections, 4 theorems, 109 equations.

Table of Contents

  1. Introduction and summary of the main results
  2. Scalar Lagrangian formulation of the Darboux system
  3. Continuous case
  4. Differential-difference case (one discrete variable)
  5. Differential-difference case (two discrete variables)
  6. Discrete case
  7. Second-order integrable Lagrangians in 3D
  8. Appendix: derivation of generating PDE of the KP hierarchy
  9. Concluding remarks

Key Result

Proposition 1

Darboux system (c1Dar) can be written as a single sixth-order PDE for $u$, Equation (c1PDE) is represented in Euler-Lagrange form corresponding to a Lagrangian $\int F \, dx^1 dx^2 dx^3$, with the Lagrangian density

Theorems & Definitions (4)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4