Research ArticleSoliton Molecules and Some Novel Types of Hybrid Solutions to(2+1)-Dimensional Variable-Coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

Shuxin Yang,1,2 Zhao Zhang,1 and Biao Li 1

1School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China2School of Foundation Studies, Zhejiang Pharmaceutical College, Ningbo 315500, China

Correspondence should be addressed to Biao Li; libiao@nbu.edu.cn

Received 13 November 2019; Accepted 9 December 2019; Published 7 January 2020

Academic Editor: Laurent Raymond

Copyright © 2020 Shuxin Yang 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.

Soliton molecules of the (2 + 1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation are derived byN-soliton solutions and a new velocity resonance condition. Moreover, soliton molecules can become asymmetric solitonswhen the distance between two solitons of the molecule is small enough. Finally, we obtained some novel types of hybridsolutions which are components of soliton molecules, lump waves, and breather waves by applying velocity resonance, moduleresonance of wave number, and long wave limit method. Some figures are presented to demonstrate clearly dynamics featuresof these solutions.

1. Introduction

Solitons as localized nonlinear waves exhibit many interest-ing properties [1]. In particular, solitons can form stablebound states known as soliton molecules, which have beenobserved experimentally in some fields [2–8]. For the firsttime, soliton molecules were experimentally observed indispersion-managed optical fibers [2]. In 2017, the authorsin [5] resolved the evolution of femtosecond soliton mole-cules in the cavity of a few-cycle mode-locked laser by meansof an emerging timestretch technique. In 2018, Liu et al. haveexperimentally observed the real-time dynamics of the entirebuildup process of stable soliton molecules for the first time[6]. These soliton compounds may find important applica-tions in fiber optic communication systems to enhance theirdata-carrying capacity [9]. The existence of two-soliton boundstates in Bose-Einstein condensates with contact atomicinteractions and some dynamic phenomena involving solitonmolecules was reported in [10, 11].

As we all know, exact solutions for some local or nonlocalnonlinear evolution equations (NLEEs) have been applied inthe nonlinear science fields. Solitons and rational solutions of

many NLEEs have been investigated by researches [12–16].Among these rational solutions, lump solutions, breatherwave solutions, and rogue wave solutions are hot point allthe time. Recently, the hybrid solutions of lump solutionswith other types of solutions draw a lot of attention, whichinclude lump-soliton [17–19], lump-kink solution [20], reso-nance stripe solitons [21–23], and some hybrid solutions[24–26]. Very recently, Lou [27] introduced a new possiblemechanism, the velocity resonant, to form soliton moleculesand asymmetric solitons of three (1 + 1)-dimensional fluidmodels: fifth-order KdV, SK equation, and KK equation. Tothe best of our knowledge, soliton molecules interacting withlump waves and breather waves has not been studied yet. Inthis paper, we will extend the velocity resonant method tosoliton molecules of (2+ 1)-dimensional systems and furtherexplore some novel type hybrid solutions among soliton mol-ecules, lump waves, and breather waves.

In many physical situations, with the inhomogeneities ofmedia and boundaries taken into account, the variable coef-ficient nonlinear evolution equations might be more realisticthan the constant coefficient ones [28–33]. In the pastdecades, the study of the nonlinear evolution of variable-

HindawiAdvances in Mathematical PhysicsVolume 2020, Article ID 2670710, 9 pageshttps://doi.org/10.1155/2020/2670710

coefficient has attracted the attention of mathematiciansand physicists. In modeling, a variety of complex nonlin-ear phenomena of physical and engineering fields and manyphysical and mechanical situations are governed by variable-coefficient equations. Therefore, seeking for soliton mole-cules and hybrid solitons of variable-coefficient equationsare also of great significance.

In this paper, we will focus on a generalized ð2 + 1Þ-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation such as the one givenbelow:

ut + a1uxxxxx + a2uxuxx + a3uuxxx + a4u2ux

+ a5uxxy + a6

ðuyydx + a7ux

ðuydx

+ a8uuy + a9u = 0,

ð1Þ

where u is a function of fx, y, tg and ai = aiðtÞ, i = 1,⋯, 9, areanalytic functions with respect to t. If the parameters are spe-cially chosen, a series of equations can be obtained, which canbe integrable [34, 35] or used to describe physical phenom-ena as the interaction between a water and a floating ice coverthe gravity-capillary waves [36].

Referring to [37], the bilinear form of Eq. (1) as follows:

DxDt + a1D6x + 5c1a1D3

xDy − 5c21a1D2y

� �f · f = 0, ð2Þ

under the following transformation:

u = 2c0 exp −ða9dt

� �ln fð Þxx , ð3Þ

where D is the Hirota’s bilinear differential operator, c0 andc1 are arbitrary constants, f = f ðx, y, tÞ is a real function of

variables fx, y, tg, and aiði = 1,⋯,9Þ satisfies the followingconditions:

a2 = a3 =15a1c0

exp −ða9dt

� �,

a4 =45a1c20

exp −ða9dt

� �,

a5 = 5c1a1,a6 = −5c21a1,

a7 = a8 =15c1a1c0

exp −ða9dt

� �,

ð4Þ

with fa1, a9g being the arbitrary functions of t and fc0 ≠ 0,c1g the arbitrary constants.

It is well known that the bilinear equation (2) includesseveral other forms: the usual (2+1)-dimensional fifth-orderKdV equation for selections fa1, c1gfKdVg = fð1/36Þ, 1g, the(2 + 1)-dimensional B-type Kadomtsev-Petviashvili equa-tion (BKP) model by taking fa1, c1gfBKPg = f1,−1g, and the(2 + 1)-dimensional Sawada-Kotera equation (KP) model bytaking fa1, c1gfKPg = f−1, 1g.

Based on the Hirota’s bilinear theory, the N-soliton solu-tions for Eq. (1) can be constructed as

u = 2c0 exp −ða9dt

� �

� ln 〠μ=0,1

exp 〠N

1≤i<jμiμ jAij + 〠

N

j=1μjηj

!" #( )xx

,ð5Þ

with

where kj, pj and ϕjðj = 1, 2,⋯,NÞ being arbitrary constants,∑μ=0,1 indicates a summation over all possible combinationsof μj = 0, 1ðj = 1, 2,⋯,NÞ.

The remainder of this paper is organized as follows.First, we aim to introduce a new velocity resonant conditionin Section 2, then soliton molecules are obtained based on N-soliton formula with applying the velocity resonant condi-

tion, and further we explore their fascinating dynamicalbehaviors. In Section 3, partial parameters are handled withthe velocity resonant condition, module resonance of wavenumber and long wave limit method, and some novel typesof interaction solutions including soliton molecules, lumpwaves, and breather waves are derived. Finally, the conclu-sions are summarized in Section 4.

ηj = kjx + pjy + ωj tð Þ + ϕj,

ωj tð Þ = −k6j + 5c1k3j pj − 5c21p2j

kj

ða1dt,

eAij =ki − kj� �

c1kik2j pi 2ki − kj� �

+ c1k2i kjpj ki − 2kj

� �+ k2i k

2j k2i − kikj + k2j� �

ki − kj� �h i

+ c21 kipj − kjpi� �2

ki + kj� �

c1kik2j pi 2ki + kj� �

+ c1k2i kjpj ki + 2kj

� �+ k2i k

2j k2i − kikj + k2j� �

ki + kj� �h i

+ c21 kipj − kjpi� �2 ,

ð6Þ

2 Advances in Mathematical Physics

2. Soliton Molecules and Asymmetric Solitons

To find nonsingular analytical resonant excitation fromEq. (5), we apply a novel type of resonant conditions(ki ≠ ±kj, pi ≠ ±pj), the velocity resonance,

kikj

= pipj

= ωi ′ tð Þωj ′ tð Þ

: ð7Þ

Combining Eq. (6), we can get the following expressions:

ki =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi−kj3 + 5c1 pj

kj

s,

pi =pjkj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi−kj3 + 5c1 pj

kj

s,

ð8Þ

or

ki = −

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi−kj3 + 5c1 pj

kj

s,

pi = −pjkj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi−kj3 + 5c1 pj

kj

s:

ð9Þ

It can be known under the resonance conditions (Eq. (8)or Eq. (9)) that two solitons for N = 2 in Eq. (5) are boundedto generate a soliton molecule. From Eqs. (5)–(7), we candeduce that when a1 is an arbitrary constant and a9 = 0, theamplitudes and the velocities of solitons molecules all remainunchanged during the evolutions. While a1 is a function oft and a9 ≠ 0, the amplitudes and the velocities of them varywith time. So, we will explore some dynamics of soliton mol-ecules from the above two cases.

Figure 1 displays the molecule structure with the param-eter selections

p1 = 1,

k1 = −56 ,

k2 = −ffiffiffiffiffiffiffi191

p

6 ,

p2 =ffiffiffiffiffiffiffi191

p

5 ,

ϕ1 = 0,ϕ2 = 6,c0 = c1 = a1 = 1,a9 = 0:

ð10Þ

From Figure 1, one can find that two solitons in themolecule are different because k1 ≠ k2 though their velocitiesare same.

If we change values ϕ1 and ϕ2, the distance between twosolitons of the molecule will change, respectively. When thedistance of two solitons is close enough to have an interactionwith each other, the soliton molecule will become an asym-metric soliton. Figure 2 is the plots of the asymmetric solitonsolutions with the parameters (10) except for ϕ2 = −2. FromFigure 2, one can see that the soliton molecule keeps its asym-metric shape and velocity during the evolution with times.Because asymmetric solitons are just special case of solitonmolecules, so we will not investigate these asymmetric soli-tons in the following paper.

Figure 3 shows the propagation of one-soliton moleculevia the parameter selections (10) except for fa1 = cos ðtÞ,a9 = 0:01t, ϕ2 = 16g. From Figures 3(b)–3(e), one can see that(i) the amplitudes of two solitons in the soliton molecules arechanged with the function exp ð− Ð a9dtÞ = exp ð−0:005t2Þ;(ii) the velocities of two solitons are periodically changedwith the function a1 = cos ðtÞ at the same time.

Two-soliton molecules can be generated from four soli-tons, k1, p1,w1 and k2, p2,w2 satisfy Eq. (8) and k3, p3,w3and k4, p4,w4 satisfy Eq. (8) or Eq. (9) at the same time.Figure 4 displays the elastic interaction property for the solu-tion (5) with N = 4 and with parameter selections

k1 = −54 ,

k2 =ffiffiffiffiffiffiffi103

p

4 ,

k3 = −23 ,

k4 = −ffiffiffiffiffi41

p

3 ,

p1 = 2,

p2 = −2ffiffiffiffiffiffiffi103

p

5 ,

p3 = −23 ,

p4 =ffiffiffiffiffi41

p

3 ,

ϕ1 = 10,ϕ2 = 0,ϕ3 = −3,ϕ4 = 12,a1 = c0 = c1 = 1,a9 = 0:

ð11Þ

As can be seen from Figure 4, the height of wave peaksand the velocities of wave peaks do not change except forthe phase after the collision of one-soliton molecule andanother soliton molecule. It is necessary to point out that iftaking a1 as a function of t and a9 ≠ 0, the heights and veloc-ities of two-soliton molecules will change with time t.

3Advances in Mathematical Physics

–10–5

05

10x

–10–5

05

10y

00.5

11.5

22.5

(a)

–5 0 5 10–10

–5

0

5

10

y

–10x

(b)

Figure 1: (Color online) soliton molecule structure for Eq. (1) with the parameter selections (10) at t = 0. (a) Three-dimensional plot. (b)Density plot.

–10–5

05

10x

–10–5

05

10y

00.40.81.21.6

(a)

x = –15x = 0x = 15

00.20.40.60.8

11.21.41.6

u

–4 –3 –2 –1t

1 2 3 4

(b)

y = –20y = 0y = 20

00.20.40.60.8

11.21.41.6

u

–4 –3 –2 –1t

1 2 3 4

(c)

Figure 2: (Color online) asymmetric soliton for Eq. (1) with the parameter selections (9)) except for ϕ2 = −2. (a) Three-dimensional plot att = 0. (b) Two-dimensional plot when y = 0 at x = −15, 0, 15. (c) Two-dimensional plot when x = 0 at y = −20, 0, 20.

–20–10

010

20x

–20–10

010

20y

00.5

11.5

22.5

(a)

t = –10t = 0t = 10

0

0.51

1.52

2.5

u

–20 –10 10 20x

(b)

t = –10t = 0t = 10

–20 –10 10 20x

0

0.51

1.52

2.5u

(c)

–20

–10

0

10

20

y

–10 –5 0 5 10t

(d)

–10 –5 0 5 10–20

–10

0

10

20

x

t

(e)

Figure 3: (Color online) one-soliton molecule for Eq. (1) with the parameter selections (10) except for a1 = cos ðtÞ, a9 = 0:01t, andϕ2 = 16. (a)Three-dimensional plot at t = 0. (b) Two-dimensional plot when y = 0 at t = −10, 0, 10. (c) Two-dimensional plot when x = 0 at t = −10, 0, 10.

4 Advances in Mathematical Physics

3. Some Novel Hybrid Solutions Consisting ofSoliton Molecules, Breathers, and Lumps

In this section, some novel hybrid solutions will be investi-gated which are soliton molecules interacting with breathersand lump waves. To our knowledge, soliton molecules inter-acting with breathers and lump waves has not been studiedyet. We study the interactions between soliton moleculesand other waves via velocity resonance, module resonance,and long-wave limit method.

The long-wave limit method is a powerful technique toget lump solution; based on the N-soliton solution (5), wecan obtain interaction solutions consisting of a solitonmolecule and a lump wave. For N = 4, we are taking along wave limit on k1, k2, p1, p2ðε→ 0Þ, k3, k4, p3, p4 whichsatisfies the velocity resonance condition; the parametersare as follows:

k1 = 1 + ið Þε,k2 = 1 − ið Þε,p1 = 2ε,p2 = 2ε,ϕ1 = πi,ϕ2 = πi,

k3 = −56 ,

k4 = −ffiffiffiffiffiffiffi191

p

6 ,

p3 = 1,

p4 =ffiffiffiffiffiffiffi191

p

5 ,

ϕ3 = 0,ϕ4 = 20,

a1 tð Þ = sin tð Þ,a9 tð Þ = 0:01,

c0 = 1,c1 = 1:

ð12Þ

Figure 5 displays the interaction between a solitonmolecule and a lump wave. The collisions are also elastic.It is necessary to point out that when a9 = 0, the height ofthe lump wave and soliton molecules do not change beforeand after the collisions, but when a9 = 0:01, the height of thelump wave and soliton molecules all decrease with the func-tion exp ð− Ð a9dtÞ = exp ð−0:01tÞ. At the same time, thevelocities of the soliton molecules are synchronously periodi-cally changed with the function a1 = sin ðtÞ.

Novel hybrid solutions of a soliton molecule and breatherwave can be generated by four solitons. Furthermore, ki, piand ωiði = 1, 2Þ should satisfy the velocity resonance condi-tion (8) or (9), and the other two solitons satisfy the module

resonance condition ηi = ηj. For instance, taking the follow-ing parameters:

k1 = −45 ,

k2 =7ffiffiffi7

p ffiffiffi2

p

10 ,

k3 =25 −

25 i,

k4 =25 + 2

5 i,

p1 = −65 ,

p2 =21

ffiffiffi7

p ffiffiffi2

p

20 ,

p3 =16 + i

2 ,

p4 =16 −

i2 ,

ϕ1 = ϕ3 = ϕ4 = 0,ϕ2 = 25,

a1 tð Þ = sin tð Þ,a9 tð Þ = 0:01,

c0 = 1,c1 = −1:

ð13Þ

As shown in Figure 6, four-soliton solution (Eq. (5)) withparameter selections (Eq. (13)) exhibits the interactionbetween a soliton molecule and a breather wave under partialvelocity resonance and the partial module resonance condi-tion. The interactions of soliton molecules and breather solu-tions are also elastic. When a1 = const:,a9 = 0, the amplitudesand velocities of the soliton molecules and the breather wavesremain the same before and after the collisions. If a1 is a func-tion of t and a9 ≠ 0, their amplitudes and velocities can bechanged during evolution. Under the parameters in Figure 6,their amplitudes decrease with time due to a9 = 0:01 and theirvelocities are periodically changed due to a1 = sin ðtÞ.

More generally, we can obtain the general hybrid solutionsconsisting of m-soliton molecules, n-breather waves, and q-lump waves under the following parameter constraints:

k1k2

= p1p2

= w1w2

,⋯, k2m−1k2m

= p2m−1p2m

= w2m−1w2m

, η2m+1

= η2m+2,⋯, η2m+2 n−1 = η2m+2 n, k2m+2 n+1= K2m+2 n+1ε, k2m+2 n+2 = K2m+2 n+1ε,⋯, k2m+2 n+2 q−1

= K2m+2 n+2 q−1ε, k2m+2 n+2 q = K2m+2 n+2 q1ε, p2m+2 n+1

= P2m+2 n+1ε, p2m+2 n+2 = P2m+2 n+1ε,⋯, p2m+2 n+2 q−1

= P2m+2 n+2 q−1ε, p2m+2 n+2 q = P2m+2 n+2 q1ε, ϕ2m+2 n+1

= πi, ϕ2m+2 n+2 = πi,⋯, ϕ2m+2 n+2 q = πi, ε→ 0:

ð14Þ

5Advances in Mathematical Physics

–40–2002040 x

–20–10

010

20

y

00.5

11.5

22.5

u

(a) t = 0

–40–2002040 x

–20–10

010

20

y

00.5

11.5

22.5

u

(b) t = 8

–40–2002040 x

–20–10

010

20y

00.5

11.5

22.5

u(c) t = 12

–20

–10

0

10

20

30

x

–10 –5 0 5 10t

(d) y = 4

–20

–10

0

10

20

30

x

–10 –5 0 5 10t

(e) x = −8

Figure 5: (Color online) interaction of a soliton molecule and a lump wave for the (2 + 1)-dimensional variable-coefficient CDGKS equationwith parameter (12).

–20–10

010

2030

x

–30 –20 –10 0 10 20

y

0

2

4

u

(a) ½t = 0�

–30

–20

–10

0

10

20

30

x

–15 –10 –5 0 5 10 15t

(b) ½y = 0�

–15 –10 –5 0 5 10 15t

–20

–10

0

10

20

y

(c) ½x = 0�

Figure 6: (Color online) interaction of a soliton molecule and a breather wave for the (2 + 1)-dimensional variable-coefficient CDGKSequation with parameter (13).

–30–20

–100

1020

30

x

–30–20

–100

1020

30

y

0

2u

(a)

–30

–20

–10

0

10

20

30

y

–30 –20 –10 0 10 20 30x

(b)

Figure 4: (Color online) two-soliton molecules for the solution (5) of Eq. (1) with the parameter selections (11) at t = 0. (a) Three-dimensional plot. (b) Density plot.

6 Advances in Mathematical Physics

Then, we can get interactions between soliton molecules,lumps, and breathers by taking ε→ 0. To clearly describe theinteraction between them, let us take a simple example ofN = 6. Next, we set fk1, k2, p1, p2g which satisfies the velocityresonant condition, fk3, k4, p3, p4g, which satisfies moduleresonance condition, then taking a long wave limit on fk5,k6, p5, p6gðε→ 0Þ, taking parameters as follows:

c0 = 1,

c1 = −1,

k1 = −45 ,

k2 =7ffiffiffi7

p ffiffiffi2

p

10 ,

k3 =25 −

25 i,

k4 =25 + 2

5 i,

k5 =12 + i� �

ε,

k6 =12 − i� �

ε,

p1 = −65 ,

p2 =21

ffiffiffi7

p ffiffiffi2

p

20 ,

p3 =16 + i

2 ,

p4 =16 −

i2 ,

p5 = −2 ε,

p6 = −2 ε,

ϕ1 = 0,

ϕ2 = 25,

ϕ3 = 0,

ϕ4 = 0,

ϕ5 = iπ,

ϕ6 = iπ,

a1 = 1,

a9 = 0:

ð15Þ

Figure 7 displays the interaction between a soliton mole-cule, a lump wave, and a breather wave descried by Eq. (5)with the parameter selections in Eq. (15). The interactionsbetween these waves are also elastic.

4. Conclusion

In this paper, soliton molecules and asymmetric solitons of(2 + 1)-dimensional variable-coefficient Caudrey-Dodd-Gib-bon-Kotera-Sawada equation are theoretically obtained byvelocity resonance. By introducing a new velocity resonancecondition, we obtained the soliton molecules from the gen-eral N-soliton expression; see Figures 1, 3, and 4. When tak-ing suitable values of ϕ, the soliton molecules can change toasymmetric soliton; see Figure 2. It is necessary to point outthat when a1 = const: and a9 = 0, the height of soliton mole-cules do not change before and after the collisions; but whena1 is a function of t and a9 ≠ 0, the height of soliton moleculesare changed with the function exp ð− Ð a9dtÞ. At the sametime, the velocities of the soliton molecules are synchro-nously changed with the function a1ðtÞ. Taking a long wavelimit on part of parameters and employing resonance condi-tion on others, the new hybrid solutions consisting solitonmolecules and lump wave can be obtained; see Figure 5. Byemploying velocity resonance condition and module reso-nance condition on wave numbers, we can get a new hybridsolution consisting soliton molecules and breather waves;see Figure 6. By using velocity resonance, module resonance

–20–10

010

2030

x

–30–20

–100

10

y

0

2u

(a) ½t = 0�

–40–30

–20–10

010

x

–30–20

–100

10

y

0

2u

(b) ½t = −1�

–50–40

–30–20

–100

x

–30–20–10 0 10 20 30 40

y

0

2u

(c) ½t = −2�

Figure 7: (Color online) elastic interaction property between a solitonmolecule, a lump wave, and a breather wave for Eq. (1) described by Eq.(5) with parameters selections Eq. (15).

7Advances in Mathematical Physics

and long-wave limit method to different parts of wave num-bers, general hybrid solutions consisting of soliton mole-cules, lump waves, and breather waves are obtained; seeFigure 7. These interaction phenomena may have not beenstudied. At last, we give the general restrictions to derivethese novel interaction solutions containing m-soliton mole-cules, n-breather waves, and q-lump waves, and their interac-tions are elastic. The method to construct soliton moleculesand some novel types of hybrid solutions would be suitableto investigate in other models in mathematical physics andengineering. Meanwhile, we hope that our results will providesome valuable information in the study of nonlinear science.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by National Natural Science Founda-tion of China under Grant Nos. 11775121, 11805106, and11435005, and K.C. Wong Magna Fund in Ningbo Univer-sity. The authors would like to express their sincere thanksto Professor S.Y. Lou for his guidance and encouragement.

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