Complex physical phenomena of a generalized (3+1)-dimensional Yu-Toda-Sasa-Fukuyama equation in a two-layer heterogeneous liquid

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.


Introduction
Inhomogeneous two-layer liquid phenomena is widely occurring in nature. Two-layer liquid models were used currently to describe certain nonlinear phenomena in fluid mechanics, thermodynamics, medical and environmental sciences. Some phenomena in the lower atmosphere may be approximately described by motion of inhomogeneous lower layer surmounted by a deep upper layer of non-uniform density. Further applications are in intracellular freezing, snow precipitation, mineralization, and metal solidification. Up to date, all works existing in the literature were concerned mainly with the homogeneous case. A model equation that describes the dynamics of homogeneous two-layer liquid , which is the 3D-YTSFE, was derived [1]. It is an integro-differential equation, so that it describes a two-layer fluid. A generalized 3D-YTSFE with constant and time-dependent coefficients was studied in the literature. In [2], the behavior of mixed waves solutions in two-layer-liquid, via the study of solutions of 3D-YTSFE with time-dependent coefficients, with dispersive waveguide, was investigated. These combined multi-wave solutions were obtained in polynomial types. In a two-layer liquid (or crystal), a generalized 3D-YTSFE was studied with symbolic computation, where the Hirota method, bilinear form and bilinear auto-Bäcklund transformation were used [3]. A (2 + 1)-dimensional reduced YTSFE two-layer liquid equation was studied in [4]. A generalized 3D-YTSFE was studied in [5] for the objective of depicting multi-breather solutions. The direct bilinear method and the formula of N-soliton solutions of the generalized 3D-YTSF equation were obtained via the long-wave limit method [6]. The 3D-YTSFE was considered by employing the extended homoclininc test approach and Hirota bilinear method for deriving lump solutions [7]. In [8], the accuracy of novel lump solutions of the 3D potential YTSFE was investigated, where the solutions were obtained by using the extended simplest equation and modified Kudryashov schemes. Exact analytical solutions of YTSFE were presented by invoking Lie symmetry analysis technique and the power series expansion method [9]. The Hirota a e-mail: mtantawymath@gmail.com (corresponding author) 0123456789().: V,-vol bilinear method (HBM) and symbolic computation were employed to study the dynamics two types of interaction solutions for the potential YTSFE [10]. The periodic-wave and semi-rational solutions in determinant form for the 3D-l YTSFE were investigated via the Kadomtsev-Petviashvili hierarchy reduction [11]. By using a transformation of unknown function, the system of potential 3D-YTSFE was converted into the combined equation of differently two bilinear forms, where kink multi-soliton solutions were obtained [12]. Kinky breather-wave solutions for the potential 3D-YTSFE were obtained by using the extended homoclinic test technique [13]. The generalized bilinear differential operators to create a 3D-YTSFE like equation linked with prime number p 3 was implemented [14]. The Lie symmetry reduction was used to understand the behavior of pulses solutions of the generalized 3D-YTSFE [15]. General high-order rogue waves of the potential 3D-YTSFE were derived by employing the HBM [16]. The bilinear Bäcklund transformation was used to obtain the N-kink soliton solutions, in the Gramian, in the cases of elastic and inelastic oblique kink solitons for the 3D-YTSFE [17]. The Lie symmetry analysis combined with symbolic computations was used to construct systematically the group invariant solutions associated with some 1D sub-algebras [18]. The 3D-YTSFE was studied by computing its Lie point symmetries and then performing symmetry reductions [19]. A homoclinic (heteroclinic) breather limit method for seeking rogue wave solution to the (3+1)-dimensional YTSFE was used [20]. The lump wave and the lumpoff solutions were studied. Further special rogue waves were shown to be generated by the collision of the lump wave and a couple of stripe soliton waves [21]. A (2+1)-dimensional generalized YTSFE was analytically investigated via the engagement of Lie symmetry reductions alongside with direct integration techniques [22]. A (2+1)-dimensional potential YTSFE was studied, where mixed solutions were obtained by using the Hirota bilinear method and the long-wave limit approach [23]. The HBM was used to investigate the high-order breathers and interaction solutions between solitons and breathers of the (2+1)-dimensional YTSFE [24]. Cross-soliton solution, breather soliton, periodic solitary solution, and doubly periodic solution were obtained by using an extended homoclinic test approach [25]. Multiple rogue wave solutions of (2+1)-dimensional YTSFE were studied by applying the traveling wave transformation and the HBM. [26]. The symbolic computation was used to present an improved (G /G)-expansion method, which was applied to YTSFE [27]. Abundant exact solutions including the lump and interaction solutions to the (2 + 1)-dimensional YTSFE were studied [28]. The HBM was employed and acquired a type of the lump solution to the 3D-YTSFE [29]. Periodic-wave solutions of potential-3D-YTSF were constructed by using the bilinear form [30]. An improved homoclinic test technique and the extended homoclinic test technique together with the symbolic calculation were used to find solutions of YTSFE [31]. N-soliton and solutions and homoclinic breather waves of YTSFE were obtained via the HBM and a bilinear Bäcklund transformation [32]. The symmetry reduction method was used to reduce YTSFE to nonlinear ordinary differential equation, which has advantage to provide semi analytical solution [33]. The behavior of mixed two-soliton solutions of the3D-YTSFE with variable coefficients in two-layer liquid medium was shown [34]. Exact traveling wave solutions of the potential-3D-YTSFE by using two methods: namely, improved and modified (G /G) -expansion methods [35]. The periodic-wave solutions for the potential-3D-YTSFE were obtained using the Hirota operator [36]. Also, there is huge development in analytical methods as well as numerical methods in the field of mathematical modeling and applied sciences, see Ref. [37][38][39][40][41][42][43][44][45].
In the present work, we investigate the behavior of the dynamics of waves generated in two-layer heterogeneous liquid which was not studied in the literature. We think that the problem considered here is novel. Here, the extended unified method (EUM) [46][47][48][49][50][51][52] is used to obtain the exact solutions of inhomogeneous-nonautonomous 3D-YTSFE. The existence of the exact solutions is in the sense of existence of solutions in polynomial or rational forms in an auxiliary function. It is worthy to mention that the EUM is an alternative technique to the use of Lie symmetries of nonlinear partial differential equations (NLPDEs).
The outlines of this paper are follows as: Sect. 2 is devoted to the model equation and outlines of the EUM. Polynomial solutions of the 3D-YTSFE are found in Sect. 3. Rational solutions are obtained in Sect. 4. Sect. 5 is devoted to conclusions.

The outlines of the EUM
Consider the coupled nonlinear PDEs We introduce the transformations u(x, y, z, t) U (ξ, z, t), ξ α x + β y, thus Eq. (5) is rewritten, The EUM asserts that the solutions of Eq. (6) are expressed in polynomial and rational forms in an auxiliary function that satisfies suitable auxiliary equations. It is worthy to mention that the EUM can be considered as an alternative technique to the use of Lie group symmetries of NLPDEs. We have found that, in the application, the use of the EUM is of lower time cost than the Lie symmetry in symbolic computations, so we think that it prevails the use of Lie symmetries as the later technique requires a long hierarchy of steps. On the other hand, it provides a wide class of solutions.

Polynomial forms
The polynomial solutions of Eq. (6) are expressed in the form The solutions in Eq. (7) of Eq. (4) exist if there exist integers n, m, and k. Thus, our objective is to show that there exist integersn, m, and k. For achieving this, we use two conditions: the balance and consistency conditions. We consider the case when p 1.By inserting Eq. (7) into (4), we find that the balance condition gives rise to n m 2(k − 1). For the consistency condition, we need to calculate: (i) The number of equations that results from inserting Eq. (7) into Eq. (4) and by setting the coefficients of g(ξ, z, t) i , i 0, 1, 2, .. equal to zero (say r(k)). (ii) The number of arbitrary functions and parameters in Eq. (7), namely c i , a i (z, t), b i (z, t), (say s(k)).
Together with using the condition r (k)−s(k) ≤ q, where q is the highest order derivative in Eq. (4) (q 4) and the last conditions leads to 1 ≤ k ≤ 3.
When p 2, the same results hold. We mention that the solutions of Eq. (7) are hyperbolic functions when p 1.When p 2, they are periodic or elliptic functions.

Rational forms
The rational solutions of Eq. (6) are expressed in the form It is worth mentioning that the use of the EUM is an alternative approach to the use of Lie symmetries for PDEs. The present method is of lower time cost (than Lie symmetries) in symbolic computation, so that it could prevail the application of Lie symmetries which requires certain long hierarchy steps.