Deep- and shallow-water convergence of the generalized Gibbs measures for the intermediate long wave equation
[Limites en eaux profondes et peu profondes des équilibres statistiques pour l’équation des ondes longues intermédiaires]
Andreia Chapouto1 ; Guopeng Li2 ; Tadahiro Oh3
1CNRS and Laboratoire de mathématiques de Versailles, UVSQ, Université Paris-Saclay, CNRS, 45 avenue des États-Unis, 78035 Versailles Cedex, France
2Department of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China
3The University of Edinburgh and, The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom
Journées équations aux dérivées partielles (2025), Exposé no. 1, 16 p.

This note is based on a talk given by the first author at the conference Journées Équations aux dérivées partielles 2025. We consider the intermediate long wave equation (ILW), modeling the internal wave propagation of the interface in a stratified fluid of finite depth, connecting the deep-water regime (= the BO regime) and the shallow-water regime (= the KdV regime). Exploiting the complete integrability of ILW, we provide a detailed description of its polynomial conservation laws, construct the associated invariant generalized Gibbs measures, and lastly show their convergence to those of BO and KdV. In the shallow-water regime, we establish a novel 2-to-1 collapse of ILW conservation laws to those of KdV (and also a 2-to-1 collapse of the associated generalized Gibbs measures), which exhibits the singular nature of the shallow-water convergence.

Cette note est basée sur une présentation donnée par la première autrice lors de la conférence Journées Équations aux dérivées partielles 2025. Nous nous intéressons à l’équation intermédiaire des ondes longues (ILW), qui modélise la propagation des ondes internes à l’interface dans un fluide stratifié de profondeur finie. Cette équation relie le régime des eaux profondes (= le régime BO) et le régime des eaux peu profondes (= le régime KdV). En exploitant l’intégrabilité complète de l’ILW, nous fournissons une description détaillée de ses lois de conservation polynomiales, construisons les mesures de Gibbs généralisées invariantes associées, puis montrons leur convergence vers celles des régimes BO et KdV. Dans le régime des eaux peu profondes, nous établissons un nouveau recoupement 2-à-1 des lois de conservation de l’ILW avec celles de KdV (ainsi qu’un recoupement 2-à-1 des mesures généralisées de Gibbs associées), qui met en évidence la nature singulière de la convergence des eaux peu profondes.

Publié le :
DOI : 10.5802/jedp.692
Classification : 35Q35, 35Q53, 37K10, 60H30
Keywords: intermediate long wave equation, generalized Gibbs measures, complete integrability, Benjamin–Ono equation, Korteweg-de Vries equation

Andreia Chapouto  1   ; Guopeng Li  2   ; Tadahiro Oh  3

1 CNRS and Laboratoire de mathématiques de Versailles, UVSQ, Université Paris-Saclay, CNRS, 45 avenue des États-Unis, 78035 Versailles Cedex, France
2 Department of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China
3 The University of Edinburgh and, The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom 
Andreia Chapouto; Guopeng Li; Tadahiro Oh. Deep- and shallow-water convergence of the generalized Gibbs measures for the intermediate long wave equation. Journées équations aux dérivées partielles (2025), Exposé no. 1, 16 p.. doi: 10.5802/jedp.692
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[1] L. Abdelouhab; J. L. Bona; M. Felland; J.-C. Saut Nonlocal models for nonlinear, dispersive waves, Phys. D: Nonlinear Phenom., Volume 40 (1989) no. 3, pp. 360-392 | Zbl | DOI | MR

[2] N. Barashkov; M. Gubinelli A variational method for Φ 3 4 , Duke Math. J., Volume 169 (2020) no. 17, pp. 3339-3415 | DOI | MR | Zbl

[3] A. Bényi; T. Oh; O. Pocovnicu On the probabilistic Cauchy theory for nonlinear dispersive PDEs, Landscapes of time-frequency analysis (Applied and Numerical Harmonic Analysis), Birkhäuser/Springer, 2019, pp. 1-32 | MR | Zbl

[4] B. Berntson; E. Langmann; J. Lenells Non-chiral intermediate long-wave equation and inter-edge effects in narrow quantum Hall systems, Phys. Rev. B, Volume 102 (2020), 155308, 14 pages | DOI

[5] M. Boué; P. Dupuis A variational representation for certain functionals of Brownian motion, Ann. Probab., Volume 26 (1998) no. 4, pp. 1641-1659 | DOI | MR | Zbl

[6] J. Bourgain Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV equation, Geom. Funct. Anal., Volume 3 (1993) no. 3, pp. 209-262 | DOI | Zbl | MR

[7] J. Bourgain Periodic nonlinear Schrödinger equation and invariant measures, Commun. Math. Phys., Volume 166 (1994) no. 1, pp. 1-26 | MR | Zbl | DOI

[8] J. Bourgain Gibbs measures, quasi-periodic solutions and nonlinear partial differential equations, The Legacy of Norbert Wiener: A Centennial Symposium (Cambridge, MA, 1994) (Proceedings of Symposia in Pure Mathematics), Volume 60, American Mathematical Society, 1997, pp. 53-63 | DOI | MR | Zbl

[9] A. Chapouto; J. Forlano; G. Li; T. Oh; D. Pilod Intermediate long wave equation in negative Sobolev spaces, Proc. Am. Math. Soc., Ser. B, Volume 11 (2024), pp. 452-468 | DOI | MR | Zbl

[10] A. Chapouto; B. Harrop-Griffiths; G. Li; T. Oh On the singular nature of shallow-water convergence of the intermediate long wave equation on the real line (2026) | arXiv | Zbl

[11] A. Chapouto; G. Li; T. Oh Deep-water and shallow-water limits of statistical equilibria for the intermediate long wave equation. (2024) | arXiv | Zbl

[12] A. Chapouto; G. Li; T. Oh; D. Pilod Deep-water limit of the intermediate long wave equation in L 2 , Math. Res. Lett., Volume 31 (2024) no. 6, pp. 1655-1692 | DOI | MR | Zbl

[13] A. Chapouto; G. Li; T. Oh; T. Zhao Shallow-water convergence of the intermediate long wave equation in L 2 (2025) | arXiv | Zbl

[14] D. R. Christie; K. J. Muirhead; A. L. Hales On solitary waves in the atmosphere, J. Atmos. Sci., Volume 35 (1978), pp. 805-825 | DOI

[15] Y. Deng; N. Tzvetkov; N. Visciglia Invariant measures and long time behaviour for the Benjamin–Ono equation III, Commun. Math. Phys., Volume 339 (2015) no. 3, pp. 815-857 | DOI | MR | Zbl

[16] J. Forlano; G. Li; T. Zhao Unconditional deep-water limit of the intermediate long wave equation in low-regularity, NoDEA, Nonlinear Differ. Equ. Appl., Volume 32 (2025) no. 2, 28, 31 pages | DOI | MR | Zbl

[17] L. Gassot; T. Laurens Global well-posedness for the ILW equation in H s (𝕋) for s>-1 2. (2025) | arXiv | Zbl

[18] J. Gibbons; B. Kupershmidt A linear scattering problem for the finite depth equation, Phys. Lett. A, Volume 79 (1980) no. 1, pp. 31-32 | DOI | MR

[19] B. Harrop-Griffiths; R. Killip; M. Vişan A priori bounds and equicontinuity of orbits for the intermediate long wave equation (2025) | arXiv | Zbl

[20] M. Ifrim; J.-C. Saut The lifespan of small data solutions for intermediate long wave equation (ILW), Commun. Partial Differ. Equations, Volume 50 (2025) no. 3, pp. 258-300 | DOI | MR | Zbl

[21] R. I. Joseph Solitary waves in a finite depth fluid, J. Phys. A. Math. Gen., Volume 10 (1977) no. 12, pp. 225-227 | Zbl | DOI | MR

[22] S. Kakutani On equivalence of infinite product measures, Ann. Math. (2), Volume 49 (1948), pp. 214-224 | DOI | MR | Zbl

[23] C. Klein; J.-C. Saut Nonlinear dispersive equations—inverse scattering and PDE methods, Applied Mathematical Sciences, 209, Springer, 2021, xx+580 pages | DOI | MR | Zbl

[24] Y. Kodama; M. J. Ablowitz; J. Satsuma Direct and inverse scattering problems of the nonlinear intermediate long wave equation, J. Math. Phys., Volume 23 (1982) no. 4, pp. 564-576 | DOI | MR | Zbl

[25] Y. Kodama; J. Satsuma; M. J. Ablowitz Nonlinear intermediate long-wave equation: analysis and method of solution, Phys. Rev. Lett., Volume 46 (1981) no. 11, pp. 687-690 | DOI | MR

[26] C. Gary Koop; Gerald Butler An investigation of internal solitary waves in a two-fluid system, J. Fluid Mech., Volume 112 (1981), pp. 225-251 | DOI | MR | Zbl

[27] T. Kubota; D. R. S Ko; L. D. Dobbs Weakly-nonlinear, long internal gravity waves in stratified fluids of finite depth, J. Hydronaut., Volume 12 (1978), pp. 157-165 | DOI

[28] B. Kupershmidt Involutivity of conservation laws for a fluid of finite depth and Benjamin–Ono equations, Libertas Math., Volume 1 (1981), pp. 125-132 | MR | Zbl

[29] D. R. Lebedev; A. O. Radul Generalized internal long waves equations: construction, Hamiltonian structure, and conservation laws, Commun. Math. Phys., Volume 19 (1983) no. 4, pp. 543-555 | DOI | MR | Zbl

[30] J. L. Lebowitz; H. A. Rose; E. R. Speer Statistical mechanics of the nonlinear Schrödinger equation, J. Stat. Phys., Volume 50 (1988) no. 3-4, pp. 657-687 | DOI | MR | Zbl

[31] G. Li Deep-water and shallow-water limits of the intermediate long wave equation, Nonlinearity, Volume 37 (2024) no. 7, 075001, 44 pages | DOI | MR | Zbl

[32] G. Li; T. Oh; G. Zheng On the deep-water and shallow-water limits of the intermediate long wave equation from a statistical viewpoint, Trans. Lond. Math. Soc., Volume 12 (2025) no. 1, e70005, 83 pages | DOI | MR | Zbl

[33] F. Linares; G. Ponce On decay and asymptotic properties of solutions to the intermediate long wave equation (2024) | arXiv | Zbl

[34] Y. Matsuno Bilinear transformation method, Mathematics in Science and Engineering, 174, Academic Press Inc., 1984, viii+223 pages | MR | Zbl

[35] L. Molinet; D. Pilod; S. Vento On well-posedness for some dispersive perturbations of Burgers’ equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 35 (2018) no. 7, pp. 1719-1756 | DOI | MR | Numdam | Zbl

[36] L. Molinet; J.-C. Saut; N. Tzvetkov Ill-posedness issues for the Benjamin–Ono and related equations, SIAM J. Math. Anal., Volume 33 (2001) no. 4, pp. 982-988 | DOI | MR | Zbl

[37] L. Molinet; S. Vento Improvement of the energy method for strongly nonresonant dispersive equations and applications, Anal. PDE, Volume 8 (2015) no. 6, pp. 1455-1495 | DOI | MR | Zbl

[38] C. Muñoz; G. Ponce; J.-C. Saut On the long time behavior of solutions to the intermediate long wave equation, SIAM J. Math. Anal., Volume 53 (2021) no. 1, pp. 1029-1048 | DOI | MR | Zbl

[39] A. Nahmod; T. Oh; L. Rey-Bellet; G. Staffilani Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS, J. Eur. Math. Soc., Volume 14 (2012), pp. 1275-1330 | DOI | MR | Zbl

[40] T. Oh White noise for KdV and mKdV on the circle, Harmonic analysis and nonlinear partial differential equations (RIMS Kôkyûroku Bessatsu), Volume B18, Research Institute for Mathematical Sciences, Kyoto, 2010, pp. 99-124 | MR

[41] T. Oh; P. Sosoe; L. Tolomeo Optimal integrability threshold for Gibbs measures associated with focusing NLS on the torus, Invent. Math., Volume 227 (2022) no. 3, pp. 1323-1429 | DOI | MR | Zbl

[42] A. R. Osborne; T. L. Burch Internal Solitons in the Andaman Sea, Science, Volume 208 (1980), pp. 451-460 | DOI

[43] P. M. Santini Bi-Hamiltonian formulations of the intermediate long wave equation, Inverse Probl., Volume 5 (1989) no. 2, pp. 203-225 | DOI | MR | Zbl

[44] J. Satsuma; M. J. Ablowitz; Y. Kodama On an internal wave equation describing a stratified fluid with finite depth, Phys. Lett. A, Volume 73 (1979) no. 4, pp. 283-286 | DOI | MR

[45] J.-C. Saut Benjamin-Ono and intermediate long wave equations: modeling, IST and PDE, Nonlinear dispersive partial differential equations and inverse scattering (Fields Institute Communications), Volume 83, Springer, 2019, pp. 95-160 | MR | DOI | Zbl

[46] H. Segur; J. L. Hammack Soliton models of long internal waves, J. Fluid Mech., Volume 118 (1982), pp. 285-304 | DOI | MR | Zbl

[47] N. Tzvetkov; N. Visciglia Gaussian measures associated to the higher order conservation laws of the Benjamin–Ono equation, Ann. Sci. Éc. Norm. Supér. (4), Volume 46 (2013) no. 2, pp. 249-299 | DOI | MR | Numdam | Zbl

[48] N. Tzvetkov; N. Visciglia Invariant measures and long-time behavior for the Benjamin–Ono equation, Int. Math. Res. Not., Volume 2014 (2014) no. 17, pp. 4679-4714 | DOI | MR | Zbl

[49] N. Tzvetkov; N. Visciglia Invariant measures and long time behaviour for the Benjamin–Ono equation II, J. Math. Pures Appl. (9), Volume 103 (2015) no. 1, pp. 102-141 | DOI | MR | Zbl

[50] A. S. Üstünel Variational calculation of Laplace transforms via entropy on Wiener space and applications, J. Funct. Anal., Volume 267 (2014) no. 8, pp. 3058-3083 | DOI | MR | Zbl

[51] P. E. Zhidkov Invariant measures for the Korteweg-de Vries equation that are generated by higher conservation laws, Mat. Sb., Volume 187 (1996) no. 6, pp. 21-40 | DOI | MR

[52] P. E. Zhidkov Korteweg-de Vries and nonlinear Schrödinger equations: qualitative theory, Lecture Notes in Mathematics, 1756, Springer, 2001, vi+147 pages | MR | Zbl

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