# New Exact Solutions of Nonlinear (3 + 1)-Dimensional Boiti-Leon-Manna-Pempinelli Equation

Authors: Mohamed R. Ali; Wen-Xiu Ma
Journal: Advances in Mathematical Physics
 (2019)

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## Abstract

Based on Hirota’s bilinear structure, we evolute a new protuberance type arrangement of the (3+1)-dimensional Boiti-Boiti-Leon-Manna-Pempinelli equation, which depicts nonlinear wave spreads in incompressible fluid. New lump arrangement is built by applying the bilinear strategy and picking appropriate polynomial. Under various parameter settings, this lump arrangement has three sorts of numerous irregularity waves, blended arrangements including lump waves and solitons are additionally developed. Association practices are seen between lump soliton and soliton. Research demonstrates that soliton can somewhat swallow or release lump waves. The shape and highlights for these subsequent arrangements are portrayed by exploiting the three-dimensional plots and comparing shape plots by picking suitable parameters. The physical significance of these charts is given.


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## Body

## 1. Introduction

Numerous analysts in the ongoing years considered numerous kinds of advancement equations portraying distinctive cases in liquid and plasma fields. A wide range of techniques are utilized to examine development equations in (3+1) measurements, for example, Hirota’s strategy [1, 2] exponential function method [3, 4], tanh-coth methods and sine-cosine [5, 6], and numerous different techniques. One of the notable equations is the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equation which depicts the liquid engendering and can be considered as a model for incompressible liquid [7–12]. This equation composes as;(1)vyt+vzt+vxxxy+vxxxz-3vxvxy+vxz-3vxxvy+vz=0This equation presented by Darvishi et al. [7] as an augmentation of the (2+1)-dimensional equation depicts the (2+1)-dimensional connection of the Riemann wave along they-axis with a long wave proliferated along thex-axis. BLMP equation concentrated by alternate points of view, for example, the utilization of the exponential function method [7, 8]. The precise three-wave arrangements and various soliton arrangements were found by applying the Hirota bilinear technique [9–12]. Lump waves, as exceptional nonlinear wave wonders, have been seen in numerous fields [13–18]. It is huge to most likely find and anticipate protuberance-like waves in applications [19–21]. As of late, exploration on the lump arrangements has pulled in an ever-increasing number of considerations [22–29]. Therefore, theoretically, researches on lump waves are helpful to better understand and predict possible extremes for nonlinear evolution systems. Bilinear technique is a powerful representative calculation to develop numerous solitons. Very as of late, Ma et al. stretched out this technique to look for lump arrangements, where its basic thought is to pick appropriate polynomial capacities in bilinear frames [30–38]. We are roused to investigate new dynamical properties through examining irregularity answer for nonlinear frameworks. Here, we examine some of irregularity soliton arrangements, their elements, and the vulnerability of communication with different sorts of arrangements utilizing Hirota strategy for (1). By utilizing the complex strategy with two-term truncated arrangement, gathering the coefficients of the eigenfunction and comparing them to zero, we infer the equivalent ansatz in [39, 40]:(2)vx,y,t,z=-2ln⁡ςx,y,t,zxThat is called Cole-Hopf change, whereς is an assistant or test work that will be assumed later. Beginning by substituting from (2) in (1),(3)4ςxςyςt+4ςxςzςt+4ςxxxςxςy-12ςxxςxςxy+4ςxxxςxςz-12ςxxςxςxz-2ςςyςxxxx-2ςςxtςy-2ςςytςx-2ςςxyςt-2ςςyzςt-8ςxxxyςxς-2ςςtzςx-2ςςtxςz-2ςxtςzς+4ςςxxxςxy+12ςxxyςxςx-2ςςzςxxxx-8ςςxςxxxz+4ςςxxxςxz+12ςxxzςxςx+2ςtxyςς+2ςςςtxz+2ςςςyxxxx+2ςςςzxxxx=0The change expands the nonlinearity; however, it approves us to expect the test work. In [13], Zhang utilized the Hirota bilinearity with Bell polynomials hypotheses to create some lump soliton, lump kink arrangements, and irregularity with one-stripe soliton and with two-stripe lump solitons for (1).


## 2. Lump Soliton Solutions for BLMP Equation

To create lump arrangement, we deem that(4)ς=g2+h2+ε11,g=ε1x+ε2y+ε3t+ε4z+ε5,h=ε6x+ε7y+ε8t+ε9z+ε10.where εi, i=1…11, are genuine obscure that will be resolved consequently. By direct substitution from (4) in (3) and gathering the coefficients of polynomials in x,y,t, and z, we acquire a nonlinear algebraic system in εi; by solving those equations with aid of Maple, we get some sets of solutions as follows: (5)ε1=ε1,ε2=-ε12ε9+ε62ε9+2ε62ε72ε1ε6,ε3=0,ε4=ε9ε12-ε622ε1ε6,ε5=0,ε6=ε6,ε7=ε7,ε8=0,ε9=ε9,ε10=0,ε11=ε11So,(6)ς=ε1x-ε12ε9+ε62ε9+2ε62ε72ε1ε6y+ε9ε12-ε622ε1ε6z2+ε6x+ε7y+ε9z2+ε11Utilizing (2), the solution of (1) has the form(7)v=-22ε1x-ε12ε9+ε62ε9+2ε62ε7/2ε1ε6y+ε9ε12-ε62/2ε1ε6zε1+2ε6x+ε7y+ε9zε6ςSubstituting (5) and (6) in (7) forms lump-kink solution as shown in Figure 1 with ε1=ε9=ε6=2,ε7=1, and ε8=1.Figure 1
Top: 3D plots for (7) at (a) z=0, (b) at z=5, and (c) at z=10. Nethermost: contour plots for (7) at (d) z=0, (e) at z=5, and (f) at z=10.


## 3. Interaction Solutions

### 3.1. Lump Soliton with One-Stripe Wave

Assume that the test work is a confederation of quadratic function with exponential function as follows:(8)ς=g2+h2+ε11+eξ1x+ξ2y+ξ3t+ξ4z+ξ5,g=ε1x+ε2y+ε3t+ε4z+ε5,h=ε6x+ε7y+ε8t+ε9z+ε10.where εi,i=1…11 and ξj,j=1..5, are r genuine obscure constants that will be resolved later on. Utilizing the ansatz in (2),(9)v=22ε1g+2ε6h+k1eξ1x+ξ2y+ξ3t+ξ4z+ξ5ςComplicated algebraic system is driven by substituting from (8) in (3) and collecting the coefficients of polynomials in x,y,t, and z. We solve the constructed system utilizing Maple and snaffle the following assortment of solutions: (10)ε1=0,ε2=-ε4,ε3=ε3,ε4=ε4,ε5=ε5,ε6=0,ε7=-ε9,ε8=ε8,ε9=ε9,ε10=ε10,ε11=-ε3ε5ε102ε5ε3+ε8ε10,ξ1=ξ1,ξ2=-ξ4,ξ3=-ξ13,ξ4=ξ4,ξ5=ξ5To provide the singularity and promote the wave to localize in all directions, the following stipulation must be possessed in consideration:(11)ξ1ε1ε6≠0Substituting from (10) in (8),(12)ς=-ε4y+ε3t+ε4z+ε52+-ε9y+ε8t+ε9z+ε102-ε3ε5ε102ε5ε3+ε8ε10+eξ1x+ξ4y-ξ13t+ξ4z+ξ5From (12) and (9), we produce lump arrangement with stripe (solitary wave) arrangement. By taking an estimations of arbitrary constants as ε1=1,ε4=0,ε7=ε6=1,ε7=3,ε5=1,ε10=0,ε8=2, and ξ1=ξ2=1, we plot the outcomes in Figure 2 for various estimations of y.Figure 2
Upper plots: 3D plots for (9) at (a) y=0, at (b) y=2, and at (c) y=5 and z=0. Contour plots (nethermost plots) for (9) at (d) y=0, at (e) y=2, and at (f) y=5 and z=0.


### 3.2. Interaction of Lump Solution with Rough Wave (Two-Stripe Solitons)

We presume that the new ansatz is a collection of quadratic function and hyperbolic function as follows:(13)ς=g2+h2+ε11+cosh⁡ξ1x+ξ2y+ξ3t+ξ4z+ξ5,g=ε1x+ε2y+ε3t+ε4z+ε5,h=ε6x+ε7y+ε8t+ε9z+ε10.Switching from (13) in (2), we embezzle a variety arrangement of answers for (1):(14)v=22ε1x-ε4y+ε4zε1+sinh⁡ξ1x-ξ4y-ξ13t+ξ4z+ξ5ξ1ςMore intense computations were finished utilizing Maple software to get the obscure constants in accordance with representation form (13) in (3) and emulate the coefficients of x,y,t, and z to zero. Settling the subsequent nonlinear framework produces a few instances of the compelled parameters. For each situation, we back substitute in (13) as follows:(15)ε1=ε1,ε2=-ε4,ε3=0,ε4=ε4,ε5=0,ε6=0,ε7=ε7,ε8=0,ε9=ε9,ε10=ε10,ε11=ε11,ξ1=ξ1,ξ2=-ξ4,k3=-ξ13,ξ4=ξ4,ξ5=ξ5Substituting from (15) into (13),(16)ς=ε1x-ε4y+ε4z2+ε7y+ε9z+ε102+ε11+cosh⁡ξ1x-ξ4y-ξ13t+ξ4z+ξ5Through a similar system, we get the arrangements of (1) and plot in Figure 3 for ε2=-1,ε6=1,ε7=2,ε5=1,ε10=0,ε8=1,ε11=1,ξ1=1,ξ2=1,ξ5=1.Figure 3
Upper plots: 3D plots for (14) at (a) t=0, at (b) t=1, and at (c) t=2 and z=0. Contour plots (nethermost plots) for (14) at (d) t=0, at (e) t=1, and at (f) t=2 and z=0.


## 3.1. Lump Soliton with One-Stripe Wave

Assume that the test work is a confederation of quadratic function with exponential function as follows:(8)ς=g2+h2+ε11+eξ1x+ξ2y+ξ3t+ξ4z+ξ5,g=ε1x+ε2y+ε3t+ε4z+ε5,h=ε6x+ε7y+ε8t+ε9z+ε10.where εi,i=1…11 and ξj,j=1..5, are r genuine obscure constants that will be resolved later on. Utilizing the ansatz in (2),(9)v=22ε1g+2ε6h+k1eξ1x+ξ2y+ξ3t+ξ4z+ξ5ςComplicated algebraic system is driven by substituting from (8) in (3) and collecting the coefficients of polynomials in x,y,t, and z. We solve the constructed system utilizing Maple and snaffle the following assortment of solutions: (10)ε1=0,ε2=-ε4,ε3=ε3,ε4=ε4,ε5=ε5,ε6=0,ε7=-ε9,ε8=ε8,ε9=ε9,ε10=ε10,ε11=-ε3ε5ε102ε5ε3+ε8ε10,ξ1=ξ1,ξ2=-ξ4,ξ3=-ξ13,ξ4=ξ4,ξ5=ξ5To provide the singularity and promote the wave to localize in all directions, the following stipulation must be possessed in consideration:(11)ξ1ε1ε6≠0Substituting from (10) in (8),(12)ς=-ε4y+ε3t+ε4z+ε52+-ε9y+ε8t+ε9z+ε102-ε3ε5ε102ε5ε3+ε8ε10+eξ1x+ξ4y-ξ13t+ξ4z+ξ5From (12) and (9), we produce lump arrangement with stripe (solitary wave) arrangement. By taking an estimations of arbitrary constants as ε1=1,ε4=0,ε7=ε6=1,ε7=3,ε5=1,ε10=0,ε8=2, and ξ1=ξ2=1, we plot the outcomes in Figure 2 for various estimations of y.Figure 2
Upper plots: 3D plots for (9) at (a) y=0, at (b) y=2, and at (c) y=5 and z=0. Contour plots (nethermost plots) for (9) at (d) y=0, at (e) y=2, and at (f) y=5 and z=0.


## 3.2. Interaction of Lump Solution with Rough Wave (Two-Stripe Solitons)

We presume that the new ansatz is a collection of quadratic function and hyperbolic function as follows:(13)ς=g2+h2+ε11+cosh⁡ξ1x+ξ2y+ξ3t+ξ4z+ξ5,g=ε1x+ε2y+ε3t+ε4z+ε5,h=ε6x+ε7y+ε8t+ε9z+ε10.Switching from (13) in (2), we embezzle a variety arrangement of answers for (1):(14)v=22ε1x-ε4y+ε4zε1+sinh⁡ξ1x-ξ4y-ξ13t+ξ4z+ξ5ξ1ςMore intense computations were finished utilizing Maple software to get the obscure constants in accordance with representation form (13) in (3) and emulate the coefficients of x,y,t, and z to zero. Settling the subsequent nonlinear framework produces a few instances of the compelled parameters. For each situation, we back substitute in (13) as follows:(15)ε1=ε1,ε2=-ε4,ε3=0,ε4=ε4,ε5=0,ε6=0,ε7=ε7,ε8=0,ε9=ε9,ε10=ε10,ε11=ε11,ξ1=ξ1,ξ2=-ξ4,k3=-ξ13,ξ4=ξ4,ξ5=ξ5Substituting from (15) into (13),(16)ς=ε1x-ε4y+ε4z2+ε7y+ε9z+ε102+ε11+cosh⁡ξ1x-ξ4y-ξ13t+ξ4z+ξ5Through a similar system, we get the arrangements of (1) and plot in Figure 3 for ε2=-1,ε6=1,ε7=2,ε5=1,ε10=0,ε8=1,ε11=1,ξ1=1,ξ2=1,ξ5=1.Figure 3
Upper plots: 3D plots for (14) at (a) t=0, at (b) t=1, and at (c) t=2 and z=0. Contour plots (nethermost plots) for (14) at (d) t=0, at (e) t=1, and at (f) t=2 and z=0.


## 4. Conclusions

In this work, we constructed lump solutions and mixed solution involving lump waves and solitons for the incompressible fluid system (1) via bilinear method and symbolic computation. Starting form Cole-Hopf transformation which is that investigated by Singular Manifold method with two-term truncated series, we drive some of the new and novel lump-solitons: lump-kink, lump interacted with one-stripe soliton or kink and interacted lump with two-stripe soliton, or kink wave after many complicated calculations utilizing Maple software. The three-dimensional plots and contour plots are presented for all solutions that we obtained. Via our intensive search, there is no one to investigate these types of solutions for (1). Our work is important for an interaction between lumps and to better get these frameworks.



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*Source: 9801638-2019-06-03.xml*