New lump solutions and several interaction solutions and their dynamics of a generalized (3+1)-dimensional nonlinear differential equation

Yexuan Feng and Zhonglong Zhao

School of Mathematics,North University of China,Taiyuan,Shanxi 030051,China

Abstract In this paper,we mainly focus on proving the existence of lump solutions to a generalized(3+1)-dimensional nonlinear differential equation.Hirota’s bilinear method and a quadratic function method are employed to derive the lump solutions localized in the whole plane for a(3+1)-dimensional nonlinear differential equation.Three examples of such a nonlinear equation are presented to investigate the exact expressions of the lump solutions.Moreover,the 3d plots and corresponding density plots of the solutions are given to show the space structures of the lump waves.In addition,the breath-wave solutions and several interaction solutions of the(3+1)-dimensional nonlinear differential equation are obtained and their dynamics are analyzed.

Keywords: lump solutions,generalized (3+1)-dimensional nonlinear differential equation,Hirota’s bilinear method,quadratic function method,interaction solutions

1.Introduction

Lump solutions are special analytical rational function solutions in soliton theory,which were first found in the study of the Kadomtsev–Petviashvili (KP) equation [1].Lump solutions have been widely applied in the fields of Bose–Einstein condensation,ocean waves,optics,shallow water waves and so on [2–8],and have been paid increasing attention by mathematicians in recent years[9–14].Therefore,it is of great significance to study lump solutions of nonlinear differential equations.To get lump solutions of nonlinear differential equations,many methods have been proposed,such as Darboux transformation [15],the long-wave limit method[16,17],Hirota’s bilinear method [18],the trigonometric function series method [19] and the Jacobi elliptic function expansion method[20].The long-wave limit method was first proposed by Ablowitz et al [16,17].However,by taking the long-wave limit,one can only get lump solutions which are localized in the all-plane of the nonlinear differential equations less than (2+1)-dimensional,while the lump solutions localized in the all-plane of the (3+1)-dimensional nonlinear differential equations cannot be guaranteed.

It is always a hot topic to seek the exact solutions,such as breath-wave solutions [21,22],interaction solutions [23] and lump solutions [24–26],of nonlinear differential equations,and many exact solutions of nonlinear differential equations have been obtained by some scholars in recent years.For instance,Chen et al derived the breather solutions and the interaction solutions of the (3+1)-dimensional generalized Camassa–Holm KP equation [27].The mixed lump-stripe solutions and the mixed rogue wave-stripe solutions of the(3+1)-dimensional nonlinear wave equation were obtained by Wang et al [28].Li and Jiang obtained the lump solutions of the(2+1)-dimensional Hietarina-like equation[29].Based on the long-wave limit method,Liu obtained the lump solutions of the (2+1)-dimensional generalized Calogero–Bogoyavlenshii–Schiff (CBS) equation [30],while Cao et al obtained the lump solutions to the potential KP equation and(2+1)-dimensional Sawada–Kotera equation using the same method,respectively [31,32].With the aid of Hirota’s bilinear method,Pu et al obtained the lump solutions of the(3+1)-dimensional soliton equation and the expanded Jimbo–Miwa equation[33,34].In[35],Ma first proposed a quadratic function method for constructing lump solutions of nonlinear differential equations,which can gain the lump solutions of high-dimensional nonlinear differential equations localized in the whole plane.Subsequently,Zhou et al derived the lump solutions of the(3+1)-dimensional generalized CBS equation and reduced the (3+1)-dimensional nonlinear evolution equation using a quadratic function method[36,37].Inspired by the study of Ma et al,this paper aims to derive the lump solutions localized in the whole plane of a more generalized(3+1)-dimensional nonlinear differential equation

and a,b,c,d,g and h are any nonzero constants.Through the transformation

equation (1) can be transformed into the following Hirota’s bilinear form

When different parameters are selected for the coefficients a,b,c,d,g and h,equation (1) can be reduced to many classical integrable equations.Here,we present the following examples.

When a=b=1,c=d=g=h=0 and z=x,equation(1)is reduced to the Korteweg–de Vries (KdV) equation

When a=1,b=h1,c=h5,g=0,d+h=0 and z=x,equation (1) is reduced to the (2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation[38]

for h2=6h1,h6=3h5,h3=h4=h7=h8=0,wherehi(i=1,2,…,8)is an arbitrary real constant.When a=1,b=α,c=β,d=γ1,g=γ2,h=0 and z=x,equation (1) is reduced to the(2+1)-dimensional generalized Bogoyavlensky–Konopelchenko (BK) equation [39]

for γ3=0,where α,β,γ1,γ2and γ3are constants.Equation(1)can be widely used in ocean dynamics and other related fields,and the study of equation(1)is helpful for us to find the lump solutions of KdV-type equations.In the following,the lump solutions localized in the full plane are shortened to lump solutions.

The structure of this paper is organized as follows.In section 2,a lump solution is obtained using the quadratic function method.In section 3,we briefly introduce the amplitude of the lump wave and its propagation velocities along the x,y and z axes,respectively,and a theorem describing the amplitude and propagation velocity of this lump wave is established.In section 4,we show 3D plots and the corresponding density plots of the lump waves from three examples.In section 5,we obtain the breath-wave solutions of equation (1).In section 6,several interaction solutions of equation (1) are studied.Finally,some conclusions are given in the last section.

2.Lump solutions in (3+1)-dimensional of equation (1)

In this section,we introduce the method of finding the lump solutions.Based on the study of [35],we know that the positive quadratic function solution of equation (1) can be expressed as

where fj=d1jx+d2jy+d3jz+d4jt+d5jwith d1j,d2j,d3j,d4jandd5j(j=1,2,3)being constants.It is difficult to obtain the sum of three squares of f by equation (5) to get a lump solution in (3+1)-dimensional.Therefore,we use the following method to find the lump solution.A useful lemma is introduced as follows.

Lemma 1.Letβ=(β1,…,βN)T∊RNbe a fixed vector and consider the following quadratic function

where the real matrixB=(bij)N×Nis symmetric andq∊R is a constant.The function f defined by equation (6) is a solution to the general Hirota bilinear equation

where P is a polynomial of N variables andD=(D1,D2,…DN),Bidenotes the ith column vector of the symmetric matrix B for 1 ≤i≤N,pijandpijklare the coefficients of the quadratic and quartic terms,respectively [35].

whereα=(x,y,z,t)T,B=[bij]is a 4×4 symmetric matrix,and q is a constant.We introduce two matrices

then define

whereP(i,j) andB(i,j) are used to represent the ith row and jth column elements of matrices P and B,respectively.From lemma 1,the function f defined by equation(7)is a solution to equation (1) if and only if

Since equation(3)contains the second-order Hirota derivative terms,theorem 3.6 in [35] shows that equation (3) has a positive quadratic function solution when∣B∣=0.Thus,to satisfy∣B∣=0,we setRank(B) =3,then we get m=0.Substituting m=0 into equation (10),we obtain thatq(q>0) is an arbitrary constant andb1(bb3+cb2)=0.Since B is a semi-positive definite matrix,all the principal entries of B are non-negative.If b1=0,then

hence b2=0.Furthermore,we have b3=b4=0.Then,we get a trivial solution to f that does not depend on x and,from theorem 2 in [36],equation (1) has no lump solution.Therefore,we have to set bb3+cb2=0,namelyThen,the non-trivial solutions of equation(11)can be written as

where a,d,g and h are any nonzero constants,while b7,b8,b9and b10are free variables such that B ≥0 andRank(B)=3.

Substituting equation (12) into B in equation (8),we get the matrix B

and the function f defined by equation(5)can be expressed as

Taking the appropriate parameters,such that

then function f is positive definite.

By symbolic computations,we obtain

from equation (13).If the appropriate parameters are chosen so that

then function f is positive definite.If equation(16)holds,then function f depends on x,y,z and t,thus

Therefore,according to theorem 2 in [36],the function f defined by equation (15) produces a lump solution of equation (1).

Remark 1.Lump solutions localized in the whole plane of any (3+1)-dimensional nonlinear evolution equation cannot be obtained by taking the long-wave limit of the two-soliton solutions,but can be obtained via the quadratic function method [36].

3.The amplitude and propagation velocity of the lump waves

In this section,we obtain the amplitude of the lump produced by equation (15) and its propagation velocities along the x,y and z axes.Substituting equation (15) into equation (2),a lump solution

is obtained.

Taking partial derivatives of x,y and z,respectively,in equation (15),we have

Based on the above discussion,we give the following theorem.

Theorem 1.For a bilinear equation of the form

4.Three examples

In this section,we will give three examples of the lump solutions to equations like equation (1) based on theorem 1.

Case 1.We take a=b=c=d=g=1,h=-1,b7=-2,b8=6,b9=1,b10=10 and q=1.Then the matrix

is positive semi-definite.By considering equation (15),we know

Then the function

is a lump solution to the equation

This lump has an amplitude of 28,and the propagation velocities of this lump along the x,y,z axes are 1,1,-1,respectively,where the negative sign is the direction of propagation of the lump wave.Figure 1 shows the 3D plots of equation (19) for (a) x=0,(b) y=0,(c) z=0 when t=0,and the corresponding density plots of equation (19) for (d)x=0,(e) y=0,(f) z=0 when t=0.

Figure 1.The 3D plots of equation(19)for(a)x=0,(b)y=0,(c)z=0 when t=0,and the corresponding density plots of equation(19)for(d) x=0,(e) y=0,(f) z=0 when t=0.

Case 2.We take a=-1,b=c=2,d=1,g=h=-2,b7=-1,b8=5,b9=-2,b10=12 and q=2.Then the matrix

is positive semi-definite.By considering equation (15),we have

Then the function

is a lump solution to the equation

The amplitude of this lump is 12,and the propagation velocities of this lump along the x,y,z axes are -1,2,2,respectively.Figure 2 shows the 3D plots of equation(20)for(a) x=0,(b) y=0,(c) z=0 when t=0,and the corresponding density plots of equation (20) for (d) x=0,(e) y=0,(f) z=0 when t=0.

Figure 2.The 3D plots of equation(20)for(a)x=0,(b)y=0,(c)z=0 when t=0,and the corresponding density plots of equation(20)for(d) x=0,(e) y=0,(f) z=0 when t=0.

Case 3.We take a=b=d=2,c=1,g=-3,h=0,b7=1,b8=8,b9=1,b10=15,q=3,and z=x.Then the matrix

is positive semi-definite.By considering equation (15),we have

Then the function

is a lump solution to the equation

which is a bilinear equation of the (2+1)-dimensional generalized BK equation (4) for γ3=0.The amplitude of this lump is 18,and the propagation velocities of this lump along the x and y axes are 1 andrespectively.Figure 3 shows the 3D plots and corresponding density plots of the projection of the lump solution,equation (21),when t=0,t=5 and t=10.

Figure 3.The 3D plots of equation(21)when(a)t=0,(b)t=5,(c)t=10,and the corresponding density plots of equation(21)when(d)t=0,(e) t=5,(f) t=10.

5.The breath-wave solutions of equation (1)

In this section,we focus on finding the breath-wave solutions of equation (1).We set

where μ1,μ2,δ1and δ2are nonzero constants and

withεi(i=1,…,10)being an undetermined constant.Substituting equation (22) into bilinear equation (3),we get

where a,b,c,d,g,h,ε1,ε3,ε5,ε6,ε7,ε8,ε10,μ1,μ2and δ1are arbitrary nonzero constants.Substituting equation (23)into equation (22) yields

and then,via the transformation,equation (2),we get

We take a=b=c=d=g=h=1.Figure 4 shows the density plots of equation(24)under different parameters.As seen in figure 4(b),when the ε5decreases toand the ε7increases to 2,the breath waves become denser than those in figure 4(a).In figure 4(c),we find that when the parameter ε1is reduced tothe twist angle of the breath waves are changed.

6.The interaction solutions of equation (1)

In this section,we derive several interaction solutions of equation (1) and show corresponding plots to observe the structures of these solutions.

6.1.The mixed lump–soliton solutions of equation (1)

andri(i=1,…,16)is an undetermined constant.By substituting equation (25) into equation (3),we obtain

where a,b,c,d,g,h,r1,r2,r6,r8,r12and r14are arbitrary nonzero constants.Substituting equation (26) into equation (25) generates

From the transformation,equation (2),we have

We take the xoy-plane as an example to analyze the relationship between the lump and the soliton in solution (28).Let z=0,and take the derivative of equation (27) with respect to x and y;then it can be concluded from fx=fy=0 that the central point coordinate of the lump is

It is easy to conclude from the above equation that the distance between the lump and the soliton depends only on the choice of parameters and is independent of time.

We take a=b=c=d=g=h=1 as an example to observe the dynamic behavior of equation (28).Figure 5 shows the 3D plots of equation(28)with different parameters.As seen in figure 5,the soliton moves up to the left while the lump moves down to the left.In figures 5(a)–(c),we can see that when r10=20,the distance between the lump and the soliton is alwaysIt can be observed from figures 5(d)–(f) that when r10=1,the distance between the lump and the soliton is always 0,i.e.the lump moves on the soliton.When r10decreases to-30,the lump and the soliton are completely fused,and only one soliton is shown in the plot,as shown in figures 5(g)–(i).

Figure 5.The 3D plots of equation (28) when r1=r2=r5=r6=r8=r12=r14=1,r11=e,r16=3 and (a)–(c) r10=20,(d)–(f) r10=1,and (g)–(i) r10=-30 in the xoy-plane.

Figure 7.Density plots of equation (34) when η2=10,η3=η5=η8=η10=η13=η15=η16=δ4=δ5=1,η7η11=2,δ at (a)t=-5,(b) t=0 and (c) t=5 in the xoy-plane.

6.2.The mixed rogue-wave–soliton solutions of equation (1)

To derive the mixed rogue-wave–soliton solutions of equation (1),we assume that

where

andwi(i=1,…,16)is a constant to be determined.We substitute equation (25) into equation (3) and then obtain

where a,b,c,d,g,h,w1,w3,w5,w7,w9,w10,w11,w12,w15and w16are arbitrary nonzero constants such that bg-ch ≠0.Substituting equation (30) into equation (29) gives

Through the transformation,equation (2),we have

We still analyze the position relationship between the lump and the two solitons taking the xoy-plane as an example.Similarly to the previous analysis,if z=0,we take the derivative of equation(27)with respect to x and y,and then fx=fy=0 leads to the coordinate of the center point of the lump being

6.3.The periodic cross-kink solutions of equation (1)

We assume that

where a,b,c,d,g,h,η2,η3,η5,η6,η7,η8,η10,η11,η13,η15and η16are arbitrary nonzero constants.By substituting equation (33) into equation (32),we get

then,by virtue of the transformation,equation (2),we have

We take a=b=c=d=g=h=1.It can be observed in figure 7 that the solution,equation (34),appears as two intersecting solitons.Over time,the narrower soliton moves upward to the left,while the wider soliton moves upward to the right.And the intersection of the two solitons moves with time along the positive semi-axis of x and y.

7.Conclusion

In this paper,we first consider a generalized(3+1)-dimensional equation,and derive its bilinear form using Hirota’s bilinear method.The lump solutions localized in the whole plane are obtained via the quadratic function method.To analyze the dynamical behavior of the lump waves,three examples are given and the 3D plots and corresponding density plots are presented.It is notable that the method used here to obtain lump solutions can be extended to other (3+1)-dimensional equations.However,the lump solutions derived by this method are localized in the whole plane,but only as single-lump solutions,while the lump solutions obtained via methods such as the long-wave limit are multi-lump solutions,but not necessarily localized in the whole plane.Moreover,the breath-wave solutions,the mixed lump–soliton solutions,the mixed rogue-wave–soliton solutions and the periodic cross-kink solutions are derived.In future work,we will explore whether this method can derive multi-lump solutions localized in the whole plane of the(3+1)-dimensional equations.Recent studies have proved the existence of line rogue waves in some(2+1)-dimensional evolution equations[40,41].We will investigate the existence of line rogue waves in(3+1)-dimensional equations in future work.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Nos.12101572 and 12371256),2023 Shanxi Province Graduate Innovation Project (No.2023KY614) and the 19th Graduate Science and Technology Project of North University of China (No.20231943).