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dc.contributor.author
Riley Casper, W.
dc.contributor.author
Grünbaum, Francisco Alberto
dc.contributor.author
Yakimov, Milen
dc.contributor.author
Zurrián, Ignacio Nahuel
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dc.date.available
2021-02-08T13:02:28Z
dc.date.issued
2019-09-10
dc.identifier.citation
Riley Casper, W.; Grünbaum, Francisco Alberto; Yakimov, Milen; Zurrián, Ignacio Nahuel; Reflective prolate-spheroidal operators and the KP/KdV equations; National Academy of Sciences; Proceedings of the National Academy of Sciences of The United States of America; 116; 37; 10-9-2019; 18310-18315
dc.identifier.issn
0027-8424
dc.identifier.uri
http://hdl.handle.net/11336/125068
dc.description.abstract
Commuting integral and differential operators connect the topics of signal processing, random matrix theory, and integrable systems. Previously, the construction of such pairs was based on direct calculation and concerned concrete special cases, leaving behind important families such as the operators associated to the rational solutions of the Korteweg-de Vries (KdV) equation. We prove a general theorem that the integral operator associated to every wave function in the infinite-dimensional adelic Grassmannian Grad of Wilson always reflects a differential operator (in the sense of Definition 1 below). This intrinsic property is shown to follow from the symmetries of Grassmannians of Kadomtsev-Petviashvili (KP) wave functions, where the direct commutativity property holds for operators associated to wave functions fixed by Wilson's sign involution but is violated in general. Based on this result, we prove a second main theorem that the integral operators in the computation of the singular values of the truncated generalized Laplace transforms associated to all bispectral wave functions of rank 1 reflect a differential operator. A 90◦ rotation argument is used to prove a third main theorem that the integral operators in the computation of the singular values of the truncated generalized Fourier transforms associated to all such KP wave functions commute with a differential operator. These methods produce vast collections of integral operators with prolate-spheroidal properties, including as special cases the integral operators associated to all rational solutions of the KdV and KP hierarchies considered by [Airault, McKean, and Moser, Commun. Pure Appl. Math. 30, 95-148 (1977)] and [Krichever, Funkcional. Anal. i Priložen. 12, 76-78 (1978)], respectively, in the late 1970s. Many examples are presented.
dc.format
application/pdf
dc.language.iso
eng
dc.publisher
National Academy of Sciences
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dc.rights
info:eu-repo/semantics/restrictedAccess
dc.rights.uri
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.subject
PROLATE-SPHEROIDAL INTEGRAL OPERATORS
dc.subject
RATIONAL SOLUTIONS OF THE KDV AND KP EQUATIONS
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REFLECTIVITY
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WILSON'S ADELIC GRASSMANNIAN
dc.subject.classification
Matemática Pura
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dc.subject.classification
Matemáticas
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dc.subject.classification
CIENCIAS NATURALES Y EXACTAS
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dc.title
Reflective prolate-spheroidal operators and the KP/KdV equations
dc.type
info:eu-repo/semantics/article
dc.type
info:ar-repo/semantics/artículo
dc.type
info:eu-repo/semantics/publishedVersion
dc.date.updated
2020-11-19T21:17:31Z
dc.identifier.eissn
1091-6490
dc.journal.volume
116
dc.journal.number
37
dc.journal.pagination
18310-18315
dc.journal.pais
Estados Unidos
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dc.journal.ciudad
Washington DC
dc.description.fil
Fil: Riley Casper, W.. State University of Louisiana; Estados Unidos
dc.description.fil
Fil: Grünbaum, Francisco Alberto. University of California at Berkeley; Estados Unidos
dc.description.fil
Fil: Yakimov, Milen. State University of Louisiana; Estados Unidos
dc.description.fil
Fil: Zurrián, Ignacio Nahuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina
dc.journal.title
Proceedings of the National Academy of Sciences of The United States of America
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dc.relation.alternativeid
info:eu-repo/semantics/altIdentifier/url/http://www.pnas.org/lookup/doi/10.1073/pnas.1906098116
dc.relation.alternativeid
info:eu-repo/semantics/altIdentifier/doi/http://dx.doi.org/10.1073/pnas.1906098116
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