A discussion on the confusion surrounding the classical Stokes expansion and different forms present in the literature.
This work was inspired from Wise 2025 (last year) discussions with Yinyu Liu and Thomas Adcock (who raised the question) - and I must admit the quality of the paper benefitted tremendously from reviews (anonymous and by Milan!)
Dear Pieter,
thank you for sharing your recent work, which nicely summarizes about Stokes expansions.
The small notes in attachment describe the essence of canonical and non-canonical expansions for the simple cubic oscillator.
Noncanonical_vs_Canonical_Removal_of_Non_Resonant_Terms.pdf (167.6 KB)
I asked this question to Zakharov in 2012 and he was very clear about it. Hasselmann derived the kinetic equation from the Euler equations in physical variables, which are not canonically conjugate. Zakharov instead derived his kinetic equation from the symplectic form of the Euler equations in the variables eta and the potential evaluated at the surface, which are canonically conjugate. The associated Stokes expansions are diffferent ( Hasselman beta=1/8, Zakharov beta =-3/8 following your paper). Moreover, the associated kernels at the core of the SNL term in WAVEWATCH are different. However, Dyachenko and Lvov proved that they are identical on the resonant manifold.
Moreover, for 1D waves there are trivial resonant quartets (kernel =0) that Zakharov ‘removed’ by a canonical transformation obtaining the compact Zakharov equation, a third order model,but exact in spectral bandwidth, beyond NLS and Dysthe equations.
So, what expansion to use? it’s all about consistency and what variables (physical vs. canonical) one wants to work with. Very often canonical variables are hard to interpret physically, and that’s why it is convenient to work with physical variables directly. The price to pay is the loss of the symplectic structure. For example, the Dysthe equation is not Hamiltonian, because it has been derived in terms of physical variables. However, it can be made symplectic via a canonical transformation.
One cannot separate the Stokes non-resonant expansion from the resonant part. They go together. Fortunately, to second order the non-resonant Stokes expansions are the same.
Hope this helps
cheers,
Francesco
Thanks for sharing Francesco! Very useful - and insightful.
I had deliberately focussed the work very narrowly on the Stokes type expansion; not because the insight was new (it wasn’t - even in that narrow context) but because surprisingly even in that context the literature was not all that explicit (and sometimes confused) about it.
I also kept it narrow because - frankly - my time is rather constrained these days; not a good excuse - but reality.
That said; had I been aware of connections you point out - it would have warranted discussion.
That aside; one thing that I found genuinely interesting that you may have a better take on. In a deterministic setting I can swap variables at will (at the price of complextiy/different expressions). However, once we go to stochastic formulations we typically add an assumption of quasi-gausianity for the leading order perturbation amplitudes (or Rayleigh distribution/random phase if you work with magnitude/phase).
Because the amplitudes are related by a non-linear transformations you pick up non-gaussian contributions (cumulants) in the transformed amplitudes. This begs the question - where is it most natural to assume gaussian behavior? I suspect it is for the canonical variables; but frankly I don’t have a sound argument for it.
I forgot to attach the paper I wrote with Denys Dutykh about the canonical form of the Dysthe equations. Also available online
Fedele_Dutykh_Dysthe.pdf (577.3 KB)
Sure; to the leading order a higher order transformation will not alter the statistics. My questions is not so much; how does the transform affect it - but if I have a freedom - in which representation do we minimize the cumulants and still retrieve correct statistics for (say) wave height. Is that representation the canonical form?
anyhow; thanks for paper! We should catch up sometime!
my two cents: either forms should be fine as they are related by a canonical transformation (see my paper with Denys on Dysthe equation), and our data are in physical space. The estimation of the ‘linear’ spectrum from the expansion is an inverse problem. given data
aNL = aL + … higher order terms function of the spectrum of aL.
so, we know aNL and we need to solve for aL . that would be the ‘linear’ spectrum, which actually encodes nonlinear resonances, so we may call it resonant spectrum. If the sea is multidirectional broadbanded, the aL yields a Gaussian sea (equipartition of energy), but in unidirectional seas aL yields non-Gaussian fields dominated by breathers (condensates), so aL is not ‘linear’ but still symmetric ruled by fourth order cumulants. The Stokes expansion adds the third order skewness cumulant effects.
from Tayfun & Fedele OMAE2009 : to make second-order simulations of ocean waves with a realistic spectrum, the “linear” spectrum should scale as w^-8 to get the w^-4 Zakharov spectrum. Because if you start with a w^-4 linear spectrum, second order simulations give you nonlinear fields with w^-2 spectra!
however, one should consider the third order non-resonant terms in the inversion for the correct estimation of the ‘linear’ spectrum.
with AI nowadays it’s easy to design an autoencoder or a neural network to learn the transformation or a latent representation of it. Claude can code it in a few seconds with a good prompt. Probably, someone is already working on it (not me 
though).
