2+1 维二元模拟长短波共振方程的混合型孤

2+1维二元模拟长短波共振方程的混合型孤子和孤子相互作用江彦1，田播1,2，刘文军1，孙鲲1，刘立才11北京邮电大学理学院，北京　1008762北京邮电大学信息光子与光通信国家重点实验室，北京100876摘要：本文研究了一个2+1维二元模拟长短波共振方程，该方程能用于描述在两层流体中传输的一个长的界面重力波和两个短的表面重力波之间的共振相互作用。

运用双线性方法和符号计算，该方程混合型的单孤子和双孤子解被解析地得到。

基于得到的孤子解，孤子间的相互作用被解析和图像地加以分析。

例如双孤子之间的斜碰撞和平行碰撞，并且相应的参数条件也被给出。

关键词：偏微分方程，2+1维二元模拟长短波共振方程，混合型孤子解，双线性方法，符号计算中图分类号：O175;O29Mixed-type solitons and soliton interaction for the(2+1)-dimensional two-component long wave-short wave resonance interaction equationsin a two-layerfluidJiang Yan1,Tian Bo1,2,Liu Wen-Jun1,Sun Kun1,Liu Li-Cai1 1School of Science,Beijing University of Posts and Telecommunications,Beijing100876 2State Key Laboratory of Information Photonics and Optical Communications,Beijing University of Posts and Telecommunications,Beijing,100876Abstract:Under investigation in this paper are the(2+1)-dimensional two-component long wave-short wave resonance interaction equations,which can be used to describe the resonance interaction between a long interfacial gravity wave and two short surface gravity packets propagating in a two-layerfluid.Via the Hirota method and symbolic computation,analytic mixed-type one-and two-soliton solutions are derived.Soliton interactions are investigated 基金项目：National Natural Science Foundation of China(60772023)，Fundamental Research Funds for the Central Universities of China(2011BUPTYB02)，National Basic Research Program of China(973Program)(2010CB923200)，Spe-cialized Research Fund for the Doctoral Program of Higher Education(200800130006).作者简介：Jiang Yan（1984-），male，postgraduate student，major research direction：nonlinear science；Liu Wen-Jun（1983-），male，lecturer，major research direction：nonlinear science；Sun Kun（1985-），male，postgraduate student，major research direction：nonlinear science；Liu Li-Cai（1985-），male，postgraduate student，major research direction：nonlinear science.Correspondence author：Tian Bo（1963-），female，professor，major research direction：nonlinear science.analytically and graphically.Oblique and parallel interactions between two solitons are observed.Relevant parameter conditions are presented.Key words:Partial differential equation,(2+1)-dimensional two-component long wave-short wave resonance interaction equations,Mixed-typed soliton solution,Hirota method,Symbolic computation0IntroductionSince the observation of soliton interaction for the Korteweg-de Vries (KdV)equation,theoretical and experimental studies on the solitons and relevant nonlinear evolution equations (NLEEs)have been carried out due to their attraction and application in fluid dynamics,plasma physics,optical fibers and Bose-Einstein condensates [1].For example,KdV equation describes the propagation of the one-dimensional small amplitude and surface gravity waves in a shallow channel of water [1,2],whereas the Kadomtsev-Petviashvili (KP)equation,a two-spatial-dimensional analogue of the KdV one,can be used to model the two-dimensional small amplitude long waves with weakly nonlinear restoring forces and frequency dispersion [1,3].Coupled NLEEs need to be taken into account,to model the interaction of several waves in some physical situations [1,4].Long wave-short wave resonance interaction equations (LWSWRIEs)[5,6,7,8],which arise on the condition that the group velocity of the short wave is equal to the phase velocity of the long wave,have been investigated analytically and numerically in plasma physics,hydrodynamics and molecular crystals [9,10,11,12,13,14,15,16,17,18,19,20].The (1+1)-dimensional LWSWRIEs have the following form [5,6,7]:iE t +β1E xx =δF E ,(1a)F t +α1(|E |2)x =0,(1b)where E is the envelope of the short wave,F is the amplitude of the long wave,the subscripts x and t respectively denote the partial derivatives with respect to space and time,the real constants α1,β1and δrepresent the self-interaction of the short wave,dispersion of the short wave and nonlinear interaction of the long wave with the short wave,respectively.Periodic and soliton solutions,orbital stability and Cauchy problem for Eqs.(1)have been carried out in Refs.[6,7,8]and references therein.The discrete analogue of nonautonomous Eqs.(1)on an infinite lattice has been studied in Ref.[9]as well.There exist two generalizations of Eqs.(1)[10,11,12,13,14,15]:One is the (1+1)-dimensional two-component LWSWRIEs [10]:iϕt +α2ϕxx =β2uϕ,(2a)iψt +α2ψxx =β2uψ,(2b)u t =±β2(|ϕ|2+|ψ|2)x ,(2c)where ϕ,ψand u respectively represent two short waves and a long wave,α2and β2are the real constants.Ref.[10]has presented two numerical methods to solve Eqs.(2)and compared those methods from a computational efficiency viewpoint.The other is the following set of (2+1)-dimensional LWSWRIEs [11,12,13,14,15]:i (S t +S y )−S xx +LS =0,(3a)L t =2(|S |2)x ,(3b)where S and L respectively represent a short surface gravity packet and a long interfacial gravity wave,and the subscripts x ,y and t respectively denote the partial derivatives with respect to x direction,y direction and time.Solitonic,periodic and localized solutions,Painlevéproperty,Lie algebraic structures and interaction properties for Eqs.(3)have been studied in Refs.[11,12,13,14]and references therein.Besides,Ref.[15]has proved the existence of a global attractor and an asymptotic smoothing effect of the solution for the nonautonomous Eqs.(3),which possess the external sources,.In addition,Refs.[16,17,18,19,20]have presented a (2+1)-dimensional two-component LWSWRIEsi (S (1)t +S (1)y )−S (1)xx +LS (1)=0,(4a)i (S (2)t +σS (2)y )−S (2)xx +LS (2)=0,σ=±1,(4b)L t =2(|S (1)|2+|S (2)|2)x ,(4c)where S (1),S (2)and L respectively represent two short surface gravity packets and a long interfacial gravity wave.Eqs.(4)can be used to describe the resonance interaction between a long interfacial gravity wave and two short surface gravity packets propagating in a two-layer fluid [16,17,18,19].Ref.[16]has given the derivation for Eqs.(4),obtained the Wronskian bright soliton solutions for Eqs.(4)with σ=1,and graphically analyzed the soliton property and interaction.Ref.[17]has obtained the one-soliton solutions for Eqs.(4)with σ=−1and the Wronskian multi-soliton solutions for the integrable modification of Eqs.(4)with σ=−1.Ref.[18]has found that Eqs.(4)with σ=1possess the Painlevéproperty,localized solutions for S (1)and S (2),and the line soliton solutions for L .Besides,the interaction dynamics of the multimode dromions has been investigated in Ref.[18].Ref.[19]has derived the bright soliton solutions in a Gram determinant form for Eqs.(4)with σ=1.Refs.[19,20]have foundthat the solitons in the short wave components can be amplified with a pulse compression by reducing the width of the soliton in the long wave component,and discussed the shape-changing interactions in the short wave components and elastic interaction in the long wave component.However,to our knowledge,the dynamics of the mixed-type solitons[the bright soliton for S(1),dark soliton for S(2)and line soliton for L]for Eqs.(4)has not been investigated.In Section2of this paper,we will present the analytic mixed-type one-and two-soliton solutions for Eqs.(4)through the Hirota method[21]and symbolic computation[22,23].Section3will investigate the soliton interactions analytically and graphically,and give the relevant parameter conditions for such soliton phenomena as the two parallel solitons propagation with the periodic interaction,breather-type solitons,V-shape solitons and semi-breather-type solitons.Section 4will be our conclusions.1Bilinear form and soliton solutions1.1Bilinear formThrough the dependent variable transformations[16,17,19]S(1)=gf,S(2)=hf,L=−2(log f)xx,(5)where g and h are the complex differentiable functions with respect to x,y and t,and f is a real one,the bilinear form for Eqs.(4)is as follows:[i(D t+D y)−D2x ]g·f=0,(6a)[i(D t+σD y)−D2x ]h·f=0,(6b)(D t D x−2λ)f·f+2(gg∗+hh∗)=0,(6c)with∗as the complex conjugate andλas a constant to be determined.Hereby,D x,D y and D t are the bilinear operators[21]defined byD lx D myD nta(x,y,t)·b(x,y,t)=(∂∂x−∂∂x′)l(∂∂y−∂∂y′)m×(∂∂t−∂∂t)na(x,y,t)b(x′,y′,t′)|x′=x,y′=y,t′=t.Bilinear Form(6)will respectively reduce to the ones in Refs.[16,17,19]on the different conditions therein.It is worth noting thatλneeds to be determined in this paper,while it is considered as zero in Refs.[16,17,19].1.2Soliton solutionsIn order to obtain the mixed-type soliton solutions,g,h and f are expanded with respect to a formal parameterεas follows:g=εg1+ε3g3+···,h=h0(1+ε2h2+ε4h4+···),f=1+ε2f2+ε4f4+···,(7)where g j’s(j=1,3,5,...),h k’s and f k’s(k=0,2,4,6,...)are the differentiable functions with respect to x,y and t.Substituting Expressions(7)into Bilinear Forms(6)and collecting the coefficients of each order ofε,we can obtain the recursion relations for g j’s(j=1,3,5,...),h k’s and f k’s(k=0,2,4,6,...),through which the soliton solutions for Eqs.(4)will be obtained.1.2.1One-soliton solutionsTruncating Expressions(7)as g=εg1,h=h0(1+ε2h2)and f=1+ε2f2,and substituting them into Bilinear Forms(6),we will derive the mixed-type one-soliton solutions as follows:S(1)=g11+f2,S(2)=h0(1+h2)1+f2,L=−2[log(1+f2)]xx,(8)whereh0=χeη,g1=γeθ,f2=βeθ+θ∗,h2=αβeθ+θ∗,λ=χχ∗,η=i(a x+b y+ρt),ρ=a2−b,θ=k x+l y+s t,s=−ik2−σl,β=γγ∗(2−α−α∗)λ−(k+k∗)(l+l∗),α=2(a+ik)(k+k∗)+(l+l∗)(σ−1)2(a−ik∗)(k+k∗)+(l+l∗)(σ−1),withχ,γ,k and l as the complex constants and a and b as the real ones.1.2.2Two-soliton solutionsTruncating Expressions(7)as g=εg1+ε3g3,h=h0(1+ε2h2+ε4h4)and f=1+ε2f2+ε2f4, and substituting them into Bilinear Forms(6),we will have the mixed-type two-soliton solutions as follows:S(1)=g1+g31+f2+f4,S(2)=h0(1+h2+h4)1+f2+f4,L=−2[log(1+f2+f4)]xx,(9)whereh 0=χe η,g 1=γ1e θ1+γ2e θ2,g 3=δ1e θ1+θ2+θ∗1+δ2e θ1+θ2+θ∗2,f 2=β1e θ1+θ∗1+β2e θ2+θ∗1+β3e θ1+θ∗2+β4e θ2+θ∗2,f 4=τe θ1+θ2+θ∗1+θ∗2,h 2=α1β1e θ1+θ∗1+α2β2e θ2+θ∗1+α3β3e θ1+θ∗2+α4β4e θ2+θ∗2,h 4=ζτe θ1+θ2+θ∗1+θ∗2,λ=χχ∗,η=i (a x +b y +ρt ),ρ=a 2−b ,θj =k j x +l j y +s j t ,s j =−ik 2j −σl j ,(j =1,2),β1=γ1γ∗1(2−α1−α∗1)λ−(k 1+k ∗1)(l 1+l ∗1),β2=γ2γ∗1(2−α2−α∗3)λ−(k 2+k ∗1)(l 2+l ∗1),β3=γ1γ∗2(2−α3−α∗2)λ−(k 1+k ∗2)(l 1+l ∗2),β4=γ2γ∗2(2−α4−α∗4)λ−(k 2+k ∗2)(l 2+l ∗2),α1=2(a +ik 1)(k 1+k ∗1)+(l 1+l ∗1)(σ−1)2(a −ik ∗1)(k 1+k ∗1)+(l 1+l ∗1)(σ−1),α2=2(a +ik 2)(k 2+k ∗1)+(l 2+l ∗1)(σ−1)2(a −ik ∗1)(k 2+k ∗1)+(l 2+l ∗1)(σ−1),α3=2(a +ik 1)(k 1+k ∗2)+(l 1+l ∗2)(σ−1)2(a −ik ∗2)(k 1+k ∗2)+(l 1+l ∗2)(σ−1),α4=2(a +ik 2)(k 2+k ∗2)+(l 2+l ∗2)(σ−1)2(a −ik ∗2)(k 2+k ∗2)+(l 2+l ∗2)(σ−1),δ1=(k 1−k 2)(β2γ1k 1+k ∗1−β1γ2k 2+k ∗1),δ2=(k 1−k 2)(β4γ1k 1+k ∗2−β3γ2k 2+k ∗2),τ=β1β4(k 1−k 2)(k ∗1−k ∗2)(k 1+k ∗2)(k 2+k ∗1)−β2β3(k 1−k 2)(k ∗1−k ∗2)(k 1+k ∗1)(k 2+k ∗2),ζ=α1α4,with χ,γj ’s,k j ’s and l j ’s (j =1,2)as the complex constants and a and b as the real ones.It is worth noting that the following conditions must be satisfied for Solutions (9)when σ=−1:l 1R =µk 1R ,l 2R =µk 2R ,l 1I −l 2I =µ(k 1I −k 2I ),(10)where the subscripts R and I respectively denote the real and imaginary parts,and µis a real constant.2Dynamics of the mixed-type solitonsIn this section,we will analytically investigate the dynamics of the mixed-type solitons through Solutions (8)and (9),and present the relevant parameters conditions for some soliton phenomena.Via Solutions (8),we can obtain that the amplitudes for S (1),S (2)and L are respectively |γ|/(2√β),(|χ|√2−α−α∗)/2and (k +k ∗)2/2,which show that the mixed-type solitons via Solutions (8)can propagate ,as seen in Figs.1and 2.Besides,we can notice that the darkS 1S 2L Fig.1:Mixed-type solitons on the x-y plane att=0S 1S 2L Fig.2:Mixed-type solitons on the x-y plane at t=4soliton for S(2)has two limit cases:the black and gray solitons[24],whose critical condition satisfies:2k R(a−k I)+l R(σ−1)=0,(11)i.e.,the black(gray)solitons arise when the parameters satisfy(dissatisfy)Condition(11). Next,we will discuss the interactions between the two solitons.On the basis of Solutions(9),we can obtain the wave vectors k j’s,frequenciesωj’s and vector velocities v j’s on the x-y plane(j=1,2)for the two solitons as follows:k j=(k jR,l jR),ωj=σl jR−2k jR k jI,v j=k jωj|k j|2=(k jRωjk2jR+l2jR,l jRωjk2jR+l2jR).(12)Hereby,there are two aspects to pay attention to.On one hand,we canfind that the wave vectors k j’s(j=1,2)determine the two types of the interactions between the two solitons[see Figs.3-6],i.e.,the one is oblique when the wave vectors are not parallel(k1=ϑk2),and the other is parallel when the wave vectors are parallel(k1=ϑk2),whereϑis a real constant.We can notice that the two-soliton solutions for Eqs.(4)withσ=−1must satisfy Condition(10), which makes the wave vectors and soliton interactions always parallel.On the other hand,we can observe that there exist three different cases of the parallel interactions under appropriate conditions:(i)the overtaking interaction whenω1ω2/ϑ>0and k1I=k2I;(ii)the head-onS 1S 2LFig.3:Oblique interaction between two solitons on the x-y plane att=0S 1S 2LFig.4:Oblique interaction between two solitons on the x-y plane at t=4interaction whenω1ω2/ϑ<0;(iii)the interaction between moving and stationary solitons whenω1ω2=0andω1+ω2=0.3ConclusionIn this paper，we have investigated the(2+1)-dimensional two-component LWSWRIEs[i.e.,Eqs.(4)]through the Hirota method and symbolic computation.We have obtainedtheS 1S 2LFig.5:Parallel interaction between two solitons on the x-y plane at t=0S 1S 2LFig.6:Parallel interaction between two solitons on the x-y plane at t=4analytic mixed-type one-and two-soliton solutions[see Solutions(8)and(9)].Solutions(9)need to satisfy Condition(10)whenσ=−1.On the basis of those solutions,we have analytically investigated the dynamics of mixed-type solitons，and given the relevent parameter conditions.Acknowledgements(Optional)Thanks to Prof.Tian Bo and all the members of our discussion group for their 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