Hamiltonian formulation of SL(3) Ur-KdV equation

TL;DR

This paper develops a Hamiltonian framework for the SL(3) Boussinesq equation, unifying it with related KdV equations and introducing a new method to derive non-local operator-free equations.

Contribution

It introduces a novel procedure to obtain Ur-KdV equations without non-local operators and extends this to the SL(3) Boussinesq equation, deriving its Hamiltonian structure.

Findings

Explicit construction of the Hamiltonian operator for Ur-Bsq system

Unified view of SL(2) and SL(3) KdV-type equations

New method to derive non-local operator-free equations

Abstract

We give a unified view of the relation between the $SL(2)$ KdV, the mKdV, and the Ur-KdV equations through the Fr\'{e}chet derivatives and their inverses. For this we introduce a new procedure of obtaining the Ur-KdV equation, where we require that it has no non-local operators. We extend this method to the $SL(3)$ KdV equation, i.e., Boussinesq(Bsq) equation and obtain the hamiltonian structure of Ur-Bsq equationin a simple form. In particular, we explicitly construct the hamiltonian operator of the Ur-Bsq system which defines the poisson structure of the system, through the Fr\'{e}chet derivative and its inverse.