Soliton solutions of the Klein–Gordon–Zakharov equations with power law nonlinearity

Triki, Houria; Boucerredj, N

doi:10.1016/j.amc.2013.10.093

Cited by 12 publications

(8 citation statements)

References 19 publications

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“…In the Ref. [20], the Klein-Gordon-Zakharov equations with power law nonlinearity are considered as ( )…”

Section: The Klein-gordon-zakharov Equations With the Positive Fracti...mentioning

confidence: 99%

The Klein–Gordon–Zakharov equations with the positive fractional power terms and their exact solutions

Zhang

2016

Pramana - J Phys

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Abstract:In this paper, the famous Klein-Gordon-Zakharov equations are firstly generalized, the new special types of Klein-Gordon-Zakharov equations with the positive fractional power terms (gKGZE) are presented. In order to derive the exact solutions of new special gKGZE, the subsidiary higher order ordinary differential equations (sub-ODEs) with the positive fractional power terms are introduced, and with the aids of the Sub-ODE, the exact solutions of three special types of the gKGZE are derived, which are the bell-type solitary wave solution, the algebraic solitary wave solution, the kink-type solitary wave solution and the sinusoidal traveling wave solution, provided that the coefficients of gKGZE satisfy certain constraint conditions.

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“…In the Ref. [20], the Klein-Gordon-Zakharov equations with power law nonlinearity are considered as ( )…”

Section: The Klein-gordon-zakharov Equations With the Positive Fracti...mentioning

confidence: 99%

The Klein–Gordon–Zakharov equations with the positive fractional power terms and their exact solutions

Zhang

2016

Pramana - J Phys

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“…Of course, if the solution of the KGZ is smooth enough, we can always rise the energy space for error functions to

H^{2} \times H^{1}

, under a stronger regularity assumption than(A). Then there will be no need to assume the stability (or CFL‐type) condition . Remark The EWI‐SP method and the analysis can be easily extended to to the study the generalized KGZ system ,

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

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

for some general function

f (\cdot), g (\cdot)

defined on

ℝ

. All the results presented in this work remain valid, provided

f, g

are smooth enough. Remark By the convergence theorem, we claim that the CFL‐type condition required in is not needed for the proposed EWI‐SP method.…”

Section: Convergence Analysismentioning

confidence: 99%

“…Remark 3.7. The EWI-SP method and the analysis can be easily extended to to the study the generalized KGZ system [4,10],…”

Section: B)mentioning

confidence: 99%

“…The cubic nonlinearity

λ | ψ |^{2} ψ

in describes the nonlinear self‐interactions of the electric field similarly as the Klein–Gordon equation , with

λ > 0

for the defocusing case and

λ < 0

for focusing case. Nonlinearities as higher order power laws, that is,

λ | ψ |^{2 p} ψ, Δ (| ψ |^{2 q})

for integers

p, q > 1

are also considered in literature for generalizations .…”

Section: Introductionmentioning

confidence: 99%

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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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“…Because, the projected model play a substantial role in interpreting many complex and nonlinear phenomena and examined the solution using numerical and analytical algorithms, for instance, authors in [60] presented the global existence of the solutions for KGZ equations, Li in [36] evaluated the exact explicit traveling wave solutions for the proposed system with (n + 1)-dimension, authors in [33] find the 1-soliton solution for projected problem having power-law nonlinearity, the extended wave solutions have been cited in [54]. Moreover, many authors examined the nature of KGZ equations and its solution using various methods [2,9,13,28,29,41,53,58,59].…”

Section: Introductionmentioning

confidence: 99%

Analysis of fractional Klein–Gordon–Zakharov equations using efficient method

Benli

2020

Numer Methods Partial Differential Eq.

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In the present framework, the q-homotopy analysis transform method (q-HATM) we find the solution for the equation describing the interaction between Langmuir waves and the ion-acoustic waves in the plasma, called Klein-Gordon-Zakharov equations. The projected solution procedure is elegant unification of q-homotopy analysis technique with Laplace transform. The considered coupled nonlinear system analyzed with the different fractional-order to ensure the efficiency and applicability of the hired solution procedure. For distinct values of the parameters offered by the considered method and for different fractional-order, the physical behaviors of achieved solutions are captured in terms of surface and contour plots. To illustrate the accuracy and reliability of the considered method, we conducted the numerical simulation for different fractional order with the change in variables. The obtained results show that, the projected scheme is highly methodical, very effective, easy to implement and accurate to investigate the nature of fractional nonlinear coupled system exemplifying the real-world problems.

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Soliton solutions of the Klein–Gordon–Zakharov equations with power law nonlinearity

Cited by 12 publications

References 19 publications

The Klein–Gordon–Zakharov equations with the positive fractional power terms and their exact solutions

The Klein–Gordon–Zakharov equations with the positive fractional power terms and their exact solutions

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

Analysis of fractional Klein–Gordon–Zakharov equations using efficient method

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