|
SIGMA 2 (2006), 030, 19 pages math-ph/0511071
https://doi.org/10.3842/SIGMA.2006.030
Supersymmetric Representations and Integrable Fermionic Extensions of the Burgers and Boussinesq Equations
Arthemy V. Kiselev a, b and Thomas Wolf c
a) Department of Higher Mathematics, Ivanovo State Power University,
34 Rabfakovskaya Str., Ivanovo, 153003 Russia
b) Department of Physics, Middle East Technical University, 06531 Ankara, Turkey
c) Department of Mathematics, Brock University, 500 Glenridge Ave.,
St. Catharines, Ontario, Canada L2S 3A1
Received November 26, 2005, in final form February 25, 2006; Published online February 28, 2006
Abstract
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.
Key words:
integrable super-equations; fermionic extensions; Burgers equation; Boussinesq equation.
pdf (363 kb)
ps (209 kb)
tex (26 kb)
References
- Andrea S., Restuccia A., Sotomayor A., The Gardner category and
nonlocal conservation laws for N=1 Super KdV, J. Math. Phys., 2005, V.46, 103513, 11 pages, hep-th/0504149.
- Bilge A.H., On the equivalence of linearization and formal
symmetries as integrability tests for evolution equations,
J. Phys. A: Math. Gen., 1993, V.26, 7511-7519.
- Hlavatý L., The Painlevé analysis of fermionic extensions of
KdV and Burgers equations, Phys. Lett. A, 1989, V.137,
N 4-5, 173-178.
- Kersten P., Krasil'shchik I., Verbovetsky A.,
Hamiltonian operators and l*-coverings,
J. Geom. Phys., 2004, V.50, N 1-4, 273-302,
math.DG/0304245.
- Kersten P., Krasil'shchik I., Verbovetsky A.,
(Non)local Hamiltonian and symplectic structures,
recursions and hierarchies: a new approach and applications
to the N=1 supersymmetric KdV equation,
J. Phys. A: Math. Gen., 2004, V.37, 5003-5019,
nlin.SI/0305026.
- Kiselev A.V., Karasu A., Hamiltonian deformations of the
Boussinesq equations, Proc. Workshop `Quantization, Dualities,
and Integrable Systems' (January 23-27, 2006, Denizli, Turkey),
Preprint ISPUmath-1/2006, 12 pages.
- Kiselev A.V., Wolf T., On weakly non-local, nilpotent, and
super-recursion operators for N=1 homogeneous super-equations,
Proc. Int. Workshop `Supersymmetries and Quantum Symmetries -
2005' (July 26-31, 2005, Dubna, Russia), to appear,
nlin.SI/0511056.
- Kiselev A.V., Wolf T., The SSTOOLS environment for
classification of integrable super-equations, Comp. Phys. Commun., 2006, to appear.
- Krasil'shchik I.S., Kersten P.H.M., Symmetries and recursion
operators for classical and supersymmetric differential
equations, Dordrecht, Kluwer Acad. Publ., 2000.
- Krasil'shchik I.S., Lychagin V.V., Vinogradov A.M., Geometry of
jet spaces and nonlinear partial differential equations, New
York, Gordon & Breach Sci. Publ., 1986.
- Kupershmidt B.A., Singular symmetries of integrable curves and
surfaces, J. Math. Phys., 1982, V.23, 364-366.
- Kupershmidt B.A., Deformations of integrable systems, Proc. Roy. Irish Acad. A, 1983, V.83, 45-74.
- Laberge C.A., Mathieu P., N=2 superconformal algebra and
integrable O(2) fermionic extensions of the Korteweg-de Vries
equation, Phys. Lett. B, 1988, V.215, 718-722.
- Maltsev A.Ya., Novikov S.P., On the local systems Hamiltonian in
the weakly non-local Poisson brackets, Phys. D, 2001, V.156,
53-80, nlin.SI/0006030.
- Manin Yu.I., Radul A.O., A supersymmetric extension of the
Kadomtsev-Petviashvili hierarchy, Comm. Math. Phys.,
1985, V.98, 65-77.
- Mathieu P., Supersymmetric extension of the Korteweg-de Vries
equation, J. Math. Phys., 1988, V.29, 2499-2506.
- Mathieu P., Open problems for the super KdV equations. Bäcklund
and Darboux transformations. The geometry of solitons, CRM
Proc. Lecture Notes, 2001, V.29, 325-334, math-ph/0005007.
- Miura R.M., Gardner C.S., Kruskal M.D.,
Korteweg-de Vries equation and generalizations. II.
Existence of conservation laws and constants of motion,
J. Math. Phys., 1968, V.9, 1204-1209.
- Olver P.J. , 2nd ed., New York, Springer-Verlag, 1993.
- Olver P.J., Sokolov V.V., Integrable evolution equations on
associative algebras,
Comm. Math. Phys., 1998, V.193, 245-268.
Olver P.J., Sokolov V.V., Non-abelian integrable systems of the
derivative nonlinear Schrödinger type, Inverse Problems,
1998, V.14, L5-L8.
- Rozdestvenski B.L., Janenko N.N., Systems of quasilinear
equations and their applicatons to gas dynamics, Translations
of Mathematical Monographs, Vol. 55, Providence, RI, AMS, 1983.
- Sergyeyev A., Locality of symmetries generated by nonhereditary,
inhomogeneous, and time-dependent recursion operators: a new
application for formal symmetries, Acta Appl. Math., 2004,
V.83,
95-109, nlin.SI/0303033.
Sergyeyev A., Why nonlocal recursion operators produce local
symmetries: new results and applications, J. Phys. A: Math. Gen., 2005, V.38, 3397-3407, nlin.SI/0410049.
- Svinolupov S.I., On the analogues of the Burgers equation,
Phys. Lett. A, 1989, V.135, 32-36.
- Tsuchida T., Wolf T., Classification of polynomial integrable
systems of mixed scalar and vector evolution equations. I,
J. Phys. A: Math. Gen., 2005, V.38, 7691-7733, nlin.SI/0412003.
- Weiss J., Tabor M., Carnevale G., The Painlevé property for
partial differential equations, J. Math. Phys., 1983,
V.24, 522-526.
- Wolf T., Applications of CRACK in the classification of
integrable systems, CRM Proc. Lecture Notes, 2004, V.37,
283-300, nlin.SI/0301032.
- Wolf T., Supersymmetric evolutionary equations with higher order
symmetries, 2003,
http://beowulf.ac.brocku.ca/~twolf/htdocs/susy/all.html (please contact T. W. for access).
|
|