# From Stäckel systems to integrable hierarchies of PDE’s: Benenti class of separation relations

@article{Baszak2006FromSS, title={From St{\"a}ckel systems to integrable hierarchies of PDE’s: Benenti class of separation relations}, author={Maciej Błaszak and Krzysztof Marciniak}, journal={Journal of Mathematical Physics}, year={2006}, volume={47}, pages={032904} }

We propose a general scheme of constructing of soliton hierarchies from finite dimensional Stackel systems and related separation relations. In particular, we concentrate on the simplest class of separation relations, called Benenti class, i.e., certain Stackel systems with quadratic in momenta integrals of motion.

## 14 Citations

### RECIPROCAL TRANSFORMATIONS FOR ST ¨ ACKEL-RELATED LIOUVILLE INTEGRABLE SYSTEMS

- Mathematics
- 2006

We consider the Stackel transform, also known as the coupling-constant metamorphosis, which under certain conditions turns a Hamiltonian dynamical system into another such system and preserves the…

### Generalized Stäckel transform and reciprocal transformations for finite-dimensional integrable systems

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2008

We present a multiparameter generalization of the Stäckel transform (the latter is also known as the coupling-constant metamorphosis) and show that under certain conditions this generalized Stäckel…

### Stäckel systems generating coupled KdV hierarchies and their finite-gap and rational solutions

- Mathematics
- 2008

We show how to generate coupled KdV hierarchies from Stäckel separable systems of Benenti type. We further show that the solutions of these Stäckel systems generate a large class of finite-gap and…

### On Reciprocal Equivalence of Stäckel Systems

- Mathematics
- 2012

In this paper we investigate Stäckel transforms between different classes of parameter‐dependent Stäckel separable systems of the same dimension. We show that the set of all Stäckel systems of the…

### Multiparameter Generalization of the Stackel Transform, Deformations of Separation Curves and Reciprocal Transformations

- Mathematics
- 2007

We present a multiparameter generalization of the Stäckel transform, also known as the coupling-constant metamorphosis. We show that this transformation under certain conditions turns a Hamiltonian…

### Transforming St\"{a}ckel Hamiltonians of Benenti type to polynomial form

- Mathematics
- 2021

In this paper we discuss two canonical transformations that turn Stäckel separable Hamiltonians of Benenti type into polynomial form: transformation to Viète coordinates and transformation to Newton…

### Non-Homogeneous Hydrodynamic Systems and Quasi-Stackel Hamiltonians

- Mathematics
- 2017

In this paper we present a novel construction of non-homogeneous hydrodynamic equations from what we call quasi-Stackel systems, that is non-commutatively integrable systems constructed from approp…

### A Coordinate-Free Construction for a Class of Integrable Hydrodynamic-Type Systems

- Mathematics
- 2008

Using a (1,1)-tensor L with zero Nijenhuis torsion and maximal possible number (equal to thenumber of dependent variables) of distinct, functionally independent eigenvalues we deﬁne, in…

## References

SHOWING 1-10 OF 24 REFERENCES

### Restricted flows of soliton hierarchies: coupled KdV and Harry Dym case

- Mathematics, Physics
- 1991

Restricted flows of soliton hierarchies associated with the energy-dependent Schrodinger spectral problem are determined explicitly. A remarkable connection with separable potentials is used for…

### Systematic Construction of Separable Systems with Quadratic in Momenta First Integrals

- Mathematics
- 2004

Liouville integrable separable systems with quadratic in momenta first integrals are considered. Particular attention is paid to the systems generated by the so-called special conformal Killing…

### Separable Hamiltonian equations on Riemann manifolds and related integrable hydrodynamic systems

- Physics
- 2003

### Towards a Theory of Differential Constraints of a Hydrodynamic Hierarchy

- Mathematics
- 2003

Abstract We present a theory of compatible differential constraints of a hydrodynamic hierarchy of infinite-dimensional systems. It provides a convenient point of view for studying and formulating…

### THE GEOMETRY OF HAMILTONIAN SYSTEMS OF HYDRODYNAMIC TYPE. THE GENERALIZED HODOGRAPH METHOD

- Mathematics
- 1991

It is proved that there exists an infinite involutive family of integrals of hydrodynamic type for diagonal Hamiltonian systems of quasilinear equations; the completeness of the family is also…

### Intrinsic characterization of the variable separation in the Hamilton–Jacobi equation

- Mathematics
- 1997

The nonorthogonal separation of variables in the Hamilton–Jacobi equation corresponding to a natural Hamiltonian H=12gijpipj+V, with a metric tensor of any signature, is intrinsically characterized…

### NONLINEARIZATION OF THE LAX SYSTEM FOR AKNS HIERARCHY

- Mathematics
- 1990

The Lax system for the AKNS vector field is nonlinearized and becomes naturally compatible under the constraint induced by a relation (q,r) = f(ψ) between reflectionless potentials and the…

### Multi-Hamiltonian Theory of Dynamical Systems

- Physics
- 1998

Preliminary considerations elements of differential calculus for tensor fields the theory of Hamiltonian and bi-Hamiltonian systems lax representations of multi-Hamiltonian systems multi-Hamiltonian…