4,028 research outputs found
Symmetries and (Related) Recursion Operators of Linear Evolution Equations
Significant cases of time-evolution equations, the linear Schr¨odinger and the Fokker–Planck equation are considered. It is known that equations of this type can be transformed, in some cases, into a highly simplified form. The properties of these equations in their initial and their simplified form are compared, showing in particular that this transformation partially prevents a clear understanding and a full application of the (physically relevant) notion of the so-called step up/down operators. These operators are shown to be recursion operators, related to the Lie point symmetries of the equations, which are also carefully discussed
Cosymmetries and Nijenhuis recursion operators for difference equations
In this paper we discuss the concept of cosymmetries and co--recursion
operators for difference equations and present a co--recursion operator for the
Viallet equation. We also discover a new type of factorisation for the
recursion operators of difference equations. This factorisation enables us to
give an elegant proof that the recursion operator given in arXiv:1004.5346 is
indeed a recursion operator for the Viallet equation. Moreover, we show that
this operator is Nijenhuis and thus generates infinitely many commuting local
symmetries. This recursion operator and its factorisation into Hamiltonian and
symplectic operators can be applied to Yamilov's discretisation of the
Krichever-Novikov equation
Recursion Operators and Nonlocal Symmetries for Integrable rmdKP and rdDym Equations
We find direct and inverse recursion operators for integrable cases of the
rmdKP and rdDym equations. Also, we study actions of these operators on the
contact symmetries and find shadows of nonlocal symmetries of these equations
Supersymmetric Representations and Integrable Fermionic Extensions of the Burgers and Boussinesq Equations
We construct new integrable coupled systems of N=1 supersymmetric equations
and present integrable fermionic extensions of the Burgers and Boussinesq
equations. Existence of infinitely many higher symmetries is demonstrated by
the presence of recursion operators. Various algebraic methods are applied to
the analysis of symmetries, conservation laws, recursion operators, and
Hamiltonian structures. A fermionic extension of the Burgers equation is
related with the Burgers flows on associative algebras. A Gardner's deformation
is found for the bosonic super-field dispersionless Boussinesq equation, and
unusual properties of a recursion operator for its Hamiltonian symmetries are
described. Also, we construct a three-parametric supersymmetric system that
incorporates the Boussinesq equation with dispersion and dissipation but never
retracts to it for any values of the parameters.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
The Hunter-Saxton equation: remarkable structures of symmetries and conserved densities
In this paper, we present extraordinary algebraic and geometrical structures
for the Hunter-Saxton equation: infinitely many commuting and non-commuting
-independent higher order symmetries and conserved densities. Using a
recursive relation, we explicitly generate infinitely many higher order
conserved densities dependent on arbitrary parameters. We find three Nijenhuis
recursion operators resulting from Hamiltonian pairs, of which two are new.
They generate three hierarchies of commuting local symmetries. Finally, we give
a local recursion operator depending on an arbitrary parameter.
As a by-product, we classify all anti-symmetric operators of a definite form
that are compatible with the Hamiltonian operator
Why nonlocal recursion operators produce local symmetries: new results and applications
It is well known that integrable hierarchies in (1+1) dimensions are local
while the recursion operators that generate them usually contain nonlocal
terms. We resolve this apparent discrepancy by providing simple and universal
sufficient conditions for a (nonlocal) recursion operator in (1+1) dimensions
to generate a hierarchy of local symmetries. These conditions are satisfied by
virtually all known today recursion operators and are much easier to verify
than those found in earlier work.
We also give explicit formulas for the nonlocal parts of higher recursion
operators, Poisson and symplectic structures of integrable systems in (1+1)
dimensions.
Using these two results we prove, under some natural assumptions, the
Maltsev--Novikov conjecture stating that higher Hamiltonian, symplectic and
recursion operators of integrable systems in (1+1) dimensions are weakly
nonlocal, i.e., the coefficients of these operators are local and these
operators contain at most one integration operator in each term.Comment: 10 pages, LaTeX 2e, final versio
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