G. Helminck | University of Amsterdam (original) (raw)

Papers by G. Helminck

Acta Applicandae Mathematicae

Publications of the Research Institute for Mathematical Sciences, 2001

In this paper it is shown that, if you have two planes in the Sato Grassmannian that have an inte... more In this paper it is shown that, if you have two planes in the Sato Grassmannian that have an intersection of finite codimension, then the corresponding solutions of the KP hierarchy are linked by Bäcklund-Darboux (shortly BD-)transformations. The pseudodifferential operator that performs this transformation is shown to be built up in a geometric way from so-called elementary BD-transformations and is given here in a closed form. The corresponding action on the tau-function, associated to a plane in the Grassmannian, is also determined

Transactions of the American Mathematical Society, 1991

In this paper we describe how to construct convergent solutions of the multicomponent KP-hierarch... more In this paper we describe how to construct convergent solutions of the multicomponent KP-hierarchy, starting from a certain open subset of the Grassmann manifold of a special kind of Banach space, and derive an expression of its solutions in terms of Fredholm determinants. Further we show that the simplest nonscalar reduction of the present hierarchy leads to the AKNShierarchy.

In this paper one considers commutative subalgebras of the Z × Z-matrices generated by a maximal ... more In this paper one considers commutative subalgebras of the Z × Z-matrices generated by a maximal commutative subalgebra of the complex k ×k-matrices and a shift matrix that commutes with them. Key objects are parameter dependent perturbations of these algebras inside the upper triangular Z × Z-matrices such that the perturbed generators satisfy Lax equations with respect to an infinite number of commuting directions. Various appropriate geometric settings are described in which one can actually construct solutions of these hierarchies.

Theoretical and Mathematical Physics

We present an integrable hierarchy that includes both the AKNS hierarchy and its strict version. ... more We present an integrable hierarchy that includes both the AKNS hierarchy and its strict version. We split the loop space g of gl2 into Lie subalgebras g ≥0 and g<0 of all loops with respectively only positive and only strictly negative powers of the loop parameter. We choose a commutative Lie subalgebra C in the whole loop space s of sl2 and represent it as C = C ≥0 ⊕ C<0. We deform the Lie subalgebras C ≥0 and C<0 by the respective groups corresponding to g<0 and g ≥0. Further, we require that the evolution equations of the deformed generators of C ≥0 and C<0 have a Lax form determined by the original splitting. We prove that this system of Lax equations is compatible and that the equations are equivalent to a set of zero-curvature relations for the projections of certain products of generators. We also define suitable loop modules and a set of equations in these modules, called the linearization of the system, from which the Lax equations of the hierarchy can be obtained. We give a useful characterization of special elements occurring in the linearization, the so-called wave matrices. We propose a way to construct a rather wide class of solutions of the combined AKNS hierarchy.

Theoretical and Mathematical Physics

In the algebra P sΔ of pseudodifference operators, we consider two deformations of the Lie subalg... more In the algebra P sΔ of pseudodifference operators, we consider two deformations of the Lie subalgebra spanned by positive powers of an invertible constant first-degree pseudodifference operator Λ0. The first deformation is by the group in P sΔ corresponding to the Lie subalgebra P sΔ<0 of elements of negative degree, and the second is by the group corresponding to the Lie subalgebra P sΔ ≤0 of elements of degree zero or lower. We require that the evolution equations of both deformations be certain compatible Lax equations that are determined by choosing a Lie subalgebra depending on Λ0 that respectively complements the Lie subalgebra P sΔ<0 or P sΔ ≤0. This yields two integrable hierarchies associated with Λ0, where the hierarchy of the wider deformation is called the strict version of the first because of the form of the Lax equations. For Λ0 equal to the matrix of the shift operator, the hierarchy corresponding to the simplest deformation is called the discrete KP hierarchy. We show that the two hierarchies have an equivalent zero-curvature form and conclude by discussing the solvability of the related Cauchy problems.

Lecture Notes in Physics

Dedicated to the memory of J.W. de Roever, a true scientist and enthusiastic participant, who fle... more Dedicated to the memory of J.W. de Roever, a true scientist and enthusiastic participant, who flew off too soon.

Теоретическая и математическая физика, 2013

Теоретическая и математическая физика, 2010

Пусть E 0-голоморфное векторное расслоение над P 1 (C), а ∇ 0-мероморфная связность в E 0. Введен... more Пусть E 0-голоморфное векторное расслоение над P 1 (C), а ∇ 0-мероморфная связность в E 0. Введено понятие интегрируемой связности, описывающей движение полюсов связности ∇ 0 в комплексной плоскости при сохранении интегрируемости. Показано, что при достаточно слабых условиях на деформационное пространство такая деформация существует. Также показано, что если векторное расслоение E 0 тривиально, то решения соответствующих нелинейных уравнений мероморфно продолжаются на деформационное пространство. Ключевые слова: интегрируемая связность, деформационное пространство, интегрируемая деформация, логарифмический полюс.

Journal of Nonlinear Mathematical Physics, 2005

In this paper one considers the problem of finding solutions to a number of Todatype hierarchies.... more In this paper one considers the problem of finding solutions to a number of Todatype hierarchies. All of them are associated with a commutative subalgebra of the k ×k-matrices. The first one is formulated in terms of upper triangular Z×Z-matrices, the second one in terms of lower triangular ones and the third is a combination of the two foregoing types. It is shown that in an appropriate group setting solutions of the linearization of these Lax equations can be constructed by using a Birkhoff-type decomposition in the relevant group.

Journal of Physics A-mathematical and General, 2002

In this paper, we introduce the infinite-dimensional flag varieties associated with integrable sy... more In this paper, we introduce the infinite-dimensional flag varieties associated with integrable systems of the KdV- and Toda-type and discuss the structure of these manifolds. As an example we treat the Fubini-Study metric on the projective space associated with a separable complex Hilbert space and conclude by showing that all flag varieties introduced before possess a Kähler structure.

Acta Applicandae Mathematicae, 2006

Acta Applicandae Mathematicae, 1995

In this paper we present several instances where infinite dimensional flag varieties and their ho... more 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 ‫.

Theoretical and Mathematical Physics

Complex Variables and Elliptic Equations

Theoretical and Mathematical Physics, 2015

Quantized Algebra and Physics - Proceeding of the International Workshop, 2011

Lecture Notes in Physics, 1993

Acta Applicandae Mathematicae

Publications of the Research Institute for Mathematical Sciences, 2001

In this paper it is shown that, if you have two planes in the Sato Grassmannian that have an inte... more In this paper it is shown that, if you have two planes in the Sato Grassmannian that have an intersection of finite codimension, then the corresponding solutions of the KP hierarchy are linked by Bäcklund-Darboux (shortly BD-)transformations. The pseudodifferential operator that performs this transformation is shown to be built up in a geometric way from so-called elementary BD-transformations and is given here in a closed form. The corresponding action on the tau-function, associated to a plane in the Grassmannian, is also determined

Transactions of the American Mathematical Society, 1991

In this paper we describe how to construct convergent solutions of the multicomponent KP-hierarch... more In this paper we describe how to construct convergent solutions of the multicomponent KP-hierarchy, starting from a certain open subset of the Grassmann manifold of a special kind of Banach space, and derive an expression of its solutions in terms of Fredholm determinants. Further we show that the simplest nonscalar reduction of the present hierarchy leads to the AKNShierarchy.

In this paper one considers commutative subalgebras of the Z × Z-matrices generated by a maximal ... more In this paper one considers commutative subalgebras of the Z × Z-matrices generated by a maximal commutative subalgebra of the complex k ×k-matrices and a shift matrix that commutes with them. Key objects are parameter dependent perturbations of these algebras inside the upper triangular Z × Z-matrices such that the perturbed generators satisfy Lax equations with respect to an infinite number of commuting directions. Various appropriate geometric settings are described in which one can actually construct solutions of these hierarchies.

Theoretical and Mathematical Physics

We present an integrable hierarchy that includes both the AKNS hierarchy and its strict version. ... more We present an integrable hierarchy that includes both the AKNS hierarchy and its strict version. We split the loop space g of gl2 into Lie subalgebras g ≥0 and g<0 of all loops with respectively only positive and only strictly negative powers of the loop parameter. We choose a commutative Lie subalgebra C in the whole loop space s of sl2 and represent it as C = C ≥0 ⊕ C<0. We deform the Lie subalgebras C ≥0 and C<0 by the respective groups corresponding to g<0 and g ≥0. Further, we require that the evolution equations of the deformed generators of C ≥0 and C<0 have a Lax form determined by the original splitting. We prove that this system of Lax equations is compatible and that the equations are equivalent to a set of zero-curvature relations for the projections of certain products of generators. We also define suitable loop modules and a set of equations in these modules, called the linearization of the system, from which the Lax equations of the hierarchy can be obtained. We give a useful characterization of special elements occurring in the linearization, the so-called wave matrices. We propose a way to construct a rather wide class of solutions of the combined AKNS hierarchy.

Theoretical and Mathematical Physics

In the algebra P sΔ of pseudodifference operators, we consider two deformations of the Lie subalg... more In the algebra P sΔ of pseudodifference operators, we consider two deformations of the Lie subalgebra spanned by positive powers of an invertible constant first-degree pseudodifference operator Λ0. The first deformation is by the group in P sΔ corresponding to the Lie subalgebra P sΔ<0 of elements of negative degree, and the second is by the group corresponding to the Lie subalgebra P sΔ ≤0 of elements of degree zero or lower. We require that the evolution equations of both deformations be certain compatible Lax equations that are determined by choosing a Lie subalgebra depending on Λ0 that respectively complements the Lie subalgebra P sΔ<0 or P sΔ ≤0. This yields two integrable hierarchies associated with Λ0, where the hierarchy of the wider deformation is called the strict version of the first because of the form of the Lax equations. For Λ0 equal to the matrix of the shift operator, the hierarchy corresponding to the simplest deformation is called the discrete KP hierarchy. We show that the two hierarchies have an equivalent zero-curvature form and conclude by discussing the solvability of the related Cauchy problems.

Lecture Notes in Physics

Dedicated to the memory of J.W. de Roever, a true scientist and enthusiastic participant, who fle... more Dedicated to the memory of J.W. de Roever, a true scientist and enthusiastic participant, who flew off too soon.

Теоретическая и математическая физика, 2013

Теоретическая и математическая физика, 2010

Пусть E 0-голоморфное векторное расслоение над P 1 (C), а ∇ 0-мероморфная связность в E 0. Введен... more Пусть E 0-голоморфное векторное расслоение над P 1 (C), а ∇ 0-мероморфная связность в E 0. Введено понятие интегрируемой связности, описывающей движение полюсов связности ∇ 0 в комплексной плоскости при сохранении интегрируемости. Показано, что при достаточно слабых условиях на деформационное пространство такая деформация существует. Также показано, что если векторное расслоение E 0 тривиально, то решения соответствующих нелинейных уравнений мероморфно продолжаются на деформационное пространство. Ключевые слова: интегрируемая связность, деформационное пространство, интегрируемая деформация, логарифмический полюс.

Journal of Nonlinear Mathematical Physics, 2005

In this paper one considers the problem of finding solutions to a number of Todatype hierarchies.... more In this paper one considers the problem of finding solutions to a number of Todatype hierarchies. All of them are associated with a commutative subalgebra of the k ×k-matrices. The first one is formulated in terms of upper triangular Z×Z-matrices, the second one in terms of lower triangular ones and the third is a combination of the two foregoing types. It is shown that in an appropriate group setting solutions of the linearization of these Lax equations can be constructed by using a Birkhoff-type decomposition in the relevant group.

Journal of Physics A-mathematical and General, 2002

In this paper, we introduce the infinite-dimensional flag varieties associated with integrable sy... more In this paper, we introduce the infinite-dimensional flag varieties associated with integrable systems of the KdV- and Toda-type and discuss the structure of these manifolds. As an example we treat the Fubini-Study metric on the projective space associated with a separable complex Hilbert space and conclude by showing that all flag varieties introduced before possess a Kähler structure.

Acta Applicandae Mathematicae, 2006

Acta Applicandae Mathematicae, 1995

In this paper we present several instances where infinite dimensional flag varieties and their ho... more 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 ‫.

Theoretical and Mathematical Physics

Complex Variables and Elliptic Equations

Theoretical and Mathematical Physics, 2015

Quantized Algebra and Physics - Proceeding of the International Workshop, 2011

Lecture Notes in Physics, 1993