# (Non)local Hamiltonian and symplectic structures, recursions and hierarchies: a new approach and applications to the N = 1 supersymmetric KdV equation

@article{Kersten2003NonlocalHA, title={(Non)local Hamiltonian and symplectic structures, recursions and hierarchies: a new approach and applications to the N = 1 supersymmetric KdV equation}, author={P. H. M. Kersten and Iosif Krasil’shchik and Alexander Verbovetsky}, journal={Journal of Physics A}, year={2003}, volume={37}, pages={5003-5019} }

Using methods of Kersten et al (2004 J. Geom. Phys. 50 273–302) and Krasil'shchik and Kersten (2000 Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations (Dordrecht: Kluwer)), we accomplish an extensive study of the N = 1 supersymmetric Korteweg–de Vries (KdV) equation. The results include a description of local and nonlocal Hamiltonian and symplectic structures, five hierarchies of symmetries, the corresponding hierarchies of conservation laws, recursion…

## 10 Citations

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

- Mathematics
- 2006

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…

### A Geometric Study of the Dispersionless Boussinesq Type Equation

- Mathematics
- 2006

We discuss the dispersionless Boussinesq type equation, which is equivalent to the Benney–Lax equation, being a system of equations of hydrodynamical type. This equation was discussed in [4]. The…

### On Integrability of the Camassa–Holm Equation and Its Invariants

- Mathematics
- 2008

Using geometrical approach exposed in (Kersten et al. in J. Geom. Phys. 50:273–302, [2004] and Acta Appl. Math. 90:143–178, [2005]), we explore the Camassa–Holm equation (both in its initial scalar…

### New conserved quantities for N = 1 SKdV, the supersymmetric cohomology and W -algebras

- Mathematics, Physics
- 2009

An infinite sequence of new non-local conserved quantities for the N = 1 Super KdV (SKdV) equation is obtained. The sequence is constructed, via a Gardner trasformation, from a new conserved quantity…

### Conservation Laws and Nonlocal Variables

- Mathematics
- 2017

We discuss here the notion of conservation laws and briefly the theory of Abelian coverings over infinitely prolonged equations. Computation of conservation laws is also closely related to that of…

### The Symbolic Computation of Integrability Structures for Partial Differential Equations

- MathematicsTexts & Monographs in Symbolic Computation
- 2017

We present a unified mathematical approach to the symbolic computation of integrability structures of partial differential equations, like Hamiltonian operators, recursion operators for symmetries…

### A convenient criterion under which ℤ2-graded operators are Hamiltonian

- Mathematics
- 2011

We formulate a simple and convenient criterion under which skew-adjoint ℤ2-graded total differential operators are Hamiltonian, provided that their images are closed under commutation in the Lie…

### Computational Problems and Dedicated Software

- Mathematics
- 2017

In this chapter, we give an overview of the basic computational problems that arise in the study of geometrical aspects related to nonlinear partial differential equations and in the study of their…

### Supersymmetric Representations and Integrable Super-extensions of the Burgers and Boussinesq Equations

- Mathematics

New evolutionary supersymmetric systems whose right-hand sides are homogeneous differential polynomials and which possess infinitely many higher symmetries are constructed. Their intrinsic geometry…

## References

SHOWING 1-10 OF 25 REFERENCES

### An extensive study of the $N = 1$ supersymmetric KDV equation

- Physics, Mathematics
- 2002

Using recently developed methods, we accomplish an extensive study of the N = 1 supersymmetric Korteweg- de Vries equation. Our results include: five hierarchies of symmetries (including a new one),…

### The bi-Hamiltonian structure of fully supersymmetric Korteweg-de Vries systems

- Physics, Mathematics
- 1991

The bi-Hamiltonian structure of integrable supersymmetric extensions of the Korteweg-de Vries (KdV) equation related to theN=1 and theN=2 superconformal algebras is found. It turns out that some of…

### On the formalism of local variational differential operators

- Mathematics
- 2002

The calculus of local variational differential operators introduced by B. L. Voronov, I. V. Tyutin, and Sh. S. Shakhverdiev is studied in the context of jet super space geometry. In a coordinate-free…

### A Simple model of the integrable Hamiltonian equation

- Physics, Mathematics
- 1978

A method of analysis of the infinite‐dimensional Hamiltonian equations which avoids the introduction of the Backlund transformation or the use of the Lax equation is suggested. This analysis is based…

### Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations

- Mathematics
- 2000

Preface. 1. Classical symmetries. 2. Higher symmetries and conservation laws. 3. Nonlocal theory. 4. Brackets. 5. Deformations and recursion operators. 6. Super and graded theories. 7. Deformations…

### Nonlocal trends in the geometry of differential equations: Symmetries, conservation laws, and Bäcklund transformations

- Mathematics, Physics
- 1989

The theory of coverings over differential equations is exposed which is an adequate language for describing various nonlocal phenomena: nonlocal symmetries and conservation laws, Bäcklund…

### Elements of Superintegrable Systems: Basic Techniques and Results

- Mathematics
- 1987

I. Classical Superintegrable Systems.- 1. Introduction.- 2. Variational Calculus with Anticommuting Variables.- 3. SuperHamiltonian Formalism and Infinite-Dimensional Stable Lie Superalgebras.- 4.…

### Bäcklund and Darboux Transformations. The Geometry of Solitons

- Mathematics

We show that any multi-component matrix KP hierarchy is equivalent to the standard onecomponent (scalar) KP hierarchy endowed with a special infinite set of abelian additional symmetries, generated…

### Homological Methods in Equations of Mathematical Physics

- Mathematics
- 1998

These lecture notes are a systematic and self-contained exposition of the cohomological theories naturally related to partial differential equations: the Vinogradov C-spectral sequence and the…