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A. H. Zimerman

Publications and source records attributed to A. H. Zimerman.

At least 19 recordsLinked to original sources
arXiv2025

Generalized Riemann-Hilbert-Birkhoff Decomposition and a New Class of Higher Grading Integrable Hierarchies

We propose a generalized Riemann-Hilbert-Birkhoff decomposition that expands the standard integrable hierarchy formalism in two fundamental ways: it allows for integer powers of Lax matrix components in the flow equations to be increased as compared to conventional models, and it incorporates constant non-zero vacuum (background) solutions. Two additional parameters control these features. The first one defines the grade of a semisimple element that underpins the algebraic construction of the hierarchy, where a grade-one semi-simple element recovers known hierarchies such as mKdV and AKNS. The second parameter distinguishes between zero and non-zero constant background (vacuum) configurations. Additionally, we introduce a third parameter associated with an ambiguity in the definition of the grade-zero component of the dressing matrices. While not affecting the decomposition itself, this parameter classifies different gauge realizations of the integrable equations (like for example, Kaup-Newell, Gerdjikov-Ivanov, Chen-Lee-Liu models). For various values of these parameters, we construct and analyze corresponding integrable models in a unified universal manner demonstrating the broad applicability and generative power of the extended formalism.

nlin.SI↗
arXiv2025

New negative grade solitonic sector for supersymmetric KdV and mKdV hierarchies

A systematic construction for supersymmetric negative graded (non-local) flows for mKdV and KdV based on $sl(2,1)$ with a principal gradation is proposed in this paper. We show that smKdV and sKdV can be mapped onto each other through a gauge super Miura transformation, together with an additional condition for the negative flows, which ensure the supersymmetry of the negative sKdV flow. In addition, we classify both smKdV and sKdV flows with respect to the vacuum (boundary) solutions. These are classified according to zero or non-zero vacuum. Each vacuum solution is used to derive both soliton solutions and the corresponding Heisenberg subalgebra for the smKdV hierarchy. We present the new solutions corresponding to non-zero bosonic and fermionic vacuum by constructing the deformed vertex operators. Finally, the gauge Miura transformation is employed to obtain the sKdV solutions, which exhibit a rich degeneracy due to both multiple gauge super Miura transformations and multiple vacuum possibilities.

hep-th↗
arXiv2024

SKdV, SmKdV flows and their supersymmetric gauge-Miura transformations

The construction of Integrable Hierarchies in terms of zero curvature representation provides a systematic construction for a series of integrable non-linear evolution equations (flows) which shares a common affine Lie algebraic structure. The integrable hierarchies are then classified in terms of a decomposition of the underlying affine Lie algebra $\hat {\cal{G}} $ into graded subspaces defined by a grading operator $Q$. In this paper we shall discuss explicitly the simplest case of the affine $\hat {sl}(2)$ Kac-Moody algebra within the principal gradation given rise to the KdV and mKdV hierarchies and extend to supersymmetric models. It is known that the positive mKdV sub-hierachy is associated to some positive odd graded abelian subalgebra with elements denoted by $E^{(2n+1)}$. Each of these elements in turn, defines a time evolution equation according to time $t=t_{2n+1}$. An interesting observation is that for negative grades, the zero curvature representation allows both, even or odd sub-hierarchies. In both cases, the flows are non-local leading to integro-differential equations. Whilst positive and negative odd sub-hierarchies admit zero vacuum solutions, the negative even admits strictly non-zero vacuum solutions. Soliton solutions can be constructed by gauge transforming the zero curvature from the vacuum into a non trivial configuration (dressing method). Inspired by the dressing transformation method, we have constructed a gauge-Miura transformation mapping mKdV into KdV flows. Interesting new results concerns the negative grade sector of the mKdV hierarchy in which a double degeneracy of flows (odd and its consecutive even) of mKdV are mapped into a single odd KdV flow. These results are extended to supersymmetric hierarchies based upon the affine $\hat {sl}(2,1)$ super-algebra.

nlin.SI↗
arXiv2024

Two-fold degeneracy of a class of rational Painlevé V solutions

We present a construction of a class of rational solutions of the Painlevé V equation that exhibit a two-fold degeneracy, meaning that there exist two distinct solutions that share identical parameters. The fundamental object of our study is the orbit of translation operators of $A^{(1)}_{3}$ affine Weyl group acting on the underlying seed solution that only allows action of some symmetry operations. By linking points on this orbit to rational solutions, we establish conditions for such degeneracy to occur after involving in the construction additional Bäcklund transformations that are inexpressible as translation operators. This approach enables us to derive explicit expressions for these degenerate solutions. An advantage of this formalism is that it easily allows generalization to higher Painlevé systems associated with dressing chains of even period $N>4$.

nlin.SI↗
arXiv2023

Comments on the negative grade KdV hierarchy

The construction of negative grade KdV hierarchy is proposed in terms of a Miura-gauge transformation. Such gauge transformation is employed within the zero curvature representation and maps the Lax operator of the mKdV into its couterpart within the KdV setting. Each odd negative KdV flow is obtained from an odd and its subsequent even negative mKdV flows. The negative KdV flows are shown to inherit the two different vacua structure that characterizes the associated mKdV flows.

nlin.SI↗
arXiv2023

Why is my rational Painlevé V solution not unique?

Under special conditions the Painlevé V equation has more than one rational solution solving it with the same parameters. In the setting of formalism that identifies points on orbits of the fundamental shift operators of $A^{(1)}_{3}$ affine Weyl group with rational solutions we derive conditions for such non-uniqueness to occur. We identify the seed solutions from which the non-unique solutions are generated and put forward a method to systematically obtain their closed expressions from the underlying seed solutions.

nlin.SI↗
arXiv2023

On Rational Solutions of Dressing Chains of Even Periodicity

We develop a systematic approach to deriving rational solutions and obtaining classification of their parameters for dressing chains of even N periodicity or equivalently $A^{(1)}_{N-1}$ invariant Painlevé equations. This construction identifies rational solutions with points on orbits of fundamental shift operators acting on first-order polynomial solutions derived for dressing chains of even periodicity. We also obtain conditions for the existence of special function solutions that occur for a special class of first-order polynomial solutions. For the special case of the N=4 dressing chain equations the method yields all the known rational solutions of Painlevé V equation. They are obtained through action of shift operators on the two independent first-order polynomial solutions. The formalism naturally extends to N=6 and beyond as shown in the paper.

nlin.SI↗
arXiv2022

On Hamiltonian Formalism for Dressing Chain Equations of Even Periodicity

We propose a Hamiltonian formalism for $N$ periodic dressing chain with the even number $N$. The formalism is based on Dirac reduction applied to the $N+1$ periodic dressing chain with the odd number $N+1$ for which the Hamiltonian formalism is well known. The Hamilton dressing chain equations in the $N$ even case depend explicitly on a pair of conjugated Dirac constraints and are equivalent to $A^{(1)}_{N-1}$ invariant symmetric Painlevé equations.

nlin.SI↗
arXiv2022

Twisted Affine Integrable Hierarchies and Soliton Solutions

A systematic construction of a class of integrable hierarchy is discussed in terms of the twisted affine $A_{2r}^{(2)}$ Lie algebra. The zero curvature representation of the time evolution equations are shown to be classified according to its algebraic structure and according to its vacuum solutions. It is shown that a class of models admit both zero and constant (non zero) vacuum solutions. Another, consists essentially of integral non-local equations and can be classified into two sub-classes, one admitting zero vacuum and another of constant, non zero vacuum solutions. The two dimensional gauge potentials in the vacuum plays a crucial ingredient and are shown to be expanded in powers of the vacuum parameter $v_0$. Soliton solutions are constructed from vertex operators, which for the non zero vacuum solutions, correspond to deformations characterized by $v_0$.

nlin.SI↗
arXiv2021

Gauge Symmetry Origin of Bäcklund Transformations for Painlevé Equations

We identify the self-similarity limit of the second flow of $sl(N)$ mKdV hierarchy with the periodic dressing chain thus establishing % a connection to $A^{(1)}_{N-1}$ invariant Painlevé equations. The $A^{(1)}_{N-1}$ Bäcklund symmetries of dressing equations and Painlevé equations are obtained in the self-similarity limit of gauge transformations of the mKdV hierarchy realized as zero-curvature equations on the loop algebra $\widehat{sl}(N)$ endowed with a principal gradation.

nlin.SI↗
arXiv2021

Generalized Backlund transformations for Affine Toda Hierarchies

The construction of generalized Backlund transformation for the $A_n$ Affine Toda hierarchy is proposed in terms of gauge transformation acting on the zero curvature representation. Such construction is based upon the graded structure of the underlying affine algebra which induces a classification of generalized Backlund transformations. Moreover, explicit examples for $su(3)$ and $su(4)$ lead to uncover interesting composition properties of various types of Backlund transformations. The universality character of the gauge-Backlund transformation method is extended to all equations of the hierarchy. Such interesting property provides a systematic framework to construct Backlund transformations to higher flow equations. Explicit example for the simplest higher flow of the $sl(3)$ hierarchy is presented.

nlin.SI↗
arXiv2020

Coalescence, Deformation and Bäcklund Symmetries of Painlevé IV and II Equations

We extend Painlevé IV model by adding quadratic terms to its Hamiltonian obtaining two classes of models (coalescence and deformation) that interpolate between Painlevé IV and II equations for special limits of the underlying parameters. We derive the underlying Bäcklund transformations, symmetry structure and requirements to satisfy Painlevé property.

nlin.SI↗
arXiv2019

Symmetries and hamiltonians of Ince's XXXVIII and XLIX equations

We discuss symmetries of Hamiltonians of I$_{38}$ and I$_{49}$ equations that appear on Ince's list of fifty second-order differential equations with Painlevé property. This study is informed by structure of Weyl symmetries of Painlevé P$_{III}$ and mixed Painlevé P$_{III-V}$ equations and provides insights into differences between the symmetries of Painlevé equations and symmetries of solvable equations on Ince's list.

nlin.SI↗
arXiv2018

Solutions of Mixed Painlevé P$_{\mathbf{III-V}}$ Model

We review the construction of the mixed Painlevé P$_{III-V}$ system in terms of a 4-boson integrable model and discuss its symmetries. Such a mixed system consist of an hybrid differential equation that for special limits of its parameters reduces to either Painlevé P$_{III}$ or P$_{V}$. The aim of this paper is to describe solutions of P$_{III-V}$ model. In particular, we determine and classify rational, power series and transcendental solutions of P$_{III-V}$. A class of power series solutions is shown to be convergent in accordance with the Briot-Bouquet theorem. Moreover, the P$_{III-V}$ equations are reduced to Riccati equations and solved for special values of parameters. The corresponding Riccati solutions can be expressed as Whittaker functions or alternatively confluent hypergeometric and Laguerre functions and are given by ratios of polynomials of order $n$ when the parameter of P$_{III-V}$ equation is quantized by integer $n \in \mathbb{Z}$.

nlin.SI↗
arXiv2018

Recursion Operator and Bäcklund Transformation for Super mKdV Hierarchy

In this paper we consider the ${\cal N}=1$ supersymmetric mKdV hierarchy composed of positive odd flows embedded within an affine $\hat sl(2,1)$ algebra. Its Bäcklund transformations are constructed in terms of a gauge transformation preserving the zero curvature representation. The recursion operator relating consecutive flows is derived and shown to relate their Bäcklund transformations.

nlin.SI↗
arXiv2018

Defects in the supersymmetric mKdV hierarchy via Backlund transformations

The integrability of the ${\cal N}=1$ supersymmetric modified Korteweg de-Vries (smKdV) hierarchy in the presence of defects is investigated through the construction of its super Bäcklund transformation. The construction of such transformation is performed by using essentially two methods: the Bäcklund-defect matrix approach and the superfield approach. Firstly, we employ the defect matrix associated to the hierarchy which turns out to be the same for the supersymmetric sinh-Gordon (sshG) model. The method is general for all flows and as an example we derive explicitly the Bäcklund equations in components for the first few flows of the hierarchy, namely $t_3$ and $t_5$. Secondly, the supersymmetric extension of the Bäcklund transformation in the superspace formalism is constructed for those flows. Finally, this super Bäcklund transformation is employed to introduce type I defects for the supersymmetric mKdV hierarchy. Further integrability aspects by considering modified conserved quantities are derived from the defect matrix.

math-ph↗
arXiv2016

Miura and Generalized Bäcklund Transformation for KdV Hierarchy

Using the fact that Miura transformation can be expressed in the form of gauge transformation connecting the KdV and mKdV equations, we discuss the derivation of the Bäcklund transformation and its Miura-gauge transformation connecting both hierarchies.

nlin.SI↗