Solitons and almost-intertwining matrices

Kasman, Alex; Gekhtman, Michael

doi:10.1063/1.1379313

Cited by 26 publications

(25 citation statements)

References 20 publications

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“…It is known that this formula gives a tau-function of the KP hierarchy precisely when the matrix XZ − Y X has rank one [18], but as this happens to be the lower-left block of the matrix S we can now also see this as a consequence of Theorem 3.5.…”

Section: 2mentioning

confidence: 86%

“…For instance, their role in finite dimensional integrable systems can be seen in [5,12,17,28,29,37] and their role in infinite dimensional integrable systems appears in papers such as [1,6,10,18,30,35].…”

Section: Discussionmentioning

confidence: 99%

“…It is not a coincidence that both rational solutions and soliton solutions have been frequently described in terms of "rank one conditions" on finite matrices in the literature of integrable systems [5,10,17,18,28,29,37]. These rank one 5 Consider the decomposition of H into…”

Section: 2mentioning

confidence: 99%

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On KP generators and the geometry of the HBDE

Gekhtman

Kasman

2006

Journal of Geometry and Physics

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Abstract. Sato theory provides a correspondence between solutions to the KP hierarchy and points in an infinite dimensional Grassmannian. In this correspondence, flows generated infinitesimally by powers of the "shift" operator give time dependence to the first coordinate of an arbitrarily selected point, making it a tau-function. These tau-functions satisfy a number of integrable equations, including the Hirota Bilinear Difference Equation (HBDE). Here, we rederive the HBDE as a statement about linear maps between Grassmannians. In addition to illustrating the fundamental nature of this equation in the standard theory, we make use of this geometric interpretation of the HBDE to answer the question of what other infinitesimal generators could be used for similarly creating tau-functions. The answer to this question involves a "rank one condition", tying this investigation to the existing results on integrable systems involving such conditions and providing an interpretation for their significance in terms of the relationship between the HBDE and the geometry of Grassmannians.

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Section: 2mentioning

confidence: 86%

Section: Discussionmentioning

confidence: 99%

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On KP generators and the geometry of the HBDE

Gekhtman

Kasman

2006

Journal of Geometry and Physics

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“…Further developments of the Marchenko scheme are given in [11,12,78]. For application of the matrix identities to the construction of solitons see also [39].…”

Section: Introductionmentioning

confidence: 99%

On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory

Sakhnovich

2010

Math. Model. Nat. Phenom.

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Abstract. A general theorem on the GBDT version of the Bäcklund-Darboux transformation for systems depending rationally on the spectral parameter is treated and its applications to nonlinear equations are given. Explicit solutions of direct and inverse problems for Dirac-type systems, including systems with singularities, and for the system auxiliary to the N -wave equation are reviewed. New results on explicit construction of the wave functions for radial Dirac equation are obtained.

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“…Moreover, putting = T − I, after substituting n for z and log(1 + z) for x in (4.3) and (4.4), it follows that 8) with R(n, ) and S(n, ) some (positive) difference operators in (acting on functions depending on n), whose coefficients are rational functions of n. We introduce the following definition.…”

Section: )mentioning

confidence: 99%

Trigonometric Darboux Transformations and Calogero–moser Matrices

Haine¹,

Horozov²,

Илиев³

2009

Glasgow Math. J.

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Abstract. We characterize in terms of Darboux transformations the spaces in the Segal-Wilson rational Grassmannian, which lead to commutative rings of differential operators having coefficients which are rational functions of e x . The resulting subgrassmannian is parametrized in terms of trigonometric Calogero-Moser matrices.2000 Mathematics Subject Classification. 35Q53, 37K101. Introduction. The rational Segal-Wilson Grassmannian Gr rat parametrizes the soliton solutions of the Kadomtsev-Petviashvili (KP) equation. In [10], Wilson embarked upon a study of a subgrassmannian Gr ad ⊂ Gr rat that he called the adelic Grassmannian, which parametrizes the solutions of the KP equation, rational in x and vanishing as x → ∞. The adelic Grassmannian has (and is indeed characterized by) a remarkable bispectral involution V → b(V ), V ∈ Gr ad , which exchanges the role of the 'space' and the 'spectral' variables in the corresponding stationary wave functions ψ V (x, z), that is

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Solitons and almost-intertwining matrices

Cited by 26 publications

References 20 publications

On KP generators and the geometry of the HBDE

On KP generators and the geometry of the HBDE

On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory

Trigonometric Darboux Transformations and Calogero–moser Matrices

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