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On the Trace-Class Property of Hankel Operators Arising in the Theory of the Korteweg–de Vries Equation
S. M. Grudskya, A. V. Rybkinb a Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional
b University of Alaska Fairbanks
Abstract:
The trace-class property of Hankel operators (and their derivatives with respect to the parameter) with strongly oscillating symbol is studied. The approach used is based on Peller's criterion for the trace-class property of Hankel operators and on the precise analysis of the arising triple integral using the saddle-point method. Apparently, the obtained results are optimal. They are used to study the Cauchy problem for the Korteweg–de Vries equation. Namely, a connection between the smoothness of the solution and the rate of decrease of the initial data at positive infinity is established.
Keywords:
Hankel operator, trace-class operator, Korteweg–de Vries equation, inverse problem method.
DOI:
https://doi.org/10.4213/mzm12112
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English version:
Mathematical Notes, 2018, 104:3, 377–394
Bibliographic databases:
UDC:
517.957+517.984.2 Received: 05.02.2018
Citation:
S. M. Grudsky, A. V. Rybkin, “On the Trace-Class Property of Hankel Operators Arising in the Theory of the Korteweg–de Vries Equation”, Mat. Zametki, 104:3 (2018), 374–395; Math. Notes, 104:3 (2018), 377–394
Citation in format AMSBIB
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http://mi.mathnet.ru/eng/mz12112https://doi.org/10.4213/mzm12112 http://mi.mathnet.ru/eng/mz/v104/i3/p374
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