34,315 research outputs found
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
Classification of integrable quadratic Hamiltonians on e(3)
Linear Poisson brackets on e(3) typical of rigid body dynamics are
considered. All quadratic Hamiltonians of Kowalevski type having additional
first integral of fourth degree are found. Quantum analogs of these
Hamiltonians are listed.Comment: 11 page
Gardner's deformations of the N=2 supersymmetric a=4-KdV equation
We prove that P.Mathieu's Open problem on constructing Gardner's deformation
for the N=2 supersymmetric a=4-Korteweg-de Vries equation has no supersymmetry
invariant solutions, whenever it is assumed that they retract to Gardner's
deformation of the scalar KdV equation under the component reduction. At the
same time, we propose a two-step scheme for the recursive production of the
integrals of motion for the N=2, a=4-SKdV. First, we find a new Gardner's
deformation of the Kaup-Boussinesq equation, which is contained in the bosonic
limit of the super-hierarchy. This yields the recurrence relation between the
Hamiltonians of the limit, whence we determine the bosonic super-Hamiltonians
of the full N=2, a=4-SKdV hierarchy. Our method is applicable towards the
solution of Gardner's deformation problems for other supersymmetric KdV-type
systems.Comment: Extended version of the talks given by A.V.K. at 8th International
conference `Symmetry in Nonlinear Mathematical Physics' (June 20-27, 2009,
Kiev, Ukraine) and 9th International workshop `Supersymmetry and Quantum
Symmetries' (July 29 - August 3, 2009, JINR, Dubna, Russia); 22 page
Computing symmetries and recursion operators of evolutionary super-systems using the SsTools environment
At a very informal but practically convenient level, we discuss the
step-by-step computation of nonlocal recursions for symmetry algebras of
nonlinear coupled boson-fermion supersymmetric systems by using the
SsTools environment.Comment: 18 pages, accepted to Nonlinear Systems and Their Remarkable
Mathematical Structures. (N.Euler ed) CRC Press, Boca Raton FL, US
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