Primitive solutions of the Korteweg–de Vries equation

S. A. Dyachenko; P. Nabelek; D. V. Zakharov; V. E. Zakharov

doi:10.1134/S0040577920030058

Primitive solutions of the Korteweg–de Vries equation

S. A. Dyachenko, P. Nabelek, D. V. Zakharov, V. E. Zakharov

Mathematics

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

We survey recent results connected with constructing a new family of solutions of the Korteweg-de Vries equation, which we call primitive solutions. These solutions are constructed as limits of rapidly vanishing solutions of the Korteweg-de Vries equation as the number of solitons tends to infinity. A primitive solution is determined nonuniquely by a pair of positive functions on an interval on the imaginary axis and a function on the real axis determining the reflection coefficient. We show that elliptic one-gap solutions and, more generally, periodic finite-gap solutions are special cases of reflectionless primitive solutions.

Original language	English (US)
Pages (from-to)	334-343
Number of pages	10
Journal	Theoretical and Mathematical Physics(Russian Federation)
Volume	202
Issue number	3
DOIs	https://doi.org/10.1134/S0040577920030058
State	Published - Mar 1 2020

Keywords

Korteweg-de Vries equation
integrable system
primitive solution

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics

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10.1134/S0040577920030058

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title = "Primitive solutions of the Korteweg–de Vries equation",

abstract = "We survey recent results connected with constructing a new family of solutions of the Korteweg-de Vries equation, which we call primitive solutions. These solutions are constructed as limits of rapidly vanishing solutions of the Korteweg-de Vries equation as the number of solitons tends to infinity. A primitive solution is determined nonuniquely by a pair of positive functions on an interval on the imaginary axis and a function on the real axis determining the reflection coefficient. We show that elliptic one-gap solutions and, more generally, periodic finite-gap solutions are special cases of reflectionless primitive solutions.",

keywords = "Korteweg-de Vries equation, integrable system, primitive solution",

author = "Dyachenko, {S. A.} and P. Nabelek and Zakharov, {D. V.} and Zakharov, {V. E.}",

note = "Funding Information: The research of S. A. Dyachenko and D. V. Zakharov was supported by the National Science Foundation (Grant No. DMS-1716822) Funding Information: The research of V. E. Zakharov was supported by the National Science Foundation (Grant No. DMS-1715323). The results in Secs. 3-5 were obtained with support of a grant from the Russian Science Foundation (Project No. 19-72-30028) Publisher Copyright: {\textcopyright} 2020, Pleiades Publishing, Ltd.",

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AU - Dyachenko, S. A.

AU - Nabelek, P.

AU - Zakharov, D. V.

AU - Zakharov, V. E.

N1 - Funding Information: The research of S. A. Dyachenko and D. V. Zakharov was supported by the National Science Foundation (Grant No. DMS-1716822) Funding Information: The research of V. E. Zakharov was supported by the National Science Foundation (Grant No. DMS-1715323). The results in Secs. 3-5 were obtained with support of a grant from the Russian Science Foundation (Project No. 19-72-30028) Publisher Copyright: © 2020, Pleiades Publishing, Ltd.

PY - 2020/3/1

Y1 - 2020/3/1

N2 - We survey recent results connected with constructing a new family of solutions of the Korteweg-de Vries equation, which we call primitive solutions. These solutions are constructed as limits of rapidly vanishing solutions of the Korteweg-de Vries equation as the number of solitons tends to infinity. A primitive solution is determined nonuniquely by a pair of positive functions on an interval on the imaginary axis and a function on the real axis determining the reflection coefficient. We show that elliptic one-gap solutions and, more generally, periodic finite-gap solutions are special cases of reflectionless primitive solutions.

AB - We survey recent results connected with constructing a new family of solutions of the Korteweg-de Vries equation, which we call primitive solutions. These solutions are constructed as limits of rapidly vanishing solutions of the Korteweg-de Vries equation as the number of solitons tends to infinity. A primitive solution is determined nonuniquely by a pair of positive functions on an interval on the imaginary axis and a function on the real axis determining the reflection coefficient. We show that elliptic one-gap solutions and, more generally, periodic finite-gap solutions are special cases of reflectionless primitive solutions.

KW - Korteweg-de Vries equation

KW - integrable system

KW - primitive solution

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ER -

Primitive solutions of the Korteweg–de Vries equation

Abstract

Keywords

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