WAVE PULSE EVOLUTION FOR FULLY NONLINEAR SERRE EQUATION

Although the shallow-water theory is a classical subject of investigations with a huge number of papers devoted to it, it still remains very active field of research with many important applications. When one neglects dissipation effects and non-uniformity of the basin’s bottom, the interplay of nonlinearity and dispersion effects leads to quite complicated wave patterns which form depends crucially on the initial profile of the pulse. If the nonlinearity and dispersion effects are taken into account in the lowest approximation and one considers a one-directional propagation of the wave, then its dynamics is governed by the famous Korteweg-de Vries (KdV) equation. Comparison with experiments shows that the KdV approximation is not good enough and one needs to go beyond it. Therefore considerable efforts were directed to the derivation of the corresponding wave equation that was able to better describe the system. One of the most popular models was first suggested and studied in much detail by Serre (Serre, 1953). For such a model, in which evolution is described by the Serre (Su-Gardner, Green- Naghdi) equation, El made an important study of the law of conservation of the “number of waves” and its soliton analogue (El, 2006). Using El’s method one can find the laws of motion of the edges of the dispersive shock waves (DSW) in problems related with self-similar evolution of step-like initial discontinuities. In (Kamchatnov, 2018) these methods were shown that allow one to go beyond such an initial profile. In this report, we will show the application of the methods of this work to study of simple wave initial pulses evolution in the theory of the Serre equations and give an analytical solution for the laws of motion of edges of DSW formed in the process of evolution of the initial pulses. Analytical results are confirmed by numerical calculations. The reported study was funded by RFBR according to the research project №19- 01-00178 А.