Infinitely many local higher symmetries without recursion operator or master symmetry: integrability of the Foursov--Burgers system revisited

Singularity Analysis and Integrability of a Burgers-Type System of Foursov

We apply the Painleve test for integrability of partial differential equations to a system of two coupled Burgers-type equations found by Foursov, which was recently shown by Sergyeyev to possess infinitely many commuting local generalized symmetries without any recursion operator. The Painleve analysis easily detects that this is a typical C-integrable system in the Calogero sense and rediscovers its linearizing transformation.

Recursion Operators admitted by non-Abelian Burgers equations: Some Remarks

The recursion operators admitted by different operator Burgers equations, in the framework of the study of nonlinear evolution equations, are here con- sidered. Specifically, evolution equations wherein the unknown is an operator acting on a Banach space are investigated. Here, the mirror non-Abelian Burgers equation is considered: it can be written as $r_t = r_{xx} + 2r_x r$. The structural properties of the admitted recursion operator are studied; thus, it is proved to be a strong symmetry for the mirror non-Abelian Burgers equation as well as to be the hereditary. These results are proved v…

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 equatio…

The Lax integrability of a two-component hierarchy of the Burgers type dynamical systems within asymptotic and differential-algebraic approaches

The Lax type integrability of a two-component polynomial Burgers type dynamical system within a differential-algebraic approach is studied, its linear adjoint matrix Lax representation is constructed. A related recursion operator and infnite hierarchy of Lax integrable nonlinear dynamical systems of the Burgers-Korteweg-de Vries type are derived by means of the gradient-holonomic technique, the corresponding Lax type representations are presented.

Burgers方程的广义对称性

Generalized symmetries of Burgers equation

本文填补了Burgers方程广义对称性代数未被详尽描述的空白，显式给出该代数基，并证明利用两个已知递归算子与两个种子广义对称性即可生成该代数。

On weakly non-local, nilpotent, and super-recursion operators for N=1 super-equations

We consider nonlinear, scaling-invariant N=1 boson + fermion supersymmetric systems whose right-hand sides are homogeneous differential polynomials and satisfy some natural assumptions. We select the super-systems that admit infinitely many higher symmetries generated by recursion operators; we further restrict ourselves to the case when the dilaton dimensions of the bosonic and fermionic super-fields coincide and the weight of the time is half the weight of the spatial variable. We discover five systems that satisfy these assumptions; one system is transformed to the purely bosonic Burgers eq…