On the variational interpretation of the discrete KP equation

TL;DR

This paper explores the variational structure of the discrete KP equation using a pluri-Lagrangian framework, demonstrating that on higher-dimensional lattices, the Euler-Lagrange equations align with the dKP equation.

Contribution

It establishes the equivalence between Euler-Lagrange equations and the dKP equation on lattices of dimension four or higher within a pluri-Lagrangian setting.

Findings

Euler-Lagrange equations are equivalent to the dKP equation in four or more dimensions.

The variational formulation is considered on both cubic and root lattices.

The study advances understanding of the geometric structure of the discrete KP equation.

Abstract

We study the variational structure of the discrete Kadomtsev-Petviashvili (dKP) equation by means of its pluri-Lagrangian formulation. We consider the dKP equation and its variational formulation on the cubic lattice ${\mathbb Z}^{N}$ as well as on the root lattice $Q(A_{N})$. We prove that, on a lattice of dimension at least four, the corresponding Euler-Lagrange equations are equivalent to the dKP equation.