Riemann-Hilbert approach and N-soliton formula for the N-component Fokas-Lenells equations
aa r X i v : . [ n li n . S I] M a y Riemann-Hilbert approach and N -soliton formula for the N -component Fokas-Lenells equations Wei-Kang Xun and Shou-Fu Tian ∗ School of Mathematics and Institute of Mathematical Physics, China University of Mining and Technology,Xuzhou 221116, People’s Republic of China
Abstract
In this work, the generalized N -component Fokas-Lenells(FL) equations, which havebeen studied by Guo and Ling (2012 J. Math. Phys. 53 (7) 073506) for N = N , which have more complex linear relationship than the analoguesreported before. We first analyze the spectral analysis of the Lax pair associated with a( N + × ( N +
1) matrix spectral problem for the N -component FL equations. Then, akind of RH problem is successfully formulated. By introducing the special conditionsof irregularity and reflectionless case, the N -soliton solution formula of the equationsare derived through solving the corresponding RH problem. Furthermore, take N = , Keywords: N -component Fokas-Lenells equations, Riemann-Hilbert approach,Multi-soliton solutions.
1. Introduction
As we all known, the nonlinear Schr¨odinger equation (NLS) is an important in-tegrable system, which plays an improtant role in nonlinear optics, water waves andplasma physics. Futher more, the Fokas-Lenells(FL) equation is colsly related to theNLS equation in the same way as the Camassa-Holm equation associated with theKdV equation. Since the FL system is one of the important models from both math-ematical and physical considerations, obtaining a series solutions of FL system has ✩ Project supported by the Fundamental Research Fund for the Central Universities under the grant No.2019ZDPY07. ∗ Corresponding author.
E-mail addresses : [email protected] and [email protected] (S. F. Tian)
Preprint submitted to Journal of L A TEX Templates May 8, 2020 een always a focusing subject for many scholars, and much of the research has beencarried out on the coupled FL system. The single-component FL equation was firstconstructured by Fokas[1]. After that, the bi-hamiltonian structure, the Lax pair andconservation laws were constructured by Fokas and Lenells[2]. Besides, many otherscholars have obtained a series of solutions of single-component FL equation, such asdark soliton[3], algebraic geometry solution[4] and long-time asymptotic behavior ofsolutions[5]. As for multi-component FL equations, Yang has constructured the gen-eralized Darboux transformation(DT) method for the generalized two-component FLequations to get the high-order rogue wave solutions[6]. Zhang et al have obtained thesoliton, breather and rogue waves solutions for a special two-component FL equationsvia DT method[7]. With the aid of Riemann-Hilbert(RH) approach, Kang et al havesolve the two-component FL equations to get multi-soliton solutions formula[8]. Hu etal have considered the initial boundary value problem for the two-component FL(FL)equations on the half-line via RH approach[9]. In addition, the multi-soliton solutionsof a m -component FL equations with vanishing boundary conditions and nonvanishingboundary conditions are also given by bilinear transformation method[10].Of particular concern in the field of nonlinear science is to find multi-soliton so-lutions for nonlinear partial di ff erential equations, and a number of e ff ective methodshave been produced to solve this problem, such as Hirota bilinear method[11], Dar-boux and B¨acklund transformation[12], inverse scattering transformation[13–15] andRH approach. Moreover, many scholars show an increasing interest in using the RHapproach as a powerful tool to solve certain important problem. For instance, apply-ing the RH approach can slove the soliton solutions of a series of nonlinear evolutionequations[16–30], study integrable systems with non-zero boundaries[31–35], disscussthe asymptoticity of integrable system solutions[36–39] and so on. The main purposeof our work is to use RH approach which is a powerful tool to solve the multi-solitonsolutions of a new class of multi-component FL equations.In this work, we mainly consider a generalized N -component FL equations u x , t − u + i ( u x u † Au + uu † Au x ) = , (1.1)where u ( x , t ) = ( u ( x , t ) , u ( x , t ) , . . . , u N ( x , t )) T is a N -component vector function andmatrix A is Hermitian. In [40], the N -component FL equations are given as a by-product of the coupled derivative Schr¨odinger equations(cDNLS). In addition, the N -component FL equations[40] is equivalent to the coupled FL system in [7] by a gaugetransformation. Since the authors only considered the multi-soliton solutions under thesimplest non-trivial case, i.e., u x , t − u + i ( u x u † Au + uu † Au x ) = , u ( x , t ) = ( u , u ) T , A = σ , σ = ± , (1.2)2e decided to generalize the author’s result to obtain the N -soliton solutions in morecomplex case. In our work, we consider the more complex case that the FL systemis a N -component equations and the relationship matrix A is promoted to the generalHermitian matrix, i.e., u x , t − u + i ( u x u † Au + uu † Au x ) = , u ( x , t ) = ( u , u , . . . , u N − , u N ) T , (1.3)and A = a , a ∗ , a ∗ , . . . a ∗ N , a , a , a ∗ , . . . a ∗ N , a , a , a , . . . a ∗ N , ... ... ... . .. ... a N , a N , a N , . . . a N , N , (1.4)with a i , i (1 ≤ i ≤ N +
1) are real constants and a i , j ( i , j ) are complex constants. Viacarrying out the RH approach, we successfully obtain the multi-soliton solutions of thenew generalized FL equations and get some certain interesting phenomena about thesoltions.The outline of this work is as follows. In Section 2, starting from analyzing thespectral problem of the Lax pair and analyticity of scattering matrix, a RH problemfor the N -component FL equations is formulated. Next, via solving the RH problem,we obtain the explicit multi-soliton solutions of the N -component FL equations. InSection 3, we consider the solutions under the special case that N and the elements of A are taken as fixed values. Moreover, the localized structures and dynamic propaga-tion behavior of these solutions are presented vividly by some graphics. Finally, theconclusions are given in the last section.
2. Riemann-Hilbert problem
We begin our discussion by considering the Lax pair representation of the N -component FL equations Φ x = U Φ , U = i ζ σ + i ζ U , x , Φ t = V Φ , V = − ζ i σ + ζσ U − i σ U , (2.1)where σ = − × N N × I N × N , σ = × N N × − I N × N , U = v T u N × N , (2.2)3ith u = ( u , u , . . . , u N ) T and v = ( v , v , . . . , v N ) T . If taking v = A T u ∗ , Eq. (1.1)can be derived by the compatibility condition of Eq. (2.1). The Lax pair (2.1) can berewritten as the equivalent form Φ x = i ζ σ Φ + P Φ , Φ t = − i ζ σ Φ + Q Φ , (2.3)where P = i ζ U , x , Q = ζσ U − i σ U . From Eq. (2.3), when | x | → ∞ , we have Φ ∝ exp( i ζ σ x − ζ i σ t ) . (2.4)Let J = Φ exp( − i ζ σ x + ζ i σ t ), then we obtain J x = i ζ [ σ , J ] + PJ , J t = − i ζ [ σ , J ] + QJ , (2.5)where [ σ , J ] = σ J − J σ is the commutator. Moreover, we can get the followingformula d ( e − i ( ζ − ζ ) ˆ σ J ) = e − i ( ζ − ζ ) ˆ σ ( Pdx + Qdt ) J . (2.6)Now, let us construct two matrix solutions of Eq. (2.5) J − = ([ J − ] , [ J − ] , . . . , [ J − ] N + ) , J + = ([ J + ] , [ J + ] , . . . , [ J + ] N + ) , (2.7)where each [ J ± ] l denotes the l -th column of J ± . In addition, J ± are determined by J − = I + Z x −∞ e i ζ σ ( x − ξ ) P ( ξ ) J − ( ξ, ζ ) e − i ζ σ ( x − ξ ) d ξ, J + = I − Z + ∞ x e i ζ σ ( x − ξ ) P ( ξ ) J + ( ξ, ζ ) e − i ζ σ ( x − ξ ) d ξ, (2.8)which satisfy the asymptotic conditions J − → I , ζ → −∞ , J + → I , ζ → + ∞ . (2.9)It is easy to find that [ J − ] , [ J + ] , [ J + ] , . . . , [ J + ] N + are analytic for ζ ∈ D + , and[ J + ] , [ J − ] , [ J − ] , . . . , [ J − ] N + are analytic for ζ ∈ D − , where D + = ( ζ | arg ζ ∈ (0 , π ∪ ( π, π ) , D − = ( ζ | arg ζ ∈ ( π , π ) ∪ ( 3 π , π ) ) . (2.10)4ext, we pay attention to the properties of J ± . According to the Abel’s identity and tr ( U ) =
0, we obtain that det( J ± ) are independent of x and det( J ± ) =
1. Moreover,since J − E and J + E are the matrix solutions of the spectral problem, where E = e i ζ σ x ,they are linearly related by a ( N + × ( N +
1) scattering matrix S ( ζ ) = ( s k j ) ( N + × ( N + ,namely J − E = J + ES ( ζ ) , ζ ∈ R ∪ i R , (2.11)where S ( ζ ) = s , . . . s , N s , N + ... . . . ... ... s N , . . . s N , N s N , N + s N + , . . . s N + , N s N + , N + (2.12)represents the inverse scattering matrix, ζ ∈ R ∪ i R . Futhermore, it is easy to knowdet( S ( ζ )) = N -component FL equationsneed two matrix functions: one is analytic in D + and the other is analytic in D − . Let Γ = Γ ( x , ζ ) be an alalytic function of ζ Γ ( x , ζ ) = ([ J − ] , [ J + ] , [ J + ] , . . . , [ J + ] N + )( x , ζ ) , (2.13)defining in D + , with the asymptotic behavior Γ → I , ζ ∈ D + → ∞ .To formulate a Riemann-Hilbert probelm for the N -component FL equations, wealso need to consider the inverse matrices of J ± . We write the matrices J − ± as a collec-tion of rows J − ± = [ J − ± ] [ J − ± ] ... [ J − ± ] N [ J − ± ] N + . (2.14)It can be seen that J − ± satisfy K x = i ζ [ σ , K ] − KP , (2.15)and meet the boundary condition J − ± → I , x → ±∞ . The matrix function Γ ( x , ζ )which is analytic in D − , can be defined as follows: Γ ( x , ζ ) = [ J − − ] [ J − + ] [ J − + ] ... [ J − + ] N + ( x , ζ ) . (2.16)5imilar to Γ ( x , ζ ), we obtain that Γ ( x , ζ ) → I as ζ ∈ D − → ∞ . Besides, we can getthe linear relationship e − i ζ σ x J − − = R ( ζ ) e − i ζ σ x J − + , ζ ∈ R ∪ i R , (2.17)where R ( ζ ) = r , . . . r , N r , N + ... . . . ... ... r N , . . . r N , N r N , N + r N + , . . . r N + , N r N + , N + (2.18)represents the inverse scattering matrix, ζ ∈ R ∪ i R . As for the analyticity of thescattering matrix and the inverse scattering matrix, we have the following theorem, Theorem 2.1.
The spectral function Γ ( x , ζ ) , the element s , and r k , j (2 ≤ k , j ≤ N + are analytic in D + ; The spectral function Γ ( x , ζ ) , the element r , and s k , j (2 ≤ k , j ≤ N + are analytic in D − ; In addition, s , k (2 ≤ k ≤ N + and s j , (2 ≤ j ≤ N + arenot analytic in D + or D − but continuous to the real axis and image axis, and r , k (2 ≤ k ≤ N + and r j , (2 ≤ j ≤ N + are not analytic in D + or D − and not continuous tothe real axis and image axis. Based on the above analysis, the Riemann-Hilbert problem of the N -component FLequations can be formulated. Theorem 2.2.
Let’s make the convention that the limit of Γ ( x , ζ ) when ζ ∈ D + ap-proaches R ∪ i R is Γ + , and the limit of Γ ( x , ζ ) when ζ ∈ D − approaches R ∪ i R is Γ − ,then the Riemann-Hilbert problem can be set up as follows: Γ ± are analytic in D ± , Γ − Γ + = G ( x , ζ ) , ζ ∈ R , Γ ± → I , ζ → ∞ , (2.19) where G = r , e − i ζ x r , e − i ζ x . . . r , N e − i ζ x r , N + e − i ζ x s , e i ζ x . . . s , e i ζ x . . . ... ... ... . . . ... ... s N , e i ζ x . . . s N + , e i ζ x . . . , and thecanonical normalization condition of the RH problem is Γ ( x , ζ ) → I , ζ ∈ D + → ∞ , Γ ( x , ζ ) → I , ζ ∈ D − → ∞ . (2.20)6o solve the RH problem, we suppose that it is irregular, which means that det( Γ )and det( Γ ) have certain zeros in the analytic domains, respectively. According to thedefinition of them, we can obtaindet( Γ ( x , ζ )) = s , ( ζ ) , ζ ∈ D + , det( Γ ( x , ζ )) = r , ( ζ ) , ζ ∈ D − . (2.21)Because of the above important results, it is necessary to consider the characteristics ofzeros by the symmetry of U ( x , ζ ), which is helpful for classfying the soliton solutionsof the N -component FL equations. At frist, we can see that the matrix U satisfy U † = − BU B − , (2.22)where B = − A , and the symbol † represents the Hermitian transporse of onematrix. According to Eqs. (2.5) and (2.22), we obtain J †± ( ζ ∗ ) = BJ − ± ( ζ ) B − , (2.23)futhermore, the scattering matrix S ( ζ ) meet the condition B − S † ( ζ ∗ ) B = S − ( ζ ) = R ( ζ ) , (2.24)moreover, Γ † ( ζ ∗ ) = B Γ ( ζ ) B − . (2.25)Besides, the matrix U meets the relation U = − σ U σ , based on which we canconclude that J ± ( − ζ ) = σ J ± ( ζ ) σ , (2.26)and Γ ( ζ ) = σ Γ ( − ζ ) σ . (2.27)At this point, we suppose that det( Γ ) has 2 N simple zeros { ζ j } (1 ≤ j ≤ N ) in D + whcih satisfy ζ N + l = − ζ l , 1 ≤ l ≤ N . At the same time, det( Γ ) has 2 N simplezeros { ˆ ζ j } (1 ≤ j ≤ N ) in D − , where ˆ ζ j = ζ ∗ j , 1 ≤ j ≤ N .In fact, the scattering data needed to solve the RH problem include the continu-ous scattering data { s , , s , , . . . , s N + , } as well as the discrete data { ζ j , ˆ ζ j , ϑ j , ˆ ϑ j } (1 ≤ j ≤ N ), where ϑ j and ˆ ϑ j are nonzero column vectors and row vectors, respectively,satisfying Γ ( ζ j ) ϑ j = , ˆ ϑ j Γ ( ˆ ζ j ) = . (2.28)7ccording to Eqs. (2.25) and (2.28), we can reveal the relationˆ ϑ j = ϑ † j B , ≤ j ≤ N . (2.29)Similarly, from Eqs. (2.27) and (2.28), we can obtain ϑ N + j = σ ϑ j , ≤ j ≤ N . (2.30)To obtain the explicit form of ϑ j (1 ≤ j ≤ N ), we take the derivatives of the firstexpression of Eq. (2.28) with respect to x and t and get ∂ϑ j ∂ x = i ζ j σ ϑ j ,∂ϑ j ∂ t = − i ζ j σ ϑ j , (2.31)then the ϑ j and ˆ ϑ j are determined by ϑ j = e i ( ζ j x − ζ j t ) σ ϑ j , , ≤ j ≤ N ,σ e i ( ζ j − N x − ζ j − N t ) σ ϑ j − N , , N + ≤ j ≤ N , (2.32)and ˆ ϑ j = ϑ † j , e [ i ( ζ j x − ζ j t )] ∗ σ B , ≤ j ≤ N ,ϑ † j − N , e [ i ( ζ j − N x − ζ j − N t )] ∗ σ σ B , N + ≤ j ≤ N , (2.33)where ϑ j , are the complex constant vectors.It is pointed out that the Riemann-Hilbert problem examined corresponds to thereflectionless case, namely, s , = s , = · · · = s N + , =
0. We introduce a 2 N × N matrix M with elements m k , j = ˆ ϑ k ϑ j ζ j − ˆ ζ k , ≤ k , j ≤ N , (2.34)and suppose that the inverse matrix M − exists, then the solutions of the RH problemcan be given by Γ ( x , ζ ) = I − N X k = N X j = ϑ k ˆ ϑ j ( M − ) k , j ζ − ˆ ζ j , Γ ( x , ζ ) = I + N X k = N X j = ϑ k ˆ ϑ j ( M − ) k , j ζ − ζ k . (2.35)Futhermore, we take the expansion for Γ ( x , ζ ) Γ ( x , ζ ) = I + Γ (1)1 ζ + Γ (2)1 ζ + Γ (3)1 ζ + O ζ ! . (2.36)8omparing Eq. (2.35) and Eq. (2.36), we obtain Γ (1)1 = − N X k = N X j = ϑ k ˆ ϑ j ( M − ) k , j , (2.37)substituting the above expression into Eq. (2.5), the following relationship can beobtained U = i σ [ σ , Γ (1)1 ] , (2.38)more explicitly, u ( x , t ) = − i ( Γ (1)1 ) , , u ( x , t ) = − i ( Γ (1)1 ) , ,. . . . . . , u N ( x , t ) = − i ( Γ (1)1 ) N + , , (2.39)where ( Γ (1)1 ) k , j denotes the ( k , j )-entry of matrix Γ (1)1 . Besides, from Eq. (2.37), thepotential functions u j ( x , t )(1 ≤ j ≤ N ) can be recovered as follows u ( x , t ) = i N X k = N X j = ϑ k ˆ ϑ j ( M − ) k , j , , u ( x , t ) = i N X k = N X j = ϑ k ˆ ϑ j ( M − ) k , j , ,. . . . . . , u N ( x , t ) = i N X k = N X j = ϑ k ˆ ϑ j ( M − ) k , j N + , , (2.40)where ( M − ) k , j denotes the ( k , j )-entry of inverse matrix of M .
3. Multi-soliton solutions
To obtain the explicit expression of the multi-soliton solutions, we should makemore e ff orts. At first, we set ϑ j , = ( µ j , , µ j , , . . . , µ j , N + ) T , θ j = i ( ζ j x − ζ j t ) and ζ j = a j + ib j . When 1 ≤ j ≤ N , ϑ j = e θ j σ ϑ j , = e − θ j . . . e θ j . . .
00 0 e θ j . . . ... ... ... . . . ... . . . e θ j µ j , µ j , ...µ j , N µ j , N + = µ j , e − θ j µ j , e θ j ...µ j , N e θ j µ j , N + e θ j , (3.1)9hen N + ≤ j ≤ N , ϑ j = σ e θ j − N σ ϑ j − N , = . . . − . . .
00 0 − . . . ... ... ... . . . ... . . . − µ j − N , e − θ j − N µ j − N , e θ j − N ...µ j − N , N e θ j − N µ j − N , N + e θ j − N = µ j − N , e − θ j − N − µ j − N , e θ j − N ... − µ j − N , N e θ j − N − µ j − N , N + e θ j − N . (3.2)According to Eqs .(3.1) , (3.2) and (2.33), we can obtain the explicit expressions of ϑ j and ˆ ϑ j (1 ≤ j ≤ N ) needed to solve the RH problem. Inserting these data into Eq.(2.40), we have the explicit expressions of multi-soliton solutions. To observe the propagation behavior of the solutions, we take N = A = − ,i.e. v = u ∗ and v = − u ∗ , Eq. (1.1) is reduced into the following form u , xt = u − i (2 | u | u , x − u ∗ u u , x − | u | u , x ) , u , xt = u − i ( − | u | u , x − u ∗ u u , x − | u | u , x ) , (3.3)when N =
1, we can express the solution to Eq. (2.40) explicitly u ( x , t ) = i ( − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , ) , u ( x , t ) = i ( − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , ) , (3.4)where m , = −| µ , | e − θ − θ ∗ + | µ , | e θ + θ ∗ − | µ , | e θ + θ ∗ ζ − ζ ∗ , m , = −| µ , | e − θ − θ ∗ − | µ , | e θ + θ ∗ + | µ , | e θ + θ ∗ ζ − ζ ∗ , m , = −| µ , | e − θ − θ ∗ − | µ , | e θ + θ ∗ + | µ , | e θ + θ ∗ ζ + ζ ∗ , m , = −| µ , | e − θ − θ ∗ + | µ , | e θ + θ ∗ − | µ , | e θ + θ ∗ ζ + ζ ∗ , (3.5)10
20 -15 -10 -5 0 5 10 15 20 x | u1 | t=-10t=0t=10 ( a ) ( b ) ( c ) -20 -15 -10 -5 0 5 10 15 20 x | u2 | t=-10t=0t=10 ( d ) ( e ) ( f ) Figure 1.
One-hump solutions to Eq. (3.4) with parameters ζ = . + . i , µ , = . µ , = .
02, and µ , = .
21 . (a)(b)(c) : the local structure, density and wave propagation of theone-hump solution | u ( x , t ) | , (d)(e)(f) : the local structure, density and wave propagation of theone-hump solution | u ( x , t ) | . -20 -15 -10 -5 0 5 10 15 20 x | u1 | t=-10t=0t=10 ( a ) ( b ) ( c ) -20 -15 -10 -5 0 5 10 15 20 x | u2 | t=-10t=0t=10 ( d ) ( e ) ( f ) Figure 2.
One-soliton solutions to Eq. (3.4) with parameters ζ = . + . i , µ , = . + . i , µ , = . + . i , and µ , = . + . i . (a)(b)(c) : the local structure, density and wavepropagation of the one-soliton solution | u ( x , t ) | , (d)(e)(f) : the local structure, density and wavepropagation of the one-soliton solution | u ( x , t ) | . N =
2, the solutions to Eq. (2.40) can be expressed explicitly by u ( x , t ) = i ( − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , ) , u ( x , t ) = i ( − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , ) , (3.6)where m k , j = ˆ ϑ k ϑ j ζ j − ˆ ζ k , ≤ k , j ≤ , (3.7)with ζ = − ζ , ζ = − ζ , ˆ ζ j = ζ ∗ j , ≤ j ≤ . (3.8)12
50 -40 -30 -20 -10 0 10 20 30 40 50 x | u1 | t=-10t=0t=10 ( a ) ( b ) ( c ) -50 -40 -30 -20 -10 0 10 20 30 40 50 x | u2 | t=-10t=0t=10 ( d ) ( e ) ( f ) Figure 3.
Two-soliton solutions to Eq. (3.6) with parameters ζ = . + . i , ζ = . + . i , µ , = . µ , = . µ , = . µ , = . µ , = .
2, and µ , = . (a)(b)(c) : the localstructure, density and wave propagation of the two-soliton solution | u ( x , t ) | , (d)(e)(f) : the localstructure, density and wave propagation of the two-soliton solution | u ( x , t ) | . The localized structures and dynamic propagation behavior of the two soliton so-lutions are displayed in Fig. 3. From Fig. 3, it can be seen that before two solitonscolliside each other, they spread forward in directions that cross each other. After theycolliside each other, the directions of two solitons are not exchanged, but the positionsof them has been shifted and the enenry of them has been swapped.13hen N =
3, we can rewrite Eq. (2.40) as the following form u ( x , t ) = i ( − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , ) , u ( x , t ) = i ( − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , ) , (3.9)where m k , j = ˆ ϑ k ϑ j ζ j − ˆ ζ k , ≤ k , j ≤ , (3.10)with ζ = − ζ , ζ = − ζ , ζ = − ζ , ˆ ζ j = ζ ∗ j , ≤ j ≤ . (3.11)14 a ) ( b ) ( c ) Figure 4.
Three-soliton solution to Eq. (3.9) with parameters ζ = . + . i , ζ = . + . i , ζ = . + . i , µ , = . µ , = µ , = . µ , = µ , = . µ , = . µ , = µ , = . µ , = . (a) : the local structures of the three soliton solutions u ( x , t ), (b) : the density plot of u ( x , t ), (c) : the wave propagation of the three soliton solutions u ( x , t ). If we take N = v = A T u ∗ , where A = a a ∗ a ∗ a a a ∗ a a a = , when N =
1, we can express the solution to Eq. (2.40) explicitly u ( x , t ) = i ( − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , ) , u ( x , t ) = i ( − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , ) , u ( x , t ) = i ( − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , ) , (3.12)where m = −| µ | e − θ − θ ∗ + e θ + θ ∗ ( | µ | + µ µ ∗ + µ µ ∗ + µ ∗ µ + | µ | + µ µ ∗ + | µ | ) ζ − ζ ∗ , m = −| µ | e − θ − θ ∗ − e θ + θ ∗ ( | µ | + µ µ ∗ + µ µ ∗ + µ ∗ µ + | µ | + µ µ ∗ + | µ | ) ζ − ζ ∗ , m = −| µ | e − θ − θ ∗ − e θ + θ ∗ ( | µ | + µ µ ∗ + µ µ ∗ + µ ∗ µ + | µ | + µ µ ∗ + | µ | ) ζ + ζ ∗ , m = −| µ | e − θ − θ ∗ + e θ + θ ∗ ( | µ | + µ µ ∗ + µ µ ∗ + µ ∗ µ + | µ | + µ µ ∗ + | µ | ) ζ + ζ ∗ . (3.13)15 a ) ( b ) ( c )( d ) ( e ) ( f )( g ) ( h ) ( i ) Figure 5.
One-hump solutions to Eq. (3.12) with parameters ζ = . + . i , ζ = − ζ , µ , = . + . i , µ , = . + . i , µ , = . + . i and µ , = . + . i . (a)(d)(g) : thestructures of | u ( x , t ) | , Re( u ) and Im( u ), (b)(e)(h) : the structures of | u ( x , t ) | , Re( u ) and Im( u ), (c)(f)(i) : the structures of | u ( x , t ) | , Re( u ) and Im( u ). When N =
2, we can express the solutions to Eq. (2.40) explicitly u ( x , t ) = i ( − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , ) , (3.14)16 u ( x , t ) = i ( − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , ) , u ( x , t ) = i ( − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , ) , (3.15)where m k , j = ˆ ϑ k ϑ j ζ j − ˆ ζ k , ≤ k , j ≤ , (3.16)with ζ = − ζ , ζ = − ζ , ˆ ζ j = ζ ∗ j , ≤ j ≤ . (3.17) -50 -40 -30 -20 -10 0 10 20 30 40 50 x | u1 | t=-10t=0t=10t=-5t=5 ( a ) ( b ) ( c )17
50 -40 -30 -20 -10 0 10 20 30 40 50 x -0.8-0.6-0.4-0.200.20.40.6 R e ( u1 ) t=-10t=0t=10t=-5t=5 ( d ) ( e ) ( f ) -50 -40 -30 -20 -10 0 10 20 30 40 50 x -0.8-0.6-0.4-0.200.20.40.60.8 I m ( u1 ) t=-10t=0t=10t=-5t=-7t=7 ( g ) ( h ) ( i ) Figure 6.
Two-soliton solutions to Eq. (3.14) with parameters ζ = + . i , ζ = . + . i , µ , = + i , µ , = + i , µ , = + i , µ , = + i , µ , = − i , µ , = − i , µ , = − i and µ , = + i . (a)(b)(c) : the local structure, density and wave propagation of | u ( x , t ) | , (d)(e)(f) :the local structure, density and wave propagation of Re( u ), (g)(h)(i) : the local structure, densityand wave propagation of Im( u ). The localized structures, density plot and the dynamic propagation behavior of thetwo soliton solutions are presented in Fig. 6. From Fig. 6, we can learn the propagationprocess and interaction mechanism of the two solitons. To be more concrete, the energyof two solitons changes significantly before and after collision, and the direction andthe position of them has also changed to some extent.18ext, if we take N =
3, the solutions to Eq. (2.40) can be expressed explicitly by u ν ( x , t ) = i ( − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , ) , (3.18)where ν = , , , m k , j = ˆ ϑ k ϑ j ζ j − ˆ ζ k , ≤ k , j ≤ , (3.19)with ζ = − ζ , ζ = − ζ , ζ = − ζ , ˆ ζ j = ζ ∗ j , ≤ j ≤ . (3.20)( a ) ( b ) ( c ) Figure 7.
Three-soliton solution to Eq. (3.27) with parameters ζ = . + . i , ζ = . + . i , ζ = . + . i , µ , = . µ , = µ , = µ , = . µ , = µ , = . µ , = µ , = µ , = . µ , = µ , = . µ , = µ , = . µ , = . µ , = . µ , = . (a) : the local structuresof the three soliton solutions u ( x , t ), (b) : the density plot of u ( x , t ), (c) : the wave propagationof the three soliton solutions u ( x , t ). .3. Case 3: multi-soliton solutions of four-component FL equations If we take N = v = A T u ∗ , where A = a a ∗ a ∗ a ∗ a a a ∗ a ∗ a a a a ∗ a a a a = − i − i − i + i − i − i + i + i − i + i + i + i , (3.21)when N =
1, we can express the solution to Eq. (2.40) explicitly u ( x , t ) = i ( − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , ) , u ( x , t ) = i ( − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , ) , u ( x , t ) = i ( − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , ) , u ( x , t ) = i ( − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , ) , (3.22) -50 -40 -30 -20 -10 0 10 20 30 40 50 x | u1 | t=-10t=0t=10t=-5t=5 ( a ) ( b ) ( c ) -50 -40 -30 -20 -10 0 10 20 30 40 50 x -0.8-0.6-0.4-0.200.20.40.60.8 R e ( u1 ) t=-10t=0t=10t=-5t=5 ( d ) ( e ) ( f )20
50 -40 -30 -20 -10 0 10 20 30 40 50 x -0.6-0.4-0.200.20.40.60.8 I m ( u1 ) t=-10t=0t=10t=-5t=-7t=7 ( g ) ( h ) ( i ) Figure 8.
One-hump solutions to Eq. (3.22) with parameters ζ = . + . i , ζ = − ζ , µ , = . + . i , µ , = . + . i , µ , = . + . i , µ , = . + . i and µ , = . + . i . (a)(b)(c) : thelocal structure, density and wave propagation of | u ( x , t ) | , (d)(e)(f) : the local structure, densityand wave propagation of Re( u ), (g)(h)(i) : the local structure, density and wave propagation ofIm( u ). -50 -40 -30 -20 -10 0 10 20 30 40 50 x | u1 | t=-10t=0t=10t=-5t=5 ( a ) ( b ) ( c ) -50 -40 -30 -20 -10 0 10 20 30 40 50 x -0.6-0.4-0.200.20.40.6 R e ( u1 ) t=-10t=0t=10t=-5t=5 ( d ) ( e ) ( f ) -50 -40 -30 -20 -10 0 10 20 30 40 50 x -0.8-0.6-0.4-0.200.20.40.6 I m ( u1 ) t=-10t=0t=10t=-5t=-7t=7 ( g ) ( h ) ( i ) Figure 9.
Two-soliton solutions to Eq. (3.24) with parameters ζ = . + . i , ζ = . + . i , µ , = . µ , = . µ , = . µ , = . µ , = . µ , = . µ , = . µ , = . µ , = . µ , = . (a)(b)(c) : the local structure, density and wave propagation of | u ( x , t ) | , (d)(e)(f) : the local structure, density and wave propagation of Re( u ), (g)(h)(i) : the local struc-ture, density and wave propagation of Im( u ). m , = −| µ , | e − θ − θ ∗ ζ − ζ ∗ + ( a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ ) µ , e θ ζ − ζ ∗ + ( a ∗ µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ ) µ , e θ ζ − ζ ∗ + ( a ∗ µ ∗ , e θ ∗ + a ∗ µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ ) µ , e θ ζ − ζ ∗ + ( a ∗ µ ∗ , e θ ∗ + a ∗ µ ∗ , e θ ∗ + a ∗ µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ ) µ , e θ ζ − ζ ∗ , m , = −| µ , | e − θ − θ ∗ ζ − ζ ∗ − ( a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ ) µ , e θ ζ − ζ ∗ − ( a ∗ µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ ) µ , e θ ζ − ζ ∗ − ( a ∗ µ ∗ , e θ ∗ + a ∗ µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ ) µ , e θ ζ − ζ ∗ − ( a ∗ µ ∗ , e θ ∗ + a ∗ µ ∗ , e θ ∗ + a ∗ µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ ) µ , e θ ζ − ζ ∗ , m , = −| µ , | e − θ − θ ∗ ζ + ζ ∗ − ( a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ ) µ , e θ ζ + ζ ∗ − ( a ∗ µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ ) µ , e θ ζ + ζ ∗ − ( a ∗ µ ∗ , e θ ∗ + a ∗ µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ ) µ , e θ ζ + ζ ∗ − ( a ∗ µ ∗ , e θ ∗ + a ∗ µ ∗ , e θ ∗ + a ∗ µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ ) µ , e θ ζ + ζ ∗ , m , = −| µ , | e − θ − θ ∗ ζ + ζ ∗ + ( a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ ) µ , e θ ζ + ζ ∗ + ( a ∗ µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ ) µ , e θ ζ + ζ ∗ + ( a ∗ µ ∗ , e θ ∗ + a ∗ µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ ) µ , e θ ζ + ζ ∗ + ( a ∗ µ ∗ , e θ ∗ + a ∗ µ ∗ , e θ ∗ + a ∗ µ ∗ , e θ ∗ + a µ ∗ , e θ ∗ ) µ , e θ ζ + ζ ∗ . (3.23)22hen N =
2, we can express the solutions to Eq. (2.40) explicitly u ( x , t ) = i ( − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , ) , u ( x , t ) = i ( − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , ) , u ( x , t ) = i ( − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , ) , u ( x , t ) = i ( − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , − µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , + µ , µ ∗ , e θ − θ ∗ ( M − ) , ) , (3.24)23here m k , j = ˆ ϑ k ϑ j ζ j − ˆ ζ k , ≤ k , j ≤ , (3.25)with ζ = − ζ , ζ = − ζ , ˆ ζ j = ζ ∗ j , ≤ j ≤ . (3.26)In Figs. 8 and 9, we display the localized structures, density plot and the wavepropagation of the one-soliton and two-soliton solutions. It is interesting that whateverthe solutions are single-soliton or two-soliton solutions, their real part and image partare all breather-like solitons.Simliar to Case 1 and Case 2, the three soliton solutions are given by u ν ( x , t ) = i ( − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , − µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , + µ ,ν + µ ∗ , e θ − θ ∗ ( M − ) , ) , (3.27)where ν = , , , , m k , j = ˆ ϑ k ϑ j ζ j − ˆ ζ k , ≤ k , j ≤ , (3.28)with ζ = − ζ , ζ = − ζ , ζ = − ζ , ˆ ζ j = ζ ∗ j , ≤ j ≤ . (3.29)24 a ) ( b ) ( c ) Figure 10.
Three-soliton solution to Eq. (3.27) with parameters ζ = . + . i , ζ = . + . i , ζ = . + . i , µ , = . µ , = µ , = µ , = . µ , = µ , = . µ , = µ , = µ , = . µ , = µ , = . µ , = µ , = . µ , = . µ , = . (a) : the local structures of the threesoliton solutions u ( x , t ), (b) : the density plot of u ( x , t ), (c) : the wave propagation of the threesoliton solutions u ( x , t ).
4. Conclusions
Based on the previous work [40], the main purpose of our work is to investigate ageneralized N -component FL equations via the Riemann-Hilbert approach, practicallyspeaking, which is to greatly promote the results of previous work. In this work, thespectral analysis of the associated Lax pair is first carried out and a Riemann-Hilbertproblem is established. After that, via solving the presented Riemann-Hilbert problemwith reflectionless case, the N -soliton solution to the N -component FL equations areobtained at last. Furthermore, by selecting specific values of the involved parameters,a few plots of one-, two- and three- soliton solutions are made to display the localizedstructures and dynamic propagation behaviors. Acknowledgements