Darboux integration of

Ustinov, N. V.; Czachor, Marek; Kuna, Maciej; Leble, Sergey

doi:10.1016/s0375-9601(01)00013-5

Cited by 19 publications

(35 citation statements)

References 22 publications

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“…All these equations were of the form iρ = [H, f (ρ)] where [f (ρ), ρ] = 0. Their physically nontrivial solutions were found in [73] for f (ρ) = ρ 2 and recently generalized to other f 's in [74]. A link of such general f 's with nonextensive statistical mechanics was described in [75].…”

mentioning

confidence: 89%

Probing the Structure of Quantum Mechanics

Aerts¹,

Czachor²,

Durt³

2002

Probing the Structure of Quantum Mechanics

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We believe that in the decades to come quantum theory will play an increasingly important role for many different fields. One of the reasons is that technology aims at manipulating and controlling information and energy in ever smaller regions of space and windows of time. As a consequence the behavior of the entities to manipulate and control will become more and more quantum. This means not only spectacular advances of new techniques and outlooks on revolutionary applications, but also a constant stress and attention on the theory of quantum mechanics itself.It is well known that quantum mechanics has been scrutinized in all kind of ways during the past decades, but that still a lot of conceptual problems remain. The problems of standard quantum mechanics are however not only of a conceptual nature. Also the formal mathematical structure of quantum mechanics has been investigated with the aim to make the theory more operational and to found the basic concepts in direct correspondence with what happens in the laboratory. Such an operationally founded quantum mechanics may soon become of great value because of the technological * Published as: D. Aerts, M. Czachor and T. Durt, "Probing the structure of quantum mechanics", in Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality,

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mentioning

confidence: 89%

Probing the Structure of Quantum Mechanics

Aerts¹,

Czachor²,

Durt³

2002

Probing the Structure of Quantum Mechanics

Self Cite

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“…Another important property of the dynamics is that (6) is integrable in the sense of soliton theory for any f . A soliton technique of solving (6), based on Darboux transformations, was introduced in [14], and further developed in [32,33,34]. Therefore, as opposed to standard kinetic equations that typically have to be solved numerically, we can work with exact analytic solutions.…”

Section: Soliton Kinetic Equationsmentioning

confidence: 99%

Soliton Kinetic Equations with Non-Kolmogorovian Structure: A New Tool for Biological Modeling?

Aerts

Czachor

Gabora

et al. 2006

AIP Conference Proceedings

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Abstract. Non-commutative diagrams, where X → Y → Z is allowed and X → Z → Y is not, may equally well apply to Malusian experiments with photons traversing polarizers, and to sequences of elementary chemical reactions. This is why non-commutative probabilistic, logical, and dynamical structures necessarily occur in chemical or biological dynamics. We discuss several explicit examples of such systems and propose an exactly solvable nonlinear toy model of a "brain-heart" system. The model involves non-Kolmogorovian probability calculus and soliton kinetic equations integrable by Darboux transformations.

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“…There exists a class of solutions of (4) which exhibits a kind of a three-regime switching effect [23][24][25]: For times −∞ < t ≪ t 1 the dynamics looks as if there was not feedback, then in the switching regime t 1 < t < t 2 a 'sudden' transition occurs which drives the system into a new state which for times t 2 ≪ t < ∞ evolves again as if there was no feedback. Of course, the feedback is present for all times, but is 'visible' only during the switching period.…”

Section: Soliton Morphogenesismentioning

confidence: 99%

Quantum morphogenesis: A variation on Thom’s catastrophe theory

Aerts¹,

Czachor

Gabora³

et al. 2003

Phys. Rev. E

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Non-commutative propositions are characteristic of both quantum and non-quantum (sociological, biological, psychological) situations. In a Hilbert space model states, understood as correlations between all the possible propositions, are represented by density matrices. If systems in question interact via feedback with environment their dynamics is nonlinear. Nonlinear evolutions of density matrices lead to phenomena of morphogenesis which may occur in non-commutative systems. Several explicit exactly solvable models are presented, including 'birth and death of an organism' and 'development of complementary properties'.

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Darboux integration of

Cited by 19 publications

References 22 publications

Probing the Structure of Quantum Mechanics

Probing the Structure of Quantum Mechanics

Soliton Kinetic Equations with Non-Kolmogorovian Structure: A New Tool for Biological Modeling?

Quantum morphogenesis: A variation on Thom’s catastrophe theory

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