Further physical study about solution structures for the fractional equal width equation (EW) in shallow water hydrodynamics

Abstract

In this article, with the help of Matlab R2021a software, the exact solutions of the EW are successfully examined by Sub-Equation method and the unified F-expansion method. The main novelty of this paper lies in the six aspects: (1) Singular periodic wave U-shaped solition is discovered by the sub equation method which did not happen in previous papers [35–37]. (2) More and more intensive multi-periodic M-shape wave solutions are also discovered by the unified F-expansion method which did not happen in previous papers [32,34–37]. (3) The dynamic wave behaviours about the got results and the effects of fractionality α are illustrated by Figs. 1–7. The fractional parameter α masters the appearance of the solitons and makes the involved alphabetic shaped solitions due to the increase in the fractionality number α in Figs. 1–7. (4) The effect about free parameters c,k can be seen from Figs. 8–11 in this paper. If only the increase of wave velocity parameter c or wave number parameter k, the energy trend of the wave is always positive and intensive. This also never happens in previous literatures [34–37]. (5) The Jumarie’s modified Riemann–Liouville and beta fractional and conformable derivatives are tested for accuracy in Figs. 12 and 13. We seem firstly compare the relations and distinctions among different fractional order derivatives in EW model. (6) In our best cognitive, the planar dynamical system about EW model whose phase portraits are also first studied. Furthermore, phase portraits for the space–time fractional equal width equation are plotted, and the corresponding sensitivity and chaotic behaviors are analyzed. The geometrical representations of the EW provide the corporal information to explain the physically phenomenons.