Envelope Bright- and Dark-Soliton Solutions for the Gerdjikov–Ivanov Model

Nonlinear Dyn (2015) 82:1211–1220 DOI 10.1007/s11071-015-2227-6

ORIGINAL PAPER

Envelope bright- and dark-soliton solutions for the Gerdjikov–Ivanov model

Xing Lü · Wen-Xiu Ma · Jun Yu · Fuhong Lin · Chaudry Masood Khalique

Received: 20 April 2015 / Accepted: 14 June 2015 / Published online: 11 July 2015 © Springer Science+Business Media Dordrecht 2015

Abstract Within the context of the Madelung fluid which possesses bright- and dark-type (including gray description, investigation has been carried out on the and black) solitary waves due to associated parametric connection between the envelope soliton-like solu- constraints, and finally envelope solitons are found cor- tions of a wide family of nonlinear Schrödinger equa- respondingly for the Gerdjikov–Ivanov model. More- tions and the soliton-like solutions of a wide fam- over, this approach may be useful for studying other ily of Korteweg–de Vries or Korteweg–de Vries-type nonlinear Schrödinger-type equations. equations. Under suitable hypothesis for the current velocity, the Gerdjikov–Ivanov envelope solitons are Keywords Gerdjikov–Ivanov model · Madelung derived and discussed in this paper. For a motion with fluid description · Solitary wave · Envelope soliton the stationary profile current velocity, the fluid density satisfies a generalized stationary Gardner equation, Mathematics Subject Classification 35Q51 · 35Q55 · 37K40

Xing Lü (B) Department of Mathematics, Beijing Jiao Tong University, Beijing 100044, China 1 Introduction e-mail: [email protected]; [email protected] The proposal of the concept “quantum potential” is a X. Lü · W.-X. Ma · J. Yu Department of Mathematics and Statistics, University valuable and seminal contribution to quantum mechan- of South Florida, Tampa, FL 33620, USA ics [1,2]. The Madelung ﬂuid description of quantum mechanics plays an important role in a number of J. Yu applications, e.g., from stochastic mechanics to quan- Institute of Nonlinear Science, Shaoxing University, Shaoxing 312000, China tum cosmology and from the pilot waves theory to the hidden variables theory [3–5]. It has been employed F. Lin to describe quantum effects in mesoscopic systems School of Computer and Communication Engineering, and in plasma physics and discuss quantum aspects of University of Science and Technology Beijing, Beijing 100083, China beam dynamics in high-intensity accelerators (see [6,7] and references therein). It has also been applied to C. M. Khalique such situations where the quantum formalism is a International Institute for Symmetry Analysis and useful tool for describing the evolution of quantum- Mathematical Modelling, Department of Mathematical Sciences, North-West University, Maﬁkeng Campus, like systems or solving classical nonlinear evolution Private Bag X 2046, Mmabatho 2735, South Africa equations [8–24]. 123 1212 X. Lü et al.

Within the context of Madelung fluid description, cubic nonlinear Schrödinger equation can be put the complex wave function (say Ψ ) is represented in in correspondence with the standard Korteweg–de terms of modulus and phase, and submitted into the Vries equation in such a way that the soliton solu- Schrödinger equation to lead to the possible achieve- tions of the latter are the squared modulus of the ment of a pair of nonlinear fluid equations for the envelope soliton solution of the former. The condi- “density” ρ =|Ψ |2 and the “current velocity” v = tions for different types of envelope solitons (bright, ∇Arg(Ψ ): one is the continuity equation (taking into dark or gray ones) are also discussed. account probability conservation) and the other one is • In Ref. [16], a modified nonlinear Schrödinger a Navier-Stokes-like equation√ of motion [6,7,15–20]. equation with a quartic nonlinear potential in the i Θ(x,t) Substitution of Ψ(x, t) = ρ(x, t) e h¯ into the modulus of the wave function has been studied one-dimensional Schrödinger equation within the framework of Madelung fluid description, i.e., setting U |Ψ |2 = q |Ψ |2 +q |Ψ |4 with ∂Ψ(x, t) h¯ 2 ∂2Ψ(x, t) 1 2 i h¯ =− +mU(x)Ψ (x, t), (1) q and q as real constants in Eq. (3). By consider- ∂ ∂ 2 1 2 t 2 m x ing different boundary conditions, up-shifted bright results in the following pair of coupled Madelung fluid soliton, upper-shifted bright soliton, gray soliton equations and dark soliton are finally found and can be cast ∂ρ ∂ + (ρ v) = 0, (2a) into envelope solitons conversely. ∂t ∂x • In Ref. [17], the existence of envelope soliton-like √ ∂v ∂v h¯ 2 ∂ 1 ∂2 ρ ∂U solutions of a nonlinear Schrödinger equation con- + v − √ + = 0, |Ψ |−4Ψ ∂t ∂x 2 m2 ∂x ρ ∂x2 ∂x taining an anti-cubic nonlinearity ( )plus a ‘regular’ nonlinear part is investigated. In par- (2b) ticular, in the case that the regular nonlinear part ρ =|Ψ |2 v = 1 ∂Θ where is the fluid density and m ∂x consists of a sum of cubic and quintic nonlineari- 2 −4 2 is the current velocity. Equation (2a) is a continuity ties, i.e., setting U |Ψ | = Q0|Ψ | + q1|Ψ | + 4 equation for the fluid density, while Eq. (2b) is an Euler q2|Ψ | with Q0, q1 and q2 as real constants in equation or equation of motion for the fluid velocity and Eq. (3), an upper-shifted bright envelope soliton- contains a force term proportional to√ the gradient of the like solution is explicitly found. 2 ∂ ∂2 ρ “quantum potential”, h¯ √1 . • In Ref. [18], a review is given on the results of inves- 2 m2 ∂x ρ ∂x2 Madelung fluid description has been used to dis- tigations dealing with the connection between the cuss families of generalized one-dimensional nonlinear envelope soliton-like solutions of a wide family of Schrödinger equations [15–19], as follows: nonlinear Schrödinger equations and the soliton- like solutions of a wide family of Korteweg– • In Ref. [15], the following nonlinear Schrödinger- de Vries equations. In two different fluid motion like equation regimes (uniform current velocity and stationary ∂Ψ μ2 ∂2Ψ profile current velocity variation, respectively), i μ =− + U |Ψ |2 Ψ, (3) ∂ ∂ 2 t 2 x bright- and gray-/dark-soliton-like solutions of where U |Ψ |2 and μ are as an arbitrary real func- those equations are found. tional of the complex wave function Ψ and an arbi- • In Ref. [19], a similar discussion is given for a class trary real dispersion/diffraction coefficient, respec- of derivative nonlinear Schrödinger-type equations. tively, has been cast as For a motion with the stationary profile current ∂ρ ∂ velocity, the fluid density satisfies a generalized sta- + (ρv) = , ∂ ∂ 0 (4a) tionary Gardner equation, and solitary wave solu- t x ∂v ∂ρ ∂v ∂ρ tions are found. For the completely integrable cases, −ρ + v + 2 c (t) − dx these solutions are compared with existing solu- ∂t ∂t 0 ∂t ∂x tions in the literature. dU ∂ρ μ2 ∂3ρ − ρ + 2U + = 0, (4b) dρ ∂x 4 ∂x3 In the present paper, we will investigate the with c0(t) as an arbitrary function of t. Under the Gerdjikov–Ivanov envelope solitons with the frame- hypothesis of stationary fluid, it is revealed that the work of Madelung fluid description. Various optical 123 Envelope bright- and dark-soliton solutions for the Gerdjikov–Ivanov model 1213 √ envelope soliton problems have been proposed and ∂v 2 ∂v ∂ 1 ∂2 ρ + v = k √ studied in the series literature [25–34]. In Sect. 2,we ∂t k ∂x ∂x ρ ∂x2 will pay attention to the Gerdjikov–Ivanov model and ∂ 2 derive the basic equations of Madelung fluid: the conti- + ρv+ ρ2 . ∂ (6b) nuity equation for the fluid density and motion equation x k for the fluid velocity. Symbolic-computation manipulation on Eq. (6b)gives It will be found in Sect. 3 that for a motion with rise to the stationary profile current velocity, the fluid den- ∂ρ ∂v ∂ρ ∂v ∂ρ sity satisfies a generalized stationary Gardner equation. − v + ρ − ρv − ρ2 − k ρ2 ∂ ∂ 2 ∂ ∂ 2 ∂ Under suitable hypothesis for the current velocity and t t x x x k ∂3ρ ∂ρ ∂ρ ∂v due to associated parametric constraints, in Sect. 4,we − + 2 c (t) + 2 dx = 0, (6c) ∂ 3 0 ∂ ∂ ∂ will present bright- and dark-type (including gray and 2 x x x t black) solitary waves for the stationary Gardner equa- where c (t) is an arbitrary function of t. tion, and finally, associated envelope solitons will be 0 Hereby, Eqs. (6a) and (6c) constitute the basic equa- constructed correspondingly for the Gerdjikov–Ivanov tions for the subsequent discussion. model. Sect. 5 will be our conclusions.

3 Motion with the stationary profile current 2 Gerdjikov–Ivanov model and basic equations velocity ρ v The Gerdjikov–Ivanov model studied hereby is given Assuming that both the quantities and involved as [35–38] in Eqs. (6a) and (6b) are functions of the combined variable ξ = x − u0 t with u0 being a real constant, we ∗ ∂Ψ ∂2Ψ ∂Ψ 1 can cast Eq. (6a)into i + − i Ψ 2 + |Ψ |4Ψ = 0, (5) ∂t ∂x2 ∂x 2 dρ d 2 1 where ∗ denotes the complex conjugation and Ψ = −u + ρv− ρ2 = , 0 ξ ξ 0 (7) Ψ(x, t) is the complex wave function. Eq. (5)isalso d d k 2 called the derivative nonlinear Schrödinger III equation and integrate once of Eq. (7) with respect to ξ to obtain and can be transformed into the Kaup–Newell or the Chen–Lee–Liu equation by corresponding gauge trans- k 1 A0 formations [37,38]. With the help of gauge transfor- v = u0 + ρ + , (8) 2 2 ρ mation of the spectral problem, a Darboux transformation for Eq. (5) has been constructed with the achieve- by taking the integration constant as A0. Substitution ment of the explicit soliton-like solutions in Ref. [37]. of Eq. (8) into Eq. (6c) leads to In Ref. [38], the spectral problem with the associated Gerdjikov–Ivanov hierarchy of nonlinear evolu- k k dρ 3k dρ u2 − A + 2 c − u ρ tion equations is presented, and its bi-Hamiltonian 2 0 2 0 0 dξ 2 0 dξ structure, finite-dimensional integrable systems and N- 11k dρ k dρ3 fold Darboux transformation have also been studied. − ρ2 − = . √ 3 0 (9) i Θ(x,t) 4 dξ 2 dξ Introducing Ψ(x, t) = ρ(x, t) e k with k = 0 into Eq. (5), we obtain the continuity equation for the Remark (a) According to Eq. (8), it is clear that v = fluid density ρ as constant no matter A0 = 0 or not. That is to say, we can only consider the motion with the stationary profile ∂ρ ∂ 2 1 + ρv− ρ2 = 0, (6a) current velocity case and the motion with the constant ∂t ∂x k 2 current velocity case is excluded, which is different from that in Refs. [15–18]. and the equation of motion for the fluid velocity v = (b) Note that the fluid density ρ satisfies a ∂Θ ∂x as generalized stationary Gardner equation, i.e., Eq. (9), 123 1214 X. Lü et al.

1 which is not the Korteweg–de Vries or modiﬁed Setting ρ = in Eq. (11) generates Korteweg–de Vries equation, but a mixed one. ϕ dϕ 2 11 4 =− − u ϕ + u2 + c ϕ2. (12) 4 Solitary waves versus envelope solitons dξ 12 0 0 k 0 2 + 4 > To be followed, with the constraint ρ>0 and under With the constraint u0 k c0 0, the second-order suitable assumptions for the current velocity associated polynomial in the right-hand side of Eq. (12) has two ϕ ϕ with corresponding boundary conditions of ρ, we will real roots, one positive ( 2) and the other negative ( 1). investigate different types of solitary waves for Eq. (9). Because the right-hand side of Eq. (12) has to be pos- ρ = Case I: In case ρ satisﬁes the boundary conditions itive and limϕ→+∞ 0, the region of interest on ϕ ϕ ∈ (ϕ , +∞) in the ξ-space as limξ→±∞ ρ(ξ) = 0, it follows from -axis is 2 . Writing Eq. (12)as = v = k ( + 1 ρ) ϕ Eq. (8) that A0 0 and u0 . d 2 4 2 2 = (u + c0)(ϕ − ϕ2)(ϕ − ϕ1), (13) Then, Eq. (9) becomes dξ 0 k we can solve Eq. (13) and have ρ ρ ρ ρ3 k 2 d 3k d 11k 2 d k d u +2 c0 − u0ρ − ρ − 2 0 dξ 2 dξ 4 dξ 2 dξ 3 2 2 + = , 42 u k 132 kc0 0 (10) ϕ = ku0 + 0 2 2 2 ku + 8 c0 6 ku + 24 c0 which can be integrated twice with respect to ξ and take 0 0 4 the integration constant as zero to give × Cosh u2 + c (ξ + ξ ) , (14) 0 k 0 0 dρ 2 4 11 = u2 + c ρ2 − u ρ3 − ρ4. (11) dξ 0 k 0 0 12 and then the solution of Eq. (11) is obtained as

ρ = 1 = 1 , ϕ 2 2 + ku 42 u0 k 132 kc0 4 (15) 0 + Cosh u2 + c (ξ + ξ ) 2 + 2 + 0 0 0 2 ku0 8 c0 6 ku0 24 c0 k

and

⎛ ⎞ ⎜ ⎟ ⎜ ⎟ dΘ k ⎜ 1 ⎟ v = = ⎜u + ⎟ , ξ ⎜ 0 ⎟ (16) d 2 ⎜ + 2 2 ⎟ ⎝ ku 132 kc0 42 u0 k 4 ⎠ 0 + 2 + (ξ + ξ ) 2 2 Cosh u0 c0 0 4 c0 + ku 12 c0 + 3 ku k 0 0 k k k k Θ(x, t) = u ξ + ρ dξ − 2 c t − Θ = u x − u2 + 2 c t 2 0 4 0 0 2 0 2 0 0 132 c + 42 ku2 − 3 k − 0 0 k ArcTan 3 u0 2 2 11 132 c0 + 33 ku 132 c0 + 33 ku 0 0 1 4 u 4 ξ × Tanh u2 + c x − 0 u2 + c t + 0 − Θ , (17) 2 0 k 0 2 0 k 0 2 0

123 Envelope bright- and dark-soliton solutions for the Gerdjikov–Ivanov model 1215 where ξ0 and Θ0 are integration constants. Attention should be paid to the sign of ρ1, which (can Note that the solitary wave solution of the sta- be negative or positive) is different from the sign of tionary Gardner equation [i.e., Eq. (10)] possesses ρ in Case I. This inspires us to study other types of a bright-soliton proﬁle [see Expression (15) above]. solutions. Finally, the bright-soliton-type envelope solution of Integrating Eq. (18) twice with respect to ξ, taking the Gerdjikov–Ivanov model can be expressed as 1 √ the integration constants as zero and setting ρ1 = , i Θ(x,t) ϕ Ψ(x, t) = ρ(x, t) e k with ρ(x, t) and Θ(x, t) 1 we get given by Eqs. (15) and (17), respectively. 2 Case II: ξ→±∞ ρ(ξ) = dϕ 4 11 For the case of lim 0, we 1 = 2 − + − ρ − ρ2 ϕ2 u0 A0 c0 3 u0 0 0 1 denote ρ(ξ) = ρ0 + ρ1(ξ) with limξ→±∞ ρ1(ξ) = 0 dξ k 2 ρ = > v = and 0 constant 0, which determines that 11 11 k 1 A0 − u0 + ρ0 ϕ1 − . (19) u0 + (ρ0 + ρ1(ξ)) + . 3 12 2 2 ρ0 + ρ1(ξ) Substituting ρ(ξ) = ρ + ρ (ξ) into Eq. (9), we 4 11 0 1 With the constraint u2 − A + c −3 u ρ − ρ2 > have 0 0 k 0 0 0 2 0 0, the second-order polynomial in the right-hand side of Eq. (19) has two real roots, as follows, ρ + + − 2 − ρ2 2 − ρ 2 + 2 2 22 k 0 6 ku0 528 kc0 132 A0 k 242 0 k 132 0 u0 k 168 u0 k α = > 0, 48 c − 12 A k − 66 k ρ2 − 36 ku ρ + 12 ku2 0 0 0 0 0 0 ρ + − − 2 − ρ2 2 − ρ 2 + 2 2 22 k 0 6 ku0 528 kc0 132 A0 k 242 0 k 132 0 u0 k 168 u0 k β = < 0. − − ρ2 − ρ + 2 48 c0 12 A0 k 66 k 0 36 ku0 0 12 ku0

k k 3 11 dρ u2 − A + c − ku ρ − k ρ2 1 Case II-I: With the assumption ρ > 0, we take the 0 0 2 0 0 0 0 ξ 1 2 2 2 4 d solution of Eq. (19)as 3 11 dρ 11 dρ − ku + k ρ ρ 1 − k ρ2 1 2 0 2 0 1 dξ 4 1 dξ α + β α − β ϕ1 = + Cosh k dρ3 2 2 − 1 = 0, (18) ξ 3 4 11 2 d u2 − A + c − 3 u ρ − ρ2 (ξ − ξ ) 0 0 k 0 0 0 2 0 01 which is also a generalized stationary Gardner equa- ≥ α>0. tion and can be similarly discussed as that in Case I. Therefore,

1 1 ρ = = , 1 ϕ (20) 1 α + β α − β 4 11 + Cosh u2 − A + c − 3 u ρ − ρ2 (ξ − ξ ) 2 2 0 0 k 0 0 0 2 0 01 1 ρ = ρ0 + , (21) α + β α − β 4 11 + Cosh u2 − A + c − 3 u ρ − ρ2 (ξ − ξ ) 2 2 0 0 k 0 0 0 2 0 01

123 1216 X. Lü et al.

k 1 kA k 1 kA ξ Θ(x, t) = u + ρ + 0 x − u2 + ρ u + 2 c t − 0 01 − Θ 2 0 4 0 2 ρ 2 0 4 0 0 0 2 ρ 01 0 0 β 1 4 11 ArcTanh Tanh u2 − A + c − 3 u ρ − ρ2 (x − u t − ξ ) α 2 0 0 k 0 0 0 2 0 0 01 + 4 11 2 αβ(u2 − A + c − 3 u ρ − ρ2) 0 0 k 0 0 0 2 0 1 + ρ β 1 4 11 0 2 − + − ρ − ρ2 ( − − ξ ) kA0 ArcTanh Tanh u0 A0 c0 3 u0 0 0 x u0 t 01 1 + ρ0 α 2 k 2 − , 4 11 ρ (1 + ρ α)(1 + ρ β)(u2 − A + c − 3 u ρ − ρ2) 0 0 0 0 0 k 0 0 0 2 0 i Θ(x,t) Ψ(x, t) = ρ(x, t) e k , (22)

Correspondingly, via Expression (23), i.e., differ- where Θ and ξ are two integration constants. 01 01 ent forms and types of the solitary waves for station- It is clear that in Expression (20), the solution ary Gardner equation, we can investigate the associ- ρ enjoys a bright-soliton proﬁle. Based on Expres- 1 ated different types of envelope solitons for the origi- sion (21), we can discuss different cases of solutions nal Gerdjikov–Ivanov model, i.e., Eq. (5) by means of for Eq. (9), correspondingly for the original Eq. (5). Expression (22) (details ignored here). Under the condition 3 u0 + 11 ρ0 = 0, Expres- sion (21) reduces to 4 11 ρ = ρ 1 + ε Sech u2 − A + c − 3 u ρ − ρ2 (x − u t − ξ ) (23) 0 0 0 k 0 0 0 2 0 0 01

288 c k−72 A k2−396 k2 ρ2−216 k2 u ρ +72 k2 u2 ρ < with ε= 0 0 0 0 0 0 . Case II-II: With the assumption 1 0, we take the ρ − 2− ρ2 2− ρ 2+ 2 2 0 528 c0 k 132 A0 k 242 0 k 132 0 u0 k 168 u0 k solution of Eq. (19)as α + β α − β • When 0 <ε<1, Expression (23) represents a ϕ1 = − Cosh up-shifted bright soliton, whose maximum ampli- 2 2 ρ ( + ε) tude is 0 1 and up-shifted by the quantity 2 4 11 2 u − A0 + c0 − 3 u0 ρ0 − ρ (ξ − ξ02) ρ0. 0 k 2 0 • When ε = 1, Expression (23) represents a upper- ≤ β<0. shifted bright soliton, whose maximum amplitude is 2 ρ0 and up-shifted by the quantity ρ0. Thus,

1 1 ρ = = , 1 ϕ (24) 1 α + β α − β 4 11 − Cosh u2 − A + c − 3 u ρ − ρ2 (ξ − ξ ) 2 2 0 0 k 0 0 0 2 0 02 1 ρ = ρ0 + , (25) α + β α − β 4 11 − Cosh u2 − A + c − 3 u ρ − ρ2 (ξ − ξ ) 2 2 0 0 k 0 0 0 2 0 02

123 Envelope bright- and dark-soliton solutions for the Gerdjikov–Ivanov model 1217

ξ Θ( , ) = k + 1 ρ + kA0 − k 2 + 1 ρ + − kA0 02 − Θ x t u0 0 x u0 0 u0 2 c0 t 02 2 4 2 ρ0 2 4 2 ρ0 α 1 4 11 ArcTanh Tanh u2 − A + c − 3 u ρ − ρ2 (x − u t − ξ ) β 2 0 0 k 0 0 0 2 0 0 02 + 4 11 2 αβ(u2 − A + c − 3 u ρ − ρ2) 0 0 k 0 0 0 2 0 1 + ρ α 1 4 11 0 2 − + − ρ − ρ2 ( − − ξ ) kA0 ArcTanh Tanh u0 A0 c0 3 u0 0 0 x u0 t 02 1 + ρ0 β 2 k 2 − , 4 11 ρ (1 + ρ α)(1 + ρ β)(u2 − A + c − 3 u ρ − ρ2) 0 0 0 0 0 k 0 0 0 2 0 i Θ(x,t) Ψ(x, t) = ρ(x, t) e k , (26)

where Θ02 and ξ02 are two integration constants. Gardner equation, we can investigate the associated Notice that in Expression (24), the solution ρ1 enjoys a non-bright-soliton proﬁle. Based on Expression (25), different types of envelope solitons for the original we can discuss different cases of solution for Eq. (9), Gerdjikov–Ivanov model, i.e., Eq. (5) by means of correspondingly for the original Eq. (5). Expression (26) (details ignored here). Under the following condition

⎧ + ρ = , ⎨⎪ 3 u0 11 0 0 48 c − 12 A k − 66 k ρ2 − 36 ku ρ + 12 ku2 ρ > 0 0 0 0 0 0 , (27) ⎩⎪ 0 − ρ − + − 2 − ρ2 2 − ρ 2 + 2 2 22 k 0 6 ku0 528 kc0 132 A0 k 242 0 k 132 0 u0 k 168 u0 k

Expression (25) reduces to

4 11 ρ = ρ 1 − δ Sech u2 − A + c − 3 u ρ − ρ2 (x − u t − ξ ) (28) 0 0 0 k 0 0 0 2 0 0 01

288 c k−72 A k2−396 k2 ρ2−216 k2 u ρ +72 k2 u2 with δ = 0 0 0 0 0 0 . ρ − 2− ρ2 2− ρ 2+ 2 2 0 528 c0 k 132 A0 k 242 0 k 132 0 u0 k 168 u0 k 5 Conclusions • When 0 <δ<1, Expression (28) denotes a gray With the application of Madelung fluid description soliton, whose minimum amplitude is ρ (1−δ)and 0 finding solutions for nonlinear Schrödinger equations, reaches asymptotically the upper limit ρ . 0 fruitful results have been obtained. It is proven that • When δ = 1, Expression (28) denotes a black soli- the Madelung fluid description is a remarkably useful ton, whose minimum amplitude is zero and reaches approach. In particular, on the basis of the present the- asymptotically the upper limit ρ . 0 ory, we have derived bright- and dark-type (including Correspondingly, via Expression (28), i.e., different gray and black) soliton solutions for the Gerdjikov– forms and types of the solitary waves for the stationary Ivanov model, by solving the corresponding stationary 123 1218 X. Lü et al.

Table 1 Solitary wave types Solitary wave types for stationary Gardner Eq. (9) Parametric constraints 2 4 u + c0 > 0; bright-soliton-type 0 k A0 =0; ρ0 > 0; bright-soliton-type 4 11 u2 − A + c − 3 u ρ − ρ2 > 0; 0 0 k 0 0 0 2 0 ρ0 > 0; 0 <ε<1; up-shifted-bright-type 3 u0 +11ρ0 =0; 4 11 u2 − A + c − 3 u ρ − ρ2 > 0; 0 0 k 0 0 0 2 0 ρ0 > 0; ε =1; upper-shifted-bright-type 3 u0 +11ρ0 =0; 4 11 u2 − A + c − 3 u ρ − ρ2 > 0; 0 0 k 0 0 0 2 0 0 <δ<1; gray-soliton-type Expression (27); 4 11 u2 − A + c − 3 u ρ − ρ2 > 0; 0 0 k 0 0 0 2 0 δ =1; black-soliton-type Expression (27); 4 11 u2 − A + c − 3 u ρ − ρ2 > 0; 0 0 k 0 0 0 2 0

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