THE SASA-SATSUMA EQUATION · v.2.1r20190102 *2019.3.10#45033f9e DARBOUX TRANSFORMATION AND...

v.2.1r20190102 *2019.3.10#45033f9e

DARBOUX TRANSFORMATION AND HIGHER-ORDER SOLUTIONS OFTHE SASA-SATSUMA EQUATION

LIJUAN GUO1, YI CHENG1, DUMITRU MIHALACHE2, JINGSONG HE3,∗

1School of Mathematical Sciences, University of Science and Technology of China,Hefei, Anhui, 230026, P. R. China

2Horia Hulubei National Institute for Physics and Nuclear Engineering,Reactorului 30, RO-077125, P.O.B. MG-6, Bucharest-Magurele, Romania

3Department of Mathematics, Ningbo University, Ningbo, Zhejiang 315211, P. R. ChinaCorresponding author∗: [email protected]; [email protected]

Received November 1, 2018

Abstract. We derive a general n-fold Darboux transformation (DT) matrix Tn forthe integrable Sasa-Satsuma equation. The elements of the matrix Tn are expressed incompact determinant forms. Through the detailed analysis of the linear instability phe-nomenon, different kinds of solitons on a continuous wave background are obtained byusing the DT, including W -shaped first-order and second-order rational solutions, co-existing W -shaped second-order rational solutions, first-order semi-rational solutions,first-order and second-order periodic solutions, first-order half-periodic solutions, andsecond-order breather-positon solutions.

Key words: Sasa-Satsuma equation, Darboux transformation, W -shapedrational solitons, Semi-rational solitons, Periodic solutions.

1. INTRODUCTION

The study of rational soliton solutions, especially, rogue (freak) waves, has at-tracted a lot of attention during the past years. The Peregrine soliton of the nonlinearSchrodinger equation (NLSE), which describes an extreme (“giant”) wave has beenstudied in many different physical systems [1–14]. The generic NLSE, which con-tains only the terms accounting for the group-velocity dispersion and the self-phasemodulation effect is a very good dynamical model to describe doubly localized highamplitude wave wells, with duration of the order of a few picoseconds. However, thestudy of propagation in optical fibers of ultrashort pulses with duration shorter than100 fs requests the use of generalized NLSEs such as the Sasa-Satsuma equation(SSE) [15–17]:

iqt− qxx−2|q|2q+ i[6(|q|2q)x−3(|q|2)xq+ qxxx] = 0. (1)

The SSE (1) is a natural extention of the NLSE. It contains the third-order dispersionand self-steepening terms and describes the propagation of subpicosecond pulses inoptical fibers.

Romanian Journal of Physics 64, 104 (2019)

Article no. 104 Lijuan Guo et al. 2

Because the SSE (1) is completely integrable, different categories of exact solu-tions have been obtained through distinct mathematical methods, such as the inversescattering method [18–20], the Darboux transformation (DT) method [21] and the bi-nary DT method [22], the dressing operator method [23–26], and the Hirota bilinearmethod [27–30]. In particular, Ref. [21] provides the one-fold DT method through adeep reduction of the Darboux transformation to a 3×3 spectral problem.

Recently, anti-dark solitons [31] and special types of rational solutions, such asW -shaped rational solutions [32–36] have been studied. In this paper, we investigatenew types of soliton solutions of the SSE (1) and illustrate their dynamical behavior.We mainly address the following issues:I) Can we find the determinant representation Tn of the n-fold DT for the SSE with-out the use of quasi-determinants [22]? This issue is a natural extension of the one-fold DT given in Ref. [21]. In recent years, the determinant representation Tn is aconvenient and effective tool to yield rogue (freak) waves (i.e., special rational solu-tions which are doubly localized in both time and space) and other types of solutionsin a plethora of soliton equations [37–47]; see also Refs. [48–62] for recent workson rogue waves and other types of localized structures in a series of relevant phy-sical settings. Moreover, the quasi-determinant of matrices is mainly used to studynoncommutative rings [63], hence it is not necessary to use this relatively new andcomplicated tool to investigate a commutative mathematical object such as the soli-ton solution of SSE. So it is an interesting problem to give a simple representation ofthe soliton solution without the use of the quasi-determinant.II) Except for W -shaped rational solutions and other solutions given in Refs. [32–34], are there other soliton solutions showing new dynamical behavior? In this paper,by using the DT, we obtain for the generic SSE different kinds of solitons on a con-tinuous wave background, such as W -shaped first-order and second-order rationalsoliton solutions, coexisting W -shaped second-order rational soliton solutions, first-order semi-rational soliton solutions, first-order and second-order periodic solutions,first-order semi-periodic solutions, and second-order breather-positon solutions.

This paper is organized as follows: in Sec. 2, we construct the n-fold DT forthe SSE, which is a simple and compact expression in the determinant form. In Sec.3, through the linear stability analysis of the SSE, different kinds of solitons on acontinuous wave background are obtained by using the DT and we investigate twocategories of W -shaped rational soliton solutions according to the choices of Eq.(19) and the root conditions of Eq. (21). Finally, a discussion of the obtained resultsand our conclusions are given in Sec. 4.

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3 Darboux transformation and higher-order solutions of the Sasa-Satsuma eq. Article no. 104

2. DARBOUX TRANSFORMATION

Equation (1) is the compatibility condition of the following Lax pair:

Ψx = UΨ, Ψt = VΨ , (2)

where

U = λU0 +U1, V = λ3U0 +λ2U1 +λU2 +U3, (3)

with

U0 =

−2i3 0 0

0 i3 0

0 0 i3

, U1 =

0 −e−iθq −eiθq∗eiθq∗ 0 0e−iθq 0 0

, (4)

where θ =−13x+ 2

27 t and

U2 =− 1

3U0 + [U1x,U0]−U1[U0,U1],

U3 =− 1

3U1 + [U1x,U1]−U1xx+ 2U3

1 .

(5)

Here, [x,y] = xy−yx and the symbol ∗ stands for complex conjugation.From the knowledge of the Lax pair, the Darboux transformation has been

proved to be an efficient tool to construct solitons, breathers, and rogue waves. Inorder to apply the DT method, it is crucial to find a gauge transformation T such that

Ψ[1]x =U [1]Ψ[1], U [1]T = Tx+TU,

Ψ[1]t =V [1]Ψ[1], V [1]T = Tt+TU.

(6)

By cross-differentiating Eqs. (6), we obtain

U[1]t −V [1]

x + [U [1],V [1]] = T (Ut−Vx+ [U,V ])T−1. (7)

This implies that, if the SSE is invariant under the gauge transformation T , then U [1]

and V [1] have the same form as U and V , and the seed solutions (q,q∗) are mappedto the new solutions (q[1], q[1]∗).

2.1. ONE-FOLD DT

We consider the one-fold Darboux transformation in the following form [39,40, 46]:

T1 = λE+A, (8)

where E is an identity matrix and A = (aij)3×3 is a undetermined function matrixwhose entries can be expressed in terms of members in the kernel of the matrix T .

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Substituting Eq. (8) into the first of equations (6) and comparing the coefficients ofλ

U[1]1 −U1 = [A,U0], (9)

one obtains the relation between the new solution and the “seed” one:

q[1] = q− ieiθa12. (10)

We assume that Ψ1 = (f1(λ,x,t),g1(λ,x,t),h1(λ,x,t))T , (f1,g1,h1)

T (here Trepresents the transpose) is a solution of the Lax pair Eq. (2) corresponding to thespectral parameter λ1, and we choose three members of T ’s kernel as following:

λ1↔Ψ1 =

f1g1h1

,λ2 = λ∗1↔Ψ2 =

f2g2h2

=

−g∗1f∗10

,λ3 = λ∗1↔Ψ3 =

f3g3h3

=

−h∗10f∗1

.(11)

Hence, by solving the algebraic equations T1(λ)|λ=λkΨk = 0,(k = 1,2,3), the en-tries in the matrix T1 are obtained by Cramer’s rule. Finally we get,

T1 =

(T1)11 (T1)12 (T1)13(T1)21 (T1)22 (T1)23(T1)31 (T1)32 (T1)33

|D1|

(12)

where

(T1)11 =

∣∣∣∣1 0 0 −λD1 η1

∣∣∣∣ ,(T1)12 =

∣∣∣∣0 1 0 0D1 η1

∣∣∣∣ ,(T1)13 =

∣∣∣∣0 0 1 0D1 η1

∣∣∣∣ ,(T1)21 =

∣∣∣∣1 0 0 0D1 η2

∣∣∣∣ ,(T1)22 =

∣∣∣∣0 1 0 −λD1 η2

∣∣∣∣ ,(T1)23 =

∣∣∣∣0 0 1 0D1 η2

∣∣∣∣ ,(T1)31 =

∣∣∣∣1 0 0 0D1 η3

∣∣∣∣ ,(T1)32 =

∣∣∣∣0 1 0 0D1 η3

∣∣∣∣ ,(T1)33 =

∣∣∣∣0 0 1 −λD1 η3

∣∣∣∣ ,and

D1 =

f1 g1 h1f2 g2 h2f3 g3 h3

, η1 =

−λ1f1−λ2f2−λ3f3

, η2 =

−λ1g1−λ2g2−λ3g3

, η3 =

−λ1h1−λ2h2−λ3h3

Hence, the explicit form of q[1] is given by

q[1] = q+ i(λ1−λ∗1)g∗1f1

|f1|2 + |g1|2 + |h1|2exp(iθ) (13)

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2.2. n-FOLD DT

According to the form of the one-fold Darboux matrix, we set the n-fold Dar-boux matrix Tn = Tn(λ) = λnE+

∑n−1k=0 tkλ

k, where tk are undetermined matricesdepending on the members of Tn’s kernel. Considering the n-fold transformationΨ

[n]x = U [n]Ψ,U [n]Tn = Tnx+TnU , and comparing the coefficient of λn−1, we ob-

tain the relation between the n-th order new solution and the “seed” solution:

q[n] = q− i(tn−1)12 exp(iθ). (14)

Similar to the one-fold DT, we choose the kernel of Tn as the following reductioncondition:

λk↔Ψk =

fkgkhk

,λk+1 = λ∗k↔Ψk+1 =

fk+1

gk+1

hk+1

=

−g∗kf∗k0

,

λk+2 = λ∗k↔Ψk+2 =

fk+2

gk+2

hk+2

=

−h∗k0f∗k

, (15)

with k = 3l− 2, l = 1,2, · · · ,n. Solving the algebraic equations Tn(λ)|λ=λjΨj =0(j = 1,2, · · · ,3n), we can obtain the determinant representation of Tn.Theorem 1 The n-fold DT of the SSE can be expressed as

Tn =

(Tn)11 (Tn)12 (Tn)13(Tn)21 (Tn)22 (Tn)23(Tn)31 (Tn)32 (Tn)33

|Dn|

(16)

where

(Tn)11 =

∣∣∣∣λn−1 0 0 λn−2 0 0 · · · 1 0 0 −λnDn χ1

∣∣∣∣ ,(Tn)12 =

∣∣∣∣0 λn−1 0 0 λn−2 0 · · · 0 1 0 0Dn χ1

∣∣∣∣ ,(Tn)13 =

∣∣∣∣0 0 λn−1 0 0 λn−2 · · · 0 0 1 0Dn χ1

∣∣∣∣ ,(Tn)21 =

∣∣∣∣λn−1 0 0 λn−2 0 0 · · · 1 0 0 0Dn χ2

∣∣∣∣ ,(Tn)22 =

∣∣∣∣0 λn−1 0 0 λn−2 0 · · · 0 1 0 −λnDn χ2

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(Tn)23 =

∣∣∣∣0 0 λn−1 0 0 λn−2 · · · 0 0 1 0Dn χ2

∣∣∣∣ ,(Tn)31 =

∣∣∣∣λn−1 0 0 λn−2 0 0 · · · 1 0 0 0Dn χ3

∣∣∣∣ ,(Tn)32 =

∣∣∣∣0 λn−1 0 0 λn−2 0 · · · 0 1 0 0Dn χ3

∣∣∣∣ ,(Tn)33 =

∣∣∣∣0 0 λn−1 0 0 λn−2 · · · 0 0 1 −λnDn χ3

∣∣∣∣ .and

Dn =

λn−11 f1 λn−11 g1 λn−11 h1 · · · f1 g1 h1λn−12 f2 λn−12 g2 λn−12 h2 · · · f2 g2 h2λn−13 f3 λn−13 g3 λn−13 h3 · · · f3 g3 h3

......

.... . .

......

...λn−13n−2f3n−2 λn−13n−2g3n−2 λn−13n−2h3n−2 · · · f3n−2 g3n−2 h3n−2λn−13n−1f3n−1 λn−13n−1g3n−1 λn−13n−1h3n−1 · · · f3n−1 g3n−1 h3n−1λn−13n f3n λn−13n g3n λn−13n h3n · · · f3n g3n h3n

χ1 = (−λn1f1,−λn2f2,−λn3f3, · · · ,λn3n−2f3n−2,λn3n−1f3n−1,λn3nf3n)T ,

χ2 = (−λn1g1,−λn2g2,−λn3g3, · · · ,λn3n−2g3n−2,λn3n−1g3n−1,λn3ng3n)T ,

χ3 = (−λn1h1,−λn2h2,−λn3h3, · · · ,λn3n−2h3n−2,λn3n−1h3n−1,λn3nh3n)T .

,

Considering the reduction relation given by Eq. (15), we derive the relation betweenthe new solution q[n] and the ”seed“ solution q.Corollary 1 The n-th order solution q[n] is derived compactly by

q[n] = q− iFGeiθ, (17)

where G is the determinant:∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

λn−11 f1 λn−11 g1 λn−11 h1 · · · f1 g1 h1−(λn−11 g1)

∗ (λn−11 f1)∗ 0 · · · −g∗1 f∗1 0

−(λn−11 h1)∗ 0 (λn−11 f1)

∗ · · · −h∗1 0 f∗1...

......

. . ....

......

λn−13n−2f3n−2 λn−13n−2g3n−2 λn−13n−2h3n−2 · · · f3n−2 g3n−2 h3n−2−(λn−13n−2g3n−2)

∗ (λn−13n−2f3n−2)∗ 0 · · · −g∗3n−2 f∗3n−2 0

−(λn−13n−2h3n−2)∗ 0 (λn−13n−2f3n−2)

∗ · · · −h∗3n−2 0 f∗3n−2

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and F is constructed fromG through the replacing of the second column by a specialvector η:

η = (−λn+11 f1,λ

n+11 g∗1,λ

n+11 h∗1, · · · ,−λn+1

3n−2f3n−2,λn+13n−2g

∗3n−2,λ

n+13n−2h

∗3n−2)

T .

Note that G is obtained from |Dn| by applying the reduction condition givenby Eq. (15).

3. TWO KINDS OF W -SHAPED RATIONAL SOLUTIONS WITH NON-VANISHINGBOUNDARY CONDITIONS

In this Section, we start to consider different kinds of solutions with non-vanishing boundary conditions. The modulation instability (MI) phenomenon asa rogue wave generation mechanism has resulted in the obtaining of rogue wavesolutions for SSE [23, 25, 26], and the modulation stability (MS) on the continu-ous wave (cw) background has resulted in the emerging of W-shaped solitons forSSE [32–34]. However, other types of soliton solutions such as semi-periodic solu-tions, coexistingW -shaped second-order rational soliton solutions, and second-orderbreather-positon solutions have not been studied before, to the best of our knowl-edge. Here, in order to obtain such new types of soliton solutions, we first reviewthe linear stability analysis of the cw for SSE. Let us start with the cw backgroundq = cexp[i(ax+ bt)], b = a3− 6ac2 +a2− 2c2. A perturbed nonlinear backgroundcan be written as q = [c+ p(x,t)ε] exp[i(ax+ bt)], where ε is a small parameterand p satisfies a linear equation. Whenever p is x-periodic with frequency Q, i.e.,p = r(t)eiQx + s(t)e−iQx, the linear equation reduces to a 2× 2 linear differentialequation η′=iMη, where η = (r,s∗)T (here the prime denotes differentiation withrespect to t) and

M =

(M11 M12

M21 M22

).

Here

M11 =Q3 + 3Q2a+ 3Qa2−9Qc2−6ac2 +Q2 + 2Qa−2c2,

M12 =−3Qc2−6ac2−2c2,

M21 =−3Qc2 + 6ac2 + 2c2,

M22 =Q3−3Q2a+ 3Qa2−9Qc2 + 6ac2−Q2 + 2Qa+ 2c2.

The linear dispersion relation between Q and the eigenvalues m1,2 of the matrix Mis

m1,2 =(Q2 + 3a2−9c2 + 2a±

√(9a2 + 6a+ 1)Q2− c2M1

)Q (18)

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withM1 = (6a+ 2 + 3c)(6a+ 2−3c) (19)

From Eqs. (18), when M1 > 0, MI occurs. However, in the present work, we focusour attention on the MS condition, i.e., M1 ≤ 0.

Substituting the cw “seed” solution q = cexp{i[ax+(a3−6ac2 +a2−2c2)t]}into the Lax pair (2), we obtain the following forms of the three fundamental Blocheigenfunctions

ψk =R

13c

−i(λ+3a+1)+3ξk3c

i(−λ+3a+1)+3ξk

exp(ωk),k = 1,2,3, (20)

where ξk are the three roots of the cubic equation:

ξ3 + (1

9+λ2

3+a2 +

2a

3+ 2c2)ξ+

2i

27(9a2−9c2−λ2 + 6a+ 1)λ= 0, (21)

andR=diag{1,exp[i(θ−ax− bt)],exp[−i(θ−ax− bt)]},

ωk =ξkx+ [iλξ2k +1

9(9a2−36c2 + 6λ2 + 6a−2)ξk

+2iλ3

9+

2i

27(9a2 + 18c2 + 6a+ 1)λ]t

(22)

Because of linearity of the Lax pair given by Eq. (2), we combine the three funda-mental functions as

Ψ =K1ψ1 +K2ψ2 +K3ψ3 , (23)whereKj(j = 1,2,3) are arbitrary complex constants. Ψ is an eigenfunction solutionof the Lax pair (2) corresponding to the spectral parameter λ.

Remark According to Eq. (9), there is another solution, namely, q[1] = q−ia31e

iθ = q+ i(λ1−λ∗1)hf∗

|f1|2+|g1|2+|h1|2 , which implies that g∗1f1 = hf∗1 . In particular, wealways guarantee f1 =−f∗1 ,h1 =−g∗1 , i.e., λ1 is pure imaginary and ξk are real rootsin Eq. (20) by choosing some specific spectral parameters. Furthermore, throughsimply analyzing the root conditions of Eq. (21), we find that when M1 = 0, thatequation has a triple real root whereas it has two roots when M1 < 0.

3.1. THE CASE WHEN M1 = 0

In the Subsection, we consider the case when M1 = 0. Thus Eq. (21) has atriple real root, i.e., λ1 = λ2 = λ3 = λ0. In order to obtain the higher-order rationalsolutions, we need to modify the n-fold DT for the n-th order solution q[n] in Eq.(17) as in the following Theorem.

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Theorem 2 Let Ψ(λ0) = (f(λ0),g(λ0),h(λ0))T given by Eq. (23) be the eigenfunc-

tion solution of the Lax pair (2) with the spectral parameter λ0, then the n-th orderW -shaped rational solutions of SSE is given by

q[n]r = q− iF′

G′eiθ. (24)

Here q is a “seed” solution of SSE,

F ′ = (∂3(ni−1)

∂ε3(ni−1)|ε=0(F )ij(λ0 + ε3))3n×3n,

G′ = (∂3(ni−1)

∂ε3(ni−1)|ε=0(G)ij(λ0 + ε3))3n×3n.

and ni = [ i+23 ], where [i] denotes the floor function of integer i.

For simplicity, we set a= 16 , c= 1 in this Subsection. We choose the following

combination forms ofF1 =ψ1 +ψ2 +ψ3,

F2 =1

ε(ψ1 +wψ2 +w2ψ3),

F3 =1

ε2(ψ1 +w2ψ2 +wψ3),

(25)

withw=−12 +

√3i2 . By further combination of the above functions with ε-dependent

coefficients, the eigenfunction solutions Ψk of the Lax pair (2) associated with λk are

Ψk = (mF1 +nF2 +sF3)|λ=λk ,k = 1,4,7, · · · ,3n−2, (26)with

m=m1 +m2ε3 +m3ε

6 + · · ·+mjε3(j−1), n= n1 +n2ε

3 +n3ε6 + · · ·+njε

3(j−1),

s=s1 +s2ε3 +s3ε

6 + · · ·+sjε3(j−1),

(27)where mj ,nj , and sj are complex parameters.

3.1.1. The first-order solutions

When λ0 = 3√3i

2 , according to different choices of the parameters m1,n1,s1,we classify two categories of the first-order solutions involving a second-order poly-nomial and a fourth-order one, respectively.

Case 1. When m1 = 0,n1 = 1,s1 = 0, the solution involving a second-orderpolynomial is a W -shaped rational soliton as follows

q[1]r =1

2

F1

G1

e36x−641t

216i (28)

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with

F1 =10609 i√

3t2−2472 i√

3tx+ 144 i√

3x2 + 3708 it−432 ix+ 412√

3t

−48√

3x−10609 t2 + 2472 tx−144x2 + 192,

G1 =824√

3t−96√

3x+ 10609 t2−2472 tx+ 144x2 + 96.

(29)

When 103t−12x=−4√

3, the soliton |q[1]| has a maximum value 2. When 103t−12x = ±12−4

√3, the soliton |q[1]| has a minimum value

√104 . And |q[1]| goes to a

constant background 1 as x→∞, t→∞. Its dynamical evolution is shown in Fig.1(a).

The trajectory of the peak’s center of |q[1]|, i.e. the dependence of the peak’scoordinate, Tp, on t is Tp = 103t

12 +√33 , while the coordinates Tv of the centers of the

two soliton’s valleys are given by Tv = 103t12 +

√33 ±1. So we define the soliton speed

vspeed = 10312 , and the soliton width, which is defined as the distance between the two

valleys, is given by twidth = 24√10753

10753 . The plots of the soliton’s trajectory is shownin Fig. 1(b).

(a) |q[1]r | (b)

Fig. 1 – (Color online) (a) The W -shaped first-order rational soliton solution given by Eq. (28) in Case1. (b) The trajectories of the peak’s and the valleys’ centers.

Case 2. When s1 6= 0, the soliton solution involves a fourth-order polynomial.There are two different dynamical behaviours for distinct choices of the parametersm1 and n1. (I) We set n1 = 0 and s1 = 1, and if we fix the spatial coordinate x= 0,then when m1 goes from −∞ to +∞, the two W -shaped rational soliton solutionsgradually fuse to a single anti-dark soliton illustrated by the blue dashed lines in Fig.2(a). (II) We set m1 = 0 and s1 = 1, and we fix x=−20. We see that when n1 goesfrom −∞ to +∞, the two W -shaped rational solutions, illustrated by the red-dottedlines, fuse to a single anti-dark soliton and finally, two W -shaped rational solutionsemerge, which are illustrated by the blue-dashed lines in Fig. 2(b).

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(a) |q[1]r | (b) |q[1]r |

Fig. 2 – (Color online) (a) Fixing x = 0, the dynamical behaviour of q[1]r with parameters n1 = 0 ands1 = 10 is shown for m1 equal to -166 (red, dot), 0 (green, solid), and 166 (blue, dash). (b) Fixingx=−20, the dynamical behaviour of q[1]r with parameters m1 = 0 and s1 = 10 is shown for n1 equalto -126 (red, dot), 0 (green, solid), and 126 (blue, dash), as described in Case 2.

3.1.2. The second-order solutions

(a) |q[2]r | (b) (c)

Fig. 3 – (Color online) (a) Dynamical evolution of the W -shaped second-order rational solutions inCase 3, with parameters m1 = 0,n1 = 1,s1 = 0,m2 = 0,n2 = 0,s2 = 0. (b) The profile cut plot atspatial coordinate x=−4. (c) The profile cut plot at spatial coordinate x= 7.

Because of the awkward expressions of the second-order solution q[2]r in Eq.(24), we only use a graphical method to illustrate their dynamical behaviours.

Case 3. We choose n1 = 1 and we consider that the other parameters are allzero. This solution displays a complex dynamical behavior, as shown in Fig. 3(a),where we see two W -shaped solitons in the negative half plane, while there existthree W -shaped solitons in the positive half plane. In addition, their profiles can beclearly seen by making cut plots at a certain spatial location x. The soliton profilesat x = −4 (green solid line) and at x = 7 (red dashed line) clearly show this rathercomplex dynamics, see Figs. 3(b-c).

Case 4. When we consider the parameter s1 = 20 and other parameters areequal to zero, we see a more complex dynamical behavior in Fig. 4(a), where there

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(a) |q[2]r | (b) (c)

Fig. 4 – (Color online) (a) The evolution plots of theW -shaped second-order rational solutions in Case4 with parameters m1 = 0,n1 = 0,s1 = 20,m2 = 0,n2 = 0,s2 = 0. (b) The soliton profile at spatialcoordinate x=−4. (c) The soliton profile at spatial coordinate x= 5.

are three W -shaped solitons in the negative half plane, while there exist four W -shaped solitons in the positive half plane. In addition, their profiles can be clearlyobserved by plotting them at a certain spatial location x. The cut plots at x = −4(green solid line) and at x= 5 (red dashed line) clearly show this complex dynamicalbehavior, see Figs. 4(b-c).

3.2. THE CASE WHEN M1 < 0

In this subsection, we consider the case when M1 < 0. Equation (21) hastwo roots, i.e., λ1 = λ0a and λ2 = λ0b. In order to obtain the higher-order rationalsolutions of SSE, we need to modify the n-fold DT for the n-th order solution q[n] inEq. (17) as in the following Theorem.Theorem 3 Let Φ(λ0a(λ0b)) = (f(λ0a(λ0b)),g(λ0a(λ0b)),h(λ0a(λ0b))

T given byEq. (23) with K3 = 0 be two distinct eigenfunction solutions of the Lax pair (2) withthe spectral parameter λ0a(λ0b), and corresponding Taylor expansion to ml order(ml = 0,2, · · · ; l = 1,2), which satisfies the constraint condition n= 2+

∑l=1,2

ml2 ,

then the n-th order rational solution of SSE is given by

q[n]r = q− iF′

G′eiθ. (30)

Here q is a “seed” solution of SSE,{F ′ = ( ∂

3(ni−1)

∂ε3(ni−1) |ε=0(F )ij(λ0 + ε2))3n×3n,

G′ = ( ∂3(ni−1)

∂ε3(ni−1) |ε=0(G)ij(λ0 + ε2))3n×3n, i= 1,2, · · · ,3( m12 + 1),

F ′ = ( ∂3(ni−(n− m2

2 ))

∂ε3(ni−(n− m22 ))|ε=0(F )ij(λ0 + ε2))3n×3n,

G′ = ( ∂3(ni−(n− m2

2 ))

∂ε3(ni−(n− m22 ))|ε=0(G)ij(λ0 + ε2))3n×3n, i= 3(n− m2

2 ) + 1, · · · ,3n,

and ni = [ i+23 ], where [i] denotes the floor function of integer i.

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If the eigenfunctions given by Eq. (23) have an exponent function associated withthe third root of Eq. (21), the constraint K3 = 0 in Theorem 3 is removed, then Eq.(30) in Theorem 3 can produce semi-rational solutions of SSE.Corollary 2 IfK3 6= 0 in Theorem 3, then Eq. (30) yields the semi-rational solutionsq[n]1semir of SSE.

In what follows, for convenience we set a= 13 , c= 5

3 , λ0a = 2√

6i, and λ0b = 92 i

to discuss different kinds of solutions according to Theorem 3 and Corollary 2.

3.2.1. W -shaped rational solution

We set K3 = 0 in Eq. (23), and we choose different parameters K1 and K2.We obtain the following two functions:

G1 = ψ1 +ψ2, G2 =1

ε(ψ1−ψ2). (31)

By further combination of the above functions with ε-dependent coefficients, theeigenfunction solutions Ψk of the Lax pair (2) and the associated λk are

Ψk =mG1 +nG2|λ=λk ,k = 1,4,7, · · · ,3n−2, (32)

with

m=m1 +m2ε2 +m3ε

4 + · · ·+mjε2(j−1), n= n1 +n2ε

2 +n3ε4 + · · ·+njε

2(j−1),(33)

where mj and nj are complex parameters. Thus by setting distinct eigenfunctionsof the Lax pair (2) with distinct parameters λ0, we can obtain higher-order rationalsolutions.

A. The first-order rational solutionsCase 5. When m1 = 0,n1 = 1,n = 1,λ = λ0a, Eq. (30) gives a W -shaped first-order rational soliton q

[1]r whose dynamical evolution plot is shown in Fig. 5(a).

Through simple calculations, we get the maximum value of the wave |q[1]r |max = 73 ,

and the minimum value |q[1]r |min =√223 . Similar to Case 1, we plot the intensity

of the rational soliton |q[1]r | and we get the trajectory of the peak’s center, i.e., thedependence of Tp on t as Tp = 19t+

√64 , while the coordinates Tv of the centers of

the two valleys are Tv = 19t+√64 ±

3√2

4 . So we get the soliton’s speed: vspeed = 19and the soliton’s width that is defined as the distance between the corresponding twosoliton’s valleys: twidth = 3

√181

362 .Case 6. When m1 = 0,n1 = 1,n = 1,λ = λ0b, we obtain another type of first-orderW -shaped rational soliton q[1]r whose dynamical evolution plot is shown in Fig. 5(b).The maximum value of the wave intensity is |q[1]r |max = 13

3 , whereas the minimum

value is |q[1]r |min =√73 . As in the Case 5, the trajectory of the center of the intensity

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peak, i.e., the dependence of Tp on t is given by Tp = 29t+ 14 , while the coordinates

Tv of the centers of two valleys of the soliton are given by Tv = 29t+ 14±

√34 . So we

get the soliton speed vspeed = 29, and the soliton width that is defined as the distancebetween the two soliton valleys is given by twidth =

√25261684 . Comparing the Case

5 with the Case 6, we see from Figs. 5(a, b) that these two first-order W -shapedrational solitons are indeed different.

B. The second-order rational solutionsCase 7. If we choose in Theorem 3 an eigenfunction Φ(λ0a) and its correspond-ing derivative order m1 = 2, we can then obtain a W -shaped second-order ratio-nal soliton with parameters m1 = 0,n1 = 1,m2 = 0,n2 = 0 by means of a two-fold DT. Its dynamical evolution is shown in Fig. 6(a). However, if we choosethe eigenfunction Φ(λ0b) and its corresponding derivative order m2 = 2, then wealso get another W -shaped second-order rational soliton with the same parametersm1 = 0,n1 = 1,m2 = 0,n2 = 0 in Eq. (30); Fig. 6(b) displays its dynamical beha-vior. Furthermore, we see that the two soliton profiles are so different by plottingthem at x= 0, as clearly seen in Fig. 6(c) and Fig. 6(d).Case 8. In Theorem 3, choosing two distinct eigenfunctions Φ(λ0a) with parametersm1 = 0,n1 = 1 and Φ(λ0b) with parameters m1 = 0,n1 = 1, and the correspondingderivative orders m1 = 0, m2 = 0, we can obtain a novel type of rational solution.Its dynamical evolution plotted in Fig. 6(e) clearly shows the coexistence of the twodistinct W -shaped second-order rational solitons, plotted in Fig. 6(c) and Fig. 6(d).

(a) |q[1]r | (b) |q[1]r |

Fig. 5 – (Color online) Dynamical evolution of different first-orderW -shaped rational soliton solutions:(a) m1 = 0, n1 = 1, n= 1, λ= λ0a in Case 5; (b) m1 = 0, n1 = 1, n= 1, λ= λ0b in Case 6.

3.2.2. Semi-rational solution

If K3 6= 0 in Eq. (23), we modify the combination of eigenfunctions given inEq. (31) as follows:

G1 =K2ψ2 +K3ψ3, G2 =1

ε(K1ψ1 +K2ψ2) (34)

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(a) |q[2]r | (b) |q[2]r | (c) |q[2]r |

(d) |q[2]r | (e) |q[2]r |

Fig. 6 – (Color online) Dynamical evolution plots of two different second-order rational solitons. (a)λ0a =2

√6i,m1 =0,n1 =1,m2 =0,n2 =0 in Case 7; (b) λ0b =

92 i,m1 =0,n1 =1,m2 =0,n2 =0

in Case 7; (c) and (d) are the section plots at x= 0 for (a) and (b), respectively; (e) coexistence of theabove two W -shaped second-order rational solitons in Case 8.

(a) |q[1]anti| (b) |q[1]semir|

(c) |q[1]semir|

Fig. 7 – (Color online) Dynamical evolution plots of different first-order semi-rational solutions. (a)m1 = 1,n1 = 0,K2 = 1,K3 = 1,n = 1,λ = λ0a in Case 9; (b) m1 = 1,n1 = 1,K1 = I,K2 =1,K3 = 1,n = 1,λ = λ0a in Case 10; (c) the profile of the wave in (b) at t = −0.5 (red, solid) andt= 0.5 (blue, dash).

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(a) |q[1]| (b) |q[1]|

(c) |q[1]| (d) |q[1]|

Fig. 8 – (Color online) Plots of the first-order periodic solutions. (a) K1 =K2 =K3 = 1,λ0 = 5i2 in

Case 11. (b) K1 =K2 =K3 = 1,λ0 =− 5i2 in Case 11. (c) K1 =K2 = 1,K3 = 0,λ0 =

5i2 in Case

12. d) K1 =K2 = 1,K3 = 0,λ0 = 3i2 in Case 12.

(a) |q[2]| (b) |q[2]|

Fig. 9 – (Color online) (a) The dynamical evolution plot of the second-order periodic solutions atλ1 = 5i

2 and λ4 =√7i in Case 13; (b) The dynamical evolution plot of the second-order breather-

positon (“bpositon”) solution at λ1 = 5i2 in Case 14.

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By further combination of the above functions with ε-dependent coefficients, theeigenfunction solutions Ψk of the Lax pair (2) and the associated λk are

Ψk =mG1 +nG2|λ=λk ,k = 1,4,7, · · · ,3n−2, (35)

with

m=m1 +m2ε2 +m3ε

4 + · · ·+mjε2(j−1), n= n1 +n2ε

2 +n3ε4 + · · ·+njε

2(j−1),(36)

where mj and nj are complex parameters.According to Corollary 2, we take the spectral parameter λ0 = λ0a as an exam-

ple to show below two different first semi-rational solution by using one-fold DT.Case 9. Whenm1 = 1,K2 = 1,K3 = 1,n1 = 0, Eq. (30) yields a first-order anti-darksoliton solution q[1]anti. Its dynamical evolution is shown in Fig. 7(a).Case 10. When m1 = 1,n1 = 1,K1 = I,K2 = 1,K3 = 1, Eq. (30) gives a semi-rational solution q[1]semi, namely, the interaction between an anti-dark soliton and aW -shaped rational solution; see Fig. 7(b). Furthermore, we explicitly show theprofile of this solution at different values of time in Fig. 7(c).

4. PERIODIC SOLUTIONS

In this Section, through choosing different combinations of functions givenby Eq. (20) and selecting specific spectral parameters, we obtain different types ofperiodic solutions of SSE.

C. The-first order periodic solutionsCase 11. When K1 = K2 = K3 = 1 in Eq. (23) and λ1 = 5i

2 , then Eq. (17) withn = 1 yields a first-order half-periodic solution q[1] in time and space shown in Fig.8(a). If we take λ1 = −5i

2 , then the half-periodicity property will occur in the othersemiplane as illustrated in Fig. 8(b); see also Ref. [34], where a two-fold DT wasused to construct such types of soliton solutions.Case 12. When K1 = K2 = 1,K3 = 0 in Eq. (23) and λ1 = 5i

2 , in contrast to theabove case, Eq. (17) with n = 1 gives a fully-periodic solution q[1] in the wholeplane of time and space coordinates whose dynamical evolution is shown in Fig.8(c). However, if we change the value of λ1 to λ1 = 3i

2 , we get another type ofperiodic solution as shown in Fig. 8(d).

D. The second-order periodic solutionsCase 13. When choosing two distinct spectral parameters λ1 = 5i

2 and λ4 =√

7i andtheir corresponding eigenfunctions ψ1 and ψ4 given in Eq. (23) with K1 = K2 =K3 = 1, then Eq. (17) with n= 2 yields a general second-order periodic solution q[2]

as shown in Fig. 9(a).Case 14. When we choose λ1 = 5i

2 and let λ4→ λ1, similar to the DT in Theorem 2,

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i.e., just changing the special spectral parameter λ0 into λ1, we obtain a second-orderbreather-positon (“bpositon”) solution q[2]bp from Eq. (24) with n = 2. Its dynamicalevolution plot is displayed in Fig. 9(b). These special types of periodic solutions ofother relevant soliton equations have been recently studied in Refs. [47, 64, 65].

5. CONCLUSION AND DISCUSSION OF THE RESULTS

In this paper, we have constructed a general n-fold DT for the SSE (1) (see The-orem 1 and Corollary 1) that was expressed by a compact determinant representationin contrast to the representation obtained by using the dressing operator method inRefs. [23, 34]. Especially, one can directly compare Eq. (8) in the present workwith Eq. (8) in Ref. [34] and with Eq. (4) in Ref. [23]. According to the valueof M1 in Eq. (19), i.e., the multiple root condition of Eq. (21), we have consideredin this work two distinct situations: I) when M1=0, Eq. (21) has a triple real root,and we have obtained W -shaped rational solutions up to second order according toTheorem 2; II) when M1 < 0, Eq. (21) possesses two distinct roots and according toTheorem 3 and Corollary 2 we have obtained many types of soliton solutions suchas W -shaped first-order rational solutions, two distinct types of second-order ratio-nal solutions and the coexistence phenomenon of these two kinds of second-orderrational solutions, first-order semi-rational solutions, first-order half-periodic solu-tions, and second-order periodic solutions including breather-positon (“bpositon”)solutions. In Section 4, we find different types of periodic solutions of the SSE (1).Though we have derived solutions up to second order in Section 4, Eqs. ( 17), (24),and (30) can be also used to analyze higher-order solutions, and our method could beextended to other integrable soliton models.

Comparing to the results reported in Refs. [22, 23, 33, 34], we list below thenew results obtained in this paper:

• We have obtained a compact determinant representation Tn of the n-fold DT forthe SSE (1) without the use of quasi-determinant [22].

• W -shaped second-order rational solitons (see the Case 4 and the Case 7) and thecoexistence of two second-order rational solitons, which are different in shape,are first reported (see the Case 8).

• The combination between an anti-dark soliton and aW -shaped soliton is obtainedby using one-fold DT. This result is different from that illustrated in Fig. 8 of Ref.[33] where a two-fold DT was used.

• The semi-periodic solutions in Case 11 are obtained by using a one-fold DTwhereas a similar result was derived in Ref. [34] by using a two-fold DT. Differ-ent types of periodic solutions were illustrated in Cases 12, 13, 14, in particular,

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the second-order breather-soliton (“bpositon”) solution q[2]bp in Case 14 has notbeen reported before, to the best of our knowledge.

We believe that the compact determinant form Tn (see Theorem 1 and Corollary 1)can be applied to generate much more complicated patterns of mixed waveformsinvolving other types of rational solutions of the Sasa-Satsuma equation. Also, itcan be applied to generate interesting wave patterns for other physically relevantsoliton equations. We note that the n-fold DT matrix Tn given in this paper cannotgenerate rogue wave structures. Thus in order to get other interesting mixed solutionsassociated with rogue waves, breathers, and different types of rational solitons, wehave to find a new form of DT matrix, an issue that will be addressed elsewhere.

Acknowledgements. This work is supported by the NSF of China under Grant No. 11671219,the Natural Science Foundation of Zhejiang Province under Grant No. LZ19A010001, and the K.C.Wong Magna Fund in Ningbo University. We thank other members in our group at Ningbo Universityfor many discussions and suggestions on this paper.

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