<i>PT</i>
            -symmetric deformations of integrable models

Fring, Andreas

doi:10.1098/rsta.2012.0046

Cited by 46 publications

(50 citation statements)

References 51 publications

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“…Finally, it will be fascinating to introduce a PT deformation into the angular system [39,40], because it will remove the codimension-one singular loci of the potential, thereby connect the n! disjoint particle sectors and also give meaning to g<0 states, which are needed for the action of the odd Q charges in the spectrum.…”

Section: Jhep10(2015)191mentioning

confidence: 99%

The tetrahexahedric angular Calogero model

Correa

Lechtenfeld

2015

J. High Energ. Phys.

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Abstract:The spherical reduction of the rational Calogero model (of type A n−1 and after removing the center of mass) is considered as a maximally superintegrable quantum system, which describes a particle on the (n−2)-sphere subject to a very particular potential. We present a detailed analysis of the simplest non-separable case, n=4, whose potential is singular at the edges of a spherical tetrahexahedron. A complete set of independent conserved charges and of Hamiltonian intertwiners is constructed, and their algebra is elucidated. They arise from the ring of polynomials in Dunkl-deformed angular momenta, by classifying the subspaces invariant and antiinvariant under all Weyl reflections, respectively.

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Section: Jhep10(2015)191mentioning

confidence: 99%

The tetrahexahedric angular Calogero model

Correa

Lechtenfeld

2015

J. High Energ. Phys.

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“…(1.1) for ε = 0, shares the same PT -symmetry with the NLSE, as it is invariant with respect to PT :Hence HNLSEs may also be viewed as PT -symmetric extensions of the NLSE. Similarly as for many other PT -symmetric nonlinear integrable systems [13], various other PT -symmetric generalizations have been proposed and investigated by adding terms to the original equation, e.g. [14,15,16].…”

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confidence: 99%

Integrable nonlocal Hirota equations

Cen

Correa

Fring

2019

Journal of Mathematical Physics

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We construct several new integrable systems corresponding to nonlocal versions of the Hirota equation, which is a particular example of higher order nonlinear Schrödinger equations. The integrability of the new models is established by providing their explicit forms of Lax pairs or zero curvature conditions. The two compatibility equations arising in this construction are found to be related to each other either by a parity transformation P, by a time reversal T or a PT -transformation possibly combined with a conjugation. We construct explicit multi-soliton solutions for these models by employing Hirota's direct method as well as Darboux-Crum transformations. The nonlocal nature of these models allows for a modification of these solution procedures as the new systems also possess new types of solutions with different parameter dependence and different qualitative behaviour. The multi-soliton solutions are of varied type, being for instance nonlocal in space, nonlocal in time of time crystal type, regular with local structures either in time/space or of rogues wave type. Integrable nonlocal Hirota equationswith constants ε, α, β, γ ∈ R. Besides the NLSE for ε = 0, four cases are known to be integrable. When the ratio of the constants are taken to be α : β : γ = 0 : 1 : 1 or α : β : γ = 0 : 1 : 0 one obtains the derivative NLSE of type I [8] and II [9], respectively, which are in fact related to each other by a dependent variable transformation [10]. For α : β : γ = 1 : 6 : 3 one obtains the Sasa-Satsuma equation [11] and for α : β : γ = 1 : 6 : 0 the Hirota equation [12]. Variations of the latter are the subject of this manuscript.We notice that the additional term in the HNLSE when compared to the NLSE, i.e. (1.1) for ε = 0, shares the same PT -symmetry with the NLSE, as it is invariant with respect to PT :Hence HNLSEs may also be viewed as PT -symmetric extensions of the NLSE. Similarly as for many other PT -symmetric nonlinear integrable systems [13], various other PT -symmetric generalizations have been proposed and investigated by adding terms to the original equation, e.g. [14,15,16]. A further option, that will be important here, was explored by Ablowitz and Musslimani [17,18] who identified a new class of nonlinear integrable systems closely related to the NLSE by exploiting a hitherto unexplored PTsymmetry present in the zero curvature condition. Exploring this option below for the Hirota equation will lead us to new integrable systems with nonlocal properties.Our manuscript is organized as follows: In section 2 we discuss the zero curvature condition or AKNS-equation for the new class of integrable systems. The solutions to these systems involve fields at different points in space or time and reduce in certain limits to the standard Hirota equation, so that we refer to them as nonlocal Hirota equations. The equations possess two types of solutions of qualitatively different behaviour and parameter dependence. We identify the origin for this novel feature within the context of Hirota's direct method as w...

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“…The obtained in such a way superextended system can be considered on the whole real line x ∈ R, and boundary condition at x = 0 can be omitted. The systemĤ + (ξ) is PT -symmetric [85,86,87,88,89,90,91]: [P T,Ĥ + (ξ)] = 0, where P is a space reflection operator, P x = −P x, and T is the operator defined by T (x + iα) = (x − iα)T . SubsystemĤ + (ξ) has one bound eigenstate of zero eigenvalue described by quadratically integrable on the whole real line function ψ + 0 = ξ −n , which lies at the very edge of the continuos spectrum with E > 0.…”

Section: Perfectly Invisible Pt -Symmetric Zero-gap Systemsmentioning

confidence: 99%

Nonlinear Supersymmetry as a Hidden Symmetry

Plyushchay

2019

Integrability, Supersymmetry and Coherent States

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Nonlinear supersymmetry is characterized by supercharges to be higher order in bosonic momenta of a system, and thus has a nature of a hidden symmetry. We review some aspects of nonlinear supersymmetry and related to it exotic supersymmetry and nonlinear superconformal symmetry. Examples of reflectionless, finite-gap and perfectly invisible PT -symmetric zero-gap systems, as well as rational deformations of the quantum harmonic oscillator and conformal mechanics, are considered, in which such symmetries are realized.

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PT -symmetric deformations of integrable models

Cited by 46 publications

References 51 publications

The tetrahexahedric angular Calogero model

The tetrahexahedric angular Calogero model

Integrable nonlocal Hirota equations

Nonlinear Supersymmetry as a Hidden Symmetry

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