1-Soliton solution of the Klein–Gordon–Zakharov equation with power law nonlinearity

Ismail, M.S.; Biswas, Anjan

doi:10.1016/j.amc.2010.10.035

Cited by 26 publications

(22 citation statements)

References 10 publications

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“…The solutions of this equation play a vital rule to analyze the wave propagation of various types of physical phenomena in the related fields. There is an amount of paper [35][36][37][38][39][40][41][42][43][44][45][46], where the various types of nonlinear KGZ equation have been studied. Some of the KGZ equations are also appeared to describe the acoustic wave propagation in plasma physics.…”

Section: Introductionmentioning

confidence: 99%

New exact traveling wave solutions to the (1+1)-dimensional Klein-Gordon-Zakharov equation for wave propagation in plasma using the exp(-Φ(ξ))-expansion method

Hafez

Akbar

2015

Propulsion and Power Research

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The (1þ1)-dimensional nonlinear Klein-Gordon-Zakharov equation considered as a model equation for describing the interaction of the Langmuir wave and the ion acoustic wave in high frequency plasma. By the execution of the exp(-Φ(ξ))-expansion, we obtain new explicit and exact traveling wave solutions to this equation. The obtained solutions include kink, singular kink, periodic wave solutions, soliton solutions and solitary wave solutions of bell types. The variety of structure and graphical representation make the dynamics of the equations visible and provides the mathematical foundation in plasma physics and engineering.

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Section: Introductionmentioning

confidence: 99%

New exact traveling wave solutions to the (1+1)-dimensional Klein-Gordon-Zakharov equation for wave propagation in plasma using the exp(-Φ(ξ))-expansion method

Hafez

Akbar

2015

Propulsion and Power Research

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“…Chen and Zhang proposed two new different schemes for an initial‐boundary‐value problem of the coupled KGZ equations and proved stability and convergence of different solutions. Many authors obtained some explicit exact solitary wave solutions and numerical solutions for the coupled KGZ equations by using a various of different approaches ( and references therein). Note that if α = 1 and β = 0, then system reduces to the classical KGZ equations

({, \begin{matrix} u_{t t} - u_{x x} + u + u v = 0, \\ v_{t t} - v_{x x} = {(| u |^{2})}_{x x} . \end{matrix})

The system arises in the study of the interaction between a Langmuir wave and an ion sound wave in plasma.…”

Section: Introductionmentioning

confidence: 99%

Orbital stability of solitary waves of the coupled Klein–Gordon–Zakharov equations

Zheng

Shang

Peng

2016

Math Methods in App Sciences

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This paper investigates the orbital stability of solitary waves for the coupled Klein–Gordon–Zakharov (KGZ) equations utt−uxx+u+αuv+β|u|2u=0,vtt−vxx=(|u|2)xx, where α ≠ 0. Firstly, we rewrite the coupled KGZ equations to obtain its Hamiltonian form. And then, we present a pair of sech‐type solutions of the coupled KGZ equations. Because the abstract orbital stability theory presented by Grillakis, Shatah, and Strauss (1987) cannot be applied directly, we can extend the abstract stability theory and use the detailed spectral analysis to obtain the stability of the solitary waves for the coupled KGZ equations. In our work, α = 1,β = 0 are advisable. Hence, we can also obtain the orbital stability of solitary waves for the classical KGZ equations which was studied by Chen. Copyright © 2016 John Wiley & Sons, Ltd.

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“…The nondimensional Klein–Gordon–Zakharov (KGZ) system in d ‐dimensions (

d = 1, 2, 3

) reads as the following,

\partial_{t t} ψ (x, t) - Δ ψ (x, t) + ψ (x, t) + ψ (x, t) ϕ (x, t) + λ | ψ |^{2} ψ (x, t) = 0,

\partial_{t t} ϕ (x, t) - Δ ϕ (x, t) - Δ (| ψ (x, t) |^{2}) = 0, x \in ℝ^{d}, t > 0,

with given initial conditions

ψ (x, 0) = ψ^{(0)} (x), \partial_{t} ψ (x, 0) = ψ^{(1)} (x), ϕ (x, 0) = ϕ^{(0)} (x), \partial_{t} ϕ (x, 0) = ϕ^{(1)} (x

…”

Section: Introductionmentioning

confidence: 99%

On error estimates of an exponential wave integrator sine pseudospectral method for theKlein–Gordon–Zakharov system

Zhao

2015

Numerical Methods Partial

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In this article, we propose an exponential wave integrator sine pseudospectral (EWI‐SP) method for solving the Klein–Gordon–Zakharov (KGZ) system. The numerical method is based on a Deuflhard‐type exponential wave integrator for temporal integrations and the sine pseudospectral method for spatial discretizations. The scheme is fully explicit, time reversible and very efficient due to the fast algorithm. Rigorous finite time error estimates are established for the EWI‐SP method in energy space with no CFL‐type conditions which show that the method has second order accuracy in time and spectral accuracy in space. Extensive numerical experiments and comparisons are done to confirm the theoretical studies. Numerical results suggest the EWI‐SP allows large time steps and mesh size in practical computing. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 266–291, 2016

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1-Soliton solution of the Klein–Gordon–Zakharov equation with power law nonlinearity

Cited by 26 publications

References 10 publications

New exact traveling wave solutions to the (1+1)-dimensional Klein-Gordon-Zakharov equation for wave propagation in plasma using the exp(-Φ(ξ))-expansion method

New exact traveling wave solutions to the (1+1)-dimensional Klein-Gordon-Zakharov equation for wave propagation in plasma using the exp(-Φ(ξ))-expansion method

Orbital stability of solitary waves of the coupled Klein–Gordon–Zakharov equations

On error estimates of an exponential wave integrator sine pseudospectral method for theKlein–Gordon–Zakharov system

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