Bang-Xing Guo(郭幫興),Zhan-Jie Gao(高戰(zhàn)杰),and Ji Lin(林機(jī))

Department of Physics,Zhejiang Normal University,Jinhua 321004,China

1 Introduction

The technology of the ultrashort laser has been developing after the invention of laser mode locking in 1964.[1]the duration of laser pulse has been shortened from picosecond to femtosecond.We should point out that such ultrashort laser pulses with the duration of including a only few optical cycles are of much importance because they are brief enough to resolve temporal on an atomic level.Notably,the ultrashort pulses possess extensive applications to the field of light-matter interactions,highorder harmonic generation,extreme nonlinear optics,[2]and attosecond physics.[3]

In theory,the propagation of few-cycle pulses(FCP)solitons in cubic(Kerr)media can be described in a way completely free from the slowly varying envelope approximation,using completely integrable models as the modified Korteweg-de Vries(mKdV)equation,[4?5]sine-Gordon equation[6?7]and mKdV-sG equation,[8?9]even including the(2+1)-dimensional non-integrable generalized Kadomtsev–Petviashvili(KP)equation.[10]Up to now,the study of FCP solitons is restricted to the case of a cubic nonlinearity.In fact,the study of optical solitons in quadratic nonlinear media has shown many remarkable properties with respect to the cubic case.[11?12]People have found walking soliton,[13]vortex solitary waves,[14]and gap soliton[15]that exist in quadratic nonlinear media.It demonstrates that the formation of envelope solitons in quadratic media due to second harmonic generation.[16]The half-cycle optical soliton can be derived from a Korteweg-de Vries(KdV)equation in the(1+1)-dimensional quadratic media.[17]In the(2+1)-dimensional situation and the long-wave approximation,a generic KP equation[18]can describe the propagation of few-cycle pulse when the resonance frequency of the twolevel atoms is well above the inverse of the characteristic duration of the pulse.And it exists line soliton and lump soliton by the direct numerical simulations.The propagative property of light pulse can also be described by the(2+1)-dimensional Boussinesq equation(BE)in quadratic nonlinear media.[19]

Various methods have been used to study this equation.The solitary wave solution and periodic wave solution are obtained by using homogeneous balance method.[20]The solitary wave solution,two soliton solution,and resonant solution are got by using bilinear method.[21]And people applied the generalized Jacobi elliptic function expansion method to obtain many new Jacobi and Weierstrass double periodic elliptic function solutions.[22?23]In this paper,we want to seek the exact analytical solution of the(2+1)-dimensional Boussinesq equation by the consistent tanh expansion(CTE)method and truncated Painlevé expansion method which is widely applied to many kinds of integrable equations.[24?28]

This paper is organized as follows.In Sec.2,the interaction solution between solitary wave and periodic wave is obtained by using CTE method.In Sec.3,the bright soliton solution,kind bright soliton solution,double well dark soliton solution,and kink-bright soliton interaction solution are got by using truncated Painlevé expansion method.In Sec.4,a short summary and discussions are given.

2 New Analytical Solutions

In this section,we research new analytical solutions of the(2+1)-dimensional Boussinesq equation via the CTE method.We take the following truncated function expansion form by using leading order analysis

where u0,u1,u2,and f are arbitrary functions of variables(x,y,t).By substituting(2)into the Boussinesq equation(1)and vanishing coefficients of the power of tanh(f)6,tanh(f)5and tanh(f)4,we get

where

Collecting the coefficients of tanh(f)3,tanh(f)2,tanh(f)1,and tanh(f)0,weobtain the following fouroverdetermined equations which function f satisfied

We can get exact solutions of(2+1)-dimensional Boussinesq equation if we can find the solution f from Eqs.(4)–(7).According to the nonauto-BT theorem,we assume function f of(4)–(7)has the form

where F(X)is an arbitrary function with respect to X and the k0,l0,ω0,k1,l1,and ω1are all of constants to be determined.Substituting the expression(8)into(4)–(7)and obtaining the following relation

2.1 Solitary and Periodic Wave Interaction Solution

Many new typesofexactsolutionsof(2+1)-dimensional BE(1)are found due to the existence of the arbitrary function F(X).If we choose

then we can derive a periodic wave-bright soliton interaction solution of(2+1)-dimensional BE(1)

Figure 1 shows the structure of the periodic wave-bright soliton interaction solution on the(t,y)plane when x=0.It indicates that the bright soliton propagates along the negative direction of the t-axis,and there is no variation about the peak of bright soliton during the propagation.

One few-cycle solitary wave and one bright soliton interaction solution of Eq.(1)is obtained

Fig.1 The periodic wave-bright soliton interaction of(11)at x=0 with the parameters a=1/3,b=0,k0=1/3,l0= ?1/6,ω0=1/6,ω1= ?1/2.

If the function F(X)in Eq.(8)is selected as the following formation

Actually,if we choose the function F(X)as

we can obtain another variational amplitude periodic wave interaction solution of(2+1)-dimensional BE(1)

Figure 2(a)shows the structure of one few-cycle solitary wave and one bright soliton interaction solution(12)at t=0.We find this solution structure is not changed at along the time.Figure 2(b)shows the evolution of structure of periodically variable amplitude bright soliton(15)when x=y=1 along the time.The amplitude of periodic bright soliton is decreased along the time.

Fig.2 (a)The few optical periodic soliton and bright soliton interaction solution(12)at t=0,(b)the evolution of variational amplitude periodic wave interaction solution(15)at x=y=1 along the time with the parameters a=3/20,k0=1/5,l0=1/6,ω0=1/5,ω1=1/2.

A kink-periodic wave interaction solution of(2+1)-dimensional BE(1)can be obtained

by choosing

Figures 3(a)–3(c)show the propagation of kink-periodic wave interaction solution,and the periodic wave and kink soliton propagate along the negative y-axis.The amplitude of the periodic wave keeps invariable during the propagation process.Many types of exact solutions can be obtained by selecting the arbitrary function F(X).We only consider several cases in here.

Fig.3 The propagation of kink-periodic wave interaction solution(16)at x=2 and with the parameters k0=1/8,l0=1/3,ω0= ?1/3,ω1=1/3,(a)t= ?8,(b)t=0,(c)t=8.

2.2 Kind Soliton Solution and Their Interaction

In this section,we consider to obtain different types of soliton solutions of the(2+1)-dimensional BE(1)by using the truncated Painlevé expansion method.We assume the dependent variable u form as

Next,we substitute(18)into(1)and collect together the coefficients of φ?6, φ?5,and φ?4.Vanishing the coefficients and the explicit expression of u can be obtained

Finally,collecting the coefficient of φ?3,we get the following trilinear partial differential equation

We can obtain the exact solution of Eq.(1)if we find the solution φ to satisfy(20).It is easy to know the trilinear partial differential equation(20)possesses the solution

where ci(i=1,...,6)are the free constants.Substituting(21)into(19)and we obtain the solution of(2+1)-dimensional BE(1)

where θ=c2x+c3y+c4t+c1.We can obtain various types solutions of u due to including six free constants.Next,we select three groups of different parameters as special cases

Fig.4 (a)The evolution of intensity for bright soliton solution(22)with(23)at x=y=0.(b)the evolution of intensity for cowboy-hat’s kind of bright soliton(22)with(24)at x=y=0.

Figures 4(a)–4(b)show that the intensities of the bright soliton and cowboy-hat’s kind of bright soliton solutions(22)with(23)and(24)at x=y=0,respectively.Figures 5(a)–5(c)show the evolution of the intensity of double well dark soliton solution(22)with(25).This double well dark soliton has twin intensity null.It had been demonstrated that the double well dark soliton existed in quadratic medium.[28?29]Although the φ is the single form of tanh,the special form of solution u of expression(20),and it provides convenience to obtain abundant solution.In fact,we can also choose other parameters to get different types of soliton solutions.

Another special solution of(20)reads

where η1=k1x+l1y+ω1t,η2=k2x+l2y+ω2t,and a12is free constant.Substituting(26)into trilinear partial differential equation(20)and it yields

Finally,we get multi-soliton solution of(2+1)-dimensional BE(1)

where

Fig.5 The evolution plot of double well dark soliton solution(22)with(25)at y=0 and(a)t=?3,(b)t=0,(c)t=3.

Fig.6 The kink-bright soliton solution(28)at t=5 with the parameters k1=1,l1=1/3,ω1=2/3,k2= ?1,l2=1,ω2=2.

Figure 6 shows the structure of the one kink-two bright solitons solution at t=5.This structure of the kink-bright international solitons is invariant when it evolutes along the time.Actually,if one can obtain other solutions of trilinear partial differential equation(20),we can derive more different types of soliton solutions of(2+1)-dimensional BE(1).

3 Summary and Discussion

We apply for the CTE method to research new soliton solutions of the(2+1)-dimensional BE(1).We have obtained new exact analytical solutions which include one arbitrary function F(X).So we can derive various types of different periodic wave and soliton interaction solutions,such as the periodic wave-bright soliton interaction solution,few optical periodic soliton wave and bright soliton interaction solution,variational amplitude periodic wave interaction solution,and kink-periodic wave interaction solution.These solutions are explicitly expressed both analytically and graphically.Finally,we also obtain the bright soliton solution,cowboy-hat’s kind of bright soliton solution,double well dark soliton solution and one kinktwo bright solitons interaction solution by using truncated Painlevé expansion method.

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