Dispersive hydrodynamics of soliton condensates for the Korteweg-de Vries equation

Thibault Congy; Gennady El; Giacomo Roberti; Alexander Tovbis

doi:10.1007/s00332-023-09940-y

Dispersive hydrodynamics of soliton condensates for the Korteweg-de Vries equation

Thibault Congy^*, Gennady El, Giacomo Roberti, Alexander Tovbis

^*Corresponding author for this work

Mathematics, Physics and Electrical Engineering

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

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Abstract

We consider large-scale dynamics of non-equilibrium dense soliton gas for the Korteweg-de Vries (KdV) equation in the special ``condensate'' limit. We prove that in this limit the integro-differential kinetic equation for the spectral density of states reduces to the $N$-phase KdV-Whitham modulation equations derived by Flaschka, Forest and McLaughlin (1980) and Lax and Levermore (1983). We consider Riemann problems for soliton condensates and construct explicit solutions of the kinetic equation describing generalized rarefaction and dispersive shock waves. We then present numerical results for "diluted'' soliton condensates exhibiting rich incoherent behaviors associated with integrable turbulence.

Original language	English
Article number	104
Number of pages	45
Journal	Journal of Nonlinear Science
Volume	33
Issue number	6
DOIs	https://doi.org/10.1007/s00332-023-09940-y
Publication status	Published - 19 Sept 2023

Keywords

Integrability
Kinetic equation
Korteweg-de Vries equation
Soliton gas
Whitham modulation equations

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title = "Dispersive hydrodynamics of soliton condensates for the Korteweg-de Vries equation",

abstract = "We consider large-scale dynamics of non-equilibrium dense soliton gas for the Korteweg-de Vries (KdV) equation in the special ``condensate'' limit. We prove that in this limit the integro-differential kinetic equation for the spectral density of states reduces to the $N$-phase KdV-Whitham modulation equations derived by Flaschka, Forest and McLaughlin (1980) and Lax and Levermore (1983). We consider Riemann problems for soliton condensates and construct explicit solutions of the kinetic equation describing generalized rarefaction and dispersive shock waves. We then present numerical results for {"}diluted'' soliton condensates exhibiting rich incoherent behaviors associated with integrable turbulence.",

keywords = "Integrability, Kinetic equation, Korteweg-de Vries equation, Soliton gas, Whitham modulation equations",

author = "Thibault Congy and Gennady El and Giacomo Roberti and Alexander Tovbis",

note = "Funding information: The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme “Dispersive hydrodynamics: mathematics, simulation and experiments, with applications in nonlinear waves” when the work on this paper was undertaken. This work was supported by EPSRC Grant Number EP/R014604/1. GE{\textquoteright}s and GR{\textquoteright}s work was also supported by EPSRC Grant Number EP/W032759/1 and AT{\textquoteright}s work was supported by NSF Grant DMS 2009647. TC, GR and AT thank Simons Foundation for partial support. All authors thank T. Bonnemain, S. Randoux and P. Suret for numerous useful discussions.",

year = "2023",

month = sep,

day = "19",

doi = "10.1007/s00332-023-09940-y",

language = "English",

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journal = "Journal of Nonlinear Science",

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T1 - Dispersive hydrodynamics of soliton condensates for the Korteweg-de Vries equation

AU - Congy, Thibault

AU - El, Gennady

AU - Roberti, Giacomo

AU - Tovbis, Alexander

N1 - Funding information: The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme “Dispersive hydrodynamics: mathematics, simulation and experiments, with applications in nonlinear waves” when the work on this paper was undertaken. This work was supported by EPSRC Grant Number EP/R014604/1. GE’s and GR’s work was also supported by EPSRC Grant Number EP/W032759/1 and AT’s work was supported by NSF Grant DMS 2009647. TC, GR and AT thank Simons Foundation for partial support. All authors thank T. Bonnemain, S. Randoux and P. Suret for numerous useful discussions.

PY - 2023/9/19

Y1 - 2023/9/19

N2 - We consider large-scale dynamics of non-equilibrium dense soliton gas for the Korteweg-de Vries (KdV) equation in the special ``condensate'' limit. We prove that in this limit the integro-differential kinetic equation for the spectral density of states reduces to the $N$-phase KdV-Whitham modulation equations derived by Flaschka, Forest and McLaughlin (1980) and Lax and Levermore (1983). We consider Riemann problems for soliton condensates and construct explicit solutions of the kinetic equation describing generalized rarefaction and dispersive shock waves. We then present numerical results for "diluted'' soliton condensates exhibiting rich incoherent behaviors associated with integrable turbulence.

AB - We consider large-scale dynamics of non-equilibrium dense soliton gas for the Korteweg-de Vries (KdV) equation in the special ``condensate'' limit. We prove that in this limit the integro-differential kinetic equation for the spectral density of states reduces to the $N$-phase KdV-Whitham modulation equations derived by Flaschka, Forest and McLaughlin (1980) and Lax and Levermore (1983). We consider Riemann problems for soliton condensates and construct explicit solutions of the kinetic equation describing generalized rarefaction and dispersive shock waves. We then present numerical results for "diluted'' soliton condensates exhibiting rich incoherent behaviors associated with integrable turbulence.

KW - Integrability

KW - Kinetic equation

KW - Korteweg-de Vries equation

KW - Soliton gas

KW - Whitham modulation equations

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U2 - 10.1007/s00332-023-09940-y

DO - 10.1007/s00332-023-09940-y

M3 - Article

SN - 0938-8974

VL - 33

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

IS - 6

M1 - 104

ER -

Dispersive hydrodynamics of soliton condensates for the Korteweg-de Vries equation

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Keywords

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