Classical $W_3^{(2)}$ algebra and its Miura map

TL;DR

This paper demonstrates the bi-Hamiltonian structure of the fractional KdV equation via zero curvature methods, explicitly derives Hamiltonian operators related to the $W^{(2)}_3$ algebra, and constructs its Miura map.

Contribution

It provides the first explicit derivation of the Hamiltonian operators and the Miura map for the classical $W^{(2)}_3$ algebra using gauge transformations.

Findings

Verified fractional KdV as bi-Hamiltonian system.

Derived explicit Hamiltonian operators for the system.

Constructed the Miura map for $W^{(2)}_3$ algebra.

Abstract

We verify that the fractional KdV equation is a bi-hamiltonian system using the zero curvature equation in $SL(3)$ matrix valued Lax pair representation, and explicitly find the closed form for the hamiltonian operators of the system. The second hamiltonian operator is the classical version of the $W^{(2)}_3$ algebra. We also construct systematically the Miura map of $W^{(2)}_3$ algebra using a gauge transformation of the $SL(3)$ matrix valued Lax operator in a particular gauge, and construct the modified fractional KdV equaiotn as hamiltonian system.