Research ArticleMultiple Lump Solutions of the (4 + 1)-Dimensional FokasEquation

Hongcai Ma ,1,2 Yunxiang Bai ,1,2 and Aiping Deng1,2

1Department of Applied Mathematics, Donghua University, Shanghai 201620, China2Institute for Nonlinear Sciences, Donghua University, Shanghai 201620, China

Correspondence should be addressed to Hongcai Ma; hongcaima@hotmail.com and Yunxiang Bai; baiyunxiang429@163.com

Received 13 March 2020; Revised 6 May 2020; Accepted 11 May 2020; Published 10 June 2020

Academic Editor: Ming Mei

Copyright © 2020 Hongcai Ma et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, we investigate multiple lump wave solutions of the new (4 + 1)-dimensional Fokas equation by adopting asymbolic computation method. We get its 1-lump solutions, 3-lump solutions, and 6-lump solutions by using its bilinearform. Moreover, some basic characters and structural features of multiple lump waves are explained by depicting thethree-dimensional plots.

1. Introduction

Nonlinear evolution equations can be used to simulate manynonlinear phenomena in the real world, which appear inmany areas, especially in physical [1], engineering sciences[2], applied mathematics [3], chemistry, and biology [4].Recently, it is well known that rogue waves play an essentialrole in helping us apprehending the qualitative propertiesof many phenomena; it is interesting that lump functionscan provide approximate fitting prototypes to model roguewaves. In addition to appearing in the ocean [5], lump wavesalso actually appear in many other fields, such as atmosphere[6], superfluids [7], and capillary waves [8]. Further study oflump waves will help us interpret some unknown fields moredeeply. Certain ways have been arranged to solve the lumpwave solutions of some equations; they are inclusive of theHirota bilinear method [9, 10], the inverse scattering trans-formation [11], the Darboux transformation [12], the Bäck-lund transformation [13], the functional variable method[14], the reduced differential transform method [15], and soon. Many integrable equations which have lump wave solu-tions are enumerated here, for example, the (3 + 1)-dimen-sional KPI equation [16], the Davey-Stewartson I equation[17], the (3 + 1)-dimensional nonlinear evolution equation

[18, 19], and the nonlinear Schrödinger equation [20]. Gen-erally speaking, it is easier to solve the lower order rationalsolutions than to solve the multiple lump waves of the non-linear evolution equation. In this paper, we mainly work ona (4 + 1)-dimensional Fokas equation.

4utx − uxxxy + uxyyy + 12uxuy + 12uuxy − 6uzw = 0, ð1Þ

which was first derived by Fokas by the generalization of twocritical nonlinear evolution equations, which are the integra-ble KP equation and DS equation [21]. The (4 + 1)-dimen-sional Fokas equation could be applied to portraynonelastic and elastic interactions [21, 22]. In nonlinear wavetheory, KP and DS equations can be used to characterize thesurface waves and internal waves in straits or channels ofvarying depth and width, respectively [23–26]. The signifi-cance of the (4 + 1)-dimensional Fokas equation follows nat-urally from the physical applications of the KP and DSequations. Therefore, the (4 + 1)-dimensional Fokas equa-tion could be adopted to represent a number of phenomenain fluid mechanics, optical fiber communications, ocean engi-neering, and many others. More recently, (4 + 1)-dimensionalFokas equation has been discussed by some scholars. Demirayet al. obtained the exact solutions of Equation (1) by applying

HindawiAdvances in Mathematical PhysicsVolume 2020, Article ID 3407676, 7 pageshttps://doi.org/10.1155/2020/3407676

the generalized Kudryashov method (GKM) [27]. Based onthe Hirota bilinear form, two classes of lump-type solutionsof Equation (1) are studied by Cheng and Zhang [28]. In2017, two distinct methods, namely, the modified simpleequation method (MSEM) and the extended simplest equa-tion method (ESEM), are employed to look for exact trav-eling wave solutions of Equation (1) [29]. Zhang et al.used the fractional subequation method to obtain the exactanalytical solutions of Equation (1) [30]. The lump-bellsolutions of Equation (1) are obtained based on the bilinearequation and different test functions in [31]. Particularly,El-Ganaini and Al-Amr discussed the space-time fractional(4+1)-dimensional Fokas equation via the functional vari-able, the generalized Kudryashov, the Jacobi elliptic func-tion expansion, and the generalized Riccati equationmapping methods and got abundant distinct types of newexact solutions [32]. Zhang and Xia obtained soliton solu-tions, fissionable wave solutions, M-lump solutions, andinteraction solutions of the (4 + 1)-dimensional Fokas equa-tion based on the Hirota bilinear method [33]. However,Zhaqilao proposed a novel method to construct the multi-ple rogue wave solutions of nonlinear partial differentialequation; the multiple rogue wave solutions of Equation(1) have not been extracted by this new method. This paperis constructed as follows. In Section 2, the bilinear equationof Equation (1) is acquired. The 1-lump waves are alsogained by employing a new ansatz. In Section 3, 3-lumpwaves of Equation ((1)) are researched when the subscriptof f n is equal to 1 in Equation (10). In Section 4, the 6-lump waves of Equation (1) are studied when the subscriptof f n is equal to 2 in Equation (10). Section 5 is devoted toa short conclusion and discussion.

2. 1-Lump Solutions

The fundamental desire of this section is to investigate the 1-lump solutions of the new (4 + 1)-dimensional Fokasequation. Firstly, setting X = x +my + nt and Z = z + cw inEquation (1) yields

4nuXX + m3 −m� �

uXXXX + 12mu2X + 12muuXX − 6cuZZ = 0,ð2Þ

wherem, n, c are all real parameters. With the help of variabletransformation

u = u0 + m2 − 1� �

lnfð ÞXX , ð3Þ

we can convert the Equation (2) into a bilinear form thatreads

4nD2X + m3 −m� �

D4X − 6cD2

Z

� �f · f = 0, ð4Þ

where D2X f · f = f f XX − f 2X , D4

X f · f = 3f 2XX − 4f X f XXX + ff XXXX , and D2

Z f · f = f f ZZ − f 2Z , where f is a real functionwith regard to variable X, Z.D2

X ,D4X , andD

2Z are called Hirota

bilinear D operators. By applying the symbolic computationapproach, assuming

f = X − βð Þ2 + a1 Z − αð Þ2 + a0, ð5Þ

where α, β, a1, and a0 are constants to be determined.Substituting (5) into (4) and equating the coefficients of

all powers of Xi1Zi2 to 0, one has

12ca21 + 8ma1 = 0,−24cαa21 − 16mαa1 = 0,

−12ca1 − 8m = 0,24βca1 + 16βm = 0,

12a21α2c + −12β2 − 12a0� �

c + 8mα2� �

a1

+ −8β2 + 8a0� �

m + 12n3 − 12n = 0:

ð6Þ

Solving these equations, one has

a0 = −3n n2 − 1� �

4m ,

a1 = −2m3c :

ð7Þ

Therefore, we can get a solution of Equation (4) as

f = X − βð Þ2 − 2m Z − αð Þ23c −

3n n2 − 1� �

4m : ð8Þ

By using variable transformation (3), the 1-lump wavesolutions of Equation (1) read

where X = x +mt, Z = z + cw,m, n, c, α, β are arbitrary realconstants. The 1-lump wave (9) has the structure for threewave peaks. One peak is higher than the water level, and

the other two are opposite. Figure 1 presents the three dimen-sional plot, the density plot, and the corresponding contourplot of the 1-lump wave solution of Equation (1). From

u x, y, z, t,wð Þ = m2 − 1� � 2

X − βð Þ2 − 2m Z − αð Þ2� �/3c

� �− 3n n2 − 1ð Þð Þ/4mð Þ −

2X − 2βð Þ2X − βð Þ2 − 2m Z − αð Þ2� �

/3c� �

− 3n n2 − 1ð Þð Þ/4mð Þ� �2

!

,

ð9Þ

2 Advances in Mathematical Physics

Figure 1, we can see that the 1-lump wave has one center ðβ, αÞ.Furthermore, at the point ððð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

−3mnðn2 − 1Þp+ 2mβÞ/2mÞ,

αÞ in the plane ðX, ZÞ, the maximum amplitude of the 1-lump wave is ðu0 − ðð2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

−3mnðn2 − 1Þp Þ/ð3nðn2 − 1ÞÞÞÞ.

3. 3-Lump Solutions

In the section, in order to construct the multiple lump solu-tions, we propose the notation just like this:

Fn X, Zð Þ = 〠n n+1ð Þ/2

k=0〠k

i=0an n+1ð Þ−2k,2iZ

2iXn n+1ð Þ−2k,

Pn X, Zð Þ = 〠n n+1ð Þ/2

k=0〠k

i=0bn n+1ð Þ−2k,2iX

2iZn n+1ð Þ−2k,

Qn X, Zð Þ = 〠n n+1ð Þ/2

k=0〠k

i=0cn n+1ð Þ−2k,2iZ

2iXn n+1ð Þ−2k,

f X, Zð Þ = f n+1 X, Zð Þ = Fn+1 X, Zð Þ + 2αZPn X, Zð Þ+ 2βXQn X, Zð Þ + α2 + β2� �

,ð10Þ

where ai,j, bi,j, and ci,j are arbitrary constants. Then, we taken = 1,

f X, Zð Þ = F2 X, Zð Þ + 2αZP1 X, Zð Þ+ 2βXQ1 X, Zð Þ + α2 + β2� �

,ð11Þ

where

F2 X, Zð Þ = X6 + a4,0X4 + a4,2Z

2X4

+ a2,0 + a2,2Z2 + a2,4Z

4� �X2

+ a0,0 + a0,2Z2 + a0,4Z

4 + a0,6Z6,

P1 X, Zð Þ = b0,0 + b0,2X2 + b2,0Z

2,Q1 X, Zð Þ = c0,0 + c0,2Z

2 + c2,0X2:

ð12Þ

Substituting Equation (11) into Equation (4) and collect-ing all the coefficients of Xi1Zi2 , we can get a group of con-straining relationships for the parameters. Dealing withthese equations, one gets

–20–1001010

–10–200

0.51

1.52

2.5

0

20x

z

(a)

z

x

–20

–10

–10 0 10 20

0

10

20

(b)

–10 –5

–4–3–2

0

1234

5 10N

(c)

Figure 1: The 1-lump solution u of Equation (1) with the parameter selectionsm = −2, n = 2, c = 1, u0 = 0, α = 0, β = 0. (a) Perspective view ofthe wave uðX, ZÞ, (b) overhead view of the wave uðX, ZÞ, and (c) the corresponding contour plot uðX, ZÞ.

–20–20

02468

–10–10 00 1010 20 xz

(a)

z

x

–20

–10

–10 0 10 20

0

10

20

(b)

–15 –5

–8–6–4–20

2468

5 10

Z

X

(c)

Figure 2: The 3-lump solution u of Equation (1) with the parameter selections m = −3, n = 2, c = 1, u0 = 0, α = 1000, β = 1000, c2,0 = 1,b0,2 = 1. (a) Perspective view of the wave uðX, ZÞ, (b) overhead view of the wave uðX, ZÞ, and (c) the corresponding contour plotuðX, ZÞ.

3Advances in Mathematical Physics

where b0,2 and c2,0 are arbitrary constants. Thus, the 3-lumpsolution of Equation (1) is shown by

u = u0 + m2 − 1� �

lnfð ÞXX , ð14Þ

where f is given in Equation (11), in which X = x +my + nt,Z = z + cw. By increasing the value of α and β, 3-lump wavemerge and their centers form a triangle (see Figure 2). The3-lump wave is the arrangement of three 1-lump waves inthe plane ðX, ZÞ:

4. 6-Lump Solutions

To get the 6-lump solution of Equation (1), the 6-lump solu-tion waves of a (4 + 1)-dimensional Fokas equation can be

presented if we choose n = 2,

f X, Zð Þ = F3 X, Zð Þ + 2αZP2 X, Zð Þ + 2βXQ2 X, Zð Þ+ α2 + β2� �

F1 X, Zð Þ,ð15Þ

where

F3 X, Zð Þ = X12 + a10,0 + a10,2Z2� �X10

+ a8,0 + a8,2Z2 + a8,4Z

4� �X8

+ a6,0 + a6,2Z2 + a6,4Z

4 + a6,6Z6� �X6

+ a4,0 + a4,2Z2 + a4,4Z

4 + a4,6Z6 + a4,8Z

8� �X4

+ a2,0 + a2,2Z2 + a2,4Z

4 + a2,6Z6 + a2,8Z

8 + a2,10Z10� �

X2

+ a0,0 + a0,2Z2 + a0,4Z

4 + a0,6Z6

+ a0,8Z8 + a0,10Z

10 + a0,12Z12,

a0,0 = −5625n9 − 192β2c22,0m

3 − 16875n7 + 32α2cm2b20,2 + 192α2m3 + 192β2m3 + 16875n5 − 5625n3192m3 ,

a0,2 = −475n2 n4 − 2n2 + 1

� �

24cm ,

a0,4 = −17n n2 − 1� �

m

9c2 ,

a0,6 = −8m3

27c3 ,

a2,0 = −125n2 n4 − 2n2 + 1

� �

16m2 ,

a2,2 =15n n2 − 1� �

c,

a2,4 =4m2

3c2 ,

a4,0 = −−25n n2 − 1

� �

4m ,

a4,2 = −2mc,

b0,0 = −5b0,2n n2 − 1

� �

12m ,

b2,0 =2mb0,29c ,

c0,0 =c2,0n n2 − 1

� �

4m ,

c0,2 =2mc2,0

c,

c2,0 = c2,0,b0,2 = b0,2,

ð13Þ

4 Advances in Mathematical Physics

P2 X, Zð Þ = b0,0 + b2,0 + b2,2X2 + b2,4X

4� �Z2

+ b4,0 + b4,2X2� �Z4 + Z6 + b0,2X

2

+ b0,4X4 + b0,6X

6,

Q2 X, Zð Þ = c0,0 + c0,2Z2 + c0,4Z

4 + c0,6Z6

+ c2,0 + c2,2Z2 + c2,4Z

4� �X2

+ c4,0 + c4,2Z2� �X4 + X6,

F1 X, Zð Þ = X2 + a0,2Z2 + a0,0: ð16Þ

Substitute (15) into (4) and let all the coefficients of thedifferent powers of Xi1Zi2 equal to zero, we can get a set ofconstraining relations for the parameters. Figuring out theseequations, one has

a0,0 =1

36864m8 α2 + β2 + 1� � 878826025m2n17

��

− 5272956150m2n15 + 13182390375m2n13

− 17576520500m2n11 + 472392α2c7n2

− 27648β2m7n2 + 13182390375m2n9 − 472392α2c7

+ 27648m7β2 − 5272956150m2n7 + 878826025m2n5ÞnÞ,

a0,10 =464m4n n2 − 1

� �

243c5 ,

a0,12 =64m6

729c6 ,

a0,2 =1

2304m6c α2 + β2 + 1� �

� 150448375m2n15 − 752241875m2n13�

+ 1504483750m2n11 − 1504483750n9m2

+ 26244α2c7 − 1536m7β2 + 752241875m2n7

− 150448375m2n5Þ,

a0,4 =16391725n4 n8 − 4n6 + 6n4 − 4n2 + 1

� �

1728m2c2,

a0,6 =199745n3 n6 − 3n4 + 3n2 − 1

� �

162c3 ,

a0,8 =1445n2 n4 − 2n2 + 1

� �m2

27c4 ,

a10,0 = −49n n2 − 1� �

2m ,

a10,2 = −4mc,

a2,0 = −1

1536m7 79893275m2n15 − 399466375m2n13�

+ 798932750m2n11 − 798932750n9m2 + 26244α2c7

+ 1536m7α2 + 399466375m2n7 − 79893275m2n5Þ,

a2,10 = −64m5

81c5 ,

a2,2 = −94325n4 n8 − 4n6 + 6n4 − 4n2 + 1

� �

64cm3 ,

a2,4 =1225n3 n6 − 3n4 + 3n2 − 1

� �

12c2m ,

a2,6 = −17710n2 n4 − 2n2 + 1

� �m

27c3 ,

a2,8 = −760m3n n2 − 1

� �

27c4 ,

a4,0 = −5187875n4 n8 − 4n6 + 6n4 − 4n2 + 1

� �

768m4 ,

a4,2 =18375n3 n6 − 3n4 + 3n2 − 1

� �

8cm2 ,

a4,4 =18725n2 n4 − 2n2 + 1

� �

18c2 ,

a4,6 =2920m2n n2 − 1

� �

27c3 ,

a4,8 =80m4

27c4 ,

a6,0 = −18865n3 n6 − 3n4 + 3n2 − 1

� �

48m3 ,

a6,2 = −4655n2 n4 − 2n2 + 1

� �

6cm ,

a6,4 = −1540mn n2 − 1

� �

9c2 ,

a6,6 = −160m3

27c3 ,

a8,0 =735n2 n4 − 2n2 + 1

� �

16m2 ,

a8,2 =115n n2 − 1

� �

c,

a8,4 =20m2

3c2 ,

b0,0 =169785n3 n6 − 3n4 + 3n2 − 1

� �c3

512m6 ,

b0,2 =17955c3n2 n4 − 2n2 + 1

� �

128m5 ,

b0,4 =2835c3n n2 − 1

� �

32m4 ,

b0,6 = −135c38m3 ,

5Advances in Mathematical Physics

b2,0 = −2205c2n2 n4 − 2n2 + 1

� �

64m4 ,

b2,2 =855c2n n2 − 1

� �

8m3 ,

b2,4 = −45c24m2 ,

b4,0 = −21cn n2 − 1

� �

8m2 ,

b4,2 =27c2m ,

c0,0 = −12005n3 n6 − 3n4 + 3n2 − 1

� �

192m3 ,

c0,2 = −535n2 n4 − 2n2 + 1

� �

24cm ,

c0,4 = −5mn n2 − 1

� �

c2,

c0,6 = −40m3

27c3 ,

c2,0 = −245n2 n4 − 2n2 + 1

� �

16m2 ,

c2,2 = −115n n2 − 1

� �

3c ,

c2,4 = −20m2

9c2 ,

c4,0 = −13n n2 − 1� �

4m ,

c4,2 =6mc: ð17Þ

By this mean, we arrive at the 6-lump solution of Equa-tion (1) represented by

u x, y, z, t,wð Þ = lnfð ÞXX , ð18Þ

where f is given in Equation (15). Figure 3 shows that the 3Dplot of the 6-lump wave consists of a central peak and five 1-lump waves in a ring. One can observe that the six peaks tendto the same height as ∣α ∣ and ∣β ∣ increase.

5. Conclusions

In this work, we have analytically established and analyzednovel multiple lump solutions of a (4 + 1)-dimensional Fokasequation based on the bilinear equation and a new ansatz. Aseries of rational solutions including the 1-lump wave solu-tions, the 3-lump wave solutions, and the 6-lump wave solu-tions are obtained. The 1-lump wave has one positive peakand two negative peaks. In order to search the 3-lump andthe 6-lump solutions, three polynomial functions Fn, Pn,and Qn are utilized. It is notable that these lump waves allhave the properties limx→±∞u = u0, limy→±∞u = u0, andlimz→±∞u = u0. The 3-lump and 6-lump waves consist ofthree and six independent single 1-lump waves, respectively.All the peaks of the multiple lump waves tend to the sameheight when α and β are large enough. The results of thispaper enrich the types of solutions of the (4 + 1)-dimen-sional Fokas equation. Comparing with the existing resultsin the literature, our results are new. We expect these resultsto provide some values for researching the dynamics of mul-tiple waves in the deep ocean and nonlinear optical fibers.And it is very helpful for us to obtain the soliton moleculesin the future.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work is supported by the National Natural Science Foun-dation of China (project Nos. 11371086, 11671258, and11975145), the Fund of Science and Technology Commissionof Shanghai Municipality (project No. 13ZR1400100), theFund of Donghua University, Institute for Nonlinear

–20–20

00.51

1.52

2.5

–10–10 00 1010 20xz

(a)

z

x

–20

–10

–10 0 10 20

0

10

20

(b)

–15 –5

–5

–10

0

5

10

5 10 15

Z

X

(c)

Figure 3: The 6-lump solution u of Equation (1) with the parameter selections m = −2, n = 2, c = 1, u0 = 0, α = 80000, β = 80000. (a)Perspective view of the wave uðX, ZÞ, (b) overhead view of the wave uðX, ZÞ, and (c) the corresponding contour plot uðX, ZÞ.

6 Advances in Mathematical Physics

Sciences, and the Fundamental Research Funds for the Cen-tral Universities.

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