Solutions of Konopelchenko-Dubrovsky Equation by Traveling Wave Hypothesis and Lie Symmetry Approach

Kumar, Sachin; Hama, Amadou; Biswas, Anjan

doi:10.12785/amis/080406

Cited by 17 publications

(8 citation statements)

References 13 publications

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“…Wazwaz [28] obtained some solitary wave, periodic wave, and kink solutions to this equation by using the tanh-sech method and the cosh-sinh scheme. The traveling wave hypothesis and the lie group approach were used on the Konopelchenko-Dubrovsky equations by Kumar et al [29] and some solitary wave, periodic singular waves, and cnoidal and snoidal solutions were reported. Song and Zhang [30] employed the extended Riccati equation rational expansion method to (1) and obtained some rational formal hyperbolic function solutions, rational formal triangular periodic solutions, and rational solutions.…”

Section: Resultsmentioning

confidence: 99%

“…Various analytical approaches have been used in obtaining the exact solutions to the Konopelchenko-Dubrovsky equations. Sheng [27] used the improved F-expansion method in addressing (1), Wazwaz [28] employed the tanhsech method, the cosh-sinh method, and the exponential functions method for obtaining the analytical solutions to (1), and Kumar et al [29] solved the KD equations by traveling wave hypothesis and lie symmetry approach. Song and Zhang [30] utilized the extended Riccati equation rational expansion method and Feng and Lin [31] applied the improved Riccati mapping approach.…”

Section: Introductionmentioning

confidence: 99%

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New Exponential and Complex Traveling Wave Solutions to the Konopelchenko-Dubrovsky Model

Düşünceli

2019

Advances in Mathematical Physics

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The Konopelchenko-Dubrovsky (KD) system is presented by the application of the improved Bernoulli subequation function method (IBSEFM). First, The KD system being Nonlinear partial differential equations system is transformed into nonlinear ordinary differential equation by using a wave transformation. Last, the resulting equation is successfully explored for new explicit exact solutions including singular soliton, kink, and periodic wave solutions. All the obtained solutions in this study satisfy the Konopelchenko-Dubrovsky model. Under suitable choice of the parameter values, interesting two- and three-dimensional graphs of all the obtained solutions are plotted.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

New Exponential and Complex Traveling Wave Solutions to the Konopelchenko-Dubrovsky Model

Düşünceli

2019

Advances in Mathematical Physics

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“…In 2014 Kumar et. al [16] used Lie symmetry analysis as well as travelling wave hypothesis to extract solutions to KD equations. In 2019 Alfalqi et al [17] applied the modified simplest equation method and B-spline method to KD-equation.…”

Section: Introductionmentioning

confidence: 99%

A Study of (2+1)-Dimensional Konopelchenko-Dubrovsky (KD) System: Closed-Form Solutions, Solitary Waves, Bifurcation Analysis and Quasi-Periodic Solution

Kumar

Mann

Kharbanda

2022

Preprint

Self Cite

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Nonlinear evolution equations (NLEEs) are extensively used to establish the elementary propositions of natural circumstances. In this work, we study the Konopelchenko-Dubrovsky (KD) equation which depicts non-linear waves in mathematical physics with weak dispersion. The considered model is investigated using the combination of generalized exponential rational function (GERF) method and dynamical system method. The GERF method is utilized to generate closedform invariant solutions to the (2+1)-dimensional KD model in terms of trigonometric, hyperbolic, and exponential forms with the assistance of symbolic computations. Moreover, three-dimensional graphics are displayed to depict the behavior of obtained solitary wave solutions. The model is observed to have single and multiple soliton profiles, kink-wave profiles, and periodic oscillating nonlinear waves. These generated solutions have never been published in the literature. All the newly generated soliton solutions are checked by putting them back into the associated system with the soft computation via Wolfram Mathematica. Moreover, the system is converted into a planer dynamical system using a certain transformation and the analysis of bifurcation is examined. Furthermore, the quasi-periodic solution is investigated numerically for the perturbed system by inserting definite periodic forces into the considered model. With regard to the parameter of the perturbed model, two-dimensional and three-dimensional phase portraits are plotted.

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“…The gardener equation, Kadomtsev -Petviashvili (KP), modified (KP) equations are their special cases that obtained through variation in m and n parameters, like m = 0 the governing couple equation reduced into Kadomtsev Petviashvili KP equation, for n = 0 it can convert into modified (KP) model and U y = 0 it behaves like gardener equation [16].…”

Section: Introductionmentioning

confidence: 99%

Dynamical behavior of the fractional coupled Konopelchenko-Dubrovsky and (3+1)-dimensional modified Korteweg-de Veries- Zakharov- Kuznestsov equations

Aslam

Majeed

Kamran

et al. 2022

Preprint

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This work studies, soliton solutions of time fractional coupled Konopelchenko-Dubrovsky and (3+1)-dimensional modified Korteweg-de Varies-Zakharov-Kuznestsov (mKdVZK) equations. These model are used to define the physical phenomena of oceans dynamic, plasma physics, and soliton theory. The unified method is used to solve these fractional models analytically. Conformable and Local M derivatives are used to tackle the time fractional part. Fractional wave transformation is used to transforme fractional partial differential equation to ordinary differential equation. Using proposed scheme soliton solutions are obtained in polynomial and rational forms. The behaviour of soliton solution is also analyzed at different fractional parameter. The results shows that proposed scheme is straight forward and easy to handle all kinds of time fractional nonlinear homogenous evolution equations that arised in different fields of science.

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Solutions of Konopelchenko-Dubrovsky Equation by Traveling Wave Hypothesis and Lie Symmetry Approach

Abstract: Abstract:In this paper, the Konopelchenko-Dubrovsky equation will be studied by the aid of traveling wave hypothesis and Lie symmetry analysis. The traveling wave hypothesis will extract the 1-soliton solution while the Lie symmetry approach will retrieve other solutions to this equation.

Cited by 17 publications

References 13 publications

New Exponential and Complex Traveling Wave Solutions to the Konopelchenko-Dubrovsky Model

New Exponential and Complex Traveling Wave Solutions to the Konopelchenko-Dubrovsky Model

A Study of (2+1)-Dimensional Konopelchenko-Dubrovsky (KD) System: Closed-Form Solutions, Solitary Waves, Bifurcation Analysis and Quasi-Periodic Solution

Dynamical behavior of the fractional coupled Konopelchenko-Dubrovsky and (3+1)-dimensional modified Korteweg-de Veries- Zakharov- Kuznestsov equations

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