The Darboux coordinates for a new family of Hamiltonian operators and linearization of associated evolution equations

TL;DR

This paper introduces Darboux coordinates for a new family of Hamiltonian operators and provides a method to linearize associated evolution equations, facilitating their analysis and solution.

Contribution

It presents a differential substitution to reduce linear combinations of these operators to constant coefficients and linearizes bi-Hamiltonian evolution equations.

Findings

Darboux coordinates for the operators are derived for all odd N ≥ 3.

A differential substitution linearizes bi-Hamiltonian evolution equations.

The method simplifies the analysis of complex Hamiltonian systems.

Abstract

A. de Sole, V. G. Kac, and M. Wakimoto (arXiv:1004.5387) have recently introduced a new family of compatible Hamiltonian operators of the form $H^{(N,0)}=D^2\circ((1/u)\circ D)^{2n}\circ D$, where $N=2n+3$, $n=0,1,2,...$, $u$ is the dependent variable and $D$ is the total derivative with respect to the independent variable. We present a differential substitution that reduces any linear combination of these operators to an operator with constant coefficients and linearizes any evolution equation which is bi-Hamiltonian with respect to a pair of any nontrivial linear combinations of the operators $H^{(N,0)}$. We also give the Darboux coordinates for $H^{(N,0)}$ for any odd $N\geqslant 3$.