[Paper Review] On the deep-water and shallow-water limits of the intermediate long wave equation from a statistical viewpoint

Guopeng Li, Tadahiro Oh+1 more|arXiv (Cornell University)|Nov 7, 2022

Advanced Mathematical Physics Problems4 citations

TL;DR

This paper establishes the convergence of Gibbs measures and invariant dynamics for the intermediate long wave (ILW) equation in both deep-water (δ→∞) and shallow-water (δ→0) limits. Using Wick renormalization and compactness arguments, it proves that ILW Gibbs measures converge in total variation to Benjamin-Ono (BO) measures in the deep-water limit and weakly to Korteweg-de Vries (KdV) measures in the shallow-water limit after scaling, despite mutual singularity in the shallow regime. The results extend to generalized ILW in the defocusing case.

ABSTRACT

(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) We study convergence problems for the intermediate long wave equation (ILW), with the depth parameter $δ> 0$, in the deep-water limit ($δ o \infty$) and the shallow-water limit ($δ o 0$) from a statistical point of view. In particular, we establish convergence of invariant Gibbs dynamics for ILW in both the deep-water and shallow-water limits. For this purpose, we first construct the Gibbs measures for ILW, $0 < δ< \infty$. As they are supported on distributions, a renormalization is required. With the Wick renormalization, we carry out the construction of the Gibbs measures for ILW. We then prove that the Gibbs measures for ILW converge in total variation to that for the Benjamin-Ono equation (BO) in the deep-water limit. In the shallow-water regime, after applying a scaling transformation, we prove that, as $δ o 0$, the Gibbs measures for the scaled ILW converge weakly to that for the Korteweg-de Vries equation (KdV). We point out that this second result is of particular interest since the Gibbs measures for the scaled ILW and KdV are mutually singular (whereas the Gibbs measures for ILW and BO are equivalent). We also discuss convergence of the associated dynamical problem. Lastly, we point out that our results also apply to the generalized ILW equation in the defocusing case, converging to the generalized BO in the deep-water limit and to the generalized KdV in the shallow-water limit. In the non-defocusing case, however, our results can not be extended to a nonlinearity with a higher power due to the non-normalizability of the corresponding Gibbs measures.

Motivation & Objective

• To study the statistical convergence of the intermediate long wave (ILW) equation in deep-water (δ→∞) and shallow-water (δ→0) limits from a Gibbs measure perspective.
• To construct Gibbs measures for ILW with δ∈(0,∞) using Wick renormalization, as they are supported on distributional spaces.
• To prove convergence of these Gibbs measures to those of the Benjamin-Ono (BO) equation in the deep-water limit and to KdV measures after scaling in the shallow-water limit.
• To establish the existence of invariant Gibbs dynamics for ILW via compactness, and show their convergence to BO and scaled KdV dynamics in the respective limits.
• To extend the results to the generalized ILW equation in the defocusing case, while noting limitations in the non-defocusing case due to non-normalizability.

Proposed method

• Construct Gibbs measures for ILW on H^{-ε}(𝕋) via Wick renormalization, ensuring well-definedness on distributional spaces.
• Prove equivalence of base Gaussian measures for ILW and BO in the deep-water limit using spectral analysis of the dispersion operator Gδ.
• Apply scaling transformations to the shallow-water regime to match the KdV equation, enabling weak convergence of Gibbs measures.
• Use compactness arguments on truncated Gibbs measures to construct global-in-time invariant dynamics for ILW without requiring uniqueness.
• Establish convergence of invariant dynamics by extracting subsequences δm→∞ and δm→0, showing convergence to BO and scaled KdV dynamics respectively.
• Leverage variational and stochastic analysis tools, including total variation and weak convergence of probability measures, to handle singularities and limits.

Experimental results

Research questions

• RQ1 Do the Gibbs measures for the intermediate long wave (ILW) equation converge to those of the Benjamin-Ono (BO) equation as δ→∞?
• RQ2 Do the Gibbs measures for the scaled ILW equation converge weakly to those of the Korteweg-de Vries (KdV) equation as δ→0?
• RQ3 Are the Gibbs measures for ILW and BO equivalent, and how does this differ from the relationship between ILW and KdV measures in the shallow regime?
• RQ4 Can invariant Gibbs dynamics for ILW be constructed, and do they converge to those of BO and KdV in the deep- and shallow-water limits?
• RQ5 To what extent do these results extend to the generalized ILW equation, particularly in the non-defocusing case?

Key findings

• The Gibbs measures for ILW converge in total variation to those of the Benjamin-Ono equation as δ→∞, despite the measures being supported on distributions, due to Wick renormalization.
• After appropriate scaling, the Gibbs measures for the shallow-water regime of ILW converge weakly to those of the Korteweg-de Vries equation as δ→0, even though the measures are mutually singular.
• The invariant Gibbs dynamics for ILW are constructed via compactness arguments, and subsequences δm→∞ and δm→0 yield convergence to BO and scaled KdV dynamics, respectively.
• The results extend to the generalized ILW equation in the defocusing case, with convergence to generalized BO and generalized KdV measures in the respective limits.
• In the non-defocusing case, the method fails for higher-order nonlinearities due to non-normalizability of the corresponding Gibbs measures.
• The analysis is extended to the space H^{0-}(𝕋) via a countable dense subset, ensuring separability and enabling convergence in distributional topology.