Lian-li Feng and Zuo-nong Zhu

School of Mathematical Sciences,Shanghai Jiao Tong University,800 Dongchuan Road,Shanghai,200240,China

Abstract

As is well known,the Sasa-Satsuma equation is an important integrable high order nonlinear Schr?dinger equation.In this paper,a two-component generalized Sasa-Satsuma(gSS) equation is investigated.We construct the n-fold Darboux transformation for the two-component gSS equation.Based on the Darboux transformation,we obtain some interesting solutions,such as a breather soliton solution,kink solution,anti-soliton solution and a periodic-like solution.

Supplementary material for this article is available online

Keywords: two-component generalized Sasa-Satsuma equation,Darboux transformation,soliton solutions

1.Introduction

A higher order nonlinear Schr?dinger (NLS) equation

was proposed by Kodama and Hasegawa [1,2],where βjare real constants,and ?is a small parameter.It is originally presented as a model for the femtosecond pulse propagation in a monomode fiber.In the general case,equation (1) is not integrable.But when choosing some appropriate parameters,it can be shown that equation (1) is integrable by inverse scattering transform.Equation(1)can be converted to several integrable equations,such as the derivative NLS equation,the Hirota equation and the Sasa-Satsuma equation [3-6].

Let us write the Sasa-Satsuma equation

By introducing variable transformations

Equation (2) changes into a complex modified KdV-type equation

which is also called a Sasa-Satsuma equation.Equation (4)has been extensively studied by different methods,such as inverse scattering transform [7,8],Darboux transformation[9-14] and Hirota bilinear method [15,16].

Based on equation (4),a two-component Sasa-Satsuma equation is proposed in [17,18],

Soliton solutions of equation (5) are obtained by Darboux transformation and the Riemann-Hilbert approach.

In this paper,inspired by equation (5),we introduce a two-component generalized Sasa-Satsuma (gSS) equation

where u(x,t)is a complex function,a is a real constant,b is a complex constant,and* denotes the complex conjugate.In equation(6),v(x,t)can be either a complex function or a real function.If the reduction is taken asthen equation (6) changes to

Equation (7) is a gSS equation.We thus can see that it is interesting to study the equation (6).It is obvious that when b=0,the equation (6) is reduced to equation (5).

In this paper,we will show that the two-component gSS equation (6) is Lax integrable.We will construct its n-fold Darboux transformation.Soliton solutions including a breather soliton solution,kink solution and periodic-like solution will also be constructed.

2.Lax pair for the two-component gSS equation

In this section,we give the following Lax pair of equation(6):

where

It can directly verify that the zero-curvature equation Ut-Vx+[U,V]=0 yields the two-component gSS equation (6).

We takeφ=(φ1,φ2,φ3,φ4)Tis an eigenfunction of Lax pair (8) at λ.Note thatψ=(φ*1,φ*3,φ*2,φ*4)Tis an eigenfunction of Lax pair (8) at-λ*if v is a real function.Thus we can construct the matrix solution of Lax pair (8)

where

3.Darboux transformation of the two-component gSS equation

In this section,we construct the Darboux transformation to the two-component gSS equation (6).Firstly,the adjoint problem of Lax pair (8) is

Assuming that θ1(x,t) is an eigenfunction of the Lax pair (8)at λ=λ1,we can verify thatθ1?(x,t)Mis an eigenfunction of equation (12) atλ=-λ*1,where

Based on the [9-12],we take the transformation

where

with

It is obvious that under the transformation(14)linear spectral equation (8) changes to

where

We can show the matrix U[1] and V[1] have the same structures with the matrix U and V,that is

where the relation between the new potential and the old one is

Let us show the conclusion that the matrix U[1]and V[1]have the same structure as the matrix U and V.The matrix T can be rewritten as

where

We hope to check that the following equations hold

Supposing

and substituting equation (27) into equation (25),we obtain

where

Thus matrices U[1]and U have the same structures.Since the structure of V[1] is too complex,we directly show the equation (26) holds.We have the following equations:

where

According to the coefficient of λ4,we have

which is equivalent to equation (20).By substituting equation(31)into the coefficients of λ3,λ2,we can know that the coefficients of λ3,λ2are zero.From the coefficients of λ3we can know the representation of S3[1].Substituting the S3[1] into the coefficients of λ1,λ0,and with the help of maple,we can verify that the coefficients of λ1,λ0are also zero.

Further,we can derive n-fold DT:

where

In fact,we have

Here we use the two formulas of Θ?MΦ=(λI-Λ*)Ω(Φ,Θ) and Θ?MΘ=Ω(Θ,Θ)Λ-Λ*Ω(Θ,Θ).Because the computation is too complicated,a detailed proof of φ[n]t=V[n] φ[n]is not given here.

4.Soliton solutions of the gSS equation

In this section,we consider different types of soliton solutions of equation (6) from zero seed solution and nonzero seed solution,respectively.From equation (20) we have

with

4.1.Soliton solution from zero seed solution

Setting zero seed solutions u=0,v=0 of the gSS equation(6)and solving the Lax pair(8)at λ=λ1,we obtain

where ck(k=1,2,3,4) are all complex constants.Substituting the equation (38) into solution (36),we obtain

where

王翔介紹，為了更好地滿足種植戶的需求，巴斯夫未來將推出一體化農(nóng)業(yè)解決方案，同時也對迫在眉睫的問題進(jìn)行持續(xù)的應(yīng)對。所謂一體化的農(nóng)業(yè)解決方案，就是從農(nóng)藥到種子，再到性狀產(chǎn)品以及數(shù)字方案的應(yīng)用，從而全方位地幫助農(nóng)戶構(gòu)建解決方案。真正能夠在全球?qū)用嫔辖鉀Q種植戶面臨的挑戰(zhàn)，最終幫助種植戶解決三大問題，即幫助種植戶提高種植效率，提高作物單產(chǎn)，最終提高種植戶的利潤率以及他們的收益。

Solution(39)represents the breather solution which is shown in figure 1.When ξ=0,the breather solution becomes to soliton solution

where

The solutions u[1] and v[1] given by (40) represent soliton solutions that propagate with the same velocity 4η2.The amplitude of |u[1]| isand it is localized at the lineThe amplitude of v[1]isand it is localized at the lineIt should be pointed out that the case of a=b,ck?R(k=1,2,3,4)cannot appear simultaneously,otherwise the solutions have singularity.Because the plots of |u[1]| and v[1] are similar,we only exhibit the evolution of soliton solution |u[1]| given by (40)in figure 2.

4.2.Soliton solution from nonzero seed solution

Substituting nonzero seed solutions u=s,v=r,where s and r are two constants,into the Lax pair (8),we get

where

Let us consider the following two cases where we sets=r=1,a=-.

Case 1: τ ?R,i.e.2a∣s∣2+2 Re(b*s2)-r2-λ12>0.

(i) Letd3=d4=1,We have

with

whereR1=We can see from figure 3 that the soliton solution (43) describes the propagation process of one soliton splitting into two solitons.

(ii) Letd2=1,d4=0,We have

where

The solution|u[1]|given by(44)is an anti-soliton soliton that propagates along the lineThe solution v[1] is a kink solution (see figure 4(a)-4(b)).

When b=0,we have

whereR3=(x+13t).The solution|u[1]|given by(45)is a soliton solution that propagates along the linex=-7t+ln(see figure 4(c)),and v[1] is still a kink solution.

We note that the solution |u[1]| given by (44) is an antisoliton solution where b ≠0 and the solution |u[1]| given by(45)is a soliton solution where b=0.This shows that there is a difference between equation (6) and equation (5).

Case 2:If 2a∣s∣2+2 Re(b*s2)-r2-λ12<0,Re(τ)=0,depending on the different choices of d2and d4,we can obtain the following two solutions.

(i) Letd2=1,d3=,d4=0,b=.We have

where

with

The solution (46) is a breather which is shown in figure 5.When t→±∞,|u[1]|→1,v[1]→1.

(ii) Letd2=0,d3=,d4=1,b=.We have

where

with

The solution (47) is a periodic-like solution,which is a plane on the left and a periodic wave on the right (see figure 6).

5.Two-soliton solutions of the gSS equation

By using the two-fold Darboux transformation,we have

where

with

and X11and Y11are represented by equation (37).

5.1.Two-soliton solutions from zero seed solution

Taking zero seed solutions u=0,v=0,and solving the Lax pair (8),we obtain

where c4k-3,c4k-2,c4k-1,c4k(k=1,2) are all complex constants.Substituting equation (49) into equation (48),and takinga=-,b=,c1=c2=c4=c5=c6=c8=1,c3=c7=1+,we get

where

When ξj≠0,the solution(50)is a two-breather solution.We plot them in figure 7.When ξj=0,we obtain a two-soliton solution from solution (50),

where

Furthermore,we get the following asymptotic property of solutions (51):

(1) When κ1～O(1),we have

(2) When κ2～O(1),we have

The solution (51) represents the two-soliton solution.Figure 8 depicts the evolution of the two-soliton solution u[2]with the parameters:c1=c2=c4=c5=c6=c8=1,c3=

5.2.Two-soliton solutions from nonzero seed solution

Choosing the seed solutions u=s,v=r,and solving the linear spectral equation yields

where

Setd3=d7=,d4=d8=0,d1=d2=d5=d6=1,s=r=1.We consider the following two cases of solutions.

Case 1:Re(τ1)=0,Re(τ2)=0.In this case,we have

where ?1=2(x+4t),?4=(x+3t).The solution describes the interaction of two breather solutions (see figure 9).

Case 2:Ifτ2?R,Re(τ1)=0,we have

Figure 1.The evolution of the breather solution (39) with the parameters:ξ=,η=,c1=c2=c4=1,c3=1+,a=-,b=.(a)-(b) show the |u[1]|.(c)-(d) show the v[1].

Figure 2.The evolution of the soliton solution (40) with the parameters :η=,c1=c2=c4=1,c3=1+,a=-,b=.

Figure 3.The evolution of the soliton solution (43) with parameter:λ1=.(a)-(b) show the |u[1]|.(c)-(d) show the v[1].

Figure 4.The evolution of solutions (44) and (45) with parameter:λ1=.(a) shows the anti-soliton solution |u[1]|.(b) shows the kink solution v[1].(c) shows the soliton solution |u[1]| of the case b=0.

Figure 5.The evolution of the breather solution (46) with parameterλ1= (a)-(b) show the |u[1]|.(c)-(d) show the v[1].

Figure 7.The evolution of two-breather solution(50)with the parameters:λ1=+i,λ2=1+i.(a)-(b)show the|u[2]|.(c)-(d)show the v[2].

Figure 8.The evolution of the two-soliton solution |u[2]| given by (51) with the parameters:η1=-,η2=1.

Figure 9.The evolution of two breather solutions with the parameters:λ1=i,λ2=,a=-,b=.(a)-(b)show the|u[2]|.(c)-(d)show the v[2].

Figure 10.The evolution of solutions |u[2]| and v[2] with the parameters: λ1=i,λ2=,a=-,b=.(a)-(b) show the |u[2]|describing the interaction of a breather and a soliton.(c)-(d) show v[2] describing the interaction of a breather and a kink.

6.Conclusion

In this paper,we have introduced and studied a two-component gSS equation.A Darboux transformation of the twocomponent gSS equation has been constructed from its Lax pair.By applying the Darboux transformation,we have obtained its various solutions,including a breather solution,kink solution,anti-soliton solution and periodic-like solution.We should stress that there exists a difference in the soliton solutions between the two-component Sasa-Satsuma equation (5) and our two-component gSS equation (6),e.g.an anti-soliton solution does not appear for equation (5).

Acknowledgments

The work of ZNZ is supported by the National Natural Science Foundation of China under Grant No.12071286,and by the Ministry of Economy and Competitiveness of Spain under contract PID2020-115273GB-I00(AEI/FEDER,EU).