Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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.

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