A FRACTIONAL KdV HIERARCHY (original) (raw)
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Applied Mathematics Letters, 2019
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A Sato-Krichever Theory for Fractional Differential Operators
arXiv (Cornell University), 2021
Fractional differential (and difference) operators play a role in a number of diverse settings: integrable systems, mirror symmetry, Hurwitz numbers, the Bethe ansatz equations. We prove extensions of the three major results on algebras of commuting (ordinary) differentials operators to the setting of fractional differential operators: (1) the Burchnall-Chaundy theorem that a pair of commuting differential operators is algebraically dependent, (2) the classification of maximal commutative algebras of differential operators in terms of Sato's theory and (3) the Krichever correspondence constructing those of rank 1 in an algebro-geometric way. Unlike the available proofs of the Burchnall-Chaundy theorem which use the action of one differential operator on the kernel of the other, our extension to the fractional case uses bounds on orders of fractional differential operators and growth of algebras, which also presents a new and much shorter proof of the original result. The second main theorem is achieved by developing a new tool of the spectral field of a point in Sato's Grassmannian, which carries more information than the widely used notion of spectral curve of a KP solution. Our Krichever type correspondence for fractional differential operators is based on infinite jet bundles.
SSRN Electronic Journal
We prove that each member of the non-commutative nonlinear Schrödinger and modified Korteweg-de Vries hierarchy is a Fredholm Grassmannian flow, and for the given linear dispersion relation and corresponding equivalencing group of Fredholm transformations, is unique in the class of odd-polynomial partial differential fields. Thus each member is linearisable and integrable in the sense that time-evolving solutions can be generated by solving a linear Fredholm Marchenko equation, with the scattering data solving the corresponding linear dispersion equation. At each order, each member matches the corresponding non-commutative Lax hierarchy field which thus represent odd-polynomial partial differential fields. We also show that the cubic form for the noncommutative sine-Gordon equation corresponds to the first negative order case in the hierarchy, and establish the rest of the negative order non-commutative hierarchy. To achieve this, we construct an abstract combinatorial algebra, the Pöppe skew-algebra, that underlies the hierarchy. This algebra is the non-commutative polynomial algebra over the real line generated by compositions, endowed with the Pöppe product-the product rule for Hankel operators pioneered by Ch. Pöppe for classical integrable systems. Establishing the hierarchy members at non-negative orders, involves proving the existence of a 'Pöppe polynomial' expansion for basic compositions in terms of 'linear signature expansions' representing the derivatives of the underlying non-commutative field. The problem boils down to solving a linear algebraic equation for the polynomial expansion coefficients, at each order.
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Annals of Physics, 1996
Within a group-theoretical approach to the description of (2+1)-dimensional anyons, the minimal covariant set of linear differential equations is constructed for the fractional spin fields with the help of the deformed Heisenberg algebra (DHA), [a − , a + ] = 1 + νK, involving the Klein operator K, {K, a ± } = 0, K 2 = 1. The connection of the minimal set of equations with the earlier proposed 'universal' vector set of anyon equations is established. On the basis of this algebra, a bosonization of supersymmetric quantum mechanics is carried out. The construction comprises the cases of exact and spontaneously broken N = 2 supersymmetry allowing us to realize a Bose-Fermi transformation and spin-1/2 representation of SU(2) group in terms of one bosonic oscillator. The construction admits an extension to the case of OSp(2|2) supersymmetry, and, as a consequence, both applications of the DHA turn out to be related. A possibility of 'superimposing' the two applications of the DHA for constructing a supersymmetric (2+1)-dimensional anyon system is discussed. As a consequential result we point out that osp(2|2) superalgebra is realizable as an operator algebra for a quantum mechanical 2-body (nonsupersymmetric) Calogero model.
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Nuclear Physics B, 2001
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Journal of Mathematical Physics, 1997
One of the cornerstones of the theory of integrable systems of KdV type has been the remark that the n-GD ͑Gel'fand-Dickey͒ equations are reductions of the full Kadomtsev-Petviashvilij ͑KP͒ theory. In this paper we address the analogous problem for the fractional KdV theories, by suggesting candidates of the ''KP theories'' lying behind them. These equations are called ''KP (m) hierarchies,'' and are obtained as reductions of a bigger dynamical system, which we call the ''central system.'' The procedure allowing passage from the central system to the KP (m) equations, and then to the fractional KdV n m equations, is discussed in detail in the paper. The case of KdV 3 2 is given as a paradigmatic example.
Fractional systems and fractional Bogoliubov hierarchy equations
Physical Review E, 2005
We consider the fractional generalizations of the phase volume, volume element and Poisson brackets. These generalizations lead us to the fractional analog of the phase space. We consider systems on this fractional phase space and fractional analogs of the Hamilton equations. The fractional generalization of the average value is suggested. The fractional analogs of the Bogoliubov hierarchy equations are derived from the fractional Liouville equation. We define the fractional reduced distribution functions. The fractional analog of the Vlasov equation and the Debye radius are considered.
Journal of Mathematical Sciences and Modelling
Our review is devoted to Lie-algebraic structures and integrability properties of an interesting class of nonlinear dynamical systems called the dispersionless heavenly equations, which were initiated by Plebański and later analyzed in a series of articles. The AKS-algebraic and related R-structure schemes are used to study the orbits of the corresponding co-adjoint actions, which are intimately related to the classical Lie-Poisson structures on them. It is demonstrated that their compatibility condition coincides with the corresponding heavenly equations under consideration. Moreover, all these equations originate in this way and can be represented as a Lax compatibility condition for specially constructed loop vector fields on the torus. The infinite hierarchy of conservations laws related to the heavenly equations is described, and its analytical structure connected with the Casimir invariants, is mentioned. In addition, typical examples of such equations, demonstrating in detail their integrability via the scheme devised herein, are presented. The relationship of a fascinating Lagrange-d'Alembert type mechanical interpretation of the devised integrability scheme with the Lax-Sato equations is also discussed. We pay a special attention to a generalization of the devised Lie-algebraic scheme to a case of loop Lie superalgebras of superconformal diffeomorphisms of the 1|N-dimensional supertorus. This scheme is applied to constructing the Lax-Sato integrable supersymmetric analogs of the Liouville and Mikhalev-Pavlov heavenly equation for every N ∈ N\{4; 5}.