Abundant Lump-Type Solutions and Interaction Solutions for a Nonlinear (3+1) Dimensional Model

1Department of Mathematics and Physics, Faculty of Engineering, Zagazig University, Egypt 2Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA 3Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia 4College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, China 5International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa


Introduction
It is very important to control the physical mechanisms of rough waves and interaction waves specially with lumptype waves. The significance of nonlinear waves of these types appears from natural disasters. Many physical phenomena need analytical approaches to classify the physical dynamics of nonlinear evaluation equations. The Darboux transformation (DT) and the Lie symmetry (LS) method [1][2][3] are efficient approaches to obtaining closed-form solutions. However, some problems occur in applying those methods, such as how to find Lax pairs in the DT method and how to carry out the back-substitution procedure in the LS method. There are also new types of closed-form solutions, for example, positions and complexions [4][5][6][7][8], and even new collision phenomena including fissions and fusions [9][10][11][12][13][14]. The Hirota bilinear method plays an influential role in discovering all the mentioned types of solutions to overcome a lot of analytic problems. Most studies apply the Hirota method to completely integrability nonlinear problems as in [10,[15][16][17][18][19][20][21][22][23]. We would like to demonstrate that the Hirota method can be used to explore various types of closed-form solutions: interaction solutions of lumps with solitons, kinks, line-solitons, resonance solutions, and one-or two-stripe solitons; and two classes of breather solutions (time periodic or space periodic solutions). Our analysis will show that those solutions can predicate the characteristics and physical significance of nonlinear problems.
Consider the following generalized (3+1) SWL equation [24,25]: There are a few studies on this equation. For example, Tian et al. in [26] generated a traveling wave solution via the tanh method. In 2010, Zayed [27] used ℎ 耠 / method to obtain some traveling wave solutions by reducing the independent variables using the linear D'lambert transformation.
In what follows, we investigate lump soliton solutions and their dynamics and the susceptibility of their interactions 2 Advances in Mathematical Physics with other types of solutions using the Hirota method for (1). By using the singular manifold method (SMM) with twoterm truncated series, one derives the same ansatz in [24,25] ( , , , ) = 2( ln ( ( , , , )) 푥 .
(2) This is called the Cole-Hopf transformation, where is an auxiliary or test function that will be determined later. Starting by substituting (2) into (1), one gets The transformation increases the nonlinearity but allows us to work with the test function. In [24], Zhang used Bell polynomial theories to generate lump-kink solutions, lumps with one-stripe solitons and lumps with two-stripe solitons for (1), but he supposed that = to minimize the number of independent variables and so studied the equation in a (2+1)dimensional domain.

Lump Soliton Solutions
To generate single-lump solutions, we suppose that where 푖 , = 1 . . . .11, are real unknowns that will be found subsequently. We carry out a direct substitution of (4) into (3) and gather the coefficients of the resulting polynomial in , , , and , to obtain a nonlinear algebraic system in 푘 . By solving this system of nonlinear algebraic equations with the aid of Maple, we acquire some sets of solutions for the parameters. Avoiding the redundancy, we surpass one studying case as follows: , 10 = 0, Using the aggregation equation (4), one can represent the auxiliary function as where û = 1 2 By using (2), the solution of (1) has the form Incorporating (6) and (5) into (8), one gets a class of lump solutions of (1) depicted in Figure 1.

Interaction Solutions
. . Lump Solitons with One-Stripe Waves. Suppose that the test function is a confederation of a quadratic function withan  exponential function as follows: = 2 + 2 + 11 + , = 1 + 2 + 3 + 4 + 5 , where 푖 , = 1 . . . .11 and 푗 , = 1..5, are real unknown constants that will be determined subsequently. Using the ansatz in (2), 4 Advances in Mathematical Physics Inserting (9) into (3), gathering the coefficients of the resulting polynomial in , , , and , and equaling these coefficients to zero, we explore a complicated algebraic system on the unknown constants. We then solve the obtained system using Maple and snaffle the following assortment of solutions: To avoid the singularity and promote the wave to localize in all directions, the following stipulation must be taken into consideration: Substituting (11) into (9), we obtain Introducing (13) into (10), we generate a class of interaction solutions with stripe soliton (solitary wave) solutions. The results have been plotted in Figure 2 for different values of times.
Through the same procedure, we get a class of solutions of (1) and plot a special solution in Figure 3.

Conclusions
Starting from the Cole-Hopf transformation, investigated in the SMM with a two-term truncated series, we derived novel lump solitons, lump-kinks, interacted lumps with onestripe solitons or kinks, and interacted lumps with two-stripe solitons or kink waves, after some complicated calculations using the Maple software. The presented three-dimensional plots of the interaction solutions show that the lump solitons are coalesced or spliced up by the stripe solitons. To the best of our knowledge, those types of solutions for (1) are presented for the first time. Our solutions are localized in the four dimensional space ( , , , and ), but in [24], the authors assumed that = and generated only one lump solution.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The authors declare that they have no conflicts of interest.