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

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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 den- sity 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 quan- tum 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 descrip- tion, 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 transforma- tion for Eq. (5) has been constructed with the achieve- by taking the integration constant as A0.
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