Are there any general statements to refer to, concerning this?
For example so-called acoustic pressure occurs both in the water and in structures in the water, so that is one case.
Other solutions, for example if a field fulfil Laplace equation or Poisson's equation, it has more nonzero derives than a linear field, so it could maybe occur also at a fine scale. Or the reverse; a piecewise linear field is discontinous and therefore enters on fine scales.
It would be possible to handle the previous, especially if the solution is for a coordinate that could be transformed between levels.
Lena, I think that you are looking for some literature of this sort: http://www.morkniga.ru/p828903.html . Asymptotical methods for solution of complex equations were well developed by applied mathematicians in the former USSR. Today almost nobody from young generation understands this, as well as analytical solutions. Many start from numerical methods, but they also have limited power. It would be difficult perhaps to find books online and in English, but if you know Russian and can have access to paper books, this can help.
For discontinuity, there is method of boundary layer.
I was also using variables of different scales (slow- fast) to derive equations in the past. But this is more on macro scales with application to tsunami waves; see https://www.researchgate.net/publication/275581998_Evolution_of_long_nonlinear_waves_on_shelves
If you are doing acoustics, there is also good book: В. М. Бабич, В. С. Булдырев, И. А. Молотков, Пространственно-временной лучевой метод. Линейные и нелинейные волны, Изд-во Ленинградского ун-та, Ленинград, 1985 .
Its translation to English also exists: https://www.amazon.com/Space-Time-Ray-Method-Nonlinear-Mathematics/dp/0521622018
Conference Paper Evolution of long nonlinear waves on shelves
Thank you.
Yes, such wave is the formation of a larger scale space, so it will cover the question.
Asymptotical, I don't know exactly at the moment.
If one assume a general wave that is known to amplify, one could expand the argument (caustic or characteristics, or what is the term), in terms of coordinates?
Before it becomes solitary (soliton), it may rise due to an incompressibility condition, but after the effective density is more like a mass.
There are asymptotical methods in variational calculus, which gives many results.
Three invariants could be derived for the variational problem, from this, and the Hamiltonian is well known. I don't know if VI Arnold denotes it differently, but the analysis is there.
( I derived an integral that simplifies Korteveg de Vries, and after that also found it in an Appendix of Arnold's book.)
Lena, there are many different questions here. But it seems that younger generation knows only how to deal with computer. I was doing similar things 25-30 years ago, but now I am completely in economics.
As for solitons, there is interesting nonlinear theory. On one hand, people do it with Hamiltonians, then transformations of Darboux type to find exact solutions of nonlinear equations. On the other hand, perturbation methods allow to study propagation of solitons in heterogeneous media and get the laws of their transformation.
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