scholarly journals On the Effect of Electron Streaming and Existence of Quasi-Solitary Mode in a Strongly Coupled Quantum Dusty Plasma—Far and Near Critical Nonlinearity

Plasma ◽  
2021 ◽  
Vol 4 (3) ◽  
pp. 408-425
Author(s):  
Shatadru Chaudhuri ◽  
Asesh Roy Chowdhury
Keyword(s):  
Dusty Plasma ◽  
Solitary Wave Solution ◽  
Kdv Equation ◽  
Elliptic Functions ◽  
Mkdv Equation ◽  
Perturbation Approach ◽  
Strongly Coupled ◽  
Critical Situation ◽  
The Kdv Equation ◽  
Reductive Perturbation

As strongly coupled quantum dusty plasma consisting of electrons and dust with the ions in the background is considered when there is a streaming of electrons. It is observed that the streaming gives rise to both the slow and fast modes of propagation. The nonlinear mode is then analyzed by the reductive perturbation approach, resulting in the KdV-equation. In the critical situation where non-linearity vanishes, the modified scaling results in the MKdV equation. It is observed that both the KdV and MKdV equations possess quasi-solitary wave solution, which not only has the character of a soliton but also has a periodic nature. Such type of solitons are nowadays called nanopteron solitons and are expressed in terms of cnoidal-type elliptic functions.

Nonlocal Symmetries, Explicit Solutions, and Wave Structures for the Korteweg–de Vries Equation

2016 ◽  
Vol 71 (8) ◽  
pp. 735-740
Author(s):  
Zheng-Yi Ma ◽  
Jin-Xi Fei
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Elliptic Functions ◽  
Group Method ◽  
Physical Interaction ◽  
Nonlocal Symmetries ◽  
Interaction Solutions ◽  
The Kdv Equation ◽  
De Vries ◽  
Exact Invariant

AbstractFrom the known Lax pair of the Korteweg–de Vries (KdV) equation, the Lie symmetry group method is successfully applied to find exact invariant solutions for the KdV equation with nonlocal symmetries by introducing two suitable auxiliary variables. Meanwhile, based on the prolonged system, the explicit analytic interaction solutions related to the hyperbolic and Jacobi elliptic functions are derived. Figures show the physical interaction between the cnoidal waves and a solitary wave.

scholarly journals Solutions and Painlevé Property for the KdV Equation with Self-Consistent Source

2018 ◽  
Vol 2018 ◽  
pp. 1-16
Author(s):  
Yali Shen ◽  
Ruoxia Yao
Keyword(s):  
Kdv Equation ◽  
Traveling Wave Solutions ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Hyperbolic Function ◽  
Group Invariant ◽  
Painlevé Property ◽  
The Kdv Equation ◽  
Self Consistent ◽  
Group Invariant Solutions

In this paper, the polynomial solutions in terms of Jacobi’s elliptic functions of the KdV equation with a self-consistent source (KdV-SCS) are presented. The extended (G′/G)-expansion method is utilized to obtain exact traveling wave solutions of the KdV-SCS, which finally are expressed in terms of the hyperbolic function, the trigonometric function, and the rational function. Meanwhile we find the Lie point symmetry and Lie symmetry group and give several group-invariant solutions for the KdV-SCS. Finally, we supplement the results of the Painlevé property in our previous work and get the Bäcklund transformations of the KdV-SCS.

scholarly journals Rigorous Asymptotics of a KdV Soliton Gas

2021 ◽  
Author(s):  
M. Girotti ◽  
T. Grava ◽  
R. Jenkins ◽  
K. D. T.-R. McLaughlin
Keyword(s):  
Hilbert Problem ◽  
Kdv Equation ◽  
Broad Class ◽  
Elliptic Functions ◽  
Jacobi Elliptic Functions ◽  
Cnoidal Wave ◽  
Large Space ◽  
The Kdv Equation ◽  
Long Time ◽  
Riemann Hilbert Problem

AbstractWe analytically study the long time and large space asymptotics of a new broad class of solutions of the KdV equation introduced by Dyachenko, Zakharov, and Zakharov. These solutions are characterized by a Riemann–Hilbert problem which we show arises as the limit $$N\rightarrow + \infty $$ N → + ∞ of a gas of N-solitons. We show that this gas of solitons in the limit $$N\rightarrow \infty $$ N → ∞ is slowly approaching a cnoidal wave solution for $$x \rightarrow - \infty $$ x → - ∞ up to terms of order $$\mathcal {O} (1/x)$$ O ( 1 / x ) , while approaching zero exponentially fast for $$x\rightarrow +\infty $$ x → + ∞ . We establish an asymptotic description of the gas of solitons for large times that is valid over the entire spatial domain, in terms of Jacobi elliptic functions.

Periodic and localized structures in dusty plasma with Kaniadakis distribution

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Muhammad Khalid ◽  
Mohsin Khan ◽  
Muddusir ◽  
Ata-ur-Rahman ◽  
Muhammad Irshad
Keyword(s):  
Dusty Plasma ◽  
Kdv Equation ◽  
Mkdv Equation ◽  
Reductive Perturbation Method ◽  
Periodic Waves ◽  
Cnoidal Wave ◽  
Reverse Effect ◽  
Cnoidal Waves ◽  
Localized Structures ◽  
De Vries

Abstract The propagation of electrostatic dust-ion-acoustic nonlinear periodic waves is investigated in dusty plasma wherein electrons follow Kaniadakis distribution. The Korteweg–de Vries (KdV) and modified Korteweg–de Vries (mKdV) equations are derived by employing reductive perturbation method and their cnoidal wave solutions are analysed. The effect of relevant parameters (viz., κ-deformed parameter κ and dust concentration β) on the dynamics of cnoidal structures is discussed. Further it is found that amplitude of compressive cnoidal waves increases with increasing values of β, while reverse effect is observed in case of rarefactive cnoidal structures with rising values of β. Also κ-deformed parameter κ bears no effect on cnoidal waves associated with KdV equation, whereas κ-deformed parameter κ significantly affects the cnoidal waves associated with mKdV equation.

Longitudinal dust acoustic solitary waves in a strongly coupled complex (dusty) plasma

2015 ◽  
Vol 81 (3) ◽  
Author(s):  
Nikhil Chakrabarti ◽  
Samiran Ghosh
Keyword(s):  
Dusty Plasma ◽  
Acoustic Waves ◽  
Kdv Equation ◽  
Low Frequency ◽  
Strongly Coupled ◽  
Weakly Nonlinear ◽  
Localized Structures ◽  
Dust Acoustic ◽  
Linear Damping ◽  
Nonlocal Nonlinear

The dynamics of the weakly nonlinear and weakly dispersive low frequency longitudinal dust acoustic waves (LDAWs) in a strongly coupled complex (dusty) plasma are investigated using generalized hydrodynamic (GH) model. In presence of strong correlation, the nonlinear wave is shown to be governed by a Korteweg–de Vries (KdV) equation with a nonlocal nonlinear forcing and a linear damping terms. This novel equation is solved numerically to show the competition between nonlinear forcing and dissipative damping in the formation of the localized structures.

scholarly journals Modified Korteweg–de Vries solitons at supercritical densities in two-electron temperature plasmas

2016 ◽  
Vol 82 (2) ◽  
Author(s):  
Frank Verheest ◽  
Carel P. Olivier ◽  
Willy A. Hereman
Keyword(s):  
Soliton Solutions ◽  
Detailed Comparison ◽  
Mkdv Equation ◽  
Perturbation Approach ◽  
Electron Population ◽  
Positive Ions ◽  
Reductive Perturbation ◽  
De Vries ◽  
Temperature Electron ◽  
Two Temperature

The supercritical composition of a plasma model with cold positive ions in the presence of a two-temperature electron population is investigated, initially by a reductive perturbation approach, under the combined requirements that there be neither quadratic nor cubic nonlinearities in the evolution equation. This leads to a unique choice for the set of compositional parameters and a modified Korteweg–de Vries equation (mKdV) with a quartic nonlinear term. The conclusions about its one-soliton solution and integrability will also be valid for more complicated plasma compositions. Only three polynomial conservation laws can be obtained. The mKdV equation with quartic nonlinearity is not completely integrable, thus precluding the existence of multi-soliton solutions. Next, the full Sagdeev pseudopotential method has been applied and this allows for a detailed comparison with the reductive perturbation results. This comparison shows that the mKdV solitons have slightly larger amplitudes and widths than those obtained from the more complete Sagdeev solution and that only slightly superacoustic mKdV solitons have acceptable amplitudes and widths, in the light of the full solutions.

scholarly journals Collisions of acoustic solitons and their electric fields in plasmas at critical compositions

2019 ◽  
Vol 85 (1) ◽  
Author(s):  
Frank Verheest ◽  
Willy A. Hereman
Keyword(s):  
Electric Field ◽  
Electric Fields ◽  
Soliton Solution ◽  
Electrostatic Potential ◽  
Kdv Equation ◽  
Mkdv Equation ◽  
Perturbation Scheme ◽  
Cubic Nonlinearity ◽  
Analytic Treatment ◽  
Reductive Perturbation

Acoustic solitons obtained through a reductive perturbation scheme are normally governed by a Korteweg–de Vries (KdV) equation. In multispecies plasmas at critical compositions the coefficient of the quadratic nonlinearity vanishes. Extending the analytic treatment then leads to a modified KdV (mKdV) equation, which is characterized by a cubic nonlinearity and is even in the electrostatic potential. The mKdV equation admits solitons having opposite electrostatic polarities, in contrast to KdV solitons which can only be of one polarity at a time. A Hirota formalism has been used to derive the two-soliton solution. That solution covers not only the interaction of same-polarity solitons but also the collision of compressive and rarefactive solitons. For the visualization of the solutions, the focus is on the details of the interaction region. A novel and detailed discussion is included of typical electric field signatures that are often observed in ionospheric and magnetospheric plasmas. It is argued that these signatures can be attributed to solitons and their interactions. As such, they have received little attention.

BÄCKLUND TRANSFORMATION AND ANALYTIC SOLUTIONS FOR A GENERALIZED VARIABLE-COEFFICIENT MODIFIED KORTEWEG–DE VRIES MODEL FROM FLUID DYNAMICS AND PLASMAS

2008 ◽  
Vol 19 (11) ◽  
pp. 1659-1671 ◽  
Author(s):  
FU-WEI SUN ◽  
YI-TIAN GAO ◽  
CHUN-YI ZHANG ◽  
XIAO-GE XU
Keyword(s):  
Fluid Dynamics ◽  
External Force ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Solitary Wave Solution ◽  
Kdv Equation ◽  
Analytic Solutions ◽  
Mkdv Equation ◽  
Backlund Transformation ◽  
De Vries

We investigate a generalized variable-coefficient modified Korteweg–de Vries model with perturbed factor and external force (vc-GmKdV) describing fluid dynamics and space plasmas. In this paper, we propose an extended variable-coefficient balancing-act method (Evc-BAM), which is concise and straightforward, to obtain the generalized analytic solutions including solitary wave solution of the vc-GmKdV model with symbolic computation. Meanwhile, using the Evc-BAM, we obtain an auto-Bäcklund transformation for the vc-GmKdV model on the relevant constraint conditions of the coefficient functions. Using the given auto-Bäcklund transformation, the solutions of special equations for the vc-GmKdV model are also obtained as the variable-coefficient Korteweg–de Vries (vc-KdV) equation, the generalized KdV equation with perturbed factor and external force (GKdV), the variable-coefficient modified Korteweg–de Vries (vc-mKdV) equation, and the variable-coefficient cylindrical modified Korteweg–de Vries (vc-cmKdV) equation, respectively.

A Car-Following Model Considering Effects of Weather and Road Conditions

2014 ◽  
Vol 488-489 ◽  
pp. 1289-1294
Author(s):  
Lu Jing ◽  
Peng Jun Zheng
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Mkdv Equation ◽  
Reductive Perturbation Method ◽  
Neutral Stability ◽  
Traffic Jam ◽  
Car Following ◽  
The Kdv Equation ◽  
Car Following Model ◽  
The Stability

In this paper, a modified car-following model is proposed, in which, the weather and road conditions are taken into account. The stability condition of the model is obtained by using the control theory method. We investigated the property of the model using linear and nonlinear analyses. The Kortewegde Vries equation near the neutral stability line and the modified Kortewegde Vries equation around the critical point are derived by applying the reductive perturbation method. The traffic jam could be thus described by the KdV soliton and the kinkanti-kink soliton for the KdV equation and mKdV equation, respectively. Numerical simulations are carried out to verify the model, and good results are obtained with the new model.

Exact solutions of nonlinear evolution equations of the AKNS class

1989 ◽  
Vol 30 (3) ◽  
pp. 313-325 ◽  
Author(s):  
W. L. Chan ◽  
Yu-Kun Zheng
Keyword(s):  
Exact Solutions ◽  
Evolution Equations ◽  
Kdv Equation ◽  
Mkdv Equation ◽  
Nonlinear Evolution Equations ◽  
Sine Gordon Equation ◽  
First Order ◽  
Linear Eigenvalue Problem ◽  
The Kdv Equation ◽  
Akns System

AbstractThe problem of obtaining explicit and exact solutions of soliton equations of the AKNS class is considered. The technique developed relies on the construction of the wave functions which are solutions of the associated AKNS system; that is, a linear eigenvalue problem in the form of a system of first order partial differential equations. The method of characteristics is used and Bäcklund transformations are employed to generate new solutions from the old. Thus, families of new solutions for the KdV equation, the mKdV equation, the sine-Gordon equation and the nonlinear Schrôdinger equation are obtained, avoiding the solution of some Riccati equations. Our results in the KdV case include those obtained recently by other investigators.

Sign in / Sign up

Export Citation Format

Share Document