Complex physical phenomena of a generalized (3+1)-dimensional Yu-Toda-Sasa-Fukuyama equation in a two-layer heterogeneous liquid
Department of Basic Sciences, Faculty of Engineering at October 6 University, Giza, Egypt
2 Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
Accepted: 12 August 2022
Published online: 3 September 2022
Inhomogeneous liquid may be argued to inhomogeneous density or induced by an external field. It is a type of the commonly seen fluids. Heterogeneous medium, which stands to, by heterogeneous medium (HM). As a realistic example, the Earth’s atmosphere, as a whole, it is blue a heterogeneous mixture. Further, the liquid formed from oil and water, which is with non-uniform composition, is immiscible HM. The study of the dynamics of clouds, as HM fluid, is of great interest in depicting many natural phenomena. It is recognized that petroleum pollutants were being discharged in marine waters worldwide, from oil spills. So, methods for assessing petroleum load and a discussion about the concerns of these loads were presented. Due to the wide spread of the applications of the heterogeneous fluid (or liquid) in nature, this motivated us to study, here, a prototype example. The model equation that describes the interaction of two-layer liquid was constructed by (3+1)-dimensional Yu-Toda-Sasa-Fukuyama (3D-YTSFE), which is an integro-differential equation. A generalized 3D-YTSFE with constant or time-dependent coefficients was intensively studied the literature. Here, we are concerned with the study of the dynamics of two-layer heteroogeneous liquid with space and time-dependent coefficients. That is, model equation constructed here is inhomogeneous-non-autonomous generalized 3D-YTSFE. The problem considered, in the present work, is completely novel and was not studied previously. This may be argued to the fact that it cannot be amenable by the known methods in the literature. On the other hand, the derivations are not straightforward. We solve the equations obtained, which contain arbitrary functions and their space and time derivatives. So, compatibility equations are needed, that will be illustrated, here, in detail. Exact solutions of the proposed model equation are found via the extended unified method. A variety of similarity solutions are found in polynomial and rational forms in an auxiliary function. They are evaluated numerically and are represented in graphs. It is shown that they reveal abundant novel waves geometric structures. They are classified as cylindrical soliton, molar soliton, soliton with support and double branches, dromian structure, lattice wave with tunneling, capillary wave, and chaotic solutions.
© The Author(s) 2022
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.