Rational and elliptic solutions of the korteweg‐de vries equation and a related many‐body problem

Abstract. We consider 2D and 3D complex Schrödinger equations with Abelian potentials and a fixed energy level. The potential, wave function, and the spectral Bloch variety are calculated in terms of the Kleinian hyperelliptic functions associated with a genus two hyperelliptic curve. In the special case in 2D when the curve covers two elliptic curves, exactly solvable Schrödinger equations are constructed in terms of the elliptic functions of these curves. The solutions obtained are illustrated by a number of plots.

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“…Therefore we have proved the following proposition (a proof in compressed form was given in [9], see also [2], pg. 144) Proposition 5.1.…”

Section: The Two Dimensional Schrödinger Equation With An Elliptic Pomentioning

confidence: 88%

“…The solution of the form (5.5) was introduced for the first time by Dubrovin and Novikov [16]; the general case was investigated by Airault et al [2]. We use this elliptic KdV solution below to construct a special solution of the 2D problem.…”

Section: The Two Dimensional Schrödinger Equation With An Elliptic Pomentioning

confidence: 99%

Multidimensional Schrödinger equations with Abelian potentials

Buchstaber

Eilbeck

Enolskii

et al. 2002

Journal of Mathematical Physics

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“…This could give a geometrical explicit explanation of the misterious connection between Calogero-Sutherland systems and KdV hierarchy [21].…”

Section: Theorem 42 the Algebra Of The Hierarchy Flows And Of The VImentioning

confidence: 98%

Hidden local, quasi-local and non-local symmetries in integrable systems

Fioravanti¹,

Stanishkov²

2000

Nuclear Physics B

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The knowledge of non usual and sometimes hidden symmetries of (classical) integrable systems provides a very powerful setting-out of solutions of these models. Primarily, the understanding and possibly the quantisation of intriguing symmetries could give rise to deeper insight into the nature of field spectrum and correlation functions in quantum integrable models. With this perspective in mind we will propose a general framework for discovery and investigation of local, quasi-local and non-local symmetries in classical integrable systems. We will pay particular attention to the structure of symmetry algebra and to the rôle of conserved quantities. We will also stress a nice unifying point of view about KdV hierarchies and Toda field theories with the result of obtaining a Virasoro algebra as exact symmetry of Sine-Gordon Model. *

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“…Such solutions are well known and have been studied in previous papers (cf. [1,6,8,16,20]). However, one should especially compare the present approach with that in [17] where these "vanishing rational KP solutions" are produced from matrix pairs (X, Z) satisfying rank(XZ − ZX + I) = 1.…”

Section: 23mentioning

confidence: 99%

Solitons and almost-intertwining matrices

Kasman

Gekhtman

2001

Journal of Mathematical Physics

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Abstract. We define the algebraic variety of almost intertwining matrices to be the set of triples (X, Y, Z) of n × n matrices for which XZ = Y X + T for a rank one matrix T . A surprisingly simple formula is given for tau-functions of the KP hierarchy in terms of such triples. The tau-functions produced in this way include the soliton and vanishing rational solutions. The induced dynamics of the eigenvalues of the matrix X are considered, leading in special cases to the Ruijsenaars-Schneider particle system.

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Rational and elliptic solutions of the korteweg‐de vries equation and a related many‐body problem

Cited by 522 publications

References 12 publications

Multidimensional Schrödinger equations with Abelian potentials

Multidimensional Schrödinger equations with Abelian potentials

Hidden local, quasi-local and non-local symmetries in integrable systems

Solitons and almost-intertwining matrices

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