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Exactly Solvable Model of Wave-Mean Field Interaction in Integrable Turbulence

T. Congy1,*, G. A. El1, and M. A. Hoefer2

  • *Contact author: thibault.congy@northumbria.ac.uk
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Phys. Rev. Lett. 136, 147201 – Published 6 April, 2026

DOI: https://doi.org/10.1103/d2br-hl5x

Abstract

The kinetic theory of soliton gases (SG) is used to develop a solvable model for wave-mean field interaction in integrable turbulence. The waves are stochastic soliton ensembles that scatter off a critically dense SG or soliton condensate—the mean field. The derived two-fluid kinetic-hydrodynamic equations admit exact solutions predicting an induced mean field and SG filtering. The obtained SG statistical moments agree with ensemble averages of numerical simulations. The developed theory readily generalizes, with applications in fluids, nonlinear optics, and condensed matter.

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References (43)

  1. B. Doyon, S. Gopalakrishnan, F. Møller, J. Schmiedmayer, and R. Vasseur, Generalized hydrodynamics: A perspective, Phys. Rev. X 15, 010501 (2025).
  2. A. Costa, A. R. Osborne, D. T. Resio, S. Alessio, E. Chrivì, E. Saggese, K. Bellomo, and C. E. Long, Soliton turbulence in shallow water ocean surface waves, Phys. Rev. Lett. 113, 108501 (2014).
  3. I. Redor, E. Barthélemy, H. Michallet, M. Onorato, and N. Mordant, Experimental evidence of a hydrodynamic soliton gas, Phys. Rev. Lett. 122, 214502 (2019).
  4. P. Suret, A. Tikan, F. Bonnefoy, F. Copie, G. Ducrozet, A. Gelash, G. Prabhudesai, G. Michel, A. Cazaubiel, E. Falcon, G. El, and S. Randoux, Nonlinear spectral synthesis of soliton gas in deep-water surface gravity waves, Phys. Rev. Lett. 125, 264101 (2020).
  5. I. Redor, H. Michallet, N. Mordant, and E. Barthélemy, Experimental study of integrable turbulence in shallow water, Phys. Rev. Fluids 6, 124801 (2021).
  6. L. Fache, F. Bonnefoy, G. Ducrozet, F. Copie, F. Novkoski, G. Ricard, G. Roberti, E. Falcon, P. Suret, G. El, and S. Randoux, Interaction of soliton gases in deep-water surface gravity waves, Phys. Rev. E 109, 034207 (2024).
  7. A. E. Kraych, D. Agafontsev, S. Randoux, and P. Suret, Statistical properties of the nonlinear stage of modulation instability in fiber optics, Phys. Rev. Lett. 123, 093902 (2019).
  8. G. Marcucci, D. Pierangeli, A. J. Agranat, R.-K. Lee, E. DelRe, and C. Conti, Topological control of extreme waves, Nat. Commun. 10 (2019).
  9. P. Suret, M. Dufour, G. Roberti, G. El, F. Copie, and S. Randoux, Soliton refraction by an optical soliton gas, Phys. Rev. Res. 5, L042002 (2023).
  10. P. Suret, S. Randoux, A. Gelash, D. Agafontsev, B. Doyon, and G. El, Soliton gas: Theory, numerics, and experiments, Phys. Rev. E 109, 061001 (2024).
  11. V. E. Zakharov, Turbulence in integrable systems, Stud. Appl. Math. 122, 219 (2009).
  12. S. B. Pope, Turbulent Flows (Cambridge University Press, Cambridge, England, 2000).
  13. O. Bühler, Waves and Mean Flows, Cambridge Monographs on Mechanics, 2nd ed. (Cambridge University Press, New York, 2014).
  14. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, 1981).
  15. S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method (Springer Science & Business Media, New York, 1984).
  16. V. Zakharov, Kinetic equation for solitons, Sov. Phys. JETP 33, 538 (1971).
  17. G. A. El and A. M. Kamchatnov, Kinetic equation for a dense soliton gas, Phys. Rev. Lett. 95, 204101 (2005).
  18. G. A. El, Soliton gas in integrable dispersive hydrodynamics, J. Stat. Mech. (2021) 114001.
  19. G. El, The thermodynamic limit of the Whitham equations, Phys. Lett. A 311, 374 (2003).
  20. T. Bonnemain, B. Doyon, and G. El, Generalized hydrodynamics of the KdV soliton gas, J. Phys. A 55, 374004 (2022).
  21. A. C. Newell and B. Rumpf, Wave turbulence, Annu. Rev. Fluid Mech. 43, 59 (2011).
  22. S. Nazarenko, Wave Turbulence, Lecture Notes in Physics (Springer, Berlin, Heidelberg, 2011), Vol. 825.
  23. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, Condensation of classical nonlinear waves, Phys. Rev. Lett. 95, 263901 (2005).
  24. T. Congy, G. A. El, G. Roberti, and A. Tovbis, Dispersive hydrodynamics of soliton condensates for the Korteweg–de Vries equation, J. Nonlinear Sci. 33, 104 (2023).
  25. B. Doyon, J. Dubail, R. Konik, and T. Yoshimura, Large-scale description of interacting one-dimensional Bose gases: Generalized hydrodynamics supersedes conventional hydrodynamics, Phys. Rev. Lett. 119, 195301 (2017).
  26. L. Landau, Theory of the superfluidity of helium II, Phys. Rev. 60, 356 (1941).
  27. T. Congy, H. T. Carr, G. Roberti, and G. A. El, Riemann problem for polychromatic soliton gases: A testbed for the spectral kinetic theory, Wave Motion 134, 103480 (2025).
  28. B. Doyon, Lecture notes on generalised hydrodynamics, SciPost Phys. Lect. Notes 18 (2020).
  29. H. Spohn, Hydrodynamic Scales of Integrable Many-Body Systems (World Scientific, Singapore, 2023).
  30. G. El and M. A. Hoefer, Dispersive shock waves and modulation theory, Physica (Amsterdam) 333D, 11 (2016).
  31. G. Biondini, G. A. El, X.-D. Luo, J. Oregero, and A. Tovbis, Breather gas fission from elliptic potentials in self-focusing media, Phys. Rev. E 111, 014204 (2025).
  32. M. D. Maiden, D. V. Anderson, N. A. Franco, G. A. El, and M. A. Hoefer, Solitonic dispersive hydrodynamics: Theory and observation, Phys. Rev. Lett. 120, 144101 (2018).
  33. M. J. Ablowitz, J. T. Cole, G. A. El, M. A. Hoefer, and X.-D. Luo, Soliton–mean field interaction in Korteweg–de Vries dispersive hydrodynamics, Stud. Appl. Math. 151, 795 (2023).
  34. G. A. El, A. M. Kamchatnov, M. V. Pavlov, and S. A. Zykov, Kinetic equation for a soliton gas and its hydrodynamic reductions, J. Nonlinear Sci. 21, 151 (2011).
  35. See Supplemental Material at http://link.aps.org/supplemental/10.1103/d2br-hl5x for additional details, which includes Refs. [36–38].
  36. L. N. Trefethen, Spectral Methods in matlab, Software, Environments and Tools (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2000) [Reference [4] in Supplemental Material not already in paper].
  37. N.-N. Huang, Darboux transformations for the Korteweg-de-Vries equation, J. Phys. A 25, 469 (1992) [Reference [5] in Supplemental Material not already in paper].
  38. A. A. Gelash and D. S. Agafontsev, Strongly interacting soliton gas and formation of rogue waves, Phys. Rev. E 98, 042210 (2018) [Reference [7] in Supplemental Material not already in paper].
  39. P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (SIAM, Philadelphia, PA, 1973).
  40. S. P. Tsarev, On Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type, Sov. Math. Dokl. 31, 488 (1985).
  41. S. P. Tsarev, The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method, Math. USSR Izvestija 37, 397 (1991).
  42. A. Kuijlaars and A. Tovbis, On minimal energy solutions to certain classes of integral equations related to soliton gases for integrable systems, Nonlinearity 34, 7227 (2021).
  43. H. Flaschka, M. G. Forest, and D. W. McLaughlin, Multiphase averaging and the inverse spectral solution of the Korteweg—de Vries equation, Commun. Pure Appl. Math. 33, 739 (1980).

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Phys. Rev. Lett. 136, 147201– Published 6 April, 2026
  • Received 15 August 2025
  • Revised 9 January 2026
  • Accepted 20 February 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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