Rigorous Asymptotics of a KdV Soliton Gas

M Girotti; T Grava; R Jenkins; K D T-R McLaughlin

doi:10.1007/s00220-021-03942-1

Rigorous Asymptotics of a KdV Soliton Gas

M Girotti, T Grava, R Jenkins, K D T-R McLaughlin

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Abstract

We analytically study the long time and large space asymptotics of a new broadclass of solutions of the KdV equation introduced by Dyachenko, Zakharov, and Zakharov.These solutions are characterized by a Riemann–Hilbert problem which we show arises as thel imit N→+∞of a gas of N-solitons. We show that this gas of solitons in the limit N→∞is slowly approaching a cnoidal wave solution forx→−∞(u p to terms of order O(1/x), while approaching zero exponentially fast for x→+∞. We establish an asymptotic description of the gas of solitons for large times that is valid over the entire spatial domain, in terms of Jacobi elliptic functions.

Original language	English
Pages (from-to)	733-784
Number of pages	52
Journal	Communications in Mathematical Physics
Volume	384
Issue number	2
DOIs	https://doi.org/10.1007/s00220-021-03942-1
Publication status	Published - 25 Apr 2021

Bibliographical note

Funding Information:
Open access funding provided by Scuola Internazionale Superiore di Studi Avanzati - SISSA within the CRUI-CARE Agreement.

Funding Information:
T.G. acknowledges the support of IDNAM. T.G. and M.G. acknowledges the support of the H2020-MSCA-RISE-2017 PROJECT No. 778010 IPADEGAN. K.M. was supported in part by the National Science Foundation under grant DMS-1733967. Part of the work of M.G. and K.M. was done during their visits at SISSA and while T.G., K.M. and R.J. were visiting CIRM, Luminy, France. We acknowledge SISSA and CIRM for excellent working conditions and generous support. We wish to thank Marco Bertola, and Alexander Minakov for useful feedback in constructing the matrix Riemann?Hilbert problem outer parametrix of KdV. In particular in Sect.?3.4 we have implemented the suggestions by Alexander Minakov.

Funding Information:
T.G. acknowledges the support of IDNAM. T.G. and M.G. acknowledges the support of the H2020-MSCA-RISE-2017 PROJECT No. 778010 IPADEGAN. K.M. was supported in part by the National Science Foundation under grant DMS-1733967. Part of the work of M.G. and K.M. was done during their visits at SISSA and while T.G., K.M. and R.J. were visiting CIRM, Luminy, France. We acknowledge SISSA and CIRM for excellent working conditions and generous support. We wish to thank Marco Bertola, and Alexander Minakov for useful feedback in constructing the matrix Riemann–Hilbert problem outer parametrix of KdV. In particular in Sect. we have implemented the suggestions by Alexander Minakov.

Publisher Copyright:
© 2021, The Author(s).

Keywords

Korteweg de Vries equation
Soliton Gas

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10.1007/s00220-021-03942-1

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title = "Rigorous Asymptotics of a KdV Soliton Gas",

abstract = "We analytically study the long time and large space asymptotics of a new broadclass of solutions of the KdV equation introduced by Dyachenko, Zakharov, and Zakharov.These solutions are characterized by a Riemann–Hilbert problem which we show arises as thel imit N→+∞of a gas of N-solitons. We show that this gas of solitons in the limit N→∞is slowly approaching a cnoidal wave solution forx→−∞(u p to terms of order O(1/x), while approaching zero exponentially fast for x→+∞. We establish an asymptotic description of the gas of solitons for large times that is valid over the entire spatial domain, in terms of Jacobi elliptic functions.",

keywords = "Korteweg de Vries equation, Soliton Gas",

author = "M Girotti and T Grava and R Jenkins and McLaughlin, {K D T-R}",

note = "Funding Information: Open access funding provided by Scuola Internazionale Superiore di Studi Avanzati - SISSA within the CRUI-CARE Agreement. Funding Information: T.G. acknowledges the support of IDNAM. T.G. and M.G. acknowledges the support of the H2020-MSCA-RISE-2017 PROJECT No. 778010 IPADEGAN. K.M. was supported in part by the National Science Foundation under grant DMS-1733967. Part of the work of M.G. and K.M. was done during their visits at SISSA and while T.G., K.M. and R.J. were visiting CIRM, Luminy, France. We acknowledge SISSA and CIRM for excellent working conditions and generous support. We wish to thank Marco Bertola, and Alexander Minakov for useful feedback in constructing the matrix Riemann?Hilbert problem outer parametrix of KdV. In particular in Sect.?3.4 we have implemented the suggestions by Alexander Minakov. Funding Information: T.G. acknowledges the support of IDNAM. T.G. and M.G. acknowledges the support of the H2020-MSCA-RISE-2017 PROJECT No. 778010 IPADEGAN. K.M. was supported in part by the National Science Foundation under grant DMS-1733967. Part of the work of M.G. and K.M. was done during their visits at SISSA and while T.G., K.M. and R.J. were visiting CIRM, Luminy, France. We acknowledge SISSA and CIRM for excellent working conditions and generous support. We wish to thank Marco Bertola, and Alexander Minakov for useful feedback in constructing the matrix Riemann–Hilbert problem outer parametrix of KdV. In particular in Sect. we have implemented the suggestions by Alexander Minakov. Publisher Copyright: {\textcopyright} 2021, The Author(s).",

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AU - Girotti, M

AU - Grava, T

AU - Jenkins, R

AU - McLaughlin, K D T-R

N1 - Funding Information: Open access funding provided by Scuola Internazionale Superiore di Studi Avanzati - SISSA within the CRUI-CARE Agreement. Funding Information: T.G. acknowledges the support of IDNAM. T.G. and M.G. acknowledges the support of the H2020-MSCA-RISE-2017 PROJECT No. 778010 IPADEGAN. K.M. was supported in part by the National Science Foundation under grant DMS-1733967. Part of the work of M.G. and K.M. was done during their visits at SISSA and while T.G., K.M. and R.J. were visiting CIRM, Luminy, France. We acknowledge SISSA and CIRM for excellent working conditions and generous support. We wish to thank Marco Bertola, and Alexander Minakov for useful feedback in constructing the matrix Riemann?Hilbert problem outer parametrix of KdV. In particular in Sect.?3.4 we have implemented the suggestions by Alexander Minakov. Funding Information: T.G. acknowledges the support of IDNAM. T.G. and M.G. acknowledges the support of the H2020-MSCA-RISE-2017 PROJECT No. 778010 IPADEGAN. K.M. was supported in part by the National Science Foundation under grant DMS-1733967. Part of the work of M.G. and K.M. was done during their visits at SISSA and while T.G., K.M. and R.J. were visiting CIRM, Luminy, France. We acknowledge SISSA and CIRM for excellent working conditions and generous support. We wish to thank Marco Bertola, and Alexander Minakov for useful feedback in constructing the matrix Riemann–Hilbert problem outer parametrix of KdV. In particular in Sect. we have implemented the suggestions by Alexander Minakov. Publisher Copyright: © 2021, The Author(s).

PY - 2021/4/25

Y1 - 2021/4/25

N2 - We analytically study the long time and large space asymptotics of a new broadclass of solutions of the KdV equation introduced by Dyachenko, Zakharov, and Zakharov.These solutions are characterized by a Riemann–Hilbert problem which we show arises as thel imit N→+∞of a gas of N-solitons. We show that this gas of solitons in the limit N→∞is slowly approaching a cnoidal wave solution forx→−∞(u p to terms of order O(1/x), while approaching zero exponentially fast for x→+∞. We establish an asymptotic description of the gas of solitons for large times that is valid over the entire spatial domain, in terms of Jacobi elliptic functions.

AB - We analytically study the long time and large space asymptotics of a new broadclass of solutions of the KdV equation introduced by Dyachenko, Zakharov, and Zakharov.These solutions are characterized by a Riemann–Hilbert problem which we show arises as thel imit N→+∞of a gas of N-solitons. We show that this gas of solitons in the limit N→∞is slowly approaching a cnoidal wave solution forx→−∞(u p to terms of order O(1/x), while approaching zero exponentially fast for x→+∞. We establish an asymptotic description of the gas of solitons for large times that is valid over the entire spatial domain, in terms of Jacobi elliptic functions.

KW - Korteweg de Vries equation

KW - Soliton Gas

U2 - 10.1007/s00220-021-03942-1

DO - 10.1007/s00220-021-03942-1

M3 - Article (Academic Journal)

SN - 0010-3616

VL - 384

SP - 733

EP - 784

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -

Rigorous Asymptotics of a KdV Soliton Gas

Abstract

Bibliographical note

Keywords

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