Nonlinear Dyn (2020) 101:2413–2422https://doi.org/10.1007/s11071-020-05881-3

ORIGINAL PAPER

Exact solutions with elastic interactions for the(2+1)-dimensional extended Kadomtsev–Petviashviliequation

Jutong Guo · Jingsong He · Maohua Li ·Dumitru Mihalache

Received: 29 June 2020 / Accepted: 3 August 2020 / Published online: 14 August 2020© Springer Nature B.V. 2020

Abstract In this work, the (2 + 1)-dimensionalextendedKadomtsev–Petviashvili equation,whichmod-els the surface waves and internal waves in straitsor channels, is investigated via the Hirota bilinearmethod. N-soliton and high-order breather solutionsare obtained analytically. Furthermore,mixed solutionsconsisting of first-order breathers and solitons are alsoderived, and the corresponding dynamic behaviors areshown by three-dimensional plots. Additionally, basedon the long-wave limit, we obtain line rogue waves,lumps and semi-rational solutions composed of lumps,line rogue waves and solitons. It is noteworthy thatthe semi-rational solutions derived in this paper exhibitelastic interactions.

Keywords Extended Kadomtsev–Petviashvili equa-tion · Rational solutions · Semi-rational solutions ·Bilinear method

J. Guo · M. LiSchool of Mathematics and Statistics, Ningbo University,Ningbo 315211, Zhejiang, People’s Republic of China

J. He (B)Institute for Advanced Study, Shenzhen University,Shenzhen 518060, Guangdong, People’s Republic of Chinae-mails: hejingsong@szu.edu.cn; jshe@ustc.edu.cn

D. MihalacheDepartment of Theoretical Physics, Horia Hulubei NationalInstitute for Physics and Nuclear Engineering, 077125Bucharest-Magurele, Romania

1 Introduction

During the past several decades, the research activityon solitons or more properly, solitary waves, has con-tinued to grow. Many soliton phenomena have beenobserved in different fields, such as physics, chemistry,biology and even sociology and finance. Awide varietyof nonlinear evolution equations (NLEEs) have beenused tomodel a series of interesting nonlinear phenom-ena and many mathematical methods have been pro-posed for seeking exact soliton solutions of the NLEEs,such as the inverse scattering transform (IST) [1–4], theDarboux transform (DT) [5–7] and the Hirota bilinearmethod [8–10]. We point out that the Hirota bilinearmethod is a direct, powerful method in soliton the-ory and has some advantages over other much com-plicated mathematical techniques such as the IST andthe DT, which are used to obtain families of exactsolutions of NLEEs. Moreover, with the wide appli-cation and continuous improvement of these mathe-matical methods, people not only stick to the solitonsolutions, but also focus on other types of exact solu-tions of NLEEs [11–25]. Particularly, “rogue wave”(RW) or “freak wave” as a kind of exact rational solu-tion has been extensively studied. RW is a special wavewhose amplitude will change dramatically in a shortperiod of time [26] and will cause great damage toships and other useful facilities built on the ocean.The mathematical research on RWs begins with non-linear Schrödinger (NLS) equation and its first-order

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RW solution was obtained by Peregrine [27]. Subse-quently, the high-order RWs solutions of the NLS-typeequationswere derived [28–31].However, thoughmostof the investigated equations are (1 + 1)-dimensionalones, the RWs that form in the ocean must be describedby (2+ 1)-dimensional (2D) models; thus, it is worth-while to study RWs of 2D NLEEs. As one of thefamous 2D NLEEs, the Kadomtsev–Petviashvili (KP)equation [32,33] is widely used to describe the non-linear dynamical systems. For example, it can be usedto model surface waves and internal waves in straitsor channels of varying depth and width [34,35]. Likethe Korteweg–de Vries (KdV) equation [36,37], theKP equation is completely integrable, and it has N-soliton solutions and other types of solutions [38–43].

Recently, the great interest in KP equation hasled to the construction and the study of many exten-sions to the KP equation [40]. Lately, a new extendedKadomtsev–Petviashvili (eKP) equation was intro-duced [44],

(ut + 6uux + uxxx )x − uyy + αutt + βuty = 0, (1)

where u = u(x, y, t) is a differentiable function andα and β are nonzero constants. The lump solution ofthe eKP equation (1) was obtained by Manukure [44].Also, Ahmed et al. [45] have derived the mixed lump-type solitons and breather solutions. However, the RWsolutions and semi-rational solutions of the eKP equa-tion (1) have never been reported, to the best of ourknowledge. Inspired by the above open problems, inthis work, we will focus on RW and semi-rational solu-tions of the eKP equation.

The organization of this paper is as follows. InSect. 2, in termsof theHirota bilinearmethod,we inves-tigate the solitons and breather solutions of the eKP

equation (1). In Sect. 3, rogue waves and semi-rationalsolutions are constructed by taking the correspondinglong-wave limit. Themain results of this paper are sum-marized in Sect. 4.

2 N-soliton and breather solutions of the eKPequation

By using the following logarithmic transformation,

u = 2(log f )xx , (2)

the eKP equation (1) has the bilinear form,

(Dx Dt + D4x − D2

y + αD2t + βDyDt ) f · f = 0,

(3)

where D is Hirota bilinear differential operator [8]:

Dnx a · b ≡

(∂

∂x− ∂

∂y

)n

a(x) b(y) |y=x ,

Dmt Dn

x a · b ≡ ∂m

∂sm∂n

∂yna(t + s, x + y)

b(t − s, x − y) |s=0,y=0, (4)

wherem and n are nonnegative integers. The N -solitonsolutions u of the eKP equation can be generated byusing the Hirota bilinear method [8], in which the func-tion f has the following expression

f =∑

exp

⎡⎣ N∑

j=1

μ jη j +(N )∑j<k

A jkμ jμk

⎤⎦ , (5)

where μ = 0, 1, η j and A jk are as follows

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Exact solutions with elastic interactions 2415

η j = Pj x + Q j y + Ω j t + η0j ,

Ω j =−βQ j − Pj +

√−4αP4

j + P2j + 2βPj Q j + Q2

j (β2 + 4α)

2α,

A jk =

{(C jCk + 8α Pj

3Pk − 12 α Pj2Pk2 + (

8α Pk3 − β Qk − Pk)Pj

−Q j(β Pk + Qk

(β2 + 4α

)))

}{ (

C jCk + 8α Pj3Pk + 12 α Pj

2Pk2 + (8α Pk3 − β Qk − Pk

)Pj

− Q j(β Pk + Qk

(β2 + 4α

)))} ,

C j =√

−4α Pj4 + Pj

2 + 2 β Pj Q j + Q j2(β2 + 4α

).

(6)

Here, Pj , Q j and η0j are arbitrary complex parameters.

−4α Pj4 + Pj

2 + 2 β Pj Q j + Q j2(β2 + 4α

) ≥ 0 isrequired. When N = 1, combining (2), (5) and (6), theexpression of the one-soliton u1s solution is as follows

u1s = P21

1 + cosh(P1x + Q1y + Ω1t + η01). (7)

From the above expression, we can obtain that themax-imum value P2

1 appears at P1x+Q1y+Ω1t+η01 = 0.At the same time, we also give three-dimensional plotsof two-, three- and four-soliton solutions; see Fig. 1.The value of all parameters η0j ( j = 1 · · · 4) are takenas 0, and α, β are taken as α = −β = 1. The addi-tional terms in (1) only affect the dispersion relation,so we can consider that α utt and β uty do not “kill”the integrability of this equation.

Following the previous works [46–49], we can getthe breather solutions by taking

N = 2n, P∗j = Pj+1, Q

∗j = Q j+1, α = −β = 1. (8)

The first-order breather solution u1b is generated bytaking N = 2, P∗

1 = P2 = i2 , Q

∗1 = Q2 = 1 + i

2 ,η10 = η20 = 0 in Eq. (5):

u1b =

⎧⎪⎨⎪⎩

[2 (28

√481 − 628) (cosh(ζ1) + sinh(ζ1))

+ (√481 − 553

25 ) (100 (cosh(ζ2) + sinh(ζ2))+ 50 (cosh(ζ3) + sinh(ζ3)))] cos(ξ1)

⎫⎪⎬⎪⎭

{(2 (−25 + √

481) (cosh(ζ3) + sinh(ζ3)) cos(ξ1)+(

√481 − 31) (cosh(ζ2) + sinh(ζ2)) + √

481 − 25)2

} ,

(9)

where

ζ1 = 3

8t

√30 + 2

√481 + 3

2t + 3 y;

ζ2 = 1

4t

√30 + 2

√481 + t + 2 y;

ζ3 = 1

8t

√30 + 2

√481 + t

2+ y;

ξ1 = 1

8t

√−30 + 2

√481 + y

2+ x

2.

(10)

From the above equations,we can see that the trajectoryof the first-order breather is ξ1 = 0. u1b is periodicon the x-axis, and the period is 4π ; see Fig. 2. Whentaking N = 4, α = −β = 1, P∗

1 = P2 = i2 , Q

∗1 =

Q2 = 1 + i2 , P

∗3 = P4 = − i

2 , Q∗3 = Q4 = 1 + i

2and η01 = η02 = −η03 = −η04 = 2π , we obtain thesecond-order breather solutions; see Fig. 3.

In addition to the soliton solutions and the breathersolutions, for N > 2, we obtain the mixed solutionsconsistingof solitons andbreathers.Whenwe take N =3, P∗

1 = P2 = i2 , Q∗

1 = Q2 = 1 + i, P3 = Q3 =1 in Eq. (5), mixed solutions consisting of first-orderbreathers and one-soliton solutions are generated; seeFig. 4a, b. For higher-order N , we have also derivedthe mixed solutions consisting of first-order breatherand two or more solitons. For example, taking N =4, P∗

1 = P2 = i2 , Q∗

1 = Q2 = 1 + i, P3 = Q3 =1, P4 = −Q4 = 1 in Eq. (5), we get themixed solutionconsisting of a first-order breather and a two-solitonsolution, which is shown in Fig. 4c, d.

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Fig. 1 Soliton solutions of the eKP equation with parameters a,e N = 1, P1 = −Q1 = 1, b, f N = 2, P1 = P2 = 1, Q1 =−Q2 = 1

2 , c,g N = 3, P1 = Q1 = 1, P2 = −Q2 = 1, P3 =

12 , Q3 = − 1

4 , d, h N = 4, P1 = Q1 = 1, P2 = −Q2 =1, P3 = P4 = 1

2 , Q3 = −Q4 = 14

Fig. 2 First-order breather solution u1b of the eKP equation in the (x, y) plane at t = 0

3 Rational and semi-rational solutions of the eKPequation

According to the long-wave limit [11,46–49], in orderto obtain the rational solutions, we take the parametersin Eq. (4) as follows

N = 2, Q1 = λ1P1, Q2 = λ2P2, η01 = η02 = iπ,

(11)

and then, we should perform the limit P1, P2 → 0.Thus, f can be expressed as a polynomial function

f = θ1θ2 + a12, (12)

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Exact solutions with elastic interactions 2417

Fig. 3 Second-orderbreather solutions of theeKP equation in the (x, y)plane at t = 0

Fig. 4 a,b Mixed solutionsconsisting of first-orderbreather and one-solitonsolution given by Eq. (5) att = 0, c,d mixed solutionsconsisting of first-orderbreather and two-solitonsolution given by Eq. (5) att = 0

where

θ1 = (−2 yλ1 − 2 x) α − (B1 − β λ1 − 1) t

2α,

a12 = 24α

−B2 B1 + (β + λ2

(β2 + 4α

))λ1 + β λ2 + 1

,

Bj =√2 β λ j + 1 + λ j

2(β2 + 4α

)γ j , ( j = 1, 2).

(13)

The obtained rational solutions can be classified into(a) roguewaves (RWs) and (b) lump solutions by takingdifferent values of the parameter λ j . When λ j is real,we get a RW solution, whereas when λ j is complex,we obtain a lump solution.

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Fig. 5 Time evolution in the (x, y) plane of the first-order line RW solution urw

Fig. 6 Lump solution ulumpgiven by Eq. (15) in the(x, y) plane at t = 0

In order to obtain RW solutions, we take α = β =−1, λ1 = λ2 = 1, γ1 = −γ2 = 1. Thus, the RWsolution of Eq. (1) can be expressed as follows

urw = 4t2 − (x + y)2 + 3

(t2 + (x + y)2 + 3)2. (14)

From this expression of the RW solution and as shownin Fig. 5, the first-order line RW rises and decays withtime, and the height of the linewave at infinity along the

(x, y)-plane tends to zero. For a fixed time variable t ,themaximumvalue of this lineRWoccurs at x+y = 0.

Taking α = −β = 1, λ∗1 = λ2 = −i, γ1 = γ2 = 1,

we obtain a lump solution:

ulump = 8 (√5 − 3)

H1√−4 − 2 i + H∗

1

√−4 + 2i + H01

(H2√−4 − 2 i + H∗

2

√−4 + 2i + H02 )2

,

(15)

where

H1 = t ((−3 + i) t + (5 − i) x + (−1 + i) y) ,

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Exact solutions with elastic interactions 2419

Fig. 7 Time evolution ofsemi-rational solutionsconsisting of a first-orderrogue wave and aone-soliton solution givenby Eq. (18) in the (x, y)plane with parameters α =β = −1, γ1 = −γ2 = 1

H2 = t (2 t + (−5 + i) x + (−1 + i) y) ,

H01 =

(2 t2 + (2 x + 2 y) t − 2 x2 + 2 y2

) √5 − 6 t2

+ (−6 x − 6 y) t + 6 x2 − 6 y2 − 24,

H02 =

(−2 t2 + (−2 x + 2 y) t + 2 x2 + 2 y2

) √5

+ 2 t2 + (6 x − 6 y) t − 6 x2 − 6 y2 − 24.

(16)

We can also calculate the critical points of the lumpsolution at t = 0:

A1 = (0, 0), A2 =(3√10

2+ 3

√2

2, 0

),

A3 =(

−3√10

2− 3

√2

2, 0

). (17)

We see that the maximum value of lump solution is

1−√53 at A1, and the minimum value is

√5

24 − 18 at A2

and A3; see Fig. 6.

Next, we will obtain the semi-rational solution ofEq. (1). Here, we mainly discuss the case of N = 3.Setting

N = 3, Q1 = λ1P1, Q2 = λ2P2, η01 = η02 = iπ,

(18)

and letting P1, P2 → 0, then the functions f definedin (5) will become a combination of polynomial andexponential functions:

f = (a13 a23 + a13 θ2 + a23 θ1

+ θ1 θ2 + a12) eη3 + θ1 θ2 + a12,

(19)

where a12 and θ j ( j, k = 1, 2) are the same as before,and a j3( j = 1, 2) is as follows

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2420 J. Guo et al.

Fig. 8 Time evolution in the (x, y) plane of the two distinct patterns of semi-rational solutions given by Eq. (18) with parametersα = β = 1, γ1 = γ2 = 1, consisting of a first-order lump and a single soliton

a j3 = 24P23 α

Bj C3 + (−β2Q3 − 4α Q3 − β P3)λ j + 8α P33 − β Q3 − P3

. (20)

Substituting f into (2), we can obtain the semi-rational solutions. Similar to the case of the first-orderrational solution, when N = 3,we can get two differentclasses of semi-rational solutions for different valuesof λ1, λ2. In the first case, taking λ1 = λ2 = 1, P3 =Q3 = 1, η03 = 0, we get the semi-rational solution con-sisting of a line RW and a soliton. As shown in Fig. 7,the line RW suddenly appears and quickly disappears.In the second case, taking λ1 = −λ2 = 2i, η03 = 0, twodifferent types of semi-rational solutions consisting oflumps and line solitons are generated by selecting dif-ferent values of P3 and Q3. If we take P3 = 1

2 , Q3 = 1(see Fig. 8d–f), the lump move along the direction ofthe peak amplitude of the line soliton wave. However,if we take P3 = 1

2 , Q3 = −1 (see Fig. 8a–c), the lump

will pass through the soliton, and the peak will becomehigher when the lump intersects with the soliton. Wepoint out here that the interaction between the lump andthe soliton is elastic, whereas that studied in Ref. [45]is an inelastic one.

4 Summary and discussion

In this paper, a new extended Kadomtsev–Petviashvili(eKP) equation is studied, which describe the surfacewaves and internal waves in straits or channels. TheHirota bilinear method has been employed to constructanalytical N-soliton solutions and high-order breathersolutions of the eKP equation. Under certain parame-ter choices, themixed solutions consisting of first-order

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Exact solutions with elastic interactions 2421

breathers and solitons are generated; seeFig. 4. Further-more, the line rogue waves and the semi-rational solu-tions consisting of line rogue waves and single solitonsolutions are derived by taking the long-wave limit; seeFigs. 5 and 7. Additionally, we have also constructedtwo distinct patterns of semi-rational solutions, con-sisting of a first-order lump and a single soliton; seeFig. 8. To the best of our knowledge, the semi-rationalsolutions given in this paper for the eKP equationhave never been reported before in the literature. Theresults obtained in this workmight be useful in explain-ing some real-world physical phenomena occurring innature and help us to deeply understand the key prop-erties of nonlinear evolution equations describing non-linear dynamical systems.

Funding This work is supported by the NSF of China underGrantNo. 11671219, theNatural ScienceFoundation ofZhejiangProvince underGrantNo.LY15A010005 and theNatural ScienceFoundation of Ningbo under Grant No. 2018A610197.

Compliance with ethical standards

Conflict of interest We declare we have no conflict of interest.

Ethical standard The authors declare that they comply withethical standards.

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