Lump waves, solitary waves and interaction phenomena to the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation

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Lump waves, solitary waves and interaction phenomena to the (2+1)-dimensional Konopelchenko–Dubrovsky equation Wenhao Liu, Yufeng Zhang ∗ , Dandan Shi School of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu, 221116, People’s Republic of China

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i n f o

a b s t r a c t

Article history: Received 30 August 2018 Received in revised form 15 October 2018 Accepted 17 October 2018 Available online xxxx Communicated by R. Wu Keywords: The (2+1)-dimensional Konopelchenko–Dubrovsky equation Lump waves Solitary waves Interaction waves

In this paper, we investigated the (2+1)-dimensional Konopelchenko–Dubrovsky equation. The lump waves, solitary waves as well as interaction between lump waves and solitary waves are presented based on the Hirota bilinear form of this equation. It is worth noting that the rational solutions are obtained by taking a long wave limit, and we also discussed the lump solutions and rogue wave solutions. Moreover, all these solutions are presented via 3-dimensional plots and density plots with choosing some special parameters to show the dynamic graphs. © 2018 Elsevier B.V. All rights reserved.

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1. Introduction Nonlinear evolution equations (NLEEs) and their solutions have been used to describe some nonlinear physical phenomena and play an important role in almost all the branches of physics [1,2], such as fluid mechanics, solid state physics, signal processing, hydrodynamics, nonlinear control, image processing and so on [3–7]. The research on soliton solutions has attracted more and more attention, which is of great significance in explaining some physical phenomena. Many methods to obtain soliton solutions are proposed on the basis of the Hirota bilinear method with the deepening of research [8], including the sine-cosine method [9], the tanh–coth method [10–12], inverse scattering transformation [13], the symbol calculation method [14,15] and so on. During the past few years, interaction phenomena has become a hot topic [16,17], and many theoretical and experimental studies of lump waves have been mentioned [18–22], which those results set off a huge academic wave. Therefore, it is necessary to study interaction between lump waves and solitary waves. In this paper, we consider the following (2+1)-dimensional Konopelchenko–Dubrovsky equation [23]

ut − u xxx − 3(∂x−1 u y ) y +

3 2

φ 2 u 2 u x + 3φ u x ∂x−1 u y − 6ρ uu x = 0, (1.1)

*

Corresponding author. E-mail address: [email protected] (Y. Zhang).

https://doi.org/10.1016/j.physleta.2018.10.018 0375-9601/© 2018 Elsevier B.V. All rights reserved.

where ∂x ∂x−1 = ∂x−1 ∂x = 1, and ∂x−1 (·) = −∞ (·)ds. When φ = 0, the (2+1)-dimensional Kadomtsev–Petviashvili like equation can be derived. Similarly, if the ρ = 0, the (2+1)-dimensional KD equation is converted to modified KP equation. In 1984, Konopelchenko and Dubrovsky first got the Konopelchenko–Dubrovsky equations [24]. And many aspects of the (2+1)-dimensional Konopelchenko– Dubrovsky equation were described in other papers, including some multi-soliton solutions are obtained to the equation with the Bäcklund transformation by And Lin [25] as well as some soliton wave solutions and soliton-like solutions are presented in [26–28], but there is no same type articles are found when compared with this paper. The rest of this paper is structured as follows. In section 2, the bilinear form of the KD equation is obtained by utilizing transformation u = ρ2 ln( f )xx . And the theoretical analysis of the interaction solutions is introduced. In section 3, we discussed the lump waves by making k1 = k2 = 0 based on the analysis of section 2. In section 4, the N-soliton solutions are given and the solitary wave solutions is obtained. What is more, we also have discussed the rational solutions of Eq. (1.1) by taking a long wave limit. In section 5, we analyze the interaction between lump wave and solitary wave of the (2+1)-dimensional Konopelchenko–Dubrovsky equation. At present, there is still a lot of research to be done of the (2+1)-dimensional Konopelchenko–Dubrovsky equation, including Bäcklund transformation, Lax pair and so on. We will try to finish it in the future work.

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2. Mathematical analysis

13 − 3m2 = 0,

In the section, we will make the theoretical analysis of the interaction solutions. But before that, utilize the following transformation

which yields broad categories of lump waves Eq. (2.1). If we want to present lump solutions specifically, H = 2 can be considered. Therefore, l = (l1 , l2 ), m = (m1 , m2 ), n = (n1 , n2 ) and λ = (λ1 , λ2 ). And then, the Eq. (3.1) can be written as

u=

2

ρ

ln( f )xx ,

(2.1)

we can convert the KD equation into a bilinear form as follows

Eq. (1.1) =( D x D t −



D 4x

− 3D 2y ) f

l1 = l1 , A=−

where f is real function with respect to variables x, y and t. D x D t , D 4x and D 2y are called Hirota bilinear D operators. Now, we consider the interaction solutions of the KD equation Eq. (1.1) has the following form

(2.3)

in which

f =A+

n 

ξ = l 0 x + m0 y + n0 t ,

where l0 , m0 and n0 are arbitrary scalar parameters to be determined later, and l = (l1 , l2 , ..., l H ), m = (m1 , m2 , ..., m H ), n = (n1 , n2 , ..., n H ), λ = (λ1 , λ2 , ..., λ H ), H is positive integers. Then the interaction solutions u can be written as follows

ρ

f

+ 3l41l22

m1 = m1 ,

+ 3l21l42

+ l62 , )2

λ1 = λ1 , m2 = m2 ,

(l1m2 − l2m1 3(l1 m21 − l1 m22 + 2l2 m1m2 ) l21 + l22 3(2l1m1m2 − l2 m21 + l2 m22 ) l21 + l22

λ2 = λ2 , (3.4)

, .



2

2l21 + 2l22

ρ

f

−

f2



= 2l1 (l1 x + m1 y + +2l2 (l2 x + m2 y +



(3.5)

,

3(l1m21 − l1 m22 + 2l2 m1m2 ) l21 + l22

3(2l1m1m2 − l2 m21 + l2 m22 ) l21 + l22

t + λ1 )

2 t + λ2 )

(2.5)

χ = lx + m y + nt + λ,

u=

l2 = l2 ,

l61

Substituting Eq. (3.3) and Eq. (3.4) into Eq. (2.1), the lump wave solutions of Eq. (1.1) can be expressed

with

where (∂ xi ) means a series of partial derivatives operations to xi (i = 1, 2, ..., n − 1), and χ = ni=1 xi Pi . A, k1 and k2 are free scalar parameters. If putting

2l2 + k1l20 e ξ + k2l20 e −ξ

n2 =

(2.4)

i , j =0



n1 =

u=

p i j xi x j + k1 e ξ + k2 e −ξ

= A + χ 2 + k1 e ξ + k2 e −ξ ,

2

f = A + (l1 x + m1 y + n1 t + λ1 )2 + (l2 x + m2 y + n2 t + λ2 )2 . (3.3)

(2.2)

u = (∂ xi )lnf ,

(3.2)

In this case, we can get the relationship of parameters by calculating the Eq. (3.2)

·f

2 =2 ( f f xt − f x f t ) − (3 f xx − 4 f x f xxx + f f xxxx )  −3( f f y y − f y2 ) ,

l 4 = 0,

−

(2χ l + k1l0 e ξ − k2l0 e −ξ )2 f2

(3.6) and

(2.6) Observing equation (2.4), we can easily find that it consists of two parts: lump wave part A + χ 2 and soliton part k1 e ξ (or k2 e −ξ ). If taking k1 = k2 = 0 and χ 2 = 0, a pure lump of Eq. (2.4) can be obtained. Similarly, we can also get a pure soliton when χ 2 = 0 and k1 = 0, k2 = 0 (or k2 = 0, k1 = 0). In particular, ξ → ∞ (ξ → −∞) and the k1 e ξ (k2 e −ξ ) tend to infinity. At the same time, χ 2 and k2 e −ξ (k1 e ξ ) can be ignored.



f = l 1 x + m1 y +





.

+ l 2 x + m2 y + −

Based on the analysis of section 2, the lump waves of Eq. (1.1) have the following form 2

f = A+χ .

(3.1)

3(l1 m21 − l1 m22 + 2l2m1 m2 ) l21 + l22 3(2l1m1 m2 − l2 m21 + l2 m22 ) l21 + l22

l61 + 3l41l22 + 3l21l42 + l62

(l1m2 − l2m1 )2

2 t + λ1

2 t + λ2

(3.7)

.

In addition, if we find the critical point of the lump waves, the moving path of the lump waves can be described. Consider the case of f x = f y = 0, we have

x=−

13 l2

y=−

3. Lump wave solutions

,

23 m2

t−

01

t−

02

l2 m2

, (3.8)

.

Then, after solving the above equations, we obtain

y=−

l2 13 m2 23

x−

23 01 + 13 02 , m2 13

(3.9)

13 = l · n = h=1 lh nh , 23 = m · n = h=1 mh nh , 11 = l · l, 22 = m · m, 33 = n · n, 01 = λ · l and 02 = λ · m. Substituting

with the case of Eq. (3.4). Analytical Eq. (3.9), we know that the lump waves move along the straight line. Fig. 1 gives the dynamic graphs of lump wave solutions by selecting following appropriate values of parameters

Eq. (3.1) into Eq. (2.2), by collecting the coefficients of the same exponent, we have

ρ = 2, l1 = m1 = m2 = 1, l2 = λ1 = λ2 = 0.

Specially, we introduce the symbols 12 = l · m =

H

H

H

h=1 lh mh ,

(3.10)

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Fig. 1. The lump wave solutions of Eq. (1.1) in the (x,y) plane: a three-dimensional plot at t = 0, b density plot, c the contour plot about the progress of moving described by the straight line Eq. (3.9) at t = −2, t = 0, t = 2, respectively. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

Fig. 2. The lump solutions of Eq. (4.8) in (x,y), (x,t) (y,t) plane, respectively.

4. Solitary wave solutions In the section, the solitary wave solutions are obtained. What is more, we have also discussed the rational solutions of Eq. (1.1) by taking a long wave limit. The N-soliton solutions given in Eq. (4.1) can be ascertained by applying the Hirota method. Of these, f has the following form

f = fN =



N 

exp(

μ=0,1

μ j μk A jk +

j
N 

μ j η j ).

(4.1)

j =1

(4.2)

ρ = 1 of Eq. (2.1), we have

if putting

k2 = l2 ,

 η20 = η10 = i π ,

into Eq. (2.1) with ρ = 1, and taking the limit as help of MAPLE, we derive

(4.6)

→ 0. With the

θ0 =

4

( p1 − p2

)2

(4.7)

.

For arbitrary p i , we always have θ0 ≥ 0. Substituting Eq. (4.5) into Eq. (2.1), the rational solutions to the Eq. (1.1) can be expressed as

(k1 − k2 )2 − ( p 1 − p 2 )2 , (k1 + k2 )2 − ( p 1 − p 2 )2 ηi = ki (x + p i y + w i t ) + ηi0 , A 12 =

(4.3)

u=−

2(θ12 + θ22 − 2θ0 )

(θ1 θ2 + θ0 )2

.

(4.8)

Setting p i = ai + ib i , then, we can get the following two cases:

w i = k2i + 3p 2i .

Case I (Lump solution):

Therefore, through transformation u = 2ln( f )xx , we get



u = 2 ln(1 + e η1 + e η2 + e η1 +η2 + 12 ) where e 12 = A 12 .

(4.5)

θi = x + p i y + 3p 2i t ,

f 2 = 1 + exp η1 + exp η2 + A 12 exp(η1 + η2 ).



f = (θ1 θ2 + θ0 )l1l2 2 + o( 3 ),

k1 = l1 ,

We take N = 2, so f 2 is derived as

Let

Next, in order to find the rational solutions of Eq. (2.1), a long wave limit of function f must be taken. The function f can be rewritten as

xx

,

(4.4)

When b i = 0, the corresponding rational solutions are lump solution. By selecting the following appropriate parameters

p 1 = 1 + 2I ,

p 2 = 1 − 2I ,

(4.9)

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Fig. 3. The line rogue wave solutions of Eq. (4.8) in the (x,y) plane.

Fig. 4. The three-dimensional plots and density plots of the interaction solutions for Eq. (1.1) with the same parameters as Eq. (5.6) in the (x,y) plane.

the dynamic graphs of the lump solutions of Eq. (4.8) are shown in Fig. 2 with different planes. It is easy to see that the profile of the lump solution remains the same although it is in (x,y), (x,t) (y,t) plane, respectively. Case II (Rogue wave solution): When b i = 0, that is to say p i is real, we can paint the profile of rogue waves with parameters p 1 = 1, p 2 = −1 in Fig. 3. Obviously, Fig. 3 satisfies regular line rogue wave solutions. When 0 | t |, the line rogue wave approaches the constant background, and the much higher amplitude can be derived at some point. Moreover, if we change the coefficient of mi and ni , the pro-

file of rogue wave solution is also changed in the other plane. Here we no longer state about it. 5. Interaction between lump waves and solitary waves Based on the obtained lump wave solutions and solitary wave solutions, we will study interaction between lump waves and solitary waves of Eq. (1.1) in the section. It is very interesting that we find that lumps will be swallowed by the solitary waves by investigating the interaction between lump waves and solitary waves. In general, we must emphasize that the corresponding coefficients k1 and k2 can not be zero at the same time. (That is k1 = 0, k2 = 0 or

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Fig. 5. The three-dimensional plots and density plots of the interaction solutions for Eq. (1.1) with the same parameters as Eq. (5.7) in the (x,y) plane.

k1 , k2 = 0). Substituting Eq. (2.4) into Eq. (2.2) and collecting the coefficients, we can get a set of algebraic equations

⎧ 13 − 3m2 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − l40 + l0n0 − 3m20 = 0, ⎪ ⎨ (−4l30 + n0 )l − 6m0 m + l0 n = 0, ⎪ ⎪ ⎪ ⎪ (−4l40 + l0n0 − 3m20 )k1k2 − 3l4 = 0, ⎪ ⎪ ⎪ ⎩ − 6l20l2 + 13 − 3m2 = 0.

+ (5.1)

3m20 l0

,

l0 = l0 ,

m0 = m0 ,

m = l 0 l,

n=(

l20

− 3l20 )l +

6m0 l0

m.

m0 = m0 ,

λ2 = λ2 , n1 = ( n2 = (

3m20 l20 3m20 l20

m1 = l 0 l 2 ,

− 3l30 )l2 + 6m0l1 ,

l2 = l2 .

f

2 

ρ

− 2l1 (l1 x + m1 y + n1 t + λ1 )

+ 2l2 (l2 x + m2 y + n2 t + λ2 ) + k1l0 e ξ − k2l0 e −ξ

2

 × f −2 ,

3m20 )t. l0

5.1. Interaction between lump waves and line single-soliton waves

(5.3)

In this part, we consider the case of k1 = 0 and k2 = 0. In Fig. 4, the interaction between lump waves and line single-soliton waves of Eq. (1.1) is drawn for a particular choice of the following parameters

m2 = l 0 l 1 , l1 = l1 ,



(5.2)

λ1 = λ1 ,

− 3l30 )l1 + 6m0l2 ,

ρ

and ξ = l0 x + m0 y + (l30 +

Similarly, let us consider M = 2, then l = (l1 , l2 ), m = (m1 , m2 ), n = (n1 , n2 ) and λ = (λ1 , λ2 ). A series of constraining expressions can be derived as follows

l0 = l0 ,

2l21 + 2l22 + k1l20 e ξ + k2l20 e −ξ

where l1 , m1 , n1 , l2 , m2 , n2 , λ1 , λ2 , l0 are satisfied with Eq. (5.4),

what is more

3m20



(5.5)

Solving the above equations, we derive

n0 = l30 +

u=

2

(5.4)

According to these parameters, we can finally get the interaction solutions of Eq. (1.1)

A = 1,

l 1 = 1,

k1 = 0.1,

l2 = −1.6,

k 2 = 0,

λ1 = 0 ,

l0 = 0.7,

λ2 = 0 ,

m0 = 0.4,

ρ = 1.

(5.6)

From Fig. 4, it is easy to see that the wave consists of two parts, including lump waves and line single-soliton waves. We show the dynamics of the two waves. According to Fig. 4(a), (d), we judge that the lump waves and line single-soliton waves always separate from each other when t  0. Step by step, the two waves, as shown in Fig. 4(b), (e), met together and the lump wave begins to be swallowed by single-soliton. Finally, when t  0, the lump wave vanishes gradually, and only the soliton wave exists (see Fig. 4(c), (f)). Specially, the two cases of k1 = 0, k2 = 0 and k2 = 0, k1 = 0 can be similar.

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5.2. Interaction between lump waves and line twin-soliton waves

References

Similarly, this situation of k1 = 0 and k2 = 0 is considered. We choose the following parameters

A = 1,

l 1 = 1,

k1 = 0.1,

l2 = −1.6,

k2 = 0.1,

λ1 = 0 ,

l0 = 0.7,

λ2 = 0 ,

m0 = 0.4,

ρ = 1,

(5.7)

and the interaction solution is shown in Fig. 5. The difference between Fig. 4 and Fig. 5 is that the solitary wave is twin-soliton waves. When t = 0, that the lump waves and line twin-soliton waves separate from each other. When t → +∞ or t → −∞, the lump wave begins to be swallowed by one of twin-soliton waves and eventually disappear. 6. Conclusions and discussions In this paper, we mainly investigated the interaction between lump waves and solitary waves of the (2+1)-dimensional Konopelchenko–Dubrovsky equation. The lump solutions and the solitary wave solutions of the equation are obtained separately. The interaction phenomena that lumps will be swallowed by the solitary waves is analyzed in detail. What is more, rational solution is mentioned by taking a long wave limit. Moreover, the dynamic properties of these solutions are discussed by some 3-dimensional plots and density plots with choosing some special parameters. Interaction phenomena which describes some nonlinear physical phenomena play an important role. In the near future, more work will be done in the fields of mathematical physics and engineering. Acknowledgements This work is supported by the Fundamental Research Funds for the Central University (No. 2017XKZD11).

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