A focusing and defocusing semi-discrete complex short pulse equation and its varioius soliton solutions
aa r X i v : . [ n li n . S I] J a n A focusing and defocusing semi-discrete complex short pulse equation and its varioiussoliton solutions
Bao-Feng Feng , Liming Ling , and Zuonong Zhu School of Mathematical and Statistical Sciences,The University of Texas Rio Grande Valley Edinburg TX, 78541-2999, USA Department of Mathematics, South China University of Technology, Guangzhou 510640, China and School of Mathematical Sciences, Shanghai Jiaotong University, Shanghai 200240, China
In this paper, we are concerned with a a semi-discrete complex short pulse (CSP) equation of bothfocusing and defocusing types, which can be viewed as an analogue to the Ablowitz-Ladik (AL) lat-tice in the ultra-short pulse regime. By using a generalized Darboux transformation method, varioussolutions to this newly integrable semi-discrete equation are studied with both zero and nonzeroboundary conditions. To be specific, for the focusing CSP equation, the multi-bright solution (zeroboundary condition), multi-breather and high-order rogue wave solutions (nonzero boudanry con-ditions) are derived, while for the defocusing CSP equation with nonzero boundary condition, themulti-dark soliton solution is constructed. We further show that, in the continuous limit, all thesolutions obtained converge to the ones for its original CSP equation (see Physica D, 327 13-29 andPhys. Rev. E 93 052227)
Key words: generalized Darboux transformation, semi-discrete complex short pulse equation,bright and dark soliton, breather, rogue wave
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I. INTRODUCTION
The most recent advances in nonlinear optics include the generation and applications of ultrashort optical pulses,whose time duration is typically of the order of femtoseconds, which lead to the Nobel prize in physics 2018 [1]. It hasbeen an important topics for the mathematical models of optical pulse propagation in medium [2–4]. When the rangeof short pulse width is from 10 ns to 10 f s , both dispersive and nonlinear effects influence their shape and spectrum.The following full wave equation ∇ E − c E tt = µ P tt , (1)originates directly from Maxwell’s equation. If we assume the local medium response and only the third-order nonlineareffects (Kerr effect) the induced polarization consists of linear and nonlinear parts, P ( r , t ) = P L ( r , t ) + P NL ( r , t ).Under the assumption of quasi-monochromatic, i.e., the pulse spectrum, centered as ω , is assumed to have a spectralwidth ∆ ω such that ∆ ω/ω <<
1. Under this assumption, one can expand the frequency-dependent dielectric constant ǫ ( ω ) in terms of Taylor series up to the second order. As a result, the so-called nonlinear Schr¨odinger equationi q z + α q ττ + α | q | q = 0 , (2)can be derived to govern the slowly varying envelop of optical waves in weakly nonlinear dispersive media [5, 6].Here α represents the effect of group velocity dispersion (GVD) ( α > α < α represents the self-phase modulation (SPM)due to Kerr effect.Upon switching the spatial and temporal variables and normalization, the nonlinear Schr¨odinger (NLS) equationcan be put into a standard form i q t = q xx + 2 σ | q | q (3)which has become a generic model equation describing the evolution of small amplitude and slowly varying wavepackets in weakly nonlinear media [2–4, 7, 8]. It arises in a variety of physical contexts such as nonlinear opticsmetioned above, Bose-Einstein condesates [9], water waves [10] and plasma physics [11]. The integrability, as well asthe bright-soliton solution in the focusing case ( σ = 1), was found by Zakharov and Shabat [12, 13]. The dark solitonwas found in the defocusing NLS equation ( σ = −
1) [6], and was observed experimentally in 1988 [14, 15]. Recently,the rogue (freak) waves were discovered and the Peregrine soliton was found in focusing NLS equation [16, 17].The integrable discretization of nonlinear Schr¨odinger equationi q n,t = (cid:0) σ | q n | (cid:1) ( q n +1 + q n − ) (4)was originally derived by Ablowitz and Ladik [18, 19], so it is also called the Ablowitz-Ladik (AL) lattice equation.Similar to the continuous case, it is known that the AL lattice equation, by Hirota’s bilinear method, admits the brightsoliton solution for the focusing case ( σ = 1) [20, 21], also the dark soliton solution for the defocusing case ( σ = − ps , higher order nonlinear effects has to be taken into account, andthe NLS equation should be modified. As a result, a generalized NLS (gNLS) equation [3]i q z + α q ττ + α | q | q + i( β q τττ + β | q | q τ + β q ( | q | ) τ ) = 0 , (5)can be derived, where β , β and β are the parameters related to the third order dispersion (TOD), self-steepening (SS)and stimulated Raman scattering (SRS). Due to the complexity of the gNLS equation, the study is mainly numerical.However, in some special case, the gNLS equation becomes integrable and is available for rigorous analysis. Forexample, when β = β = 0 , the gNLS equation is called modified NLS equation, which is integrable.i q z + α q ττ + α | q | q + i β | q | q τ = 0 . (6)In addition, there are two known integrable cases when β = 0 , β = 0 , and the condition 3 β α = β α is satisfied.Under this condition, a gauge transformation brings the generalized NLS equation into the form q z + β ′ q τττ + β ′ | q | q τ + β ′ q ( | q | ) τ = 0 . (7)When β ′ : β ′ : β ′ = 1 : 6 : 0, we obtain Hirota’s equation [32]. When β ′ : β ′ : β ′ = 1 : 6 : 3, we find the Sasa-Satsumaequation [33].However, when the width of optical pulse is of the order of sub-femtosecond ( < − s), then the width of itsspectrum is of the order greater than 10 s − , being comparable with the spectral width of the optical pulses, thequasi-monochromatic assumption is not valid anymore. Instead, by using the Kramers-Kronig relation of the responsefunction one obtains a normalized model equation E zz − E tt = E + (cid:0) | E | E (cid:1) tt , (8)and further a non-integrable equation 2 E zτ = E + (cid:0) | E | E (cid:1) ττ , (9)where τ = t − z . The solitary wave solutions and their interactions for above models were studied in details in [34–38].Following the same spirit, Sch¨afer and Wayne proposed a so-called short-pulse (SP) equation [39] to describe thepropagation of ultrashort optical pulses in silicon fiber. u xt = u + 16 (cid:0) u (cid:1) xx . (10)Here, u = u ( x, t ) is a real-valued function, representing the magnitude of the optical field. It is completely integrable[40], admits periodic and soliton solutions [41]. Wave breaking phenomenon was studied in [42]. The integrablediscretization of the SP equation and its geometric formulation was studied in [43, 44].However, similar to the NLS equation, the complex representation has advantages for the description of the opticalwaves since a single complex-valued function can contains the information of amplitude and phase in a wave packetsimultaneously. Consequently, a complex short pulse (CSP) equation q xt + q + 12 σ (cid:0) | q | q x (cid:1) x = 0 (11)was proposed by the authors [45, 46]. Here q = q ( x, t ) is a complex-valued function, representing the optical wavepackets in ultrashort pulse regime, σ = ± − x and t , the CCD system becomes the complex sine-Gordon equation [53] or the AB system [54],which is the first negative flow of the AKNS system.Recently, much attention has been paid to the study of discrete integrable systems [55] though it can traced back tothe middle of 1970s when Hirota discretized various famous soliton equations such as the KdV, mKdV, and the sine-Gordon equations through the bilinear method [56]. In the past two decades, the field of discrete system has grownto prominence as an area in which numerous breakthroughs have taken place, inspriring new developments in otherareas of mathematics. As mentioned previously, Ablowitz and Ladik proposed a method of integrable discretizationssoliton equations which involve the nonlinear Schr¨odinger equations and mKdV equation, by the Lax pair method[57]. Another successful way to discretize soliton equations was proposed by Date et al via the transformation grouptheory, which gives a large number of integrable disretizations [58]. One of the most interesting examples is thediscrete KP equation, or the so-called Hirota-Miwa equation, which embodies the whole KP hierarchy [59, 60]. Surisalso developed a general Hamiltonian approach for integrable discretizations of integrable systems [61]. Following theoriginal works on the integrable discretizations, the study on discrete integrable systems has been extended to othermathematics fields such as discrete differential geometry [62].In the present paper, we are concerned with various soliton solutions to a semi-discrete complex short pulse equation ddt q n +1 − q n ∆ x n + 12 ( q n +1 + q n ) + σ x n (cid:18) | q n +1 | q n +2 − q n +1 ∆ x n +1 − | q n | q n +1 − q n ∆ x n (cid:19) = 0 . (12)where q n = q ( nx n , t ) , ∆ x n = x n +1 − x n . This lattice equation is a semi-discrete analogue of the complex short pulse(CSP) equation, where the spatial variable is discretized and the time variable remains continuous. The semi-discreteCSP equation can also be written in a coupled two-component system [63, 64] ddt ( q n +1 − q n ) = 12 ( x n +1 − x n )( q n +1 + q n ) ,ddt ( x n +1 − x n ) + 12 σ ( | q n +1 | − | q n | ) = 0 . (13)As mentioned in [63], the semi-discrete CSP equation can be constructed in a very direct way. It is shown in [46, 49]that the CSP equation is related to the so-called complex coupled dispersionless (CCD) equation q ys = qρ, ρ s + 12 σ ( | q | ) y = 0 , (14)by a reciprocal (hodograph) transformation defined by d x = ρ d y − σ | q | d s , d t = − d s . The CCD equation [46, 48]admits a Lax pair of the form Ψ y = U ( ρ, q ; λ )Ψ , Ψ s = V ( q ; λ )Ψ , (15)where U ( q, ρ ; λ ) = λ − " − i ρ − σq ∗ y q y i ρ , V ( q ; λ ) = i4 λσ + i2 Q, (16)with ∗ representing the complex conjugate, σ , the third Pauli matrix, and Q being σ = " − , Q = " − σq ∗ q , respectively. Replacing the forward-difference to the first-order derivative in the spatial part of the Lax pair, i.e.,Ψ n +1 − Ψ n a = λ − − i ρ n − σ q ∗ n +1 − q ∗ n aq n +1 − q n a i ρ n Ψ n , one yields the Lax pair for the semi-discrete CCD equationΨ n +1 = U n Ψ n , Ψ n,s = V n Ψ n , (17)where U n = − i aρ n λ − σ q ∗ n +1 − q ∗ n λq n +1 − q n λ i aρ n λ , V n = i4 λσ + i2 Q n , Q n = " σq ∗ n q n . (18)The compatibility condition gives exactly the semi-discrete CCD equation. Replacing aρ n by x n +1 − x n , one obtainsthe semi-discrete CSP equation. As an analogue to the AL lattice equation in the ultra-short pulse regime, it isimperative to study this new integrable semi-discrete CSP equation due to its potential applications in physics.However, compared with the results for AL lattice equation, the obtained results for the semi-discrete CSP equationis much less. This motivates the present work, which intends to construct various soliton solutions with vanishingand non-vanishing boundary conditions via generalized Darboux transformation method.Darboux transformation, originating from the work of Darboux in 1882 on the Sturm-Liouville equation, is apowerful method for constructing solutions for integrable systems [65]. However, the classical Darboux transformationcannot be iterated at the same spectral parameter to obtain the multi-dark, breather and higher order rogue wavesolutions. To overcome this difficulty, one of the authors generalized the classical Darboux transformation by using alimit technique[66, 67], which can be used to yield these solutions. It is noted that recently various soliton solutionshave been found for the nonlocal NLS equation [68, 69]. In this paper, we aim at finding soliton solutions for thesemi-discrete CSP equation (12) by generalized Darboux transformation (DT). It should be pointed out that the DTand soliton solutions for the semi-discrete coupled dispersionless equation has been constructed by Riaz and Hassen[70] very recently. By the study for this system, we find the discretization equation keeps Darboux transformationand solitonic solutions with the original equation. Meanwhile, these results would be useful to study the self-adaptivemoving mesh schemes for the complex pulse type equations.The outline of the present paper is organized as follows. In section II, a generalized Darboux transformationof semi-discrete CSP equation was derived through loop group method [71]. Based on the generalized Darbouxtransformation, we can obtain the general solitonic formula for semi-discrete CSP equation. Moreover, together withthe reciprocal transformation, we can construct the general solitonic formula for semi-discrete CSP equation in termsof the determinant representation. The N -bright soliton solution for the focusing case with zero boundary conditionand N -dark soliton solution for the defocusing case with nonzero boundary condition are constructed in section III andIV, respectively. In V, the multi-breather solution with nonzero boundary condition is constructed for the focusingsemi-discrete CSP and is approved to converge to the one for the original CSP equation in the continuous limit.Based on the multi-breather solution, we further derive general rogue wave solution in VI. Section VII is devoted toconclusions and discussions. II. GENERALIZED DARBOUX TRANSFORMATION FOR THE SEMI-DISCRETE CSP EQUATION
Based on the Lax pair of the semi-discrete CSP equation (17), we give the Darboux transformation by the followingproposition.
Proposition 1
The Darboux matrix T n = I + λ ∗ − λ λ − λ ∗ P n , P n = | y ,n ih y ,n | J h y ,n | J | y ,n i , J = diag(1 , σ ) , (19) can convert system (17) into a new system Ψ [1] n +1 = U n ( ρ [1] n , q [1] n ; λ )Ψ [1] n , Ψ [1] n,s = V n ( ρ [1] n , q [1] n ; λ )Ψ [1] n , where | y ,n i = ( ψ ,n , φ ,n ) T is a special solution for system (17) with λ = λ , | y ,n i † = h y ,n | . The B¨acklund transformations between ( ρ [1] n , q [1] n ) and ( ρ n , q n ) are given through ρ [1] n = ρ n − a ln s (cid:18) E ( h y ,n | J | y ,n i ) h y ,n | J | y ,n i (cid:19) ,q [1] n = q n + ( λ ∗ − λ ) ψ ∗ ,n φ ,n h y ,n | J | y ,n i , | q [1] n | = | q n | + 4 σ ln ss (cid:18) h y ,n | J | y ,n i λ ∗ − λ (cid:19) , (20) and the symbol E denotes the shift operator n → n + 1 . Proof:
Firstly, we see that the evolution part of system (17) is a standard one for the AKNS system with SU (2)symmetry. Then the Darboux transformation for the AKNS system also satisfies the system (17). The rest of theproposition is to prove the formulas (20). To derive these formulas, we borrow some idea from the classical monograph[72].Suppose there is a holomorphic solution for Lax pair equation (17) in some punctured neighborhood of infinity onthe Riemann surface, soothing depending on n and s , with the following asymptotical expansion at infinity: (cid:20) ψ φ (cid:21) = (cid:20) (cid:21) + ∞ X i =1 Ψ i λ − i ! exp (cid:18) i4 λs (cid:19) , λ → ∞ + . (21)On the one hand, from the Lax pair equation (17), we have ψ ,s = i4 λψ + i2 σq ∗ n φ ,φ ,s = i2 q n ψ − i4 λφ . (22)With the aid of above equations (22), we can obtain the following relation H s = i2 | q n | − i2 λH − i2 σH + ln t ( q ∗ n ) H, H ≡ q ∗ n φ ψ = ∞ X i =1 H i λ − i . (23)Then the coefficient H i can be determined as following: H = | q n | , H = 2i q n,s q ∗ n ,H i +1 =2i q ∗ n (cid:18) H i q ∗ n (cid:19) s − σ i − X j =1 H j H i − j , i ≥ . It follows that the first equation of (22) can be rewritten as ψ ,s = i4 λ + i2 ∞ X i =1 H i λ − i ! ψ . (24)On the other hand, substituting the asymptotic expansion (21): ψ = ∞ X i =1 Ψ [1] i λ − i ! exp (cid:18) i4 λs (cid:19) , (25)then we have Ψ [1]1 ,s = i2 σ | q n | . (26)By Darboux transformation, it follows that " ψ [1]1 φ [1]1 = I + ∞ X i =1 T [ i ] λ − i ! "(cid:20) (cid:21) + ∞ X i =1 Ψ i λ − i exp (cid:18) i4 λs (cid:19) . (27)Moreover, we can obtain that (cid:16) T [1]1 , (cid:17) s + Ψ [1]1 ,s = i2 σ | q [1] n | , (28)where the element T [1] n [ i, j ] denotes the ( i, j ) element of matrix T [1] n . Together with (26), we can obtain that | q [1] n | = | q n | − σ (cid:16) T [1] n [1 , (cid:17) s . (29)Since T n is the Darboux matrix, it satisfies the following relation T n +1 U n = U [1] n T n . It follows that q [1] n = q n + T [1] n [2 , ,aρ [1] n = aρ n + i( E − T [1] n [1 , . (30)On the other hand, since | y i s = (cid:18) i4 λ σ + i2 Q (cid:19) | y i , −h y | t J = h y | J (cid:18) i4 λ ∗ σ + i2 Q (cid:19) it follows that (cid:18) h y | J | y i λ ∗ − λ (cid:19) s = i4 ( − σ | ψ | + | φ | ) (31)we can obtain that ( E − T [1] n [1 ,
1] = ( E − (cid:18) ( λ ∗ − λ ) | ψ ,n | h y ,n | J | y ,n i (cid:19) = ( E − (cid:18) − σ ( λ ∗ − λ ) | φ ,n | h y ,n | J | y ,n i (cid:19) = ( E − (cid:18) ( λ ∗ − λ )( | ψ ,n | − σ | φ ,n | )2 h y ,n | J | y ,n i (cid:19) = 2i ln s (cid:18) E ( h y ,n | J | y ,n i ) h y ,n | J | y ,n i (cid:19) . (32)Similarly, we have ( T [1] n [1 , s = 2i ln ss (cid:18) h y ,n | J | y ,n i λ ∗ − λ (cid:19) . (33)Finally, combining the equations (29) and (39), we obtain the formulas (20). This completes the proof. (cid:3) Assume that we have N different solutions | y i,n i = ( ψ i,n , φ i,n ) T at λ = λ i ( i = 1 , , · · · , N ), then we can constructthe N -fold DT. For simplicity, we ignore the subscript n in | y i,n i and h y i,n | . Furthermore, we have the followinggeneralized Darboux matrix Proposition 2
The general Darboux matrix can be represented as T n,N = I + Y M − n D − Y † J, (34) where the first subscript in T n,N represents that the matrix is dependent with the variable n , the second one represents the N -fold DT, Y = h | y [0]1 i , | y [1]1 i , · · · , | y [ n − i , · · · , | y [0] r i , | y [1] r i , · · · , | y [ n r − r i i ,M n = M M · · · M r M M · · · M r ... ... . . . ... M M · · · M r , M ij = M [1 , ij M [1 , ij · · · M [1 ,n j ] ij M [2 , ij M [2 , ij · · · M [2 ,n j ] ij ... ... . . . ... M [ n i , ij M [ n i , ij · · · M [ n i ,n j ] ij ,D =diag ( D , D · · · , D r ) , D i = D [0] i · · · D [ n i − i . . . ... D [0] i , and | y i ( λ i + α i ǫ i ) i = n i − X k =0 | y [ k ] i i ǫ ki + O ( ǫ n i i ) , λ − λ ∗ i − α i ǫ ∗ i = n i − X k =0 D [ k ] i ǫ ∗ ki + O ( ǫ ∗ n i i ) h y i ( λ i + α i ǫ i ) | J | y j ( λ j + α j ǫ j ) i λ ∗ i − λ j + α ∗ i ǫ ∗ i − α j ǫ j = n i X k =1 n j X l =1 M [ k,l ] ij ǫ ∗ ki ǫ lj + O ( ǫ ∗ n i i , ǫ n j j ) . The general B¨acklund transformations are ρ [ N ] n = ρ n − a ln s (cid:18) E (det( M n ))det( M n ) (cid:19) ,q [ N ] n = q n + det( G n )det( M n ) , | q [ N ] n | = | q n | + 4 σ ln ss (det( M n )) (35) where G n = (cid:20) M Y † − Y (cid:21) , Y k represents the k -th row of matrix Y . Proof:
Through the standard iterated step for DT, we can obtain the N -fold DT T n,N = I + Y M − n D − Y † J, (36)where Y = [ | y i , | y i , · · · , | y N i ] , and M n = (cid:18) h y i | J | y j i λ ∗ i − λ j (cid:19) ≤ i,j ≤ N , D = diag ( λ − λ ∗ , λ − λ ∗ , · · · , λ − λ ∗ N ) . Since T n is the Darboux matrix, it satisfies the following relation T n +1 ,N U n = U [ N ] n T n,N . (37)By using the following identities φM − ψ † = (cid:12)(cid:12)(cid:12)(cid:12) M ψ † − φ (cid:12)(cid:12)(cid:12)(cid:12) / | M | , φM − ψ † = (cid:12)(cid:12)(cid:12)(cid:12) M ψ † − φ (cid:12)(cid:12)(cid:12)(cid:12) / | M | = det( M + ψ † φ )det( M ) . (38)where M is a N × N matrix, φ , ψ are a 1 × N vectors, we could derive q [ N ] n = q n + T [1] n,N [2 ,
1] = q n + det( G n )det( M n ) ,aρ [ N ] n = aρ n + i( E − T [1] n,N [1 , , | q [ N ] n | = | q n | − σ (cid:16) T [1] n,N [1 , (cid:17) s . (39)Together with the following equalities( E − T [1] n,N [1 , E − (cid:16) Y M − Y † (cid:17) = ( E − (cid:16) − σY M − Y † (cid:17) = ( E − Y M − Y † − σY M − Y † ! = 2i ln s (cid:18) E det( M n )det( M n ) (cid:19) , ( T [1] n,N [1 , s = 2i ln ss (det( M n )) , (40)we can readily obtain the formula (35) from the above N -fold DT (36). To complete the generalized DT, we set λ r +1 = λ + α ε , , | y r +1 i = | y ( λ r +1 ) i ; · · · , λ r + n − = λ + α ε ,n − , | y r + n − i = | y ( λ r + n − ) i ; λ r + n = λ + α ε , , | y r + n i = | y ( λ r + n ) i ; · · · , λ r + n − = λ + α ε ,n − , | y r + n + n − i = | y ( λ r + n − ) i ;... λ N − n r +1 = λ r + α r ε r, , | y N − n r +1 i = | y r ( λ N − n r +1 ) i ; · · · , λ N = λ r + α r ε r,n r − , | y N i = | y r ( λ N ) i . Taking limit ε i,j →
0, we can obtain the generalized DT (34) and formulas (35). (cid:3)
With the aid of the generalized DT, one can construct more general exact solutions from the trivial solution ofthe original equation. Departing from the zero seed solution, one-, multi-bright soliton solutions can be constructedfor the focusing CSP equation. Starting from the plane wave seed solution, the multi-breather and high-order roguewave solutions can be derived for the focusing CSP equation; while one-, two- and multi-dark soliton solutions can beobtained for the defocusing CSP equation. The detailed results and the explicit expressions, as well as dynamics forthese solutions, are presented in the subsequent two sections.On account of (20), the coordinates transforation between x [ N ] n and x n can be represented as x [ N ] n = x n − a ln s (det( M n )) , (41)where x n represents the original coordinates. The second equation in (20) and coordinates expressions (41) constitutesthe solutions for semi-discrete CSP equation (12). III. SINGLE AND MULTI-BRIGHT SOLUTIONS
In this section, we construct the exact solution through formula (35) as the application of DT. The general brightwill be constructed for the focusing CSP equation ( σ = 1). To this end, we start with a seed solution ρ [0] n = γ , q [0] n = 0 , γ > . The coordinates for semi-discrete CSP (12) can be obtained x n ( s ) = γ na, t = − s. Solving the Lax pair equation (17) with ( ρ n , q n ; λ ) = ( ρ [0] n , q [0] n ; λ i = α i + i β i ), β i >
0, one obtains a special solution | y i,n i = (cid:20) e θ i,n (cid:21) , θ i,n = i2 λ i s + n ln λ i − i aγ λ i + i aγ ! + a i , (42)where a i s are complex parameters. Then we can obtain that the single soliton solution through the formula (35): ρ [1] n = γ β a [tanh( θ R ,n +1 ) − tanh( θ R ,n )] > ,q [1] n = β sech( θ R ,n )e − i θ I ,n − π i2 ,x [1] n = γ na + β a tanh( θ R ,n ) , t = − s, where the superscripts R , I represent the real part and imaginary part, respectively, θ R ,n = − β s + n g + a R , θ I ,n = 12 α s + n arg λ − i aγ λ + i aγ ! + a I , g = ln α + (2 β − aγ ) α + (2 β + aγ ) ! , where 4 α + (2 β − aγ ) = 0 . The soliton | q [1] n | propagates along the line θ R ,n = 0 . The peak values | q [1] n | max = β locate at ( x, t ) = ( n, β ( ng + 2 a R )) . To obtain the smooth bright soliton for semi-discrete CSP equation, we requirethat ρ [1] n > n ∈ Z and t ∈ R . Otherwise, the bright soliton will be either cusp or soliton solution.Inserting the equation (42) into formula (35),, we can deduce the multi-bright soliton solution as follows: ρ [ N ] n = γ − a ln s (cid:18) det( M n +1 )det( M n ) (cid:19) q [ N ] n = det( G n )det( M n ) ,x [ N ] n = γ na − a ln s det( M n ) , t = − s, (43)where M n = e θ ∗ i,n + θ j,n + 1 λ ∗ i − λ j ! ≤ i,j ≤ N , G n = (cid:20) M n Y † ,n − Y ,n (cid:21) ,Y ,n = (cid:2) e θ ,n , e θ ,n , · · · , e θ N,n (cid:3) , Y ,n = (cid:2) , , · · · , (cid:3) , the expression for θ i,n is given in (42). Here we require ρ [ N ] n > ± x ) = ± x + o ( x ) , (44)we have n ln λ i − i aγ λ i + i aγ ! ≈ − n i aγλ i = − i γλ i y (45)by letting na = y in the continuous limit a to
0. Therefore θ i,n agrees with (32) in [48] by noticing the correspondence θ i,n → θ i,n and γ → − γ .In particular, we give two soliton solution explicitly through the above general formula (43): ρ [2] n = γ − a ln s det( M [2] n +1 )det( M [2] n ) ! q [2] n = det( G [2] n )det( M [2] n ) ,x [2] n = γ na − a ln s det( M [2] n ) , t = − s, (46)where M [2] n = (cid:16) e θ ,n + θ ∗ ,n + 1 (cid:17) (cid:16) e θ ,n + θ ∗ ,n + 1 (cid:17) β β − (cid:16) e θ ,n + θ ∗ ,n + 1 (cid:17) (cid:16) e θ ,n + θ ∗ ,n + 1 (cid:17) ( − α + α ) + ( β + β ) ,G [2] n = e θ ,n + θ ∗ ,n + 1( β + β ) + ( α − α )i − e θ ,n + θ ∗ ,n + 12 β ! e θ ∗ ,n + e θ ,n + θ ,n + 1( β + β ) + i( α − α ) − e θ ,n + θ ∗ ,n + 12 β ! e θ ∗ ,n . The single bright soliton is illustrated in Fig. 1(a) with the parameters γ = 1, a = 2, α = 2, β = 1, a = 0 , . Toexhibit the dynamics for the two-soliton solution as shown in Fig. 1 (b)), we choose the parameters γ = 2, a = 2, α = 2, β = 1, a = 0 , α = 1, β = 1, a = 0. It is seen that the two solitons interact with each other elastically.0 (a)Single bright soliton (b)Two bright soliton FIG. 1: (color online): Bright solitons
IV. SINGLE AND MULTI-DARK SOLITON SOLUTION
In this section, we construct the one- and multi-dark soliton solution for the defocusing CSP equation ( σ = −
1) indetail. Generally, the DT cannot apply to derive the dark solitons directly, since the spectral points of dark solitonslocate in the real axis and the Darboux matrix is trivial if λ = λ ∗ . The authors in [73] develop a method to yieldthe dark soliton and multi-dark solitons through the Darboux transformation together with the limit technique. Inwhat followings, we follow the steps in reference [73] to give the dark and multi-dark solitons for the semi-discreteCSP equation.The dark solution and multi-dark solution can be constructed from the seed solution–plane wave solution throughformula (35). We depart from the seed solution ρ [0] n = γ , q [0] n = β i θ n , θ n = bn + c s, c = aγ b )cos( b ) − , γ > , β ≥ , b = kπ, k ∈ Z . (47)Then we have the solution vector for Lax pair equation (17) with ( q n , ρ n ; λ ) = ( q [0] n , ρ [0] n ; λ i ), | y i,n i = KL i E i = K " d φ i,n β d ψ i,n , K = diag (cid:16) e − i2 θ n , e i2 θ n (cid:17) , λ i = − c + i β, (48)where | y i,n i discards a function, L i = βc + χ + i βc + χ − i , E i = (cid:20) e ω i,n α i ( ¯ λ i − λ i )e − ω i,n (cid:21) and ω i,n = i4 ξ i s + n sin( b ) (cid:0) i aγ − ξ i (cid:1) + i cos( b ) λ i sin( b ) (cid:0) i aγ + ξ i (cid:1) + i cos( b ) λ i ! + a i ,χ ± i = λ i ± ξ i , ξ i = p ( λ i + c ) − β . (49) α i s are appropriate complex parameters and a i s are real parameters. In order to derive the single dark soliton solution,we consider only λ and replace the parameter condition (49) with χ ± = β [cos( ϕ ) ± i sin( ϕ )] , ξ = i β sin( ϕ ) . (50)and 0 < ϕ i < π, a i ∈ R . By taking a limit process λ → ¯ λ similar to the one in [73], the single dark soliton solution1can be obtained as follows ρ [1] n = γ β a sin( ϕ )(tanh( Z ,n +1 ) − tanh( Z ,n )) ,q [1] n = β (cid:2) − i sin( ϕ )e − i ϕ − i sin( ϕ )e − i ϕ tanh( Z ,n ) (cid:3) e i θ n ,x [1] n = γ an + β s + β a sin( ϕ ) tanh( Z ,n ) , t = − s, (51)where Z ,n ( ϕ , a ) = n (cid:18) aγ + 2 β sin( b ) cos( b + ϕ ) aγ + 2 β sin( b ) cos( b − ϕ ) (cid:19) − β ϕ ) s + a . (52)and a is a real parameter.Next, we proceed to finding N -dark soliton solution. Based on the N -soliton solution (35) to the defocusingsemi-discrete CSP equation, it then follows q n [ N ] = β h d Y ,n M − n d Y ,n † i e i θ n = β (cid:20) det( H n )det( M n ) (cid:21) e i θ n ,x n = γ an + β s − a ln s (det( M n )) , t = − s, (53)where M n = (cid:18) h y i,n | σ | y j,n i λ i − λ j ) (cid:19) ≤ i,j ≤ N , H n = M n + Y † ,n Y ,n , d Y ,n = h d φ ,n , d φ ,n , · · · , [ φ N,n i , d Y ,n = h d ψ ,n , d ψ ,n , · · · , [ ψ N,n i . In general, the above N -soliton solution (43) is singular. In order to derive the N -dark soliton solution through theDT method, we need to take a limit process λ i → ¯ λ i , ( i = 1 , , · · · , N ). By a tedious procedure which is omitted here,we finally have the N -dark soliton solution to the defocusing semi-discrete CSP equation (12) as follows Proposition 3 ρ [ N ] n = γ − a ln s det( G n +1 )det( G n ) ,q [ N ] n = β (cid:20) det( H n )det( G n ) (cid:21) e i θ n ,x [ N ] n = γ an + β s − a ln s det( G n ) , t = − s, (54) where G n = ( g i,j ) ≤ i,j ≤ N , H n = ( h i,j ) ≤ i,j ≤ N , g i,j = δ ij + e Z i,n + Z j,n exp( − i ϕ i ) − exp(i ϕ j ) ,h i,j = δ ij + e ( Z i,n − i ϕ i )+( Z j,n − i ϕ j ) exp( − i ϕ i ) − exp(i ϕ j ) , (55) Z i,n = Z ,n ( ϕ i , a i ) and δ i,j is the standard Kronecker delta. By taking N = 2 in (54) and (55), the determinants corresponding to two-dark soliton solution can be calculatedas | G n | = 1 + e Z i,n + e Z ,n + a e Z ,n + Z ,n ) , (56) | H n | = 1 + e Z i,n − i ϕ ) + e Z ,n − i ϕ ) + a e Z ,n + Z ,n − i ϕ − i ϕ ) , (57)2where a = sin (cid:0) ϕ − ϕ (cid:1) sin (cid:0) ϕ + ϕ (cid:1) . (58)Asymptotic analysis can be easily performed for two-soliton interaction, which shows that the collision is alwayselastic. We shows an example of such two-soliton collision. If we choose the parameter a = 1, ϕ = arcsin(4 /
5) and a = 0, then the single dark soliton are shown in Figure 2 (a). To shown their interaction for two dark solitons,choosing the parameters a = 1, ϕ = arcsin(4 / ϕ = π/ a = a = 0, we arrive at the dynamics of two darksolitons (Fig. 2(b)), which show that the interaction between them is elastic. (a)Single dark soliton (b)Two dark soliton FIG. 2: (color online): Dark soliton.
Prior to the closing of this section, let us prove that the multi-dark solution converges to the its counterpart of thecontinuous CSP equation obtained in [46]. In the continuous limit, we assume a = b →
0, it then follows c → − γ .Referring to the Taylor expansion (44), Z i,n turns out to be n (cid:18) aγ + 2 β sin( b ) cos( b + ϕ i ) aγ + 2 β sin( b ) cos( b − ϕ i ) (cid:19) − β ϕ i ) s + a i ≈ n − β sin ( b ) sin( ϕ i )2 β cos( b ) sin( b ) cos( ϕ i ) + aγ − β ϕ i ) s + a i . (59)Note that, between the present paper and [46] γ → − γ . As a result Z i,n → ω i in [46] by letting nb = y and the proofis complete. V. SINGLE BREATHER AND MULTI-BREATHER SOLUTIONS
The single breather and multi-breather solution for the focusing semi-discrete CSP equation (12) ( σ = 1) can beconstructed from the seed solution–plane wave solution through formula (35). We depart from the seed solution ρ [0] n = γ , q [0] n = β i θ n , θ n = bn + c s, c = aγ b )cos( b ) − , γ > , β ≥ . (60)The coordinates for semi-discrete CSP (12) can be obtained x n ( s ) = γ na − β s, t = − s. Then we have the solution vector for Lax pair equation (17) ( σ = 1) with( q n , ρ n ; λ ) = ( q [0] n , ρ [0] n ; λ ), | y ,n i = KL E , K = diag (cid:16) e − i2 θ n , e i2 θ n (cid:17) , λ = − c + i β, (61)3where L = βc + η βc + χ , E i = (cid:20) e θ ,n (cid:21) and θ ,n = i2 ξ s + n ln sin( b ) (cid:0) i aγ − ξ (cid:1) + i cos( b ) λ sin( b ) (cid:0) i aγ + ξ (cid:1) + i cos( b ) λ ! + a ,η = λ + ξ , χ = λ − ξ , ξ = p β + ( λ + c ) . (62)The single breather solution can be constructed from the formula (35) with the technique as in reference [73]: ρ [1] n = γ − a ln s cosh( θ R ,n +1 ) cosh( ϕ R / − sin( θ I ,n +1 ) sin( ϕ I / θ R ,n ) cosh( ϕ R / − sin( θ I ,n ) sin( ϕ I / ! > ,q [1] n = β " cosh( θ R ,n − i ϕ I ) cosh( ϕ R /
2) + sin( θ I ,n + i ϕ R ) sin( ϕ I / θ R ,n ) cosh( ϕ R / − sin( θ I ,n ) sin( ϕ I / e i θ n ,x [1] n = γ na − β s − a ln s (cid:0) cosh( θ R ,n ) cosh( ϕ R / − sin( θ I ,n ) sin( ϕ I / (cid:1) , t = − s, (63)where ξ i = β cosh (cid:20)
12 ( ϕ Ri + i ϕ Ii ) (cid:21) , η i + c = β e ( ϕ Ri +i ϕ Ii ) , χ i + c = − β e − ( ϕ Ri +i ϕ Ii ) .θ R ,n = ln( g )2 n − β (cid:18) ϕ R (cid:19) sin (cid:18) ϕ I (cid:19) s − ϕ R + a R ,θ I ,n = h n + β (cid:18) ϕ R (cid:19) cos (cid:18) ϕ I (cid:19) s − ϕ I + a I , and g = β cosh (cid:0) ϕ R / (cid:1) sin (cid:0) b/ ϕ I / (cid:1) + h β sinh (cid:0) ϕ R / (cid:1) cos (cid:0) b/ ϕ I / (cid:1) + aγ b/ i β cosh (cid:0) ϕ R / (cid:1) sin (cid:0) b/ − ϕ I / (cid:1) + h β sinh (cid:0) ϕ R / (cid:1) cos (cid:0) b/ − ϕ I / (cid:1) + aγ b/ i ,h = arg sin( b ) (cid:0) i aγ − β cosh (cid:2) ( ϕ R + i ϕ I ) (cid:3)(cid:1) + i cos( b )( β sinh (cid:2) ( ϕ R + i ϕ I ) (cid:3) − c )sin( b ) (cid:0) i aγ + β cosh (cid:2) ( ϕ R + i ϕ I ) (cid:3)(cid:1) + i cos( b )( β sinh (cid:2) ( ϕ R + i ϕ I ) (cid:3) − c ) ! . Specially, if we choose the parameters such that β = γ = 1, a = 2, b = π , ϕ R = 0, ϕ I = arcsin( ), a = 0, we canobtain the explicit dynamics (Fig. 3(a)) for the breather solution which is periodical in time and localized in spaceand usually is called the Kuznetsov-Ma (K-M) breather.Furthermore, by using the generalized N -fold DT, we drive the N -breather solution through the formula (35) andsome tedious algebraic calculations as the following proposition Proposition 4
The multi-breather solution for semi-discrete CSP equation (12) can be represented as ρ [ N ] n = γ − a ln s (cid:18) det( M n +1 )det( M n ) (cid:19) ,q [ N ] n = β (cid:20) det( G n )det( M n ) (cid:21) e i θ n ,x [ N ] n = γ an − β s − a ln s det( M n ) , t = − s, (64)4 where M n = e θ ∗ m,n + θ k,n η ∗ m − η k − e θ ∗ m,n η ∗ m − χ k − e θ k,n χ ∗ m − η m + 1 χ ∗ m − χ k ! ≤ m,k ≤ N ,G n = e θ ∗ m,n + θ k,n η ∗ m − η k η ∗ m + cη k + c − e θ ∗ m,n η ∗ m − χ k η ∗ m + cχ k + c − e θ k,n χ ∗ m − η m χ ∗ m + cη k + c + 1 χ ∗ m − χ k χ ∗ m + cχ k + c ! ≤ m,k ≤ N , the parameters θ k,n , η i , χ i are given in equations (49) . Finally, we provide a proof that the above multi-breather solution will converge to the multi-breather solution to theCSP equation obtained in [48]. To this end, we assume a = b → γ → − γ incompared with the dark soliton solution in [48]. By using the Taylor expansion (44), θ i,n becomesi2 ξ i s + n ln sin( b ) (cid:0) i aγ − ξ i (cid:1) + i cos( b ) λ i sin( b ) (cid:0) i aγ + ξ i (cid:1) + i cos( b ) λ i ! + a i ≈ i2 ξ i s − n sin( b ) ξ i i cos( b ) λ i + i aγ sin( b ) + a i = i2 ξ i s + i ξ i λ i y + a i (65)by letting nb = y in the continuous limit b →
0. This shows how the multi-breather solution to semi-discrete CSPequation converges to the multi-breather solution of the CSP equation in the continuous limit.
VI. FUNDAMENTAL AND HIGH-ORDER ROGUE WAVE SOLUTION
In this section, we will derive the general rogue wave solution for the focusing semi-discrete CSP equation based onthe general breather solution obtained in the previous section. Since the solution vectors involve the square root of acomplex number, it is inconvenient to calculate. To avoid this trouble, we introduce the following transformation: λ i + c = β sinh (cid:20)
12 ( ϕ Ri + i ϕ Ii ) (cid:21) , ( ϕ Ri , ϕ Ii ) ∈ Ω , where Ω = { ( ϕ R , ϕ I ) | < ϕ I < π, and 0 < ϕ R < ∞ , or ϕ R = 0 , and π ≤ ϕ I < π } , then ξ i = β cosh (cid:20)
12 ( ϕ Ri + i ϕ Ii ) (cid:21) , η i + c = β e ( ϕ Ri +i ϕ Ii ) , χ i + c = − β e − ( ϕ Ri +i ϕ Ii ) . Actually, we can obtain the rogue wave solution and high order rogue wave solutions in this special point. Thegeneral procedure to yield these solutions was proposed in [66, 67]. If we solve the linear system (17) with ( q n , ρ n , λ ) =( q [0] n , ρ [0] n , − c + i β ), where q [0] n and ρ [0] n are given in equations (47), then the quasi-rational solution vector is obtained.With this solution vector, we could construct the first order rogue wave solution but fails to obtain the high order RWsolutions. To obtain the general high order rogue wave solution with a simple way, we must solve the linear system(17) with ( q n , ρ n , λ ) = ( q [0] n , ρ [0] n , − c + i β cos( ǫ )), where ǫ is a small parameter. Denote λ = − c + i β cos( ǫ ) , ξ = β sin( ǫ ) , η = λ + ξ = − c + i β e − i ǫ , c = − aγ cot( b . (66) Lemma 1
The following parameters can be expanded with ǫ , where ǫ is a small parameter µ = ∞ X n =0 µ [ n ]1 ǫ n +1 ,β i( η ∗ − η ) = 1e i ǫ ∗ + e − i ǫ = ∞ , ∞ X i =0 ,j =0 F [ i,j ] ǫ ∗ i ǫ j , where µ [ n ]1 = β ( − n (2 n + 1)! ,F [ i,j ] = 1 i ! j ! ∂ i + j ∂ǫ ∗ i ∂ǫ j (cid:18)h e i ǫ ∗ + e − i ǫ i − (cid:19) | ǫ ∗ =0 ,ǫ =0 , With the aid of above lemma 1, we obtain the following expansion Z ,n ≡ i β ǫ ) s + n β sin (cid:0) b − ǫ (cid:1) + β sin (cid:0) b + ǫ (cid:1) −
2i cos (cid:0) b (cid:1) aγβ sin (cid:0) b + ǫ (cid:1) + β sin (cid:0) b − ǫ (cid:1) −
2i cos (cid:0) b (cid:1) aγ ! − i ǫ ∞ X i =1 ( e i + i f i ) ǫ i − , = i ǫ ∞ X k =0 Z [2 k +1]1 ,n ǫ k , where Z [2 k +1]1 ,n = d k +1 d ǫ k +1 Z ,n | ǫ =0 . Furthermore we have e Z ,n = ∞ X i =0 S i ( Z ,n ) ǫ i , Z ,n = (cid:16) Z [1]1 ,n , Z [2]1 ,n , · · · (cid:17) , Z [2 k ]1 ,n = 0 , k ≥ S ( Z ,n ) =1 , S ( Z ,n ) = Z [1]1 ,n , S ( Z ,n ) = Z [2]1 ,n + ( Z [1]1 ,n ) , S ( Z ,n ) = Z [3]1 ,n + Z [1]1 ,n Z [2]1 ,n + ( Z [1]1 ,n ) , · · · S i ( Z ,n ) = X l +2 l + ··· + kl k = i ( Z [1]1 ,n ) l ( Z [2]1 ,n ) l · · · ( Z [ k ]1 ,n ) l k l ! l ! · · · l k ! . Since KE ,n ( ǫ ) satisfies the Lax equation (17), then KE ,n ( − ǫ ) also satisfies the Lax equation (17). To obtain thegeneral high order rogue wave solution, we choose the general special solution | y ,n i = K ǫ [ E ,n ( ǫ ) − E ,n ( − ǫ )] ≡ K " ϕ ,n βψ ,n , where E ,n = e Z ,n β e Z ,n η + c , Finally, we have β h y ,n | y ,n i λ ∗ − λ ) = β " e Z ∗ ,n + Z ,n η ∗ − η − e Z ∗ ,n − Z ,n η ∗ − χ − e − Z ∗ ,n + Z ,n χ ∗ − η + e − Z ∗ ,n − Z ,n χ ∗ − χ = ∞ , ∞ X m =1 ,k =1 M [ m,k ] n ǫ ∗ m − ǫ k − (67)and i β h y ,n | y ,n i λ ∗ − λ ) + i βϕ ,n ψ ,n = i β " e Z ∗ ,n + Z ,n η ∗ − η η ∗ + cη + c − e Z ∗ ,n − Z ,n η ∗ − χ η ∗ + cχ + c − e − Z ∗ ,n + Z ,n χ ∗ − η χ ∗ + cη + c + e − Z ∗ ,n − Z ,n χ ∗ − χ χ ∗ + cχ + c = ∞ , ∞ X m =1 ,k =1 G [ m,k ] n ǫ ∗ m − ǫ k − (68)6where χ = η ( − ǫ ) , M [ m,k ] n = m − X i =0 2 k − X j =0 F [ i,j ] S k − i − ( Z ,n ) S m − j − ( Z ∗ ,n ) ,G [ m,k ] n = m − X i =0 2 k − X j =0 F [ i,j ] S k − i − ( Z ,n + ε ) S m − j − ( Z ∗ ,n + ε ) , (69)and ε = (1 , , , · · · ) . Based on the expansion equations (67)-(68), and formulas (35)-(38), we can obtain the general rogue wave solutions:
Proposition 5
The general high order rogue wave solution for semi-discrete CSP equation (12) can be representedas ρ [ N ] n = γ − a ln s (cid:18) det( M n +1 )det( M n ) (cid:19) > ,q [ N ] n = β (cid:20) det( G n )det( M n ) (cid:21) e i θ n ,x [ N ] n = γ an − β s − a ln t det( M n ) , t = − s, (70) where M n = (cid:16) M [ m,k ] n (cid:17) ≤ m,k ≤ N , G n = (cid:16) G [ m,k ] n (cid:17) ≤ m,k ≤ N , the expressions M [ m,k ] n and G [ m,k ] n are given in equations (69) . Specially, the first order rogue wave solution can be written explicitly through formula (70) ρ [1] n = γ − a ln s + ( Z [1] n +1 ,R ) + ( Z [1] n +1 ,I + ) + ( Z [1] n,R ) + ( Z [1] n,I + ) ! ,q [1] n = β " − − Z [1] n,R + ( Z [1] n,R ) + ( Z [1] n,I + ) e i θ ,x [1] n = γ an − β s − a ln s (cid:18)
14 + ( Z [1] n,R ) + ( Z [1] n,I + 12 ) (cid:19) , t = − s, where Z [1] n,R = 4 β sin ( b ) cos( b ) na γ + 2 β sin ( b ) ,Z [1] n,I = β aγ sin ( b ) na γ + 2 β sin ( b ) + s ! − . (71)Moreover, the general second order rogue wave solution can be represented by the formula (70): ρ [2] n = γ − a ln s (cid:18) F ,n +1 + i F ,n +1 F ,n + i F ,n (cid:19) > ,q [2] n = β (cid:20) G n F ,n + i F ,n (cid:21) e i θ n ,x [2] n = γ an − β s − a ln s ( F ,n + i F ,n ) , t = − s, where F ,n = (cid:18) − Z [1] n − Z [3] n + 136 Z [1] n (cid:19) ( Z [1] n ) + (cid:18) Z [1] n − (cid:19) ( Z [1] n ) + (cid:18) − Z [1] n + 124 Z [3] n + 29288 Z [1] n (cid:19) Z [1] n + 14 Z [3] n Z [3] n − Z [3] n Z [1] n − Z [1] n + 124 Z [1] n Z [3] n + 164 , F ,n = 124 Z [1] n Z [1] n + 18 Z [1] n Z [3] n − Z [1] n Z [1] n − Z [1] n Z [3] n + 112 Z [1] n Z [1] n − Z [1] n Z [1] n + 132 Z [1] n − Z [1] n ,G n = 112 i Z [1] n Z [1] n + (cid:18)
112 i Z [1] n − Z [1] n −
14 i Z [3] n + 112 i Z [1] n (cid:19) Z [1] n + (cid:18) Z [3] n − Z [1] n −
16 i Z [1] n − Z [1] n (cid:19) Z [1] n −
14 i Z [1] n Z [3] n + 14 i Z [3] n −
112 i Z [1] n −
112 i Z [1] n ,Z [1] n = Z [1] n,R + i Z [1] n,I , the symbol overbar denotes the complex conjugation and Z [3] n = i β sin ( b ) (cid:0) aβγ sin( b ) + i a γ + 8i β sin ( b ) (cid:1) βa γ sin( b ) − β sin ( b ) − i a γ + 3i aβ γ sin ( b ) n − i βs
24 + ( e + f i) . To illustrate the dynamics of rogue waves, we firstly show the fundamental rogue wave in Fig. 3(b) with theparameters a = 2, b = π , β = 1, γ = . For the second order rogue waves, we firstly choose the parameters a = 2, b = π , β = 1, γ = , e = f = 0, then the standard second order RWs is shown in Fig. 4 (a). To exhibit the otherdynamics for the second order RWs, we choose the parameters a = 2, b = π , β = 1, γ = , e = 10, f = 0. It is seenthat the temporal-spatial distribution exhibit the triangle shape as shown in Fig. 4 (b). (a)K-M breather (b)Fundamental RW FIG. 3: (color online): Breather and Rogue waves (a)Second order RW (b)Second order RW
FIG. 4: (color online): Second order rogue waves with different dynamics
We remark here that since the higher order rogue wave solution is obtained from multi-breather solution to thesemi-discrete CSP equation which converges to its counterpart in the continuous CSP equation, thus, the high orderrogue wave solution for the semi-discrete CSP equation should converge to the one for CSP equation in the continuouslimit.8
VII. CONCLUSIONS AND DISCUSSIONS
In the present paper, we firstly drive the generalized Darbourx transformation (gDT) for the semi-discrete CSPequation (12) with the aid of discrete hodograph transformation. Based on formulas derived from the gDT, we thenconstruct the multi-bright soliton solution for the focusing CSP equation with zero boundary condition. For thenonzero boundary conditions, we construct the multi-dark soliton solution for the defocusing case, the multi-breathersolution and high-order rogue wave solution for the focusing case. We require the condition ρ n > ρ n does not keep the positive definitive property, then the cusp or loopsoliton would appear. All above solutions are shown to converge to their counterparts for the original CSP equationin the continuous limit.It is noticed that a robust inverse scattering transform method has been proposed for the NLS equation by appro-priately setting up and solving the Riemann-Hilbert problem[74, 75]. It is imperative to study the inverse scatteringtransform and Riemann-Hilbert problem for both the original and semi-discrete CSP equation. Although the multi-bright soliton solutions have been constructed by Hirota’s bilinear method in determinant form [63] and in pfaffianform [64], it would be interesting to drive other types of solutions such as dark-soliton, breather and rogue wavesolutions for the semi-discrete CSP equation.In the last, we should point out the following coupled semi-discrete CSP equation ddt ( q ,n +1 − q ,n ) = 12 ( x n +1 − x n )( q ,n +1 + q ,n ) ,ddt ( q ,n +1 − q ,n ) = 12 ( x n +1 − x n )( q ,n +1 + q ,n ) ,ddt ( x n +1 − x n ) + 12 X j =1 σ j (cid:0) | q j,n +1 | − | q j,n | (cid:1) = 0 , (72)which has been shown to be integrable recently [64]. Beside the multi-bright soliton solution implied in [64], howabout its general initial value problem and other types of soliton solutions?The method provided in this paper is also useful to the coupled semi-discrete CSP equation. We expect to obtainand report the results in the near future. As the last conment, The obtained semi-discrete equations can be served assuperior numerical schemes: the so-called self-adaptive moving mesh schemes for the CSP and coupled CSP equations. Acknowledgments
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