Nonlinear asymptotic short-wave models in fluid dynamics (Q2750770)

After a brief introduction to long- and short-wave dynamics, the author establishes necessary conditions for existence of short-wave propagation in dispersive systems. To deal with the short wave (\(\ell \to 0\), where \(\ell\) is the wavelength), the author introduces two very useful variables \(\zeta\) and \(\tau\), defined as \(\zeta = x/\varepsilon\) and \(\tau = \varepsilon t\), where \(\varepsilon\) is an artificial small parameter. Such a treatment leads to a clear local phenomenon and to its long-time behavior. The parameter \(\varepsilon\) is then used to carry out the perturbative procedure. NEWLINENEWLINENEWLINEThe author also studies modified Green-Nagdhi equations and Green-Nagdhi system with surface tension. The difference between Green-Nagdhi and modified Green-Nagdhi systems is that the later introduces a new parameter \(c_0\) in the velocity on the surface, which can be considered as the velocity of wind. This slight modification leads to a dramatic change of wave dynamics. Finally, the author suggests a general classification of classical model equations.