THE ALGEBRAIC REPRESENTATION FOR HIGH ORDER SOLUTION OF SASA-SATSUMA EQUATION

. In this paper, we reestablish the elementary Darboux transformation for Sasa-Satsuma equation with the aid of loop group method. Furthermore, the generalized Darboux transformation is given with the limit technique. As direct applications, we give the single solitonic solutions for the focusing and defocusing case. The general high order solution formulas with the determinant form are obtained through generalized DT and the formal series method.

Introducing the variable transformations [28] u(X, T ) = q(x, t) exp i x + 2 t , T = 6 t, X = x + 3 2 t, where b is a real constant, I 3 is a 3 × 3 identity matrix and the overbar represents the complex conjugation (similarly hereafter). Which is nothing but the third flow of deep reduction for coupled nonlinear Schrödinger(CNLS) hierarchy. To derive the DT for system (5), we need to use the symmetry relation. The symmetry relation can be readily obtained as and where † represents the Hermite conjugation and Through Lax pair equation (5), we can see that (4) is the third flow for the CNLS hierarchy with deep reduction. The elementary DT for CNLS hierarchy can be constructed as the following: where |y 1 = Φ 1 (x, t; λ 1 ), y 1 | = |y 1 † , J = diag(1, σ, σ), Φ 1 (x, t; λ 1 ) is a special solution for Lax pair equations (5) at λ = λ 1 .  [1] 0 0 iq [1] 0 0   and the Bäcklund transformation between old potential function and new one is here |y 1,2 represents the second component of vector |y 1 , y 1,1 | represents the first component of vector y 1 |.
Proof. We merely need to verify that
Thus F (x, t; λ) is an holomorphic function in S 2 . By the asymptotical behavior for F (x, t; λ), we have F (x, t; λ) = 0. So the equation (10a) is proved.
Comparing the coefficient of λ, we can obtain that By the first equation of (13), we have where of f represents the (1,2), (1,3) and (2,1), (3,1) elements for the matrix. On the other hand, through the second equation of (13), we have where diag represents the other elements except the of f ones. So we have Indeed, the expression of C(t) is independent with Q[1]. Thus we take P 1 = 0, and Q[1] = Q. It follows that C(t) = 0. In a similar way, we can obtain V 0 = V 0 (Q [1]). This completes the proof.
It is obviously that the geometric multiplicity of DT can not more than [m], where m is the order of spectral problem. For instance, the order of spectral problem (5) is three, then the geometric multiplicity of DT (8) must be one. In the following, we illustrate the meaning for geometric multiplicity of DT. Suppose we have two different DT for CNLS hierarchy with rank[ker(I − π)] = 1 or 2. Firstly, if rank[ker(I − π)] = 1, then we have Since T 1 is gauge transformation, we can multiply λ − λ 1 with T 1 , and denote it as T 1 , i.e.
In what following, we consider the reduction for DT of CNLS hierarchy, such that Indeed, it is readily to verify that, if DT satisfies the relation then the new potential function keeps the symmetry relation (16). We look for the reduction condition with two different cases. For the first case, when λ 1 +λ 1 = 0. With this choice, we can find that the symmetry relation (17) can be reduced as i.e. |y 1 = (ϕ 1 , ψ 1 , φ 1 ) T and φ 1φ1 +φ 1 ϕ 1 = 0. For the second case: when λ 1 +λ 1 = 0. With this choice, we can not reduce the elementary DT to satisfy symmetry relation (16). To look for the reduction, we must iterate the DT. We find the twice iterated DT satisfies the relation (16), where Based on the above two different kinds of reduction, we have the following N -fold DT for the system (5). We conclude it as the following theorem: (5) at λ = λ k ∈ iR such that ψ kφk +φ k ϕ k = 0 (k = 1, 2, · · · , n), and m ≡ N − n different solutions Ψ l for system (5) at λ = µ l ∈ iR (l = 1, 2, · · · , m), then we have the following N -fold DT: where Y = |y 1 , |y 2 , · · · , |y n , |z 1 , K|z 1 , |z 2 , K|z 2 · · · , |z m , K|z m , and |y k = Φ k (k = 1, 2, · · · , n), |z j = Ψ j (j = 1, 2, · · · , m). And the Bäcklund transformation for SSE (4) can be represented as where Y i represents the i-th row of matrix Y .

Localized wave solution and periodical solution.
In this section, we use the DT to construct the solitonic solution and periodical solution on the nonvanishing background. We depart from the seed solution Since SSE (4) possesses the scaling symmetry, we can set a = 1 for any a = 0. Through the standard method to solve the linear differential equation [8], we can obtain the fundamental solution for Lax pair equation with q = q[0] and λ = λ 1 : where and χ i (i = 1, 2, 3) are three different roots for the following cubic equation if b = 0, then χ 3 = 0. Indeed, the classification of localized wave solution is determined by the cubic equation (24). So we give the classification of roots for equation (24). Set χ = iα, λ 1 = iβ, then equation (24) becomes Then the discriminant of equation (25) is If ∆ = 0, i.e. β 2 = 1 8b 2 −2b 4 + 10σb 2 + 1 + (1 − 4σb 2 ) 3/2 , = ±1, then equation (25) possesses multiple root. If ∆ > 0 and β ∈ R, then the equation (25) possesses three different real roots. If ∆ < 0 and β ∈ R, then equation (25) possesses a pair of conjugation complex roots and a real root. Otherwise, for general β equation (25) possesses three different complex roots.
possesses a pair of conjugation complex roots and a real root. Otherwise, equation (25) possesses three different complex roots.
it follows that equation (  On the other hand, we need to analyze the positivity condition for the Hermite quadric form y 1 |J|y 1 . For the focusing case, we can see that the quadric form y 1 |J|y 1 is positivity. We merely need to analyze the Hermite quadric form y 1 |J|y 1 for the defocusing case. We need to analyze the following Hermite quadric form Indeed, the C 1 matrix is nothing but Cauchy matrix c i,j = 1 xi+yj with x 1 = y 1 = iχ 1 , x 2 = y 2 = iχ 2 , x 3 = y 3 = iχ 3 . The determinant of Cauchy matrix C 1 can be represented as . So the sign of determinant det(C 1 ) is determined by On the other hand, matrix E 1 has two negative eigenvalues and a positive one. So is the matrix C 1 . To analyze the matrix C 1 , we rearrange x i with the order Re(x 1 ) ≥ Re(x 2 ) ≥ Re(x 3 ). Similar as Lemma 1 in ref. [18], we can conclude that Re(x 1 ) > 0 and Re(x 2 ), Re(x 3 ) < 0. Thus there exists an negative principal minor in matrix C 1 . We would like to use the negative principal minor matrix to construct the non-singular solution for defocusing SSE (4).
If b = 0, then 1986 LIMING LING Similar as above, we merely consider the defocusing case σ = −1. Suppose Im(χ 1 ) > Im(χ 2 ), then we can obtain the negative submatrix Finally, the roots of cubic equation (24) can be obtained by Cardano's formula. But the formula is rather complex. To avoid this problem, we introduce the following parameter transformation to solve the cubic equation (24) automatically: • When b = 0, introducing the parameter transformation one can verify that satisfy the cubic equation (24) automatically, where • When b = 0, setting satisfy the cubic equation (24) automatically.

3.1.
Single localized wave solution and periodical solution. We use two different DTs to construct the single solitonic and periodical solution respectively. Actually, the different types of solution are determined by the roots of characteristic equations (24). The single localized wave solution and periodical solution can be classified through the Table 1. According to the table, we can obtain a whole understanding about the exact solutions for the SSE on the NVBC. What should be pointed out that, most of the single localized wave solutions have been obtain in the previous literature. For the focusing case, the breather solution [1,30], rogue wave solution [4], rational W-shape soliton [34] and degenerate resonant soliton on NVBC [32] have been obtained in the previous literature. For the defocusing case, the dark soliton and W-shape dark soliton (dark double-hump soliton) have been been obtain through bilinear method and symbolic computation [12]. Here we give the explicit expression of solution by DT, since they possess the different representation form. It is not readily to see that they are equivalent with each other directly, but we believe that they are equivalent since they possess the same dynamics.  Table 1. The relation between type of solution and spectral parameter λ 1 : Parameteres β ± and β ± are given in proposition 1, λ ± are given in equation (34), λ 0 is given in equation (35).
In this case, one uses the first kind of DT to construct the solution. So one needs to verify the following condition then we can obtain the parameters restriction c 1 , c 2 , c 3 ∈ R. Based on above analysis, this kind of solution merely appears in the focusing case. We can obtain the general soliton solution on the NVBC: where Here we merely analyze the case b = 0, the case b = 0 was given in literature [32].
In what following, we analyze the properties for above solutions. If c 3 = 0, c 1 c 2 = 0, then we can obtain a soliton solution on the plane wave background After simple calculation, we can obtain that the peak of soliton |q[1]| 2 is along the line and the peak value is e iθ 1 +e iθ 2 +2γ1β1 2+4γ1 2 . When c 1 /c 2 > 0, we can obtain the bellshape soliton; while c 1 /c 2 < 0, we can obtain the W-shape soliton. Similarly, we can obtain the other cases c 2 = 0, c 1 c 3 = 0 and c 1 = 0, c 2 c 3 = 0. Finally, we discuss the case c 1 c 2 c 3 = 0. We can obtain the resonant soliton solution which is nothing but nonlinear superposition for above three solitons. Suppose Im(λ 1 ) > 0, and Im(χ 1 + χ 2 ) < Im( When t < α1−α3 v3−v1 , there are two solitons along the lines respectively. When above two solitons collision at ( approximately, they merge into a new soliton along the line respectively, where For instance, let κ 1 = exp ( π 4 i), b = 1 4 , by formula (29) and (30), it follows that Choosing parameters c 1 = −1, c 2 = c 3 = 1, we can obtain the two W-shape soliton and a bell-shape soliton (Fig. 1). By formula (33), we can obtain that there are two W-shape soliton with height 1.86 and 3.88 respectively, and a bell-shape soliton with height 1.24.

(b) Periodical solution and half periodical solution
For the second case, if the characteristic equation (24) merely possesses one purely image root, assume χ 1 = −χ 2 , then the restricted condition (31) gives the parameters restriction c 2 =c 1 and c 3 ∈ R. For the defocusing case, we can not obtain the where X 2,I is the image part of X 2 . The period of |q[1]| 2 in x direction is π χ 2,R ; and the period in t direction is π (b 2 − 4)χ 2,R − 4λ 1,I χ 2,R χ 2,I , where R and I represent the real part and image part respectively. If c 3 = 0, then we can obtain the semi-periodical solution. The expression for this solution is given in (32) with χ 1 = −χ 2 , c 2 =c 1 and c 3 ∈ R.
(c) Dark soliton If λ 1 = 0, there is a pair of complex conjugation root for cubic equation (24). The dark soliton solution merely exists in the defocusing case. Then we can construct the the following dark soliton solution through the limit technique [20]: And its hole is along the line x+(b 2 +4)t = 0. (

d) Breather solution
This kinds of solution can be derived by the second type DT with λ 1 + λ 1 = 0. We can obtain the following breather solution For the focusing case, there are two kinds of breather solution. The first kind of breather solution is given by choosing parameters c 1 c 2 = 0, c 3 = 0 or c 2 c 3 = 0, c 1 = 0 or c 1 c 3 = 0, c 2 = 0. This kind of breather solution has been obtain in [1]. The second kind of solution is given by choosing parameters c 1 c 2 c 3 = 0. This type of solution is the resonant breather.
For the defocusing case, we can obtain the breather solution by choosing parameters c 3 = 0, c 1 c 2 = 0, Im(χ 1 ) < 0 and Im(χ 2 ) < 0. We show the figure for the dark-breather solution ( Fig. 2(a)). (f ) W-shape Dark soliton Through the limit technique, we can obtain the W-shape dark soliton solution. This type of dark soliton solution merely exist in the defocusing case. The expression can be represented as c 1 > 0 and λ 1 ∈ R/0, χ 1 ∈ R. The parameters must be satisfied the following Choosing special parameters, we can show the figure of W-shape dark soliton ( Fig.  2(b)). It is seen that the dark soliton exhibits the "W" shape.

Multi-solitonic solution.
In this subsection, we consider the multi-solitonic solution. Firstly, we give the multi-solitonic solution formula. Suppose we have N different special solution Φ i for Lax pair (5), then |y i can be constructed as where c i,k are constants, χ i,k s are three different roots for cubic equation (24) at λ = λ i , i = 1, 2, · · · , N, k = 1, 2, 3. We discuss the solution formula with two different cases.
Then we have For example: Taking n = 1 and N = 2, we can obtain that one resonant soliton and resonant breather solution by choosing special parameters (Fig. 3(a)). Similarly, one can obtain the periodical solution and breather solution pair solution. (
Then the solitonic formula can be presented as the following: There are two types of breather-dark soliton solution. One type is breather and single dark soliton pair solution. Another type is breather and W-shape dark soliton pair solution (Fig. 3b).
In the following, we consider the expansion Furthermore we have k , X [2] k , · · · .
The explicit expression of these polynomials can be are given by the elementary Schur polynomials In what following, we give the following expansion: then we can obtain that where w [i,j] (ν, τ ) can be determined through the following way: It follows that the lemma is verified.
To obtain the general high order solution, we choose the special function k , d k = 0, 1. Finally, based on above lemmas, we can obtain the following expansions: k , Y [2] k , · · · . Similarly, we have Through the symmetry relation, we have On the other hand, we have the following expansion: Proposition 4. Suppose ψ 1,+ ϕ 1 + ψ 1,− ϕ 1 = 0, it follows that the high order solitonic solution can be represented as where .
Inserting the special parameters into above formula, we can obtain the second order solitonic solution. The dynamics of second order resonant soliton is shown in figure (4 a) with special parameters choosing. The dynamics of second order semi-periodical solution is shown in figure (4 b).
Proposition 5. The high order breather can be represented as Inserting the special parameters into above formula, we can obtain the second order breather solution. The dynamics for this kind of high order solution is similar as the high order breather solution of classical NLSE (Fig. 5 a).
we can obtain another kinds of high order solution. To analyze this problem conveniently, we set b = 1 Lemma 4.5. The asymptotical series and solve the cubic equation (24). The coefficients χ 2 can be determined recursively. If γ = 1, If γ = 1, then
Since D 1 E 1 ( 1 ) satisfies the Lax pair equations (5), then D 1 E 1 (− 1 ) also satisfies them. To obtain the general high order solution, we choose the general special solution where d 2 = 0, 1, and Finally, we have Similarly, we have Through the symmetry relation, we have On the other hand, we have the following expansion: .
Substituting the special parameters into above formula, we have the second order W-shape soliton. The dynamics is similar as derivative nonlinear Schödinger equation [10].
Through the symmetry relation, D 1 E 1 (ω ) and D 1 E 1 (ω 2 ) also satisfies them. Thus, we choose the special function 1 , Y [2] 1 , · · · . In what following, we consider the expansion for the following function: Similarly, we have the following expansion:   3(i−1) .

LIMING LING
It follows that the high order W-shape soliton can be represented as There are two kinds of solutions for the high order W-shape soliton-II solution. One kind is n = 1, which corresponds to high order W-shape soliton-II solution with the figure 7(a). Another kind is n = 2, which corresponds to high order W-shape soliton-II solution with the figure 7(b).

5.
Conclusions and discussions. It is well known that, the DT is a powerful method to construct the exact solutions for the integrable equations. To the best of my knowledge, all the elementary function solutions could be derived from this method. It is naturally that how to classify or understand these solutions. The different dynamics corresponds to the different physical process. So it is necessary to classify these solutions through dynamical behavior.
In this work, we classify the single solitonic solutions through a characteristic equation (24). It follows that we give the general multi-solitonic solution formulas. To the best of our knowledge, it should be pointed out that the breather solution, dark soliton and W-shape dark soliton for defocusing SSE (4) were first derived through the DT method in this work.
On the other hand, based on the generalized DT and formal series method, we present an algebraic representation for high order solution of SSE model in detail.
Even more important, this method could be extended to the other integrable multicomponent model.
Note added: When this paper was accepted to be published, I noticed that Mu and Qin presented the high order rogue waves for the focusing Sasa-Satsuma model by the generalized Darboux transformation (or dressing transformation) [23], which were overlapped with a part of subsection 4.2 with a different form.