2015/08/31 by Alberto De Sole, Victor G. Kac, Daniele Valeri · 1 citation
Mathematics · Physics and Astronomy · #Algebra over a field #Fractional Differential Equations Solutions #Hamiltonian (control theory) #Hamiltonian system #Integrable system #Lax pair #Nonlinear Waves and Solitons #Operator (biology) #Pseudodifferential operators #Quantum Mechanics and Non-Hermitian Physics #R-matrix #math-ph #math.MP #math.RA #math.RT #msc:17B69 #msc:35Q53 #msc:37K10 #msc:37K30 #nlin.SI
paper · pdf · doi:10.1007/s00220-016-2684-x
published as Comm. Math. Phys. 347 (2016), n. 2, 449-488 · 35 pages, final version
openalex publication_date 2016/06/08 · openalex created_date 2016/06/24 · arxiv created 2018/09/05 · arxiv updated 2018/09/07 · openalex updated_date 2026/08/05
We develop a new method for constructing integrable Hamiltonian hierarchies of Lax type equations, which combines the fractional powers technique of Gelfand and Dickey, and the classical Hamiltonian reduction technique of Drinfeld and Sokolov. The method is based on the notion of an Adler type matrix pseudodifferential operator and the notion of a generalized quasideterminant. We also introduce the notion of a dispersionless Adler type series, which is applied to the study of dispersionless Hamiltonian equations. Non-commutative Hamiltonian equations are discussed in this framework as well.