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Peer-reviewedCC BYJournal of the European Mathematical Society

Deep-water and shallow-water limits of statistical equilibria for the intermediate long wave equation

  • Andreia Chapouto1,2,3,4,5
  • Guopeng Li6,2,3
  • Tadahiro Oh6,2,3
6 affiliations
  1. 1Monash University, Clayton, Australia
  2. 2The Maxwell Institute for the Mathematical Sciences, Edinburgh, UK
  3. 3The University of Edinburgh, UK
  4. 4UVSQ, Université Paris-Saclay, CNRS, Versailles, France
  5. 5University of California, Los Angeles, USA
  6. 6Beijing Institute of Technology, P. R. China
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Abstract

We study the construction of invariant measures associated with higher order conservation laws of the intermediate long wave equation (ILW) and their convergence properties in the deep-water and shallow-water limits. By exploiting its complete integrability, we first carry out detailed analysis of the construction of appropriate conservation laws of ILW at the H^{k/2} -level for each k \in \mathbb{N} , and establish their convergence to those of the Benjamin–Ono equation (BO) in the deep-water limit and to those of the Korteweg–de Vries equation (KdV) in the shallow-water limit. In particular, in the shallow-water limit, we prove a rather striking 2-to-1 collapse of the conservation laws of ILW to those of KdV. Such a 2-to-1 collapse is novel in the literature and, to our knowledge, this is the first construction of a complete family of shallow-water conservation laws with non-trivial shallow-water limits. We then construct an infinite sequence of generalized Gibbs measures for ILW associated with these conservation laws and prove their convergence to the corresponding (invariant) generalized Gibbs measures for BO and KdV in the respective limits. Finally, for k \ge 3 , we establish invariance of these measures under ILW dynamics, and also convergence in the respective limits of the ILW dynamics at each equilibrium state to the corresponding invariant dynamics for BO and KdV constructed by Deng, Tzvetkov, and Visciglia (2010–2015) and Zhidkov (1996), respectively. In particular, in the shallow-water limit, we establish a 2-to-1 collapse at the level of the generalized Gibbs measures as well as the invariant ILW dynamics. As a byproduct of our analysis, we also prove invariance of the generalized Gibbs measure associated with the H^{2} -conservation law of KdV, which seems to be missing in the literature.

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Cite

BibTeXRIS
BibTeX
@article{chapouto2026deepwater,
  title = {{Deep-water and shallow-water limits of statistical equilibria for the intermediate long wave equation}},
  author = {Andreia Chapouto and Guopeng Li and Tadahiro Oh},
  journal = {Journal of the European Mathematical Society},
  year = {2026},
  month = nov,
  doi = {10.4171/jems/1829},
  url = {https://doi.org/10.4171/jems/1829},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Deep-water and shallow-water limits of statistical equilibria for the intermediate long wave equation
AU  - Andreia Chapouto
AU  - Guopeng Li
AU  - Tadahiro Oh
PY  - 2026
DA  - 2026/11/03
JO  - Journal of the European Mathematical Society
DO  - 10.4171/jems/1829
UR  - https://doi.org/10.4171/jems/1829
AB  - We study the construction of invariant measures associated with higher order conservation laws of the intermediate long wave equation (ILW) and their convergence properties in the deep-water and shallow-water limits. By exploiting its complete integrability, we first carry out detailed analysis of the construction of appropriate conservation laws of ILW at the H^{k/2} -level for each k \in \mathbb{N} , and establish their convergence to those of the Benjamin–Ono equation (BO) in the deep-water limit and to those of the Korteweg–de Vries equation (KdV) in the shallow-water limit. In particular, in the shallow-water limit, we prove a rather striking 2-to-1 collapse of the conservation laws of ILW to those of KdV. Such a 2-to-1 collapse is novel in the literature and, to our knowledge, this is the first construction of a complete family of shallow-water conservation laws with non-trivial shallow-water limits. We then construct an infinite sequence of generalized Gibbs measures for ILW associated with these conservation laws and prove their convergence to the corresponding (invariant) generalized Gibbs measures for BO and KdV in the respective limits. Finally, for k \ge 3 , we establish invariance of these measures under ILW dynamics, and also convergence in the respective limits of the ILW dynamics at each equilibrium state to the corresponding invariant dynamics for BO and KdV constructed by Deng, Tzvetkov, and Visciglia (2010–2015) and Zhidkov (1996), respectively. In particular, in the shallow-water limit, we establish a 2-to-1 collapse at the level of the generalized Gibbs measures as well as the invariant ILW dynamics. As a byproduct of our analysis, we also prove invariance of the generalized Gibbs measure associated with the H^{2} -conservation law of KdV, which seems to be missing in the literature.
SN  - 1435-9855
SN  - 1435-9863
ER  -