On the Lagrangian multiform structure of the extended lattice Boussinesq system

Author:

F. W. Nijhoff, D. J. Zhang

Keyword:

Nonlinear Sciences, Exactly Solvable and Integrable Systems, Exactly Solvable and Integrable Systems (nlin.SI), Mathematical Physics (math-ph)

journal:

date:

2023-12-20 00:00:00

Abstract

The lattice Boussinesq (lBSQ) equation is a member of the lattice Gel'fand-Dikii (lGD) hierarchy, introduced in \cite{NijPapCapQui1992}, which is an infinite family of integrable systems of partial difference equations labelled by an integer $N$, where $N=2$ represents the lattice Kortewg-de Vries system, and $N=3$ the Boussinesq system. In \cite{Hiet2011} it was shown that, written as three-compnent system, the lBSQ system allows for extra parameters which essentially amounts to building the lattice KdV inside the lBSQ. In this paper we show that, on the level of the Lagrangian structure, this boils down to a linear combination of Lagrangians from the members of the lGD hierarchy as was established in \cite{LobbNijGD2010}. The corresponding Lagrangian multiform structure is shown to exhibit a `double zero' structure.

PDF: On the Lagrangian multiform structure of the extended lattice Boussinesq system.pdf