| 1. |
- Atai, Farrokh, et al.
(författare)
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Exact solutions by integrals of the non-stationary elliptic Calogero–Sutherland equation
- 2020
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Ingår i: Journal of Integrable Systems. - : Oxford University Press. - 2058-5985. ; 5:1
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Tidskriftsartikel (refereegranskat)abstract
- We use generalized kernel functions to construct explicit solutions by integrals of the non-stationary Schrödinger equation for the Hamiltonian of the elliptic Calogero–Sutherland model (also known as elliptic Knizhnik–Zamolodchikov–Bernard equation). Our solutions provide integral representations of elliptic generalizations of the Jack polynomials.
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| 2. |
- Görbe, Tamas, et al.
(författare)
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Quantization and explicit diagonalization of new compactified trigonometric Ruijsenaars–Schneider systems
- 2018
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Ingår i: Journal of Integrable Systems. - : Oxford University Press (OUP). - 2058-5985. ; 3:1
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Tidskriftsartikel (refereegranskat)abstract
- Recently, Fehér and Kluck discovered, at the level of classical mechanics, new compactified trigonometric Ruijsenaars–Schneider n -particle systems, with phase space symplectomorphic to the (n−1) -dimensional complex projective space. In this article, we quantize the so-called type (i) instances of these systems and explicitly solve the joint eigenvalue problem for the corresponding quantum Hamiltonians by generalising previous results of van Diejen and Vinet. Specifically, the quantum Hamiltonians are realized as discrete difference operators acting in a finite-dimensional Hilbert space of complex-valued functions supported on a uniform lattice over the classical configuration space, and their joint eigenfunctions are constructed in terms of discretized An−1 Macdonald polynomials with unitary parameters.
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| 3. |
- Lundmark, Hans, 1970-, et al.
(författare)
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Dynamics of interlacing peakons (and shockpeakons) in the Geng–Xue equation
- 2017
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Ingår i: Journal of Integrable Systems. - : Oxford University Press. - 2058-5985. ; 2:1
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Tidskriftsartikel (refereegranskat)abstract
- We consider multipeakon solutions, and to some extent also multishockpeakon solutions, of a coupled two- component integrable PDE found by Geng and Xue as a generalization of Novikov’s cubically nonlinear Camassa–Holm type equation. In order to make sense of such solutions, we find it necessary to assume that there are no overlaps, meaning that a peakon or shockpeakon in one component is not allowed to occupy the same position as a peakon or shockpeakon in the other component. Therefore one can distinguish many inequivalent configurations, depending on the order in which the peakons or shockpeakons in the two components appear relative to each other. Here we are particularly interested in the case of interlacing peakon solutions, where the peakons alternatingly occur in one component and in the other. Based on explicit expressions for these solutions in terms of elementary functions, we describe the general features of the dynamics, and in particular the asymptotic large-time behaviour (assuming that there are no antipeakons, so that the solutions are globally defined). As far as the positions are concerned, interlacing Geng–Xue peakons display the usual scattering phenomenon where the peakons asymptotically travel with constant velocities, which are all distinct, except that the two fastest peakons (the fastest one in each component) will have the same velocity. However, in contrast to many other peakon equations, the amplitudes of the peakons will not in general tend to constant values; instead they grow or decay exponentially. Thus the logarithms of the amplitudes (as functions of time) will asymptotically behave like straight lines, and comparing these lines for large positive and negative times, one observes phase shifts similar to those seen for the positions of the peakons (and also for the positions of solitons in many other contexts). In addition to these K+K interlacing pure peakon solutions, we also investigate 1+1 shockpeakon solutions, and collisions leading to shock formation in a 2+2 peakon–antipeakon solution.
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| 4. |
- Shuaib, Budor, 1981-, et al.
(författare)
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Non-interlacing peakon solutions of the Geng–Xue equation
- 2019
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Ingår i: Journal of Integrable Systems. - Oxford : Oxford University Press. - 2058-5985. ; 4:1
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Tidskriftsartikel (refereegranskat)abstract
- The aim of the present article is to derive explicit formulas for arbitrary non-overlapping pure peakon solutions of the Geng–Xue (GX) equation, a two-component generalization of Novikov’s cubically non-linear Camassa–Holm type equation. By performing limiting procedures on the previously known formulas for so-called interlacing peakon solutions, where the peakons in the two component occur alternatingly, we turn some of the peakons into zero-amplitude ‘ghostpeakons’, in such a way that the remaining ordinary peakons occur in any desired configuration. A novel feature compared to the interlacing case is that the Lax pairs for the GX equation do not provide all the constants of motion necessary for the integration of the system. We also study the large-time asymptotics of the non-interlacing solutions. As in the interlacing case, the peakon amplitudes grow or decay exponentially, and their logarithms display phase shifts similar to those for the positions. Moreover, within a group of adjacent peakons in one component, all peakons but one have the same asymptotic velocity. A curious phenomenon occurs when the number of such peakon groups is odd, namely that the sets of incoming and outgoing velocities are unequal.
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