Solitary wave solutions of (3+1)-dimensional extended Zakharov–Kuznetsov equation by Lie symmetry approach

Kumar, Sachin; Kumar, Dharmendra

doi:10.1016/j.camwa.2018.12.009

Cited by 72 publications

(24 citation statements)

References 59 publications

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“…We perform further symmetry reductions by applying Lie symmetries to (12), then (12) has the following six Lie symmetries:…”

Section: Complexitymentioning

confidence: 99%

“…e Lie symmetry method presented by Lie [9] is one of the well-known methods for obtaining exact solutions of nonlinear PDEs. Up to now, the Lie symmetry method has been applied to a number of mathematical and physical models, see [10][11][12][13][14] and references therein. is method is effective to get similarity solutions and solitary wave solutions of NPDEs.…”

Section: Introductionmentioning

confidence: 99%

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Lie Symmetry Analysis, Traveling Wave Solutions, and Conservation Laws to the (3 + 1)-Dimensional Generalized B-Type Kadomtsev-Petviashvili Equation

2020

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In this paper, the (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili(BKP) equation is studied applying Lie symmetry analysis. We apply the Lie symmetry method to the (3 + 1)-dimensional generalized BKP equation and derive its symmetry reductions. Based on these symmetry reductions, some exact traveling wave solutions are obtained by using the tanh method and Kudryashov method. Finally, the conservation laws to the (3 + 1)-dimensional generalized BKP equation are presented by invoking the multiplier method.

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“…We perform further symmetry reductions by applying Lie symmetries to (12), then (12) has the following six Lie symmetries:…”

Section: Complexitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Lie Symmetry Analysis, Traveling Wave Solutions, and Conservation Laws to the (3 + 1)-Dimensional Generalized B-Type Kadomtsev-Petviashvili Equation

2020

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“…The width of the soliton and its velocity deviate from the predictions of this equation as the amplitude of the wave increases. The pZK equation ( 1) with fractional order γ = 1 and k = 1 includes an of extra fifth-order dispersion term ξ ∂ 5 W ∂x 5 was proposed to overcome this problem (see [57][58][59]62], for more detail). The proposed δ-HPTM and q-HATM are employed to compute numerical solutions of Eq.…”

Section: Introductionmentioning

confidence: 99%

Modified homotopy methods for generalized fractional perturbed Zakharov–Kuznetsov equation in dusty plasma

2021

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We propose a new modification of homotopy perturbation method (HPM) called the δ-homotopy perturbation transform method (δ-HPTM). This modification consists of the Laplace transform method, HPM, and a control parameter δ. This control convergence parameter δ in this new modification helps in adjusting and controlling the convergence region of the series solution and overcome some limitations of HPM and HPTM. The δ-HPTM and q-homotopy analysis transform method (q-HATM) are considered to study the generalized time-fractional perturbed $(3+1)$ ( 3 + 1 ) -dimensional Zakharov–Kuznetsov equation with Caputo fractional time derivative. This equation describes nonlinear dust-ion-acoustic waves in the magnetized two-ion-temperature dusty plasmas. The selection of an appropriate value of δ in δ-HPTM and the auxiliary parameters n and ħ in q-HATM gives a guaranteed convergence of series solution, but the difference between the two techniques is that the embedding parameter p in δ-HPTM varies from zero to nonzero δ, whereas the embedding parameter q in q-HATM varies from zero to $\frac{1}{n}, n\geq{1}$ 1 n , n ≥ 1 . We examine the effect of fractional order on the considered problem and present the error estimate when compared with exact solution. The outcomes reveal complete reliability and efficiency of the proposed algorithm for solving various types of physical models arising in sciences and engineering. Furthermore, we present the convergence and error analysis of the two methods.

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“…The exact analytical solutions provide much more specific information of the complex physical problems describing such types of NPDEs. Over the last five decades, some methods have been developed successfully to obtain soliton and group-invariant solutions of NPDEs for their trustworthy processes like as tanh function method [1], notably first integral technique [2], F-expansion technique [3], Painlevé analysis [4], cosh-sinh method and tanh-sech technique [5], expfunction method [5][6][7], extended F-expansion technique [8,9], extended homotopy perturbation method [10], Riccati equation rational expansion method [11], symmetry reductions of the Lax pair [12], Bäcklund transformation [13], general direct method [14], Hirota's bilinear method [15] and Lie symmetry method [16][17][18][19][20][21][22][23][24][25][26].…”

Section: Introductionmentioning

confidence: 99%

Exact closed-form solutions and dynamics of solitons for a $$(2+1)$$-dimensional universal hierarchy equation via Lie approach

Kumar

Niwas

2021

Pramana - J Phys

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The dynamics of localised solitary wave solutions play an essential role in the fields of mathematical sciences such as optical physics, plasma physics, nonlinear dynamics and many others. The prime objective of this study is to obtain localised solitary wave solutions and exact closed-form solutions of the (2 + 1)-dimensional universal hierarchy equation (UHE) using the Lie symmetry approach. Besides, the Lie infinitesimals, all the vector fields, commutation relations of Lie algebra and symmetry reductions are derived via the Lie transformation method. Meanwhile, the universal hierarchy equation is reduced into nonlinear ODEs through two stages of symmetry reductions. The closed-form invariant solutions are attained under some parametric conditions imposed on infinitesimal generators. Because of the presence of arbitrary independent functional parameters and other constants, these group-invariant solutions are explicitly displayed in terms of arbitrary functions that are more relevant, beneficial and useful for explaining nonlinear complex physical phenomena. Furthermore, the dynamical structures of the obtained exact solutions are illustrated for suitable values of arbitrary constants through 3D-plots based on numerical simulation. Some of these localised solitary waves are double solitons, periodic lump solitons, dark solitons, five-solitons, hemispherical solitons and lump-type solitons.

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Solitary wave solutions of (3+1)-dimensional extended Zakharov–Kuznetsov equation by Lie symmetry approach

Cited by 72 publications

References 59 publications

Lie Symmetry Analysis, Traveling Wave Solutions, and Conservation Laws to the (3 + 1)-Dimensional Generalized B-Type Kadomtsev-Petviashvili Equation

Lie Symmetry Analysis, Traveling Wave Solutions, and Conservation Laws to the (3 + 1)-Dimensional Generalized B-Type Kadomtsev-Petviashvili Equation

Modified homotopy methods for generalized fractional perturbed Zakharov–Kuznetsov equation in dusty plasma

Exact closed-form solutions and dynamics of solitons for a $$(2+1)$$-dimensional universal hierarchy equation via Lie approach

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