New approaches to integrable hierarchies of topological type (original) (raw)
Related papers
Integrable Differential Systems of Topological Type and Reconstruction by the Topological Recursion
Annales Henri Poincaré, 2017
Starting from a d × d rational Lax pair system of the form ∂ x Ψ = LΨ and ∂ t Ψ = RΨ we prove that, under certain assumptions (genus 0 spectral curve and additional conditions on R and L), the system satisfies the "topological type property". A consequence is that the formal-WKB expansion of its determinantal correlators, satisfy the topological recursion. This applies in particular to all (p, q) minimal models reductions of the KP hierarchy, or to the six Painlevé systems.
Infinite-dimensional flag manifolds in integrable systems
Acta Applicandae Mathematicae, 1995
In this paper we present several instances where infinite dimensional flag varieties and their holomorphic line bundles play a role in integrable systems. As such, we give the correspondance between flag varieties and Darboux transformations for the K P-hierarchy and the n-th KdV-hierarchy. We construct solutions of the n-th M KdVhierarchy from the space of periodic flags and we treat the geometric interpretation of the Miura transform. Finally we show how the group extension connected with these line bundles shows up at integrable deformations of linear systems on ސ 1 .
Hierarchy of Higher Dimensional Integrable System
Eprint Arxiv Solv Int 9802005, 1998
Integrable equations in (1 + 1) dimensions have their own higher order integrable equations, like the KdV, mKdV and NLS hierarchies etc. In this paper we consider whether integrable equations in (2 + 1) dimensions have also the analogous hierarchies to those in (1 + 1) dimensions. Explicitly is discussed the Bogoyavlenskii-Schiff(BS) equation. For the BS hierarchy, there appears an ambiguity in the Painlevé test. Nevertheless, it may be concluded that the BS hierarchy is integrable.
Theoretical and Mathematical Physics, 2010
We develop a group theory approach for constructing solutions of integrable hierarchies corresponding to the deformation of a collection of commuting directions inside the Lie algebra of upper-triangular Z×Z matrices. Depending on the choice of the set of commuting directions, the homogeneous space from which these solutions are constructed is the relative frame bundle of an infinite-dimensional flag variety or the infinite-dimensional flag variety itself. We give the evolution equations for the perturbations of the basic directions in the Lax form, and they reduce to a tower of differential and difference equations for the coefficients of these perturbed matrices. The Lax equations follow from the linearization of the hierarchy and require introducing a proper analogue of the Baker-Akhiezer function.
Projective differential geometry of multidimensional dispersionless integrable hierarchies
Journal of Physics: Conference Series, 2014
We introduce a general setting for multidimensional dispersionless integrable hierarchy in terms of differential m-form Ωm with the coefficients satisfying the Plücker relations, which is gauge-invariantly closed and its gauge-invariant coordinates (ratios of coefficients) are (locally) holomorphic with respect to one of the variables (the spectral variable). We demonstrate that this form defines a hierarchy of dispersionless integrable equations in terms of commuting vector fields locally holomorphic in the spectral variable. The equations of the hierarchy are given by the gauge-invariant closedness equations.
Differential-algebraic and bi-Hamiltonian integrability analysis of the Riemann hierarchy revisited
Journal of Mathematical Physics, 2012
A differential-algebraic approach to studying the Lax type integrability of the generalized Riemann type hydrodynamic hierarchy is revisited, its new Lax type representation is constructed in exact form. The related bi-Hamiltonian integrability and compatible Poissonian structures of the generalized Riemann type hierarchy are also discussed by means of the gradient-holonomic and geometric methods.
Clebsch parameterization: Basic properties and remarks on its applications J. Math. Phys. 50, 113101 (2009) Studies of perturbed three vortex dynamics J. Math. Phys. 48, 065402 (2007) Point vortex motion on the surface of a sphere with impenetrable boundaries Phys. Fluids 18, 036602 (2006) Schouten tensor and bi-Hamiltonian systems of hydrodynamic type J. Math. Phys. 47, 023504 (2006) Additional information on J. Math. Phys. Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about\_the\_journal Top downloads: http://jmp.aip.org/features/most\_downloaded Information for Authors: http://jmp.aip.org/authors A differential-algebraic approach to studying the Lax integrability of the generalized Riemann type hydrodynamic hierarchy is revisited and its new Lax representation is constructed in exact form. The bi-Hamiltonian integrability of the generalized Riemann type hierarchy is discussed by means of the gradient-holonomic and symplectic methods and the related compatible Poissonian structures for N = 3 and N = 4 are constructed. C