On the Ψ-function for finite-gap potentials

Ustinov, N. V.; Brezhnev, Yu. V.

doi:10.1070/rm2002v057n01abeh000488

Cited by 8 publications

(17 citation statements)

References 3 publications

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“…As we shall see, there is no essential difference between (quadrature) integrability schemes of this example and those of pure FG-potentials. This example was already pointed out as non-standard in [54].…”

Section: Inversion Of Non-holomorphic Integralsmentioning

confidence: 70%

“…This simple recipe of getting the Ψ-function, perhaps, has no received mention in the modern literature [54]; it is, however, implicitly exploited in monograph [28]. Elimination of the last derivative Ψ x leads to equation of the curve W (µ, λ) = 0.…”

Section: 2mentioning

confidence: 99%

“…which result from properties (54). Using condition µ = 0, one easily derives that admissible transformations are…”

Section: Non-finite-gap Integrable Counterexamplesmentioning

confidence: 99%

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Spectral/quadrature duality: Picard–Vessiot theory and finite-gap potentials

Brezhnev¹

2012

Algebraic Aspects of Darboux Transformations, Quantum Integrable Systems and Supersymmetric Quantum Mechanics

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In the framework of differential Galois theory we treat the classical spectral problem Ψ ′′ − u(x)Ψ = λΨ and its finite-gap potentials as exactly solvable in quadratures by Picard-Vessiot without involving special functions; the ideology goes back to the 1919 works by J. Drach. We show that duality between spectral and quadrature approaches is realized through the Weierstrass permutation theorem for a logarithmic Abelian integral. From this standpoint we inspect known facts and obtain new ones: an important formula for the Ψ-function and Θ-function extensions of Picard-Vessiot fields. In particular, extensions by Jacobi's θ-functions lead to the (quadrature) algebraically integrable equations for the θ-functions themselves.

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Section: Inversion Of Non-holomorphic Integralsmentioning

confidence: 70%

Section: 2mentioning

confidence: 99%

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Spectral/quadrature duality: Picard–Vessiot theory and finite-gap potentials

Brezhnev¹

2012

Algebraic Aspects of Darboux Transformations, Quantum Integrable Systems and Supersymmetric Quantum Mechanics

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“…Non-trivial examples of the J were obtained in Ustinov & Brezhnev (2002) and modifications of Dubrovin's equations and trace formulae in Brezhnev (2002).…”

Section: The Q-representationmentioning

confidence: 99%

What does integrability of finite-gap or soliton potentials mean?

Brezhnev

2007

Phil. Trans. R. Soc. A.

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In the example of the Schrödinger/KdV equation, we treat the theory as equivalence of two concepts of Liouvillian integrability: quadrature integrability of linear differential equations with a parameter (spectral problem) and Liouville's integrability of finitedimensional Hamiltonian systems (stationary KdV equations). Three key objects in this field-new explicit J-function, trace formula and the Jacobi problem-provide a complete solution. The Q-function language is derivable from these objects and used for ultimate representation of a solution to the inversion problem. Relations with nonintegrable equations are also discussed.

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“…with the restriction: 4 ( g 3 − g 3 ) c 3 = 1. The details of calculations (43) are expounded in [5] and the formula (46), Ψ-function for the potentials (31,46) in [5,4,39].…”

mentioning

confidence: 99%

Elliptic solitons and Gröbner bases

Brezhnev

2004

Journal of Mathematical Physics

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We consider the solution of spectral problems with elliptic coefficients in the framework of the Hermite ansatz. We show that the search for exactly solvable potentials and their spectral characteristics is reduced to a system of polynomial equations solvable by the Gröbner bases method and others. New integrable potentials and corresponding solutions of the Sawada-Kotera, Kaup-Kupershmidt, Boussinesq equations and others are found.

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On the Ψ-function for finite-gap potentials

Cited by 8 publications

References 3 publications

Spectral/quadrature duality: Picard–Vessiot theory and finite-gap potentials

Spectral/quadrature duality: Picard–Vessiot theory and finite-gap potentials

What does integrability of finite-gap or soliton potentials mean?

Elliptic solitons and Gröbner bases

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