Most of the model equations for water waves are approximations for the long-wave propagation regimes, since most of the energy of the wave motion is concentrated in these waves. Long waves (or shallow-water waves) are defined usually as the depth to wavelength ratio δ = h/λ < 0.05. Several famous integrable nonlinear equations, like the K d V equation [1,2], are models for long waves of small amplitude. The short waves (or waves over deep water) are usually defined with δ > 0.5, and the intermediate waves (or transitional waves) - with 0.05 < δ < 0.5. The intermediate and short waves received a lot less attention, and one reason is perhaps the fact that the corresponding approximations lead to more complicated, nonlinear and nonlocal equations. In  an integral equation for surface waves has been proposed for arbitrary wavelengths and finite depth. The problem has been studied in  and model equations both for long and short waves are derived from the governing equations as well. The short-wave effects usually compete with the capillarity effects and then resonances can be observed — these have been studied quite a lot, see for example [5–12]. For the intermediate long waves or for waves on deep water the so-called Benjamin–Ono (BO) [13–15] and the Intermediate Long Wave Equation (ILWE) [16–18] are derived for the internal waves below a flat surface, which leads to some simplifications and these models are in fact integrable.
Ivanov, Rossen, "On The Modelling of Short and Intermediate Water Waves" (2023). Articles. 346.
Science Foundation Ireland under Grant number 21/FFP-A/9150.
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