# Exact Solution of (4+1)-Dimensional Boiti–Leon–Manna–Pempinelli Equation

Authors: Qili Hao
Journal: Advances in Mathematical Physics
 (2023)

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## Abstract

Based on the Hirota bilinear method, using the heuristic function method and mathematical symbolic computation system, various exact solutions of the (4+1)-dimensional Boiti–Leon–Manna–Pempinelli equation including the block kink wave solution, block soliton solution, periodic block solution, and new composite solution are obtained. Upon selection of the appropriate parameters, three-dimensional and contour diagrams of the exact solution were generated to illustrate their properties.


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## Body

## 1. Introduction

Nonlinear integrable systems are an important research topic in the field of mathematics and have received significant attention over the recent years. Additionally, several exact solutions of nonlinear integrable systems have been obtained using different methods, including the Hirota bilinear method, Backlund transform [1], and Darboux transform [2]. Although many studies on various reduced solutions of the nonlinear evolution equation have been conducted [3–5], N-soliton solutions have been studied systematically by the Hirota method and Riemann-Hilbert problems for nonlocal integrable equations. This study primarily focuses on the (4+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation [6]:
(1)μyt+μzt+μst+h1μxxxy+h1μxxxzh1μxxxs+h2μxμxy+μxz+μxs=0whereμis an undetermined real function with respect to x, y, and z and t, h1, and h2 are arbitrary constants. In [1], Equation (1) is considered an integrable nonlinear evolution equation, and the N-soliton, periodic soliton, and mixed block torsional wave solutions of Equation (1).Based on the Hirota bilinear method, this paper uses the heuristic function method and Mathematics symbolic computation system to obtain the exact and complex solutions of Equation (1).


## 2. New Exact Solution of Equation (1)

To obtain the general bilinear form of Equation (1), we use the following theorem [6].Theorem 1.
Iff=fx,y,z,t,s and g=gx,y,z,t,s satisfy the following two conditions:
(2)ify+fz+fs=0,iigy+gz+gs=0.Forf/g>0 and g≠0, the general bilinear form μ=lnf/gx=fxg−fgx/fg is the exact solution of Equation (1).Proof.
μ can be denoted as μ=wx, where w=lnf/g,w=wx,y,z,t,s. Substituting the left side of Equation (1) directly, we obtain the following:
(3)LHS=wy+wz+wst+h1wy+wz+wsxxx+h2wxxwy+wz+wsxx.
Then, from conditions (i) and (ii), the following can be obtained:(4)wy+wz+ws=fy+fz+fsg−fgy+gz+gsfg=0.
Since the left side of Equation (1) is 0, the theorem is proved.To obtain the bulk torsional wave solution of Equation (1), we chose an appropriate test function, as shown in
(5)f=∑i=1Mfi2+A0,g=∑j=1Mkjexpgj+B0,fi=aix+biy+ciz+dit+eis,gj=mjx+njy+pjz+qjt+rjs,where A0,B0,ai,bi,ci,di,ei,i=1,⋯,M,mj,nj,pj,qj,rj,kj,j=1,⋯,N denote arbitrary parameters. Using Theorem 1 and conditions (i) and (ii), the following five cases can be obtained:
(6)Condition1:bi=−ci−ei,ai=ai,ci=ci,di=di,ei=ei,A0=Ao,nj=−pj−rj,mj=mj,pj=pj,qj=qj,rj=rj,B0=Bo,kj=kj,(7)Condition2:bi=−ci−ei,ai=ai,ci=ci,di=di,ei=ei,A0=Ao,pj=−nj−rj,mj=mj,nj=nj,qj=qj,rj=rj,B0=Bo,kj=kj,(8)Condition3:bi=−ci−ei,ai=ai,ci=ci,di=di,ei=ei,A0=Ao,rj=−nj−pj,mj=mj,nj=nj,pj=pj,qj=qj,B0=Bo,kj=kj,(9)Condition4:ci=−bi−ei,ai=ai,bi=bi,di=di,ei=ei,A0=Ao,nj=−pj−rj,mj=mj,pj=pj,qj=qj,rj=rj,B0=Bo,kj=kj,(10)Condition5:ci=−bi−ei,ai=ai,bi=bi,di=di,ei=ei,A0=Ao,pj=−nj−rj,mj=mj,nj=nj,qj=qj,rj=rj,B0=Bo,kj=kj.Substituting Equation (6) into Equation (5), the bulk torsional wave solution of Equation (1) can be obtained as follows:
(11)μ1=∑i=1M2aifi∑j=1Nkjexpgi+B0−∑i=1Mfi2+A0∑j=1Nkjmjexpgi∑i=1Mfi2+A0∑j=1Nkjexpgi+B0,fi=aix−ci+eiy+ciz+dit+eis,gj=mjx−pj+rjy+pjz+qjt+rjs.The 3D and contour plots of the solutionμ1 for M=2 and N=3 are shown in Figure 1.Figure 1
The parameters ofμ1 are A0=B0=1,a1=c1=d1=e1=2,a2=c2=d2=e2=1,m1=n1=q1=r1=3,m2=n2=q2=r2=2,m3=n3=q3=r3=2,y=t=s=0. (a) Three-dimensional map. (b) Contour map.
(a)(b)To obtain the bulk soliton solution of Equation (1), we chose an appropriate test function, as follows:
(12)f=∑i=1Mfi2+A0,g=∑j=1Mkjsechgj+B0,fi=aix+biy+ciz+dit+eis,gj=mjx+njy+pjz+qjt+rjs,where A0,B0,ai,bi,ci,di,ei,i=1,⋯,M,mj,nj,pj,qj,rj,kj,j=1,⋯,N denote arbitrary parameters. Using Theorem 1 and conditions (i) and (ii), the following four cases can be obtained:
(13)Condition1:ei=−bi−ci,ai=ai,bi=bi,ci=ci,di=di,A0=Ao,nj=−pj−rj,mj=mj,pj=pj,qj=qj,rj=rj,B0=Bo,kj=kj,(14)Condition2:ei=−bi−ci,ai=ai,bi=bi,ci=ci,di=di,A0=Ao,pj=−nj−rj,mj=mj,nj=nj,qj=qj,rj=rj,B0=Bo,kj=kj,(15)Condition3:ei=−bi−ci,ai=ai,bi=bi,ci=ci,di=di,A0=Ao,rj=−nj−pj,mj=mj,nj=nj,pj=pj,qj=qj,B0=Bo,kj=kj,(16)Condition4:ei=−bi−ci,ai=ai,bi=bi,ci=ci,di=di,A0=Ao,nj=−pj−rj,mj=mj,pj=pj,qj=qj,rj=rj,B0=Bo,kj=kj.Substituting Equation (13) into Equation (12), the block soliton solution of Equation (1) can be obtained, as follows:
(17)μ2=∑i=1M2aifi∑j=1Nkjsechgi+B0−∑i=1Mfi2+A0∑j=1Nkjmjsechgitanhgi∑i=1Mfi2+A0∑j=1Nkjsechgi+B0,fi=aix+biy+ciz+dit−bi+cis,gj=mjx−pj+rjy+pjz+qjt+rjs.The 3D and contour plots of the solutionμ2 when M=2 and N=3 are shown in the figure (see Figure 2).Figure 2
The parameters ofμ2 are A0=B0=a1=c1=b2=c2=d2=q1=r1=r2=m2=m3=q3=1,b1=d1=a2=m1=n1=n2=q2=n3=r3=2,y=z=t=0. (a) Three-dimensional map. (b) Contour map.
(a)(b)


## 3. Compound Novel Solution to Equation (1)

Using the transformation in reference [1], we obtain the following relation:
(18)μ=6h1h2lnfx.Accordingly, Equation (1) can be written in the following bilinear form:
(19)DyDt+DzDt+DsDt+h1Dx3Dy+Dx3Dz+Dx3Dsf⋅f=0.To obtain the periodic block solution for Equation (1), the following equation was chosen as the test function:
(20)f=∑i=1Mkicoshfi+∑j=1Mljcosgj+A0,fi=aix+biy+ciz+dit+eis,gj=mjx+njy+pjz+qjt+rjs,where A0,B0,ai,bi,ci,di,ei,i=1,⋯,M,mj,nj,pj,qj,rj,kj,j=1,⋯,N denote arbitrary parameters. Substituting Equation (20) into bilinear Equation (19), the following five conditions are obtained:
(21)Condition1:di=−4h1ai3,bi=−ci−ei,ai=ai,ci=ci,ei=ei,ki=ki,A0=Ao,qi=4h1mj3,nj=−pj−rj,mj=mj,pj=pj,rj=rj,lj=lj,(22)Condition2:di=−4h1ai3,ci=−bi−ei,ai=ai,bi=bi,ei=ei,ki=ki,A0=Ao,qi=4h1mj3,nj=−pj−rj,mj=mj,pj=pj,rj=rj,lj=lj,(23)Condition3:di=−4h1ai3,ei=−bi−ci,ai=ai,bi=bi,ci=ci,ki=ki,A0=Ao,qi=4h1mj3,nj=−pj−rj,mj=mj,pj=pj,rj=rj,lj=lj,(24)Condition4:di=−4h1ai3,bi=−ci−ei,ai=ai,ci=ci,ei=ei,ki=ki,A0=Ao,qi=4h1mj3,pj=−nj−rj,mj=mj,nj=nj,rj=rj,lj=lj,(25)Condition5:di=−4h1ai3,ci=−bi−ei,ai=ai,bi=bi,ei=ei,ki=ki,A0=Ao,qi=4h1mj3,pj=−nj−rj,mj=mj,pj=pj,rj=rj,lj=lj.Substituting Equation (21) into Equation (20), the periodic block solution of Equation (1) is obtained as follows:
(26)μ3=6h1h2∑i=1Mkiaisinhfi+∑j=1Nljmjsingi∑i=1Mkjcoshfi+∑j=1Nljcosgi+A0,fi=aix−ci+eiy+ciz−4h1a13t+eis,gj=mjx−pj+rjy+pjz+4h1mj3t+rjs.The 3D and contour plots of the solutionμ3 for M=3 and N=2 are shown in Figure 3.Figure 3
The parameters ofμ3 are a1=−1,e3=−2,r1=p2=c3=−3,h1=h2=e1=c2=a2=r2=k1=k2=k3=1,c1=a3=p1=l1=l2=2,e2=m1=m2=3,A0=z=t=s=0. (a) Three-dimensional map. (b) Contour map.
(a)(b)To obtain the novel compound solution of Equation (1), the following equation was chosen as the test function:
(27)f=∑i=1Mkitanhfi+∑j=1Mljexpgj+A0,fi=aix+biy+ciz+dit+eis,gj=mjx+njy+pjz+qjt+rjs.where A0,ai,bi,ci,di,ei,kii=1,⋯,M,mj,nj,pj,qj,rj,lj,j=1,⋯,N denote arbitrary parameters. Substituting Equation (27)) into bilinear Equation (19)), the following 4 conditions are obtained:
(28)Condition1:di=−4h1ai3,ei=−bi−ci,ai=ai,bi=bi,ci=ci,ki=ki,A0=Ao,qi=4h1mj3,rj=−pj−nj,mj=mj,pj=pj,nj=nj,lj=lj,(29)Condition2:di=−4h1ai3,bi=−ci−ei,ai=ai,ci=ci,ei=ei,ki=ki,A0=Ao,qi=4h1mj3,rj=−pj−nj,mj=mj,pj=pj,nj=nj,lj=lj,(30)Condition3:di=−4h1ai3,ci=−bi−ei,ai=ai,bi=bi,ei=ei,ki=ki,A0=Ao,qi=−4h1mj3,rj=−pj−nj,mj=mj,pj=pj,nj=nj,lj=lj,(31)Condition4:di=−4h1ai3,ei=−bi−ci,ai=ai,bi=bi,ci=ci,ki=ki,A0=Ao,qi=−4h1mj3,pj=−nj−rj,mj=mj,nj=nj,rj=rj,lj=lj.Substituting Equation (28) into Equation (27), the novel compound solution of Equation (1) is obtained as follows:
(32)μ4=6h1h2∑i=1Mkiaitanhfisech2fi+∑j=1Nljmjexpgi∑i=1Mkjtanh2fi+∑j=1Nljexpgi+A0,fi=aix+biy+ciz−4h1a13t−bi+cis,gj=mjx+njy+pjz−4h1mj3t−pj+njs.The 3D and contour plots of the solutionμ4 for M=3 and N=2 are shown in Figure 4.Figure 4
The parameters ofμ4 are a1=c2=−1,a2=n2=−3,n1=−4,h1=h2=b2=c3=m1=p1=k1=k2=k3=l1=l2=1,b1=p2=c1=2,a3=b3=m2=3,A0=z=t=s=0. (a) Three-dimensional map. (b) Contour map.
(a)(b)


## 4. Conclusion

In this study, based on the Hirota bilinear method and heuristic function method, using the Mathematica symbolic computation system, the exact solution of the (4+1)-dimensional Boiti–Leon–Manna–Pempinelli equation is studied. The block kink wave, block soliton, periodic block, and new composite solutions were obtained. Compared with reference [1], this study obtained multiple explicit solutions of the (4+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. The dynamic properties of these solutions were analyzed by selecting different parameters. The evolution of these waves is observed using the Mathematica symbolic computation system. The results obtained in this study facilitate the understanding of the propagation processes of nonlinear waves.



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