Envelopebright-anddark-solitonsolutionsfortheGerdjikov-Ivanovmodel资源-CSDN文库

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### Gerdjikov-Ivanov模型中的亮暗包络孤子解 #### 摘要与背景本文探讨了Gerdjikov-Ivanov模型中的亮暗包络孤子解。研究基于Madelung流体描述方法，旨在探究一系列非线性薛定谔方程与其对应的Korteweg-de Vries或Korteweg-de Vries型方程之间的联系。通过对当前速度的适当假设，作者推导出了Gerdjikov-Ivanov模型的包络孤子解，并进行了深入讨论。 #### Gerdjikov-Ivanov模型简介 Gerdjikov-Ivanov模型是一种非线性薛定谔方程的推广形式，广泛应用于光学、等离子物理等领域。该模型描述了一类具有特定非线性和色散效应的波包演化行为，能够支持亮暗孤子解的存在。本文聚焦于这类模型在Madelung流体描述下的特性分析。 #### Madelung流体描述及其应用 Madelung流体描述方法将量子力学中的波函数表示为经典流体动力学方程组，其中包含了连续性方程和运动方程。这种方法不仅为量子力学提供了新的视角，也便于运用流体力学理论来分析量子系统的行为。在本文中，通过将Gerdjikov-Ivanov模型转化为Madelung流体描述下的形式，可以更直观地理解亮暗包络孤子解的本质特征。具体来说，当考虑具有固定剖面当前速度的运动时，流体密度满足一个广义的静态Gardner方程。这个方程能够支持多种类型的孤立波（包括亮暗类型、灰色和黑色），这些孤立波是根据相关的参数约束条件得出的。 #### Gerdjikov-Ivanov模型中的包络孤子 - **亮包络孤子**：亮包络孤子通常出现在波包强度高于背景水平的情况下。它们表现为波包强度在传播过程中出现局部增强的现象。 - **暗包络孤子**：与亮包络孤子相反，暗包络孤子出现在波包强度低于背景水平的地方。它们表现为波包强度在传播过程中出现局部减弱的现象。 #### 推导与讨论文章首先介绍了Madelung流体描述的基本原理，并给出了相应的连续性和运动方程。接着，通过假设特定的当前速度剖面，建立了Gerdjikov-Ivanov模型与广义静态Gardner方程之间的联系。这一过程涉及对方程组进行适当的变换，以便求解出亮暗包络孤子解。通过对广义静态Gardner方程的求解，文章得到了Gerdjikov-Ivanov模型的亮暗包络孤子解的具体形式，并对其物理意义进行了阐述。这些解不仅丰富了对Gerdjikov-Ivanov模型的理解，也为其他非线性薛定谔型方程的研究提供了新的思路。 #### 结论本文通过Madelung流体描述方法，成功地推导出了Gerdjikov-Ivanov模型中的亮暗包络孤子解。这些解的存在不仅加深了我们对该模型的理解，还为非线性光学、等离子物理等领域的研究提供了重要的数学工具。未来的研究可以进一步探索这种方法在其他非线性薛定谔型方程中的应用，以期发现更多有趣的现象和解决实际问题的方法。

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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

Abstract Within the context of the Madelung ﬂuid

description, investigation has been carried out on the

connection between the envelope soliton-like solu-

tions of a wide family of nonlinear Schrödinger equa-

tions and the soliton-like solutions of a wide fam-

ily of Korteweg–de Vries or Korteweg–de Vries-type

equations. Under suitable hypothesis for the current

velocity, t he Gerdjikov–Ivanov envelope solitons are

derived and discussed in this paper. For a motion with

the stationary proﬁle current velocity, the ﬂuid den-

sity satisﬁes a generalized stationary Gardner equation,

Xing Lü (

)

Department of Mathematics, Beijing Jiao Tong University,

Beijing 100044, China

e-mail: XLV@bjtu.edu.cn; xinglv655@aliyun.com

X. Lü · W.-X. Ma · J. Yu

Department of Mathematics and Statistics, University

of South Florida, Tampa, FL 33620, USA

J. Yu

Institute of Nonlinear Science, Shaoxing University,

Shaoxing 312000, China

F. Lin

School of Computer and Communication Engineering,

University of Science and Technology Beijing,

Beijing 100083, China

C. M. Khalique

International Institute for Symmetry Analysis and

Mathematical Modelling, Department of Mathematical

Sciences, North-West University, Maﬁkeng Campus,

Private Bag X 2046, Mmabatho 2735, South Africa

which possesses bright- and dark-type (including gray

and black) solitary waves due to associated parametric

constraints, and ﬁnally envelope solitons are found cor-

respondingly for the Gerdjikov–Ivanov model. More-

over, this approach may be useful for studying other

nonlinear Schrödinger-type equations.

Keywords Gerdjikov–Ivanov model · Madelung

ﬂuid description · Solitary wave · Envelope soliton

Mathematics Subject Classiﬁcation 35Q51 ·

35Q55 · 37K40

1 Introduction

The proposal of the concept “quantum potential” is a

valuable and seminal contribution to quantum mechan-

ics [1,2]. The Madelung ﬂuid description of quan-

tum mechanics plays an important role in a number of

applications, e.g., from stochastic mechanics to quan-

tum cosmology and from the pilot waves theory to the

hidden variables theory [ 3–5]. It has been employed

to describe quantum effects in mesoscopic systems

and in plasma physics and discuss quantum aspects of

beam dynamics in high-intensity accelerators (see [6,7]

and references therein). It has also been applied to

such situations where the quantum formalism is a

useful tool for describing the evolution of quantum-

like systems or solving classical nonlinear evolution

equations [8–24].

123

1212 X. Lü et al.

Within the context of Madelung ﬂuid description,

the complex wave function (say Ψ ) is represented in

terms of modulus and phase, and submitted into the

Schrödinger equation to lead to the possible achieve-

ment of a pair of nonlinear ﬂuid equations for the

“density” ρ =|Ψ |

and the “current velocity” v =

∇Arg(Ψ ): one is the continuity equation (taking into

account probability conservation) and the other one is

a Navier-Stokes-like equation of motion [6,7,15–20].

Substitution of Ψ(x, t) =

√

ρ(x, t) e

Θ(x ,t)

into the

one-dimensional Schrödinger equation

∂Ψ(x, t)

∂t

=−

2 m

∂

Ψ(x, t)

∂x

+mU(x)Ψ (x, t), (1)

results in the following pair of coupled Madelung ﬂuid

equations

∂ρ

∂t

∂

∂x

(ρ v) = 0, (2a)

∂v

∂t

+ v

∂v

∂x

−

2 m

∂

∂x



√

∂

√

∂x



∂U

∂x

= 0,

(2b)

where ρ =|Ψ |

is the ﬂuid density and v =

∂Θ

∂x

is the current velocity. Equation (2a) is a continuity

equation for the ﬂuid density, while Eq. (2b) is an Euler

equation or equation of motion for the ﬂuid velocity and

contains a force term proportional to the gradient of the

“quantum potential”,

2 m

∂

∂x



√

∂

√

∂x



Madelung ﬂuid description has been used to dis-

cuss families of generalized one-dimensional nonlinear

Schrödinger equations [15–19], as follows:

• In Ref. [15], the following nonlinear Schrödinger-

like equation

i μ

∂Ψ

∂t

=−

∂

∂x

+ U



|Ψ |



Ψ, (3)

where U



|Ψ |



and μ are as an arbitrary real func-

tional of the complex wave function Ψ and an arbi-

trary real dispersion/diffraction coefﬁcient, respec-

tively, has been cast as

∂ρ

∂t

∂

∂x

(

ρv

)

= 0, (4a)

−ρ

∂v

∂t

+ v

∂ρ

∂t

+ 2



(t) −



∂v

∂t



∂ρ

∂x

−



dρ

+ 2U



∂ρ

∂x

∂

∂x

= 0, (4b)

with c

(t) as an arbitrary function of t. Under the

hypothesis of stationary ﬂuid, it is revealed that the

cubic nonlinear Schrödinger equation can be put

in correspondence with the standard Korteweg–de

Vries equation in such a way that the soliton solu-

tions of the latter are the squared modulus of the

envelope soliton solution of the former. The condi-

tions for different types of envelope solitons (bright,

dark or gray ones) are also discussed.

• In Ref. [16], a modiﬁed nonlinear Schrödinger

equation with a quartic nonlinear potential in the

modulus of the wave function has been studied

within the framework of Madelung ﬂuid descrip-

tion, i.e., setting U



|Ψ |



= q

|Ψ |

with

and q

as real constants in Eq. (3). By consider-

ing different boundary conditions, up-shifted bright

soliton, upper-shifted bright soliton, gray soliton

and dark soliton are ﬁnally found and can be cast

into envelope solitons conversely.

• In Ref. [17], the existence of envelope soliton-like

solutions of a nonlinear Schrödinger equation con-

taining an anti-cubic nonlinearity (|Ψ |

−4

Ψ )plus

a ‘regular’ nonlinear part is investigated. In par-

ticular, in the case that the regular nonlinear part

consists of a sum of cubic and quintic nonlineari-

ties, i.e., setting U



|Ψ |



= Q

|Ψ |

−4

|Ψ |

with Q

, q

and q

as real constants in

Eq. (3), an upper-shifted bright envelope soliton-

like solution is explicitly found.

• In Ref. [18], a review is given on the results of inves-

tigations dealing with the connection between the

envelope soliton-like solutions of a wide family of

nonlinear Schrödinger equations and the soliton-

like solutions of a wide family of Korteweg–

de Vries equations. In two different ﬂuid motion

regimes (uniform current velocity and stationary

proﬁle current velocity variation, respectively),

bright- and gray-/dark-soliton-like solutions of

those equations are found.

• In Ref. [19], a similar discussion is given for a class

of derivative nonlinear Schrödinger-type equations.

For a motion with the stationary proﬁle current

velocity, the ﬂuid density satisﬁes a generalized sta-

tionary Gardner equation, and solitary wave solu-

tions are found. For the completely integrable cases,

these solutions are compared with existing solu-

tions in the literature.

In the present paper, we will investigate the

Gerdjikov–Ivanov envelope solitons with the frame-

work of Madelung ﬂuid description. Various optical

123

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