1 Introduction

Nonlinear evolution equations(NLEEs)are frequently used to model a wide variety of nonlinear scientific phenomena,such as marine Engineering, fluid dynamics,plasma physics,chemistry,physics and so on.Direct seeking for exact solutions to NLEEs has become one of the most exciting and extremely active areas in mathematical physics.[1−23]In the past few decades,with the development of symbolic computation,many powerful and systematic methods have been proposed,such as Hirota direct method,[24]F-expansion method,[25]exp function method,[26−28]the auxiliary equation method,[29]three wave approach[30−36]and etc.

The breaking soliton equations can be applied to describe the(2+1)-dimensional interaction of a Riemann wave propagating along the y-axis with a long wave propagating along the x-axis.[37]In this paper,we will discuss the following(2+1)-dimensional breaking soliton equation:[37−41]

总之，教师的教育教学智慧直接影响到了课程教学的长效发展和学校教育教学质量的提升。在中职学校中积极引入和融合智慧教育的理念，可以有效地激发学生的学习积极性和主动性，培养学生的创新精神和实践能力。教育教学智慧的生成，让沉闷的课堂充满生机与活力，让倦怠的教师充满自信与幸福，有利于促进教师的专业成长。

where α is an arbitrary constant. Radaha and Lakshmanan have proved that Eq.(1)has the Painlev´e property and dromion-like structures.[38]The folded solitary waves and other coherent soliton structures are discussed.[39−40]By introducing Jacobi elliptic functions in the seed solution,two families of doubly periodic propagating wave patterns are obtained.[37]Exact breathertype and periodic-type soliton solutions for the(2+1)-dimensional breaking soliton equation are derived via the extended three-wave method.[41]

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Therefore we obtain the sixth new double-periodic soliton solutions for Eq.(1)as follows:

2 New Double-Periodic Soliton Solutions and Bilinear Form

Using the following dependent variable transformation

韩国高校教师学术道德教育实施主体呈现多元化特征。首先，为防范高校教师出现学术道德失范与不端等问题，在全社会形成良性的学术研究道德风气，韩国《学术振兴法》第15条明确规定，教育部作为高校教师学术道德教育的最高决策管理机构，全面负责资助、管理、监督高校教师的学术道德教育工作①（韩）韩国国会，学术振兴法（N）：第15条第2至4项。，是韩国学术道德教育的主管部门。而作为政府层面的学术道德教育工作则由韩国教育部下设“国家科学技术人力开发院”，以及韩国未来科技部下设“韩国研究财团”共同组织实施。

Solving the system with the aid of symbolic computation software Mathematica,we have

Equation(3)is equivalent to the following equation

Supposing the solution of Eq.(4)be expressed in the form

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where θi= αix+ βiy+ δit,i=1,2,3,4 and αi,βi,and δiare constants to be determined later.Solution(5)is if rst proposed in the literature[36]for the purpose of solving the multi periodic soliton solutions of the Kadomtsev-Petviashvili equation.This method is simple and direct,and can obtain a large number of periodic solutions of the high dimensional nonlinear equations.But the amount of computation is large and the help of symbolic computing software Mathematica is needed.Substituting Eq.(5)into Eq.(4)and equating corresponding coefficients of eθ1,eθ3,eθ4,cosθ2,sinθ2,cosθ4,and sinθ4to zero,a set of algebraic equations for αi,βi,δican be derived as follows

Substituting Eq.(2)into Eq.(1),the(2+1)-dimensional breaking soliton equation can be written in the following bilinear form

Case 1

我们注意看这句话：对于x的每一值，y都有唯一的值与它对应.什么意思啊？不就是给一个x，就会出一个y！也就是给定一个数，就会得到另一个数！把前后两个数依次列举，不就是一个有序实数对吗？据此，我们可以领悟到函数的本质——不就是在某种对应关系下得到的一对一对有序实数对嘛！对初中学生而言，我们把函数本质解读成这样，解读到这个层面，是不是很通俗易懂？学生理解起来是不是会好很多？

南一环站、芜湖路站均为主体宽约13 m的岛式站台车站，两层三跨结构;覆土厚约3.2～4.1 m，底板分别位于粉细砂层和强风化泥质砂岩中。高架主桥桥墩承台高2.5 m，置于车站顶板上。水阳江路站为主体宽度为11 m的岛式站台车站，车站为双层双跨矩形框架结构;覆土厚度约3 m，底板位于黏土层，高架匝道桥桥墩置于车站顶板上。三站均位于车水马龙的城市主干道上，车站周边高楼林立，最近的高层建筑距车站基坑仅约5 m。合肥轨道交通1号线芜湖路站与高架桥同位合建单平面如图1所示。

ξ has been explained in Eq.(56).

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Therefore we obtain the fi rst new double-periodic soliton solutions for Eq.(1)as follows:

The evolution and mechanical feature of solutions(25)–(26)are shown in Figs.1–2.

Fig.1 The physical structure of solution(25),at k2= β3= −1,α1= γ1=1,γ2= γ3= γ4= β1= β2= β4= α =1,(a)x=−5,(b)x=0 and(c)x=5.

Fig.2 The physical structure of solution(26),at k2= β3= −1,α1= γ1=1,γ2= γ3= γ4= β1= β2= β4= α =1,(a)x=−5,(b)x=0 and(c)x=5.

Case 2

where α,α1,β1,β2,β3,β4,k1,γ1,γ2,γ3,and γ4are free real constants.Substituting these results into Eq.(5),we obtain

Therefore we obtain the second new double-periodic soliton solutions for Eq.(1)as follows:ξ has been explained in Eq.(28).The evolution and mechanical feature of solutions(29)–(30)are shown in Figs.3–4.

Fig.3 The physical structure of solution(29),at k1= β3= −1,α1= γ1=1,γ2= γ3= γ4= β1= β2= β4= α =1,(a)t=−5,(b)t=0 and(c)t=5.

Fig.4 The physical structure of solution(30),at k2= β3= −1,α1= γ1=1,γ2= γ3= γ4= β1= β2= β4= α =1,(a)t=−5,(b)t=0 and(c)t=5.

Fig.5 The physical structure of solution(33),at k1= β1= β3= −1,α1=1,γ1= γ2= γ3= γ4= β2= β4= α =1,(a)y=−5,(b)y=0 and(c)y=5.

where α,α3,α4,β1,β2,β4,k2,δ2,γ2,and γ3are free real constants,ϵ5= ±1.Substituting these results into Eq.(5),we obtain

where α,α1,β1,β2,β3,β4,k1,γ1,γ2,γ3,and γ4are free real constants.Substituting these results into Eq.(5),we obtain

Therefore we obtain the third new double-periodic soliton solutions for Eq.(1)as follows:ξ has been explained in Eq.(32).The evolution and mechanical feature of solutions(33)–(34)are shown in Figs.5–6.

Fig.6 The physical structure of solution(34),at k1= β1= β3= −1,α1=1,γ1= γ2= γ3= γ4= β2= β4= α =1,(a)y=−5,(b)y=0 and(c)y=5.

Fig.7 The physical structure of solution(37),at α2= γ1= β1= γ4= −1,β4= −2,k1= γ2= γ3= β3= α =1,(a)y=−5,(b)y=0 and(c)y=5.

Fig.8 The physical structure of solution(38),at α2= γ1= β1= γ4= −1,β4= −2,k1= γ2= γ3= β3= α =1,(a)y=−5,(b)y=0 and(c)y=5.

Case 4

where α,α2,β1,β3,β4,k1,γ1,γ2,γ3,and γ4are free real constants.Substituting these results into Eq.(5),we obtain

Case 6

ξ has been explained in Eq.(36).The evolution and mechanical feature of solutions(37)–(38)are shown in Figs.7–8.

Case 5

where α,α1,α2,α3,β1,k2,γ2,and γ4are free real constants,ϵ1= ±1.Substituting these results into Eq.(5),we obtain

Therefore we obtain the fi fth new double-periodic soliton solutions for Eq.(1)as follows:

ξ has been explained in Eq.(40).

Therefore we obtain the fourth new double-periodic soliton solutions for Eq.(1)as follows:

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where α,α1,α2,β1,β4,k2,γ2,and γ4are free real constants,ϵ2= ±1.Substituting these results into Eq.(5),we obtain

However,to our knowledge,double-periodic soliton solutions for Eq.(1)have not been discussed,which turn out to be the main goal of the present work.In Sec.2,new double-periodic soliton solutions for Eq.(1)are researched by virtue of the symbolic computation,bilinear form and the special ans¨atz functions.Finally,Sec.3 will be the conclusions.

ξ has been explained in Eq.(44).

Case 7

where α,α1,α3,β1,β3,β4,k2,γ2,γ3,and γ4are free real constants,ϵ3= ±1.Substituting these results into Eq.(5),we obtain

Therefore we obtain the seventh new double-periodic soliton solutions for Eq.(1)as follows:

ξ has been explained in Eq.(48).The evolution and mechanical feature of solutions(49)–(50)are shown in Figs.9–10.

Fig.9 The physical structure of solution(49),at α3= γ2= β1= γ4= −1,k2= −2,γ3=2,α1= β3= β4= α = ϵ3=1,γ2=0(a)x=−10,(b)x= −5 and(c)x=−1.

Fig.1 0 The physical structure of solution(50),at α3= γ2= β1= γ4= −1,k2= −2,γ3=2,α1= β3= β4= α =ϵ3=1,γ2=0(a)x= −10,(b)x= −5 and(c)x= −1.

Case 8

where α,α1,α2,β1,β2,β3,β4,k2,δ2,γ1,γ3,and γ4are free real constants,ϵ4= ±1.Substituting these results into Eq.(5),we obtain

Therefore we obtain the eighth new double-periodic soliton solutions for Eq.(1)as follows:

ξ has been explained in Eq.(52).

Case 9

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Case 3

最后，总结本文基于二维结构稀疏信息的CFS ISAR成像方法主要处理过程如图3所示.其中调频步进信号的平动补偿方法可以参考文献[2]，由于篇幅限制，此处不再赘述.

Therefore we obtain the ninth new double-periodic soliton solutions for Eq.(1)as follows:

where α,α1,β1,β2,β3,β4,k2,γ1,γ2,γ3,and γ4are free real constants.Substituting these results into Eq.(5),we obtain

Case 10

where α,α4,β1,β2,β3,β4,k1,k2,γ1,γ2,γ3,and γ4are free real constants.Substituting these results into Eq.(5),we obtain

空洞-绕开最短路径HBSP(Hole-Bypassing Shortest Path):令(s,t)表示空洞凸包H外的两个节点,且s与t间连线贯穿H,如图2(b)所示。令Hs1、Hs2、Ht1、Ht2分别表示s、t的VLV节点,且Hs1和Hs2在矢量st的右边,而Ht1、Ht2在在矢量st的左边。s与t间的HBSP就是和间的最小值。

Therefore we obtain the tenth new double-periodic soliton solutions for Eq.(1)as follows:

ξ has been explained in Eq.(60).The evolution and mechanical feature of solutions(61)–(62)are shown in Figs.11–12.

Fig.1 1 The physical structure of solution(61),at α3= β2= α =1,k2=3, γ2= γ3=2,k1= β4= −2,γ1= β1= β3= γ4= −1,(a)t= −10,(b)t=0 and(c)t=10.

Fig.1 2 The physical structure of solution(62),at α3= β2= α =1,k2=3, γ2= γ3=2,k1= β4= −2,γ1= β1= β3= γ4= −1,(a)t= −10,(b)t=0 and(c)t=10.

Fig.1 3 The physical structure of solution(65),at α1= β2= α = ϵ6=1,k1= −2,γ2=2,α3= γ1= β1= γ3= −1,(a)t=−10,(b)t=0 and(c)t=10.

Fig.1 4 The physical structure of solution(66),at α1= β2= α = ϵ6=1,k1= −2,γ2=2,α3= γ1= β1= −1,γ3=0,(a)t= −10,(b)t=0 and(c)t=10.

Case 11

where α,α1,α3,β1,β2,k1,γ1,γ2,and γ3are free real constants,ϵ6= ±1.Substituting these results into Eq.(5),we obtain

Therefore we obtain the eleventh new double-periodic soliton solutions for Eq.(1)as follows:

任务型阅读教学，在设置任务的时候也要相对应的考虑到调动学生的积极性，由于缺乏设备，教师应定期一起学习研讨。对于一些简单的教具，老师们可以尽可能地自己做。考虑到课堂时效的问题，学校也应该逐步实现多媒体教学。

ξ has been explained in Eq.(64). The evolution and mechanical feature of solutions(65)–(66)are shown in Figs.13–14.

3 Conclusion

In this work,the(2+1)-dimensional breaking soliton equation is investigated.With the help of the bilinear form and the special ans¨atz functions,some entirely new double-periodic soliton solutions for the(2+1)-dimensional breaking soliton equation are presented.All these solutions are brought back to the original equation with the Mathematica software,and the results are all right.Many important and interesting properties for these obtained solutions are revealed with some figures(see Figs.1–14)by the help of symbolic computation software Mathematica.The special ans¨atz functions method is simple and straightforward than the others method.If the bilinear form of nonlinear evolution equations is existent,then rich variety of periodic-soliton solutions can be derived by the special ans¨atz functions method.

References

[1]K.Hosseini,D.Kumar,M.Kaplan,and E.Yazdani Bejarbaneh,Commun.Theor.Phys.68(2017)761.

[2]W.X.Ma and Y.Zhou,Int.J.Geom.Methods Mod.Phys.13(2016)1650105.

[3]W.X.Ma,J.H.Meng,and M.S.Zhang,Math.Comput.Simulat.127(2016)166.

[4]Z.F.Zeng,J.G.Liu,Y.Jiang,and B.Nie,Fund.Inform.145(2016)207.

[5]S.Ahmad,Ata-ur-Rahman,S.A.Khan,and F.Hadi,Commun.Theor.Phys.68(2017)783.

[6]X.Lü,J.P.Wang,F.H.Lin,and X.W.Zhou,Nonlinear Dyn.,DOI:10.1007/s11071-017-3942-y(2017);X.Lü,W.X.Ma,S.T.Chen,and M.K.Chaudry,Appl.Math.Lett.58(2016)13;X.Lü and F.H.Lin,Commun.Nonlinear.Sci.32(2016)241.

[7]W.X.Ma,Z.Y.Qin,and X.Lü,Nonlinear Dyn.84(2016)923.

[8]M.Eslami,B.F.Vajargah,M.Mirzazadeh,and A.Biswas,Indian.J.Phys.88(2014)177.

[9]X.Lü,S.T.Chen,and W.X.Ma,Nonlinear Dyn.86(2016)1;X.Lü and W.X.Ma,Nonlinear Dyn.85(2016)1217;X.Lü,W.X.Ma,J.Yu,and C.M.Khalique,Commun.Nonlinear.Sci.31(2016)40.

[10]I.Aslan,Appl.Math.Comput.217(2011)6013.

[11]M.Eslami,Appl.Math.Comput.285(2016)141.

[12]F.H.Lin,S.T.Chen,Q.X.Qu,et al.,Appl.Math.Lett.78(2018)112.

[13]M.Eslami,M.Mirzazadeh,and A.Neirameh,Pramana 84(2015)1.

[14]A.Biswas,M.Mirzazadeh,M.Eslami,et al.,Frequenz 68(2014)525.

[15]A.W.Wazwaz,Chaos,Solitons&Fractals 28(2006)1005.

[16]J.G.Liu,Y.Tian,and Z.F.Zeng,AIP Adv.7(2017)105013.

[17]A.M.Wazwaz,Appl.Math.Comput.200(2008)437.

[18]A.M.Wazwaz,Appl.Math.Comput.204(2008)162.

[19]M.Sarker,B.Hosen,M.R.Hossen,and A.A.Mamun,Commun.Theor.Phys.69(2018)107.

[20]A.Qawasmeh and M.Alquran,Appl.Math.Sci.8(2014)2455.

[21]J.G.Liu and Y.He,Nonlinear Dyn.90(2017)363.

[22]A.M.Wazwaz and S.A.El-Tantawy,Nonlinear.Dyn.373(2015)1.

[23]A.M.Wazwaz,Chaos,Solitons&Fractals 76(2015)93.

[24]J.G.Liu,Y.Tian,ahd J.G.Hu,Appl.Math.Lett.79(2018)162.

[25]E.Fan,Phys.Lett.A 265(2000)353.

[26]M.Senthilvelan,Appl.Math.Comput.123(2001)381.

[27]S.Zhang,Chaos,Solitons&Fractals 30(2006)1213.

[28]M.F.El-Sabbagh and A.T.Ali,Commun.Theor.Phys.56(2011)611.

[29]M.F.El-Sabbagh,A.T.Ali,and S.El-Ganaini,Appl.Math.Inform.Sci.2(2008)31.

[30]C.J.Wang,Z.D.Dai,et al.,Commun.Theor.Phys.52(2009)862.

[31]J.G.Liu,J.Q.Du,Z.F.Zeng,and B.Nie,Nonlinear Dyn.88(2017)655.

[32]X.P.Zeng,Z.D.Dai,and D.L.Li,Chaos,Solitons&Fractals 42(2009)657.

[33]Z.D.Dai,S.L.Li,Q.Y.Dai,and J.Huang,Chaos,Solitons&Fractals 34(2007)1148.

[34]L.Wei,Appl.Math.Comput.218(2011)368.

[35]J.G.Liu,J.Q.Du,Z.F.Zeng,and G.P.Ai,Chaos 26(2016)989.

[36]Z.D.Dai,Z.J.Liu,and D.L.Li,Chin.Phys.Lett.25(2008)1151.

[37]Z.H.Zhao,Z.D.Dai,and G.Mu,Comput.Math.Appl.61(2011)2048.

[38]R.Radha and M.Lakshmanan,Phys.Lett.A 197(1995)7.

[39]J.F.Zhang,J.P.Meng,C.L.Zheng,and W.H.Huang,Chaos,Solitons&Fractals 20(2004)523.

[40]B.G.He,C.Z.Xu,and J.F.Zhang,Acta Phys.Sin.55(2006)511.

[41]W.H.Huang,Y.L.Liu,and J.F.Zhang,Commun.Theor.Phys.49(2008)268.