East Asian Journal on Applied Mathematics Vol. 9, No. 4, pp. 780-796

doi: 10.4208/eajam.310319.040619 November 2019

Dynamics of Lump Solutions, Rogue Wave

Solutions and Traveling Wave Solutions for a

(3+ 1)-Dimensional VC-BKP Equation

Ding Guo, Shou-Fu Tian∗, Xiu-Bin Wang and Tian-Tian Zhang∗

School of Mathematics, China University of Mining and Technology,

Xuzhou 221116, P.R. China.

Received 31 March 2019; Accepted (in revised version) 4 June 2019.

Abstract. The (3 + 1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili

equation is studied by using the Hirota bilinear method and the graphical representations

of the solutions. Breather, lump and rogue wave solutions are obtained and the influence

of the parameter choice is analysed. Dynamical behavior of periodic solutions is visually

shown in different planes. The rogue waves are determined and localised in time by

a long wave limit of a breather with indefinitely large periods. In three dimensions

the breathers have different dynamics in different planes. The traveling wave solutions

are constructed by the Bäcklund transformation. The traveling wave method is used

in construction of exact bright-dark soliton solutions represented by hyperbolic secant

and tangent functions. The corresponding 3D figures show various properties of the

solutions. The results can be used to demonstrate the interactions of shallow water

waves and the ship traffic on the surface.

AMS subject classifications: 35Q51, 35Q53, 35C99, 68W30, 74J35

Key words: Breather wave solutions, rogue wave solutions, lump solutions, traveling wave solu-

tions, bright and dark soliton solutions.

1. Introduction

Nonlinear evolution equations (NLEEs) are involved in complex physical phenomena

and are used to model various problems in fluid mechanics, plasma physics, optical fibers

and solid state physics [1, 2, 6, 24, 28]. In past decades such equations have been stud-

ied by mathematicians and physicists. The NLEEs demonstrate both inelastic interactions

and admit localised coherent structures [5, 20, 40, 43]. The nonlinear B-type Kadomtsev-

Petviashvili (KP) equation is an important representative of such equations — cf. [15,22].

It has a variety of analytic solutions, the most crucial of which are the rogue, lump and

bright-dark soliton solutions [9,18,29].

∗Corresponding author. Email address: TS17080009A3�umt.edu.n (D. Guo), sftian�umt.edu.n,

shoufu2006�126.om (S. F. Tian), ttzhang�umt.edu.n (T. T. Zhang)

http://www.global-sci.org/eajam 780 c©2019 Global-Science Press

Dynamics of Lump Solutions, Rogue Wave Solutions and Traveling Wave Solutions 781

The (2+ 1)-dimensional KP-type equation has the form

(ut + 6uux + ux x x)x + 3uy y = 0,

and is used to describe the shallow water waves with the weak influence of surface tension

and viscosity [14].

The (3+ 1)-dimensional KP-type equation

ut x +αux x x y + β(ux uy)x + γuxz = 0

can be viewed as a shifted (2+ 1)-dimensional BKP equation for z = x . If δ 6= 0, the BKPequation can be written as a (3+ 1)-dimensional KP-type equation — viz.

ut x +αux x x y + β(ux uy)x + γux x +δuzz = 0.

Among other phenomena, this equation describes the propagation with a uniform speed of

shallow water waves of small amplitudes in water canals of a constant depth [8,12,16,17,

21,25,30].

Here we focus on the (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petvia-

shvili (vc-BKP) equation

(ux + uy + uz)t +αux x x y + β(uxuy)x + γux x +δuzz = 0, (1.1)

where u is a complex function of variables x , y, z, t and α,β ,γ,δ are real constants. This

equation can be used to describe propagation of long waves. It also finds various applica-

tions in water percolation. If α= 1,β = χ,γ = δ = −1, the Eq. (1.1) can be reduced to theBKP equation

(ux + uy + uz)t + ux x x y +χ(ux uy)x − (ux x + uzz) = 0. (1.2)

For a specific choice of parameters, the conservation laws and the multiple-soliton solu-

tions of the Eq. (1.2) have been studied in [3, 42]. The BKP equation can be also used to

model water waves with weakly non-linear restoring forces and frequency dispersion. Nev-

ertheless, to the best of our knowledge, the Bäcklund transformation of the Eq. (1.1), the

corresponding traveling wave solutions, breather, rogue and lump waves and bright-dark

solutions have not been yet studied and we are going to address these problems here.

The rest of the paper is arranged as follows. Section 2 deals with the bilinear represen-

tation and breather wave solutions. In Section 3, lump and rogue wave solutions are de-

rived by symbolic computation [4,7,10,11,19,23,26,27,31–39,41,44,45]. The Bäcklund

transformation considered in Section 4, is used to determine traveling soliton solutions.

Section 5 is devoted to bright-dark soliton solutions. Our conclusions are presented in the

last section.

2. Bilinear Formalism and Breather Wave Solutions

2.1. Bilinear formalism

We start with a bilinear formalism for (1.1) established by the Hirota bilinear method

and the long wave limit method [13].

782 D. Guo, S.-F. Tian, X.-B. Wang and T.-T. Zhang

Theorem 2.1. The Eq. (1.1) admits the bilinear representation�

Dt Dx + Dy Dt + DzDt +αD3xDy + γD

2x+δD2

z

�

f · f = 0,

with the variable transformation

u =6α

β(ln f )x ,

where f is a real function of variables x , y, z and t.

Proof. Consider the potential transformation

u = c(t)qx , (2.1)

where c = c(t) is an undetermined function with respect to variable t. Substituting (2.1)

into the Eq. (1.1) and integrating the result with respect to x , we obtain

ct(qx + qy + qz)t +αcqx x x y + β c2qx xqx y + γcqx x + δcqzz = 0.

To find the bilinear form of the Eq. (1.1), the terms αcqx x x y and β c2qx xqx y are associated

with the known P-polynomial by

αcqx x x y + β c2qx xqx y = αc(qx x x y + (cβ/α)qx x qx y)⇔ cα(qx x x y + 3qx xqx y) = αcPx x x y .

Thus c has to be fixed as a constant — i.e. c = 3α/β and the Eq. (1.1) is transformed to

the following Bell polynomials form:

E(q) = (qx + qy + qz)t +α(qx x x y + 3qx xqx y) + γqx x +δqzz = 0.

Using the transformation q = 2 ln( f ), which is equivalent to u = (6α/β)(ln f )x , we arrive

at the equation

E(q) = Pt x + Py t + Pzt +αP3x y + γP2x +δP2z = 0.

Consequently, it follows that�

Dt Dx + Dy Dt + DzDt +αD3x Dy + γD

2x +δD

2z

�

f · f = 0, (2.2)

which completes the proof.

The D-operator is now defined by

Dmx Dny D

lt( f · g)

=

∂

∂ x−∂

∂ x ′

m ∂

∂ y−∂

∂ y ′

n ∂

∂ t−∂

∂ t′

l

f (x , y, t) · g(x ′, y ′, t′)|x=x ′,y=y′,t=t ′ , (2.3)

and so that

Dx Dt f · f = 2( fx t f − fx ft),

D2x f · f = 2( fx x f − f2x ),

Dx Dy f · f = 2( fx y f − fx f y ),

D3xDy f · f = 2 fx x x y f − 6 fx fx x y + 6 fx x fx y − 2 fx x x f y .

Dynamics of Lump Solutions, Rogue Wave Solutions and Traveling Wave Solutions 783

The bilinear equation (2.2) is equivalent to the form

fx t f − fx ft + f y t f − f y ft + fzt f − fz ft+α( f3x y f − 3 f2x y fx + 3 fx y fx x − f3x f y )

+γ( fx x f − f2x ) +δ( fzz f − f

2z ) = 0.

2.2. Breather wave solution

The soliton solutions u can be obtained by the bilinear transform method, if the function

f is sought in the form

f = 1+ eη1 + eη2 + A12eη1+η2 , (2.4)

where

ηi = ki(x + pi y + qiz +ωi t) +η0i , i = 1,2,

ωi =αk2

ipi −δq

2i− γ

1+ pi + qi, i = 1,2,

A12 =p1p2(k1 − k2)(k1p1 − k2p2) + (k1q2 − p2q1)

2 − (p1 − p2)2

p1p2(k1 + k2)(k1p1 + k2p2) + (k1q2 − p2q1)2 − (p1 − p2)2,

and ki , pi ,qi and η0i

are real parameters. Previously, a family of analytical solutions was

determined by a suitable choice of parameters — e.g. in the Davey-Stewartson (DS) systems

they are

p1 = b1, p2 = b1, q1 = a + ik, q2 = a− ik, k1 = ia1, k2 = −ia1, η0∗2= η0

1.

For simplicity, we set

p1 = p2 = 2, q∗1 = q2 = 1+ 2i, k1 = k

∗2 = i, η

02 = η

01 = 0.

In this case function f takes the form

f =2 cosh

γ+ 2α− 11δ10

t + 2z

cos

γ+ 2α−δ5

t − x − 2y − z

− 2 sinh

γ+ 2α− 11δ10

t + 2z

cos

γ+ 2α−δ5

t − x − 2y − z

+3

2cosh

γ+ 2α− 11δ5

t + 4z

−3

2sinh

γ+ 2α− 11δ5

t + 4z

+ 1, (2.5)

with real constants α,γ,δ.

As is shown in Fig. 1, the breather wave solution has qualitatively different behavior

in different planes, but the amplitudes of the excited states are bounded and are almost

the same in different spaces. For (x , t)-plane the corresponding solutions are displayed in

Fig. 2. It can be seen that periodic line waves arise from the constant background and the

amplitudes of the excited state are bounded. Besides, these periodic soliton solutions have

different periods.

784 D. Guo, S.-F. Tian, X.-B. Wang and T.-T. Zhang

–20 –10 0 10 20x –100 10

20z

–30

–20

–10

0

10

20

30

u

a)

–10 –5 0 5 10 15x –5 05 10

y

–40

–20

0

20

40

u

b)

–15–10–5051015 y

–15 –5 0 5 10 15z

–40

–20

0

20

40

u

c)

Figure 1: Breather wave solution, δ = 1,γ = 4,α = 6, k1 = k∗2= i, p1 = p2 = 2, q1 = q

∗2= 1 + 2i.

(a) Perspetive view of the real part of the wave (y = 0, t = 0). (b) Perspetive view of the real part ofthe wave (t = 0, z = 0). () Perspetive view of the real part of the wave (x = 0, t = 0).

–10–50510x

–10–5

05

10

t

–40

–20

0

20

40

u

a1)

–10–5

05

10x

–10–5

05t

–40

–20

0

20

40

u

b1)

–10–50510x

–10–5

05

10

t

–40

–20

0

20

40

u

c1)

–20

20

40

u

–10 –8 –6 –4 –2 2 4 6 8 10t

a2)

–40

–20

20

40

u

–10 –8 –6 –4 –2 2 4 6 8 10t

b2)

–40

–20

0

20

40

u

–10 –8 –6 –4 –2 2 4 6 8 10t

c2)

Figure 2: Breather wave solution, δ = 1,γ = 4,α = 6, k1 = k∗2= i, p1 = p2 = 2, q1 = q

∗2= 1 + 2i.

(a) Perspetive view of the real part of the wave (y = −6, z = 0). (b) Perspetive view of the real partof the wave (y = 0, z = 0). () Perspetive view of the real part of the wave (y = 6, z = 0).

3. Lump Solutions and Rogue Wave Solutions

In order to generate rogue wave solutions, the long wave limit of function (2.4) has to

be determined. Choosing in the Eq. (2.4) the parameters

k1 = l1ǫ, k2 = l2ǫ, η01 = η

0∗2 = iπ,

we have

f = (θ1θ2 + θ0)l1 l2ǫ2 + O (ǫ3), ǫ→ 0,

where

θ0 =2p1p2(p1 − p2)

(p1q1 − p2q1)2 − (p1 − p2)2,

θ1 =−�

δq21 + γ�

t +�

p21 + p1q1 + p1�

y +�

q21 + p1q1 + q1�

z +�

p1 + q1 + 1�

x

1+ p1 + q1,

Dynamics of Lump Solutions, Rogue Wave Solutions and Traveling Wave Solutions 785

θ2 =−�

δq22 + γ�

t +�

p22 + p2q2 + p2�

y +�

q22 + p2q2 + q2�

z +�

p2 + q2 + 1�

x

1+ p2 + q2.

Due to the gauge freedom of f , the rational solution can be represented in the form

u =6α

β

(1+ p1 + q1)θ2 + (p2 + q2 + 1)θ1

θ1θ2 + θ0. (3.1)

Setting

p2 = p∗1, q∗

1= q2 (3.2)

yields θ2 = θ∗1 and θ0 > 0, so that the (3.1) is nonsingular. Moreover, as will be seen later

on, this solution has different dynamical behavior in each plane. These two behaviors are

given in the plane as an example. We set p1 = a1 + i b1,q1 = a2 + i b2, where a1, a2, b1, b2are real constants.

Case 1. Lump solutions

If a1 6= 0, then u is a constant along the trajectory [x(t), y(t)], where x and y satisfythe equations

x + a1 y + a2z +

�

δ�

b22− a2

2

�

− γ�

(1+ a1 + a2)− 2a2 b2(b1 + b2)δ

(1+ a1 + a2)2 + (b1 + b2)

2t = 0,

b1 y + b2z +

�

δ�

a21 − b22

�

+ γ�

(b1 + b2)− 2a2 b2(1+ a1 + a2)δ

(1+ a1 + a2)2 + (b1 + b2)

2t = 0.

Moreover, for any fixed t and z the function u tends to 0 as (x , y) tends to infinity. Thus

these rational solutions are permanent lumps moving on constant backgrounds. Moreover,

they are rationally localised in all directions in the space. Considering the critical points

under the condition (3.2), we observe the existence of the lumps — viz. one global max-

imum and two global minimum of (1.1). The dynamic behavior of the lump solutions in

different planes is demonstrated in Fig. 3. We observe that periodic line waves arise from

a constant background. Besides, the amplitude of the excited state is bounded and almost

the same in different spaces.

Fig. 4 shows that the periodic solutions have qualitatively different behavior in other

planes but the amplitude of the excited state is bounded and almost the same in different

spaces.

Case 2. Rogue wave solutions

If b1 = 0 — i.e. if p1 is real, the solutions of (3.1) are line rogue waves in the plane,

whose amplitude changes with time — cf. Fig. 5. There is an obvious difference between

the line solitons and line rogue waves, which maintain a perfect profile without any decay

during their propagation. The solution u of (3.1) has the highest amplitude at t = 0, but it

approaches a constant background for |t| ≥ 0. The solution has a rational growing mode.Obviously, those solutions are localised in time and are the line rogue waves in previous

works. On the other hand, the line rogue wave solutions can be derived with a special

786 D. Guo, S.-F. Tian, X.-B. Wang and T.-T. Zhang

–40–20

020

40x

–40–20

020y

–2

–1

0

1

2

3

4

u

a)

–40–20

020

40x

–40–20

020y

–3

–2

–1

0

1

2

3

u

b)

–40–20

020

40x

–40–20

020y

–4

–3

–2

–1

0

1

2

u

c)

Figure 3: Lump solution, δ = 1,γ = 4,α = 6, k1 = k∗2= i, p1 = p2 = 3, q1 = q

∗2= 1+ 2i. (a) Perspetive

view of the real part of the wave (t = −6, z = 0). (b) Perspetive view of the real part of the wave(t = 0, z = 0). () Perspetive view of the real part of the wave (t = 6, z = 0).

–40–20

020

40y

–40–20

020t

–3

–2

–1

0

1

2

3

u

a1)

–40–20

020

40z

–40–20

020t

–3

–2

–1

0

1

2

3

u

b1)

–60–40–200204060x

–60–40

–20 020

40t

–3–2–10123

u

c1)

–4

–2

0

2

4

t

–4 –2 0 2 4y

a2)

–4

–2

0

2

4

t

–4 –2 0 2 4z

b2)

–4

–2

0

2

4

t

–4 –2 0 2 4x

c2)

Figure 4: Lump solution by hoosing suitable parameters: δ = 1,γ = 4,α = 6, k1 = k∗2= i, p1 = p2 =

3, q1 = q∗2= 1 + 2i. (a) Perspetive view of the real part of the wave (x = 0, z = 0). (b) Perspetive

view of the real part of the wave (x = 0, y = 0). () Perspetive view of the real part of the wave(y = 0, z = 0).

parameter choice — e.g. if b2 = 0, then the corresponding solutions are line rogues in

the plane. These fundamental line rogue waves are demonstrated in Fig. 6, where we set

p1 = p2 = 3 and q1 = q∗2 = 1+ 2i.

It should be pointed out, that if amplitude remains constant but the positions changes,

the lump solutions can be also obtained in the planes under the same parameters. The lump

phenomena appear in different planes, but the rogue waves can also appear if appropriate

parameters are chosen.

The graph of the solution u in Fig. 5 shows that the periodic line waves arise from the

constant background. Besides, the amplitude of the excited state is bounded and almost

the same in different spaces. Considering Fig. 6, we note that the periodic solutions have

qualitatively different behavior in different planes but the amplitude of the excited state is

bounded and almost the same in different spaces.

Dynamics of Lump Solutions, Rogue Wave Solutions and Traveling Wave Solutions 787

–100–50050100 x

–100–50 0

50100

y

–1–0.8–0.6–0.4–0.2

00.20.40.6

u

a)

–100–50050100x

–100–50

050y

–3

–2

–1

0

1

u

b)

–80–40020406080x

–80–40

040y

0

5e+13

1e+14

1.5e+14

2e+14

2.5e+14

3e+14

u

c)

Figure 5: Rogue wave solution, δ = 1,γ = 4,α = 6, k1 = k∗2= i, p1 = p2 = 3, q1 = q

∗2= 1 + 2i.

(a) Perspetive view of the real part of the wave (t = −6, z = 0). (b) Perspetive view of the realpart of the wave (t = −3, z = 0). () Perspetive view of the real part of the wave (t = 0, z = 0).

–10–50510x

–10–5

05t

–10

–5

0

5

10

u

a1)

–10–50510t

–10–5

05y

–10

–5

0

5

10

u

b1)

–10–50510z

–10–5

05t

–20

–10

0

10

20

u

c1)

–4

–2

0

2

4

t

–4 –2 0 2 4x

a2)

–4

–2

0

2

4

t

–4 –2 0 2 4y

b2)

–4

–2

0

2

4

t

–4 –2 0 2 4z

c2)

Figure 6: Rogue wave solution, δ = 1,γ = 4,α = 6, k1 = k∗2= i, p1 = p2 = 3, q1 = q

∗2= 1 + 2i.

(a) Perspetive view of the real part of the wave (y = 0, z = 0). (b) Perspetive view of the realpart of the wave (x = 0, z = 0). () Perspetive view of the real part of the wave (x = 0, y = 0).

4. Bäcklund Transformation and Traveling Wave Solutions

In this section, we construct the bilinear Bäcklund transformation and use it to deter-

mine traveling soliton solutions.

4.1. Bilinear Bäcklund transformation

In order to construct the Bäcklund transformation, we assume that there is another

operator D satisfying (2.3). Then, we have the following bilinear form�

Dt Dx + Dy Dt + Dz Dt +αD3xDy + γD

2x+δD2

z

�

f ′ · f ′ = 0.

Considering the equation

M =��

Dt Dx + Dy Dt + Dz Dt +αD3x Dy + γD

2x +δD

2z

�

f ′ · f ′�

f · f

−��

Dt Dx + Dy Dt + Dz Dt +αD3xDy + γD

2x+δD2

z

�

f · f�

f ′ · f ′, (4.1)

788 D. Guo, S.-F. Tian, X.-B. Wang and T.-T. Zhang

and setting M = 0, we obtain��

Dt Dx + Dy Dt + Dz Dt +αD3x Dy + γD

2x +δD

2z

�

f ′ · f ′�

f · f

=��

Dt Dx + Dy Dt + Dz Dt +αD3x Dy + γD

2x +δD

2z

�

f · f�

f ′ · f ′.

Thus the function f ′ can be used to find the operator D in (2.3). The change of the de-

pendent variables f to f ′ and vice versa in (4.1) does not violate the condition M = 0 but

leads to the following equations

(Di Dj f′ f ′) f f − (Di Dj f f ) f

′ f ′ = 2Dj(Di f′ f ) f f ′,

(Di Dj f′ f ) f f ′ = (Dj Di f

′ f ) f f ′,(4.2)

and

2�

D3i Dj f′ f ′�

f f − 2�

D3i Dj f f�

f ′ f ′

=Di��

3D2iDj f′ f�

f f ′ +�

3D2i

f ′ f� �

Dj f f′�

+�

6Di Dj f′ f� �

Di f f′��

+ Dj��

D3i f′ f�

f f ′ +�

3D2i f′ f� �

Dj f f′��

, (4.3)

where we used the notation i and j for variables x , y, z and t. Doing the calculations

2M =2��

Dt Dx f′ f ′�

f f −�

Dt Dx f f�

f ′ f ′�

+ 2��

Dt Dy f′ f ′�

f f −�

Dt Dy f f�

f ′ f ′�

+ 2��

Dt Dz f′ f ′�

f f −�

Dt Dz f f�

f ′ f ′�

+ 2�

D3x Dy f′ f ′�

f f −�

D3x Dy f f�

f ′ f ′�

+ 2�

D2x

f ′ f ′�

f f −�

D2x

f f�

f ′ f ′�

+ 2�

D2z

f ′ f ′�

f f −�

D2z

f f�

f ′ f ′�

=4Dx�

Dt f′ f�

f f ′ + 4Dy�

Dt f′ f�

f f ′ + 4Dz�

Dt f′ f�

f f ′

+αDx��

3D2xDy f′ f�

f f ′ +�

3D2x

f ′ f� �

Dy f f′�

+�

6Dx Dy f′ f� �

Dx f f′��

+αDy��

D3x f′ f�

f f ′ +�

3D2x f′ f� �

Dx f f′��

+ 4γDx�

Dx f′ f�

f f ′ + 4δDz�

Dz f′ f�

f f ′

=4Dx�

Dt f′ f�

f f ′ + 4Dy�

Dt f′ f�

f f ′ + 4Dz�

Dt f′ f�

f f ′

+αDx��

3D2xDy f′ f + a1Dy f

′ f + a2 f′ f�

f f ′ +�

3D2x

f ′ f + a3Dy f′ f + a4 f

′ f�

�

Dy f f′�

+�

6Dx Dy f′ f + 6a5Dx f

′ f� �

Dx f f′��

+αDy��

D3x f′ f − a1Dx f

′ f + a6 f′ f�

f f ′ +�

3D2x f′ f + a7Dx f

′ f − a4 f′ f� �

Dx f f′��

+ 4γDx�

Dx f′ f + a8Dy f

′ f + a9Dt f′ f�

f f ′ − 4αDy�

a8Dx f′ f�

f f ′

− 4αDt�

a9Dx f′ f�

f f ′

+ 4δDz�

Dz f′ f + a10Dx f

′ f + a11Dt f′ f�

f f ′ − 4δDx�

a10Dz f′ f�

f f ′

− 4δDt�

a11Dz f′ f�

f f ′

=Dx�

B1 f′ f�

f f ′ + Dx�

B2 f′ f� �

Dy f f′�

+ Dx�

B3 f′ f� �

Dx f f′�

+ Dy�

B4 f′ f�

f f ′ + Dy�

B5 f′ f� �

Dx f f′�

+ Dt�

B6 f′ f�

f f ′ + Dz�

B7 f′ f�

f f ′, (4.4)

and taking into account (4.2)-(4.4), we derive the Bäcklund transformations of (1.1) —

viz.

B1 f′ f =�

3D2xDy + a1Dy + a2 + 4γDx + a8Dy + a9Dt − 4δa10Dz

�

f ′ f = 0,

Dynamics of Lump Solutions, Rogue Wave Solutions and Traveling Wave Solutions 789

B2 f′ f =�

3D2x + a3Dy + a4�

f ′ f = 0,

B3 f′ f =�

6Dx Dy + 6a5Dx�

f ′ f = 0,

B4 f′ f =�

4Dt +αD3x −αa1Dx +αa6

�

f ′ f = 0,

B5 f′ f =�

3αD2x+αa7Dx − a4α− 4αa8Dx

�

f ′ f = 0,

B6 f′ f =�

−4αa9Dx − 4δa11Dz�

f ′ f = 0,

B7 f′ f = (4Dt + 4δDz + 4δa10Dx + 4δa11Dt) f

′ f . (4.5)

4.2. Traveling wave solution

Substituting the solution f = 1 into the Eq. (1.1) yields

Dns g f = Dns g =

∂ n

∂ sng, n≥ 1,

which is reduced to the initial variable u. The Bäcklund transformation (4.5) can be split

into the following group of linear equations:

3 f ′x x y+ a1 f

′y+ a2 f

′ + 4γ f ′x+ a8 f

′y− 4δa10 f

′z= 0,

3 f ′x x + a3 f′y + a4 f

′ = 0,

f ′x y + a5 f′x = 0,

4 f ′t + α f′x x x −αa1 f

′x +αa6 f

′ = 0,

3 f ′x x+ a7 f

′x− a4 f

′ − 4a8 f′x= 0,

αa9 f′x +δa11 f

′z = 0,

f ′t +δ f′

z +δa10 f′x +δa11 f

′t = 0.

(4.6)

The function f ′ is sought in the form

f ′ = 1+ηeφ1 x+φ2 y+φ3z−φ4 t , φ1 6= 0, (4.7)

where φ1,φ2,φ3 and φ4 are constants and η is a real parameter. Setting a2 = a4 = a6 = 0,

we obtain

φ4 =−a1αφ1 +αφ

31

4, φ3 = −

a9αφ1

δ, a3 =

−3φ21

φ2, a5 = −ψ2,

a7 =16α�

φ31φ2 + γφ

21+δφ2

3−φ3φ4 + δa11φ3φ4

�

+ 3αφ21φ2 + 4δφ3φ4 − 16φ1φ4

αφ1φ2,

a8 =4α�

a11δφ3φ4 + γφ21 −φ

31φ2 +δφ

23 −φ3φ4�

+δφ3φ4 − 4φ2φ4αφ1φ2

,

a10 =δa11φ4 −δφ3 +φ4

δφ1.

790 D. Guo, S.-F. Tian, X.-B. Wang and T.-T. Zhang

–30–100

102030

x

–30–20–1001020y

05

101520253035

u

a1)

–30–100

102030

x

–30–20–1001020y

05

101520253035

u

b1)

–30–100

102030

x

–30–20–1001020y

05

101520253035

u

c1)

0

5

10

15

20

25

30

35

u

–40 –30 –20 –10 10 20t

a2)

0

5

10

15

20

25

30

35

u

–30 –20 –10 10 20 30t

b2)

0

5

10

15

20

25

30

35

u

–20 –10 10 20 30 40t

c2)

Figure 7: Traveling wave solution, φ1 = 1,φ2 = 1.2,φ3 = 0.9,φ4 = 2.1. (a) Perspetive view of the realpart of the wave (t = −10, z = 0). (b) Perspetive view of the real part of the wave (t = 0, z = 0).() Perspetive view of the real part of the wave (t = 10, z = 0).

It follows that we have the exponential wave solution

u =6α

β

�

ln f ′�

x,

where

f ′ = 1+ηeφ1 x+φ2 y−(a9αφ1/δ)z−((−a1αφ1+αφ31)/4)t .

Representing f ′ as

f ′ = φ1 x +φ2 y +φ3z −φ4 t, (4.8)

substituting (4.8) into (4.6) and choosing ai = 0, 2 ≤ i ≤ 7, we obtain a rational solution— viz.

u =6α

β

ψ1ψ1 x +ψ2 y +ψ3z −ψ4t

, (4.9)

the graph of which is shown in Fig. 7.

Let us briefly analyse the effects of the free parameters on the amplitude and the width

of the traveling wave solution. The solution of u is presented in Fig. 7. Note that the ampli-

tude, velocity and the widths of the traveling wave stay the same during the propagation.

It shows that the amplitude of the excited state is bounded and almost the same in different

spaces.

5. Bright and Dark Soliton Solutions

In this section, we consider one-soliton solutions of the Eq. (1.1) by using the traveling

wave method with tanh−sech functions. The influence of the parameters α,β ,γ,δ on thefield profile of one-soliton solutions is graphically illustrated in figures below.

Dynamics of Lump Solutions, Rogue Wave Solutions and Traveling Wave Solutions 791

5.1. The bright soliton solutions

For the bright soliton solution of the Eq. (1.1) the following theorem holds.

Theorem 5.1. The Eq. (1.1) has a bright soliton solution of the form

U = −5mα

βsech2�

mx + ny + qz − tγm2 +δq2 + 4αm3n

m+ n+ q

�

, (5.1)

where αβ < 0 and m, n,q denote the inverse widths of the soliton.

Proof. Consider the function

U(x , y, z, t) = λ · sechp(mx + ny + qz −υt), (5.2)

where m, n,q are the inverse widths of the soliton, whereas λ and υ are, respectively, ampli-

tude and velocity of the soliton. The exponent p will be determined later on. Substituting

(5.2) into (1.1) and equating the highest exponents at p+4 and 2p+2 yield p+4= 2p+2

or p = 2. Setting sech2p+2 = sechp+4 = 0 implies

βλ2m2np2(2p+ 2) +αλm3np(p+ 1)(p+ 2)(p+ 3) = 0.

It follows that

λ= −5mα

β, (5.3)

with the additional condition αβ < 0, which is needed for the solution existence. We next

set the coefficient at sechp(mx + ny + qz −υt) to zero, so that

−p2mvλ− p2nvλ− p2qvλ+ γp2m2λ+δp2q2λ+αp4m3nλ= 0,

or

υ=γm2 +δq2 + 4αm3n

m+ n+ q, (5.4)

where α,γ,δ,β are real constants. Finally, substituting (5.3) and (5.4) into (5.2), we obtain

the bright soliton solution of the (3+ 1)-dimensional variable-coefficient KP equation.

The corresponding solution U is shown in Fig. 8. The amplitude, velocity and the

width of the bright-soliton are not changed during the propagation but the excited state

is bounded and almost the same in different spaces.

5.2. The dark soliton solutions

The dark soliton solution of the Eq. (1.1) are described by the following theorem.

Theorem 5.2. The Eq. (1.1) has the bright soliton solution of the form

U =5mα

βtanh2�

mx + ny + qz − tγm2 +δq2 − 8αm3n

m+ n+ q

�

, (5.5)

where αβ > 0 and m, n,q are the inverse widths of the soliton.

792 D. Guo, S.-F. Tian, X.-B. Wang and T.-T. Zhang

–10–5

05

10

x

–10–5

05

10

t

01234

u

a)

–10

–5

0

5

10

t

–10 –5 0 5 10x

b)

1

2

3

4

5

u

–4 –2 2 4t

c)

–4

–2

0

2

4

–4 –2 2 4

d)

–10

–5

5

10

t

–20 –10 10 20x

e)

–4

–2

0

2

4

–2 2 4

f)

Figure 8: Bright soliton solution (5.1), α = 1,β = −1, m = n = q = δ = 1. (a) Soliton solution (5.1).(b) Overhead view. () Wave propagation along the t axis. (d) Contour plot in spherial oordinates.

(e) Contour plot. (f) Contour plot in ylindrial oordinates.

Proof. Consider the function

U(x , y, z, t) = λ · tanhp(mx + ny + qz −υt), (5.6)

where m, n,q are the inverse widths of the soliton and λ and υ amplitude and velocity of the

soliton, respectively. The exponent p will be determined later on. Substituting (5.6) into

(1.1) and equating the highest exponents of p+4 and 2p+2 yield p+4= 2p+2 or p = 2.

Setting the coefficients at tanh2p+2 and tanhp+4 to zero, and collecting the coefficients at

tanhp lead to the equations

αλm3np(p + 1)(p+ 2)(p+ 3)− 2βλ2m2np2(p+ 1) = 0,

2p2mvλ+ 2p2nvλ+ 2p2qvλ− 2γp2m2λ− 2δp2q2λ+ 4αp4m3nλ= 0.

It follows

λ=5mα

β, υ =

γm2 +δq2 − 8αm3nm+ n+ q

,

with the additional condition αβ > 0 needed for the solution existence. Finally, the dark

soliton solution of the (3 + 1)-dimensional variable-coefficient KP equation when these λ

and υ are substituted into the Eq. (5.6).

The results above show that the existence conditions for bright-dark soliton solutions

are adverse to each other. The solution of U is displayed in Fig. 9. The amplitude, velocity

and width of the bright-soliton remain the same during the propagation but the excited

state is bounded and almost the same in different spaces.

Dynamics of Lump Solutions, Rogue Wave Solutions and Traveling Wave Solutions 793

–10–5

05

10

x

–10–5

05

10

t

12345

u

a)

–10

–5

0

5

10

t

–10 –5 0 5 10x

b)

0

1

2

3

4

5

u

–4 –2 2 4t

c)

–4

–2

0

2

4

–4 –2 2 4

d)

–10

–5

5

10

t

–20 –10 10 20x

e)

–4

–2

2

4

–4 –2 2 4

f)

Figure 9: Dark soliton solution (5.5), α = β = 1, m = n = q = δ = 1. (a) Soliton solution (5.5).(b) The overhead view of the wave. () Wave propagation along the t axis. (d) Contour plot inspherial oordinates. (e) Contour plot. (f) Contour plot in ylindrial oordinates.

6. Conclusions

We study the (3 + 1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili

equation by using the Hirota bilinear method and the graphical representations of the so-

lutions. The breather, lump and rogue wave solutions are obtained and the influence of

the parameter choice is analysed. Dynamical behavior of the periodic solutions is visually

shown in different planes. The rogue waves are obtained and localised in time by the long

wave limit of breather with indefinitely large periods. In three dimensions the breathers

have different dynamics in different planes. The traveling wave solutions are constructed

by the Bäcklund transformation. The traveling wave method is used in construction of

bright-dark soliton solutions represented by sech and tanh functions. The corresponding

3D figures show various properties of the solutions. These results can be used to demon-

strate the interactions of water waves in shallow water the ship traffic on the surface.

Acknowledgments

The authors would like to thank the editor and the referees for their valuable comments

and suggestions.

This work was supported by the Postgraduate Research and Practice of Educational

Reform for Graduate Students in CUMT under Grant No. 2019YJSJG046, by the Natural

Science Foundation of Jiangsu Province under Grant No. BK20181351, by the Six Tal-

ent Peaks Project in Jiangsu Province under Grant No. JY-059, by the Qinglan Project

of Jiangsu Province of China, the National Natural Science Foundation of China under

794 D. Guo, S.-F. Tian, X.-B. Wang and T.-T. Zhang

Grant No. 11975306, by the Fundamental Research Fund for the Central Universities un-

der the Grant Nos. 2019ZDPY07 and 2019QNA35, and by the General Financial Grant

from the China Postdoctoral Science Foundation under Grant Nos. 2015M570498 and

2017T100413.

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