{"title":"Deep-water and shallow-water limits of statistical equilibria for the intermediate long wave equation","authors":"Andreia Chapouto, Guopeng Li, Tadahiro Oh","doi":"arxiv-2409.06905","DOIUrl":null,"url":null,"abstract":"We study the construction of invariant measures associated with higher order\nconservation laws of the intermediate long wave equation (ILW) and their\nconvergence properties in the deep-water and shallow-water limits. By\nexploiting its complete integrability, we first carry out detailed analysis on\nthe construction of appropriate conservation laws of ILW at the $H^\\frac\nk2$-level for each $k \\in \\mathbb{N}$, and establish their convergence to those\nof the Benjamin-Ono equation (BO) in the deep-water limit and to those of the\nKorteweg-de Vries equation (KdV) in the shallow-water limit. In particular, in\nthe shallow-water limit, we prove rather striking 2-to-1 collapse of the\nconservation laws of ILW to those of KdV. Such 2-to-1 collapse is novel in the\nliterature and, to our knowledge, this is the first construction of a complete\nfamily of shallow-water conservation laws with non-trivial shallow-water\nlimits. We then construct an infinite sequence of generalized Gibbs measures\nfor ILW associated with these conservation laws and prove their convergence to\nthe corresponding (invariant) generalized Gibbs measures for BO and KdV in the\nrespective limits. Finally, for $k \\ge 3$, we establish invariance of these\nmeasures under ILW dynamics, and also convergence in the respective limits of\nthe ILW dynamics at each equilibrium state to the corresponding invariant\ndynamics for BO and KdV constructed by Deng, Tzvetkov, and Visciglia\n(2010-2015) and Zhidkov (1996), respectively. In particular, in the\nshallow-water limit, we establish 2-to-1 collapse at the level of the\ngeneralized Gibbs measures as well as the invariant ILW dynamics. As a\nbyproduct of our analysis, we also prove invariance of the generalized Gibbs\nmeasure associated with the $H^2$-conservation law of KdV, which seems to be\nmissing in the literature.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"14 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06905","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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 on
the construction of appropriate conservation laws of ILW at the $H^\frac
k2$-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 rather striking 2-to-1 collapse of the
conservation laws of ILW to those of KdV. Such 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 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.