Hongcai Ma, Xue Mao and Aiping Deng

Department of Applied Mathematics, Donghua University, Shanghai 201620, People's Republic of China

Abstract This paper aims to search for the solutions of the (2+1)-dimensional extended Boiti-Leon-Manna-Pempinelli equation.Lump solutions, breather solutions, mixed solutions with solitons,and lump-breather solutions can be obtained from the N-soliton solution formula by using the long-wave limit approach and the conjugate complex method.We use both specific circumstances and general higher-order forms of the hybrid solutions as examples.With the help of maple software, we create density and 3D graphs to summarize the dynamic properties of these solutions.Additionally,it is possible to observe how the solutions’trajectory,velocity,and shape vary over time.

Keywords: (2+1)-dimensional extended Boiti-Leon-Manna-Pempinelli equation, long-wave limit method, conjugate complex method, interaction solutions, dynamic characteristics

1.Introduction

The exact solutions of nonlinear equations are very important in many fields.Soliton solutions, lump solutions, breather solutions, and interaction solutions can completely describe nonlinear phenomena in nature.One special kind of rational solution is the lump solution.It has spatial localization and shape invariance.Satsuma and Ablowitz [1] obtained the lump solutions in their research of the Kadomtsev-Petviashvili equation by taking the long-wave limit for theN-soliton solutions.Since then, the long-wave limit method has been widely used to discover the lump solutions of various nonlinear equations.In 2015, Ma [2] presented the positive quadratic function method, which was adopted by numerous scholars to obtain lump solutions [3-5].There is a solution with periodic oscillations and localization, similar to breathing,hence the name of the breather solution.It is divided into three categories: Kuznetsov-Ma breather, Akhmediev breather, and general breather [6-8].We can derive the breather solutions by treating the parameters ofN-soliton solutions as conjugate relations, and this method is known as the complex conjugate method.In recent years, lump solutions, breather solutions, and hybrid solutions have been studied, such as (2+1)-dimensional extended shallow water wave equation [9], (2+1)-dimensional Hirota-Satsuma-Ito equation [10], (2+1)-dimensional Sawada-Kotera equation[11, 12], the combined pKP-BKP equation [13], potential Kadomtsev-Petviashvili equation [14], (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation [15, 16],(2+1)-dimensional Bogoyavlenskii-Schieff equation [17],and so on [18-26].

The (2+1)-dimensional Boiti-Leon-Manna-Pempinelli(BLMP) equation

which was developed by Gilsonet al[27].BLMP equation has been an important model in incompressible fluid [28].In recent years, numerous results of this equation have been investigated by experts.Using the B?cklund transformation generated by the modified Clarkson-Kruskal direct technique,Liet al[29] found new solutions to the equation and explained the connection between the new and old solutions.The start and step solutions for the (2+1) and (3+1)-dimensional equations were obtained by Darvishiet al[30] using the multi-exponential function method.Then the extended homoclinic test approach was used by Tanget al[31]to study the exact solutions of these two equations.Kaplanet al[32]used the homogeneous balance method to build the Auto-B?cklund transformation of the equation.Huet al[33, 34]used the Pfaffian technique and the optimized Pfaffian technique to study the solutions, respectively.

Guoet al[35]modified the above equation and obtained the (2+1)-dimensional extended BLMP (eBLMP) equation

where δ1and δ2are arbitrary constants.Equation (2) extends BLMP equation (1) by introducing two second-order derivativesuyyanduxx, which change the dispersion relations of the original equation.Due to the arbitrariness of δ1and δ2,equation(2)can explain more dispersion relations and enrich their applications in reality.In [35], the experts used the positive quadratic method to obtain the lump and lump-kink soliton solutions.Then Shenet al[36] applied the extended homogeneous test, Riemann theta function, and polynomial expansion approach to obtain the breather solutions,periodicwave solutions, and traveling-wave solutions, respectively.To our knowledge, no literature has investigated the exact solutions of the(2+1)-dimensional eBLMP equation by using the long-wave limit approach and the conjugate complex method.These two methods will be used to obtain theMlump solutions,T-order breather solutions,and the interaction solutions with solitons in this paper.

Following is a framework for this paper.The bilinear method is used in the second section to derive both the bilinear form and theN-soliton solution of the equation.In section 3, the 1-, 2-, and 3-lump solutions as well as theMlump expressions are discovered utilizing the long-wave limit method.In section 4, we discover the interactions between lump solutions and soliton solutions.In section 5, we demonstrate how to use the conjugate relations to convertNsoliton solutions toT-order breather solutions.ForT=1, 2,and 3, specific breather solutions are also presented.The breather solution’s interactions with the soliton solutions and lump solutions are shown in section 6 and 7, respectively.Using the mathematical software Maple,we draw the 3D and density plots related to each solution to observe their dynamic characteristics.Then the paper ends with a summary.

2.N-soliton solutions

First, we consider the variable transformation

It is straightforward to calculate the bilinear form of the equation (2) then, as

wheref=f(x,y,t), and the bilinear operatorsDx,Dy, andDtare defined by

Whenfis an answer to equation (4) under the variable transformation(3),we can discover the relationship:u(x,y,t)is an answer to equation (2).TheN-soliton solutions of equation (4) can be derived by applying the Hirota bilinear method.AndfNwill be expressed as

where

wherehi,ρi,andi0χare arbitrary parameters.In the expression(6), ∑μ=0,1denotes the sum of all combinations of μi=0,1(1 ≤i≤N).The symbolmeans the sum of all combinations of pairs (i,j) chosen from the set {1, 2, …,N} with the conditioni

3.M-lump solutions

By applying the long-wave limit method on theN-soliton solutions, we can obtain the multi-lump solutions of equation (2).First,we take every exp()= -1, the expression(6)can be written as

with

By simplifyingfN, the common factorizationhican be derived from it.When performing the calculation ofu= 2 (l nf)x, it is found that the presence or absence of the common factorizationhidoes not affect the result ofu.Therefore it is easy to prove thatis also a solution of the equation (2).To simplify the calculation, we eliminate the common factorizationhi.The simplifiedfNis

If we choose the conditionN=2M,ρM+i=ρ*i(i=1, 2,… ,M) ,andQij>0,where*denotes the complex conjugation, we can get a class of generalM-lump solutions.In the following, we list the 1-lump, 2-lump, and 3-lump solutions individually.

3.1.1-lump solutions

WithM=1,N=2 in equation (12), it can be simplified into

which satisfies the conditionThe equation (15) we obtain is the 1-lump solution of the equation (2).And this solution moves along a linewith the velocityalong thex-axis andalong they-axis.If we take the parameters as,δ1=1,δ2= - 1, the evolution of the 1-lump solution at timet=-10, 0, 10 can be drawn in figure 1.It can be observed from the images that the 1-lump wave maintains the same speed and shape during propagation and travels in the same direction.

Figure 3.3-lump solution with , δ1=1, δ2=-1, at (a) t=-20;(b) t=0; (c)t=20; (d) shows three lump waves travel along the line (orange), and(blue-green).

3.2.2-lump solutions

WhenM=2,N=4 in equation (12), it is reduced to

3.3.3-lump solutions

Similarly, letM=3,N=6 in the equation (12), we can obtain the functionf3with 76 terms, whereAnd we can derive the expression of 3-lump solutionuafter bringingf3into variable transformation equation (3).Lettingρ4=1*ρ=a1-b1i,ρ5=2*ρ=a2-b2i,ρ6=3*ρ=a3-b3i,3-lump solution is obtained (see figure 3).By calculation,three lump waves move along three straight linesrespectively, which can be seen in figure 3(d).The evolutions of the 3-lump solution at timet=-20, 0, 20 are shown in figures 3(a)-(c).Similarly, the three waves will collide and change their shapes during moving.However,it has the same shape after the collision as before.

Figure 4.Hybrid solution between 1-lump and 1-soliton with, δ1=1, δ2=-1, at (a)t=-10;(b) t=0; (c) t=10.

4.Hybrid solutions between lumps and solitons

In the previous section,we study theM-lump solution.And in this section,we will further investigate the interaction of lump waves with soliton waves.The method used is also the longwave limit method.These hybrid solutions are derived from the expression (6) by taking the parameters

whereMandKare the positive integers,ar,br,hs, ρsandare the real constants (s=2M+1, …2M+K).

4.1.A hybrid solution between 1-lump and 1-soliton

For instance, in order to study the interaction solution of 1-lump and 1-soliton, we need to take the parameters as

Then bring them into equation (6), we have

with

The hybrid solution between 1-lump and 1-soliton can be gained after calculatingu= 2 (lnf1,1)xand choosingwheref1,1is given by (19).The evolutions of the hybrid solution at timet=-10,0,10 are shown in figure 4.In these pictures, we find that the lump wave and soliton wave are moving in opposite directions.From -10 to 0 min, they gradually approach and collide att=0 (figure 4b).The shapes of both waves are changed by the collision and the waves move independently after separation.

4.2.A hybrid solution between 1-lump and 2-solitons

Similarly, to find the interaction solution of 1-lump and 2-solitons, we take

Then bring them into equation (6), we have

with

After bringingf1,2intou=2 (lnf)xand setting the parameterδ1=1,δ2=-1,the interaction solution between a 1-lump and 2-solitons can be painted in figure 5.By controlling the values of ρ3and ρ4,the two soliton waves can be parallel.As seen in figure 5, the two parallel soliton waves move in the positive direction of thex-axis, and the lump wave moves in the opposite direction.As a result, the phenomenon of a lump wave passing through two soliton waves separately during moving appears.

4.3.A hybrid solution between 2-lumps and 1-soliton

The hybrid solution consists of 2-lumps and 1-soliton can be generated from 5-soliton solution.To begin with,letN=5 in the equation(6)and we obtain the 5-soliton solutionf5.Then inserting the conditions,h1,h2,h3,h4→0,ρ3=1*ρ=a1-b1i andρ4=2*ρ=a2-b2i intof5.The functionf5can be rewritten as

with

Taking

Substituting equations(24)-(26)into the variable transformationu= 2 (l nf)x, we gain the hybrid solution consists of 2-lumps and 1-soliton as shown in figure 6.In figure 6,the 2-lump waves move gradually from the right side of the soliton wave to its left side.The shape and amplitude of the lump waves do not change before and after they completely cross the soliton wave.

5.T-order breather solutions

The parameters of theN-soliton solution expression in this section need to be taken as conjugate if we want to obtain theT-order breather solutions (N=2T).We set the following parameters:

Bringing these constraints into the equation(6),theN-soliton solutions will be transformed intoT-order breather solutions.

Figure 5.Hybrid solution between 1-lump and 2-solitons withδ1=1, δ2=-1, at (a) t=-10;(b) t=0; (c) t=10.

Figure 6.Hybrid solution between 2-lumps and 1-soliton with , δ1=1,δ2=-1, at (a) t=-20;(b) t=0; (c) t=20.

5.1.1st-order breather solution

LetN=2T=2 in the equation (6), the 2-soliton solution has the following form

Taking the parameters in a conjugate form:

After the calculation,f2-solitonwill be transformed into the following form of the 1-order breather solution

with

Due to the different types of parameters of {h1,h2, ρ1, ρ2} we choose, different spatial arrangements of 1st-order breather solutions exist.We discuss the parameters in the following types:

(1) When bothhiand ρiare pure imaginary numbers, which meansa=0 andc=0, the 1st-order breather solution presents periodic on thex-axis and fixed on they-axis,as shown in figure 7(a).

(2) Whenhiis a pure real number and ρiis a pure imaginary number, which meansb=0 andc=0, the 1st-order breather solution presents periodic on they-axis and fixed on thex-axis, as shown in figure 7(b).

(3) When bothhiand ρiare complex numbers,which meansa,b,c,d≠0, the 1st-order breather solution parallel to the lineax+(ac-bd)y=0, see figure 7(c).

5.2.2nd-order breather solution

WhenN=2T=4 inN-soliton solutions (6) with conditions(27), 2nd-order breather solution is determined as

Figure 7.Three types of first-order breather solutions at t=0 with χ02=(χ 10 )*=0,δ1=1,δ2=-1,(a)h2=h1*= -3i,ρ2=ρ1*= - i;(b)h2=h1*=1,ρ2== - 2i;(c)h2=h1*= 13-i,ρ2== 25- i .

Figure 8.2nd-order breather solution with δ1=1, δ2=-1,χ10=χ20=χ03=χ04= 0at (a) t=-10;(b) t=0; (c) t=10.

We assume that parameters have a conjugate relation and set δ1=1, δ2=-1,inf2-breather.The 2nd-order breather solution consisting of two breather waves is derived in figure 8.As previously stated, the 1st-order breather wave has three distinct structures.Therefore, it leads to the occurrence of multiple spatial arrangements between two breather waves.Such as parallel to thex-axis, parallel to they-axis, and intersecting,where mutual perpendicularity(see figure 8)is a special kind of intersection.Here, we make graphs and observations only for the case of two waves perpendicular to each other.

5.3.3rd-order breather solution

The case ofN=2T=6 is similar to the above discussion.We take δ1=1, δ2=-1,in formula(6).The images of the 3rd-order breather solution are shown in figure 9.

6.Hybrid solutions between breathers and solitons

Deriving the interaction solutions of theT-order breathers andK-solitons by letting

is the purpose of the study in this section, whereTandKare the positive integers,hs, ρsand χ0sare the real constants(s=2T+1, …2T+K).

6.1.A hybrid solution between 1-breather and 1-soliton

To get more detailed results of the interaction solution between 1-breather and 1-soliton, we need to take the parameters as

Figure 9.3rd-order breather solution with δ1=1, δ2=-1,h4=h1*= 1- ,ρ4=ρ*1= - ,h5=h2*= 1+ ,ρ5== - ,h6=h3*= 1, ρ6=ρ*3=-, χ10 = χ02=χ30= χ04=χ50=χ0 6= 0at (a) t=-5;(b) t=0; (c) t=5.

Figure 10.Hybrid solution between 1-breather and 1-soliton with δ1=1, δ2=-1, (a)h2=h3= -1,ρ2=ρ1*= ρ3= -1,χ10=χ02=χ30=0,t = - 5;(b)h2=h3= -1,ρ2== , ρ3=-1,χ10=χ02=χ30=0,t=0;(c)h2= ,h3= -1,ρ2=ρ1*=,ρ3= -1,χ10=χ02=χ30=0,t = 0.

Bringingf3-solitoninto u= 2 (l n f)xafter inserting parameters(34) into it, we obtain the interaction solution as shown in figure 10.Whena1c1-b1d1=a1c3, the 1st-order breather is parallel to the 1-soliton on the (x,y) plane, as shown in figure 10(a).Whena1c1-b1d1=,the 1st-order breather and the 1-soliton are perpendicular to each other on the(x,y)plane,as shown in figure 10(b).When neither of the above is satisfied, they are intersected on the (x,y) plane, as shown in figure 10(c).

6.2.A hybrid solution between 1-breather and 2-solitons

Now, we set

Bringing parameters (35) into the expressionf4-solitongiven by equation (6), an interaction solution between 1-breather and 2-solitons is obtained, which is shown in figure 11.Figure 11 shows the evolution of the hybrid solution at timet=-3, 0,3.From these figures,we can see that the breather wave and two soliton waves move in opposite directions.

Figure 11.Hybrid solution between 1-breather and 2-solitons with δ1=1, δ2=-1,, h3=1, h4=-1,, at (a) t=-3;(b) t=0; (c) t =3.

6.3.A hybrid solution between 2-breathers and 1-soliton

Next, we choose

Taking them into 5-soliton solution, then we can turn the soliton solution into the hybrid solution between 2-breathers and 1-soliton.Figure 12 shows the evolution of the hybrid solution at timet=-3, 0, 3.

Figure 12.Hybrid solution between 2-breathers and 1-soliton with δ1=1, δ2=-1,, h5=-2,ρ3== ρ4== , at (a) t=-3;(b) t=0; (c) t=3.

7.Hybrid solutions between lumps and breathers

In this section, we find the hybrid solutions composed of theT-order breathers andM-order lumps with the conditions

whereTandMare the positive integers,aj,bj,c,dandχ0jare the real constants.

7.1.A hybrid solution between 1-breather and 1-lump

If the parameters of the equation (6) satisfy

the interaction solution will be obtained, which consists of a breather and a lump solution (see figure 13).Figure 13 demonstrates that the breather wave and lump wave move in opposite directions along thex-axis.It can also be seen that the velocity and shape of the two waves after the collision are the same as before.

Figure 13.Hybrid solution between 1-breather and 1-lump with δ1=1, δ2=-1, h2=h1*= 1,ρ2=ρ*1= i ,χ02=χ10= 0, at (a) t=-10;(b) t=0; (c) t=10.

8.Conclusions

The (2+1)-dimensional eBLMP equation is the main subject of this work.Via the bilinear transformation formula, we can easily find the bilinear form and the expressions of theNsoliton solutions of the equation.Using the long-wave limit method, we can effectively find the higher-order lump solutions and analyze the situations of 1-lump, 2-lump, and 3-lump solutions.The expressions and images of solutions can describe the dynamic characteristics.We discover that the wave follows a specific trajectory with uniform velocity.When multiple waves move simultaneously, a collision happens accompanied by morphological changes.After the collision, the waves separate from each other and return to their original shapes (see figures 1-3).Taking the parameters as conjugate relations, we can construct high-order breather solutions.The 1st-order breather solutions contain three different types of spatial arrangements due to the diverse parameter settings:parallel to thex-axis,parallel to they-axis,and the general case.This result leads to multiple spatial combinations of high-order breather solutions.The evolution and collision of solutions can be seen in graphs(see figures 7-9).Furthermore, we analyze the interactions between other solutions, such as lump-soliton solutions (see figures 4-6),breather-soliton solutions (see figures 10-12), and lumpbreather solutions (see figure 13).With the help of these solutions, we can more effectively explore nonlinear phenomena.The figures show the changes caused by different kinds of solutions colliding.The methods we adopted in this article are remarkable for solving nonlinear systems and the solutions we obtained can explain physical phenomena in nature.

Funding

There is no funding supported.

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Declarations

Conflict of interest

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