Supersymmetric representations and integrable fermionic extensions of the Burgers and Boussinesq equations

We construct new integrable coupled systems of N = 1 supersymmetric equa- tions 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 proper- ties 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.

[1]  M. Tabor,et al.  The Painlevé property for partial differential equations , 1983 .

[2]  P. Mathieu,et al.  $N=2$ Superconformal Algebra and Integrable O(2) Fermionic Extensions of the Korteweg-de Vries Equation , 1988 .

[3]  Peter J. Olver,et al.  Integrable Evolution Equations on Associative Algebras , 1998 .

[4]  Y. Manin,et al.  A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy , 1985 .

[5]  T. Wolf,et al.  Classification of polynomial integrable systems of mixed scalar and vector evolution equations: I , 2004, nlin/0412003.

[6]  Why nonlocal recursion operators produce local symmetries: new results and applications , 2004, nlin/0410049.

[7]  Vladimir E. Zakharov,et al.  The Boussinesq equation revisited , 2002 .

[8]  Artur Sergyeyev.  Locality of Symmetries Generated by Nonhereditary, Inhomogeneous, and Time-Dependent Recursion Operators: A New Application for Formal Symmetries , 2003, nlin/0303033.

[9]  I. S. Krasilʹshchik,et al.  Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations , 2000 .

[10]  P. Kersten,et al.  Hamiltonian operators and ℓ*-coverings , 2003, math/0304245.

[11]  C. S. Gardner,et al.  Korteweg‐de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion , 1968 .

[12]  P. Kersten,et al.  (Non)local Hamiltonian and symplectic structures, recursions and hierarchies: a new approach and applications to the N = 1 supersymmetric KdV equation , 2003, nlin/0305026.

[13]  Valentin Lychagin,et al.  Geometry of jet spaces and nonlinear partial differential equations , 1986 .

[14]  J. Boussinesq,et al.  Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. , 1872 .

[15]  On the equivalence of linearization and formal symmetries as integrability tests for evolution equations , 1993 .

[16]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[17]  P. Kersten,et al.  Symmetries and Recursion Operators for Classical and Supersymmetric Equations , 2000 .

[18]  Thomas Wolf.  Applications of CRACK in the Classification of Integrable Systems , 2003 .

[19]  P. Olver,et al.  Non-abelian integrable systems of the derivative nonlinear Schrödinger type , 1998 .

[20]  S. I. Svinolupov.  On the analogues of the Burgers equation , 1989 .

[21]  N. N. Yanenko,et al.  Systems of Quasilinear Equations and Their Applications to Gas Dynamics , 1983 .

[22]  E. Nissimov,et al.  Bäcklund and Darboux Transformations. The Geometry of Solitons , 2022 .

[23]  Pierre Mathieu,et al.  Supersymmetric extension of the Korteweg--de Vries equation , 1988 .

[24]  Thomas Wolf,et al.  On weakly non-local, nilpotent, and super-recursion operators for N=1 homogeneous super-equations , 2005 .

[25]  C. S. Gardner,et al.  The Korteweg-de Vries equation as a Hamiltonian System , 1971 .

[26]  The Gardner category and nonlocal conservation laws for N=1 Super KdV , 2005, hep-th/0504149.