Lie Symmetry Group of the Nonisospectral Kadomtsev-Petviashvili Equation

Chen, Yong; Hu, Xiaorui

doi:10.1515/zna-2009-1-202

Cited by 11 publications

(3 citation statements)

References 6 publications

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“…The approach was first systematically addressed by Sophus Lie [5] in the second half of the nineteenth century. In the past decades, the theory of Lie symmetry and its applications have been extended to many nonlinear differential equations by mathematical physicists [6][7][8][9][10][11][12][13][14][15]. Particularly, Olver, Ibragimov and Bluman [6][7][8] are among those who have made great contributions to this field.…”

Section: Introductionmentioning

confidence: 99%

A symmetry-preserving difference scheme and analytical solutions of a generalized higher-order beam equation

Tian

Zhang

2021

Proc. R. Soc. A.

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Under investigation in this work is a generalized higher-order beam equation, which is an important physical model and describes the vibrations of a rod. By considering Lie symmetry analysis, and using the power series method, we derive the geometric vector fields, symmetry reductions, group invariant solutions and power series solutions of the equation, respectively. The convergence analysis of the power series solutions are also provided with detailed proof. Furthermore, by virtue of the multipliers, the local conservation laws of the equation are derived as well. Furthermore, an effective and direct approach is proposed to study the symmetry-preserving discretization for the equation via its potential system. Finally, the invariant difference models of the generalized beam equation are successfully constructed. Our results show that it is very useful to construct the difference models of the potential system instead of the original equation.

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Section: Introductionmentioning

confidence: 99%

A symmetry-preserving difference scheme and analytical solutions of a generalized higher-order beam equation

Tian

Zhang

2021

Proc. R. Soc. A.

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“…For example, the Kadomtsev-Petviashvili (KP) equation, a two-dimensional generalization of the well-known KdV equation, can model several significant situations such as ones arising from the plasma [3]. Recently, the nonisospectral KP equation have attracted much research attention [4][5][6][7]. The nonisospectral KP equation provide a description of surface waves in a more realistic situation than the KP equation itself.…”

Section: Introductionmentioning

confidence: 99%

New Type of Nonisospectral KP Equation with Self-Consistent Sources and its Bilinear Bäcklund Transformation

Sun

Tam

2021

JNMP

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A new type of the nonisospectral KP equation with self-consistent sources is constructed by using the source generation procedure. A new feature of the obtained nonisospectral system is that we allow y-dependence of the arbitrary constants in the determinantal solution for the nonisospectral KP equation. In order to further show integrability of the novel nonisospectral KP equation with self-consistent sources, we give a bilinear Bäcklund transformation.

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“…A loop algebra of nonlocal isovectors of the Korteweg-de Vries (KdV) equation is introduced in [10]. By inverse recursion operators, infinitely many nonlocal symmetries and the conformal invariant forms (Schwartz forms) are researched in [6,[20][21]. Based on the geometric heat flows, symmetries, invariant solutions and reduced equations for the affine case are investigated in [11,19,[25][26]38].…”

Section: Introductionmentioning

confidence: 99%

On the Lie algebras, generalized symmetries and darboux transformations of the fifth-order evolution equations in shallow water

Tian

Zhang

Feng

et al. 2015

Chin. Ann. Math. Ser. B

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By considering the one-dimensional model for describing long, small amplitude waves in shallow water, a generalized fifth-order evolution equation named the Olver water wave (OWW) equation is investigated by virtue of some new pseudo-potential systems. By introducing the corresponding pseudo-potential systems, the authors systematically construct some generalized symmetries that consider some new smooth functions··· ,n β=1,2,··· ,N depending on a finite number of partial derivatives of the nonlocal variables v β and a restriction i,α,β ∂ξ i ∂v β 2 + ∂η α ∂v β 2 = 0, i.e., i,α,β ∂G α ∂v β 2 = 0. Furthermore, the authors investigate some structures associated with the Olver water wave (AOWW) equations including Lie algebra and Darboux transformation. The results are also extended to AOWW equations such as Lax, Sawada-Kotera, Kaup-Kupershmidt, Itô and Caudrey-Dodd-Gibbon-Sawada-Kotera equations, et al. Finally, the symmetries are applied to investigate the initial value problems and Darboux transformations.

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Lie Symmetry Group of the Nonisospectral Kadomtsev-Petviashvili Equation

Cited by 11 publications

References 6 publications

A symmetry-preserving difference scheme and analytical solutions of a generalized higher-order beam equation

A symmetry-preserving difference scheme and analytical solutions of a generalized higher-order beam equation

New Type of Nonisospectral KP Equation with Self-Consistent Sources and its Bilinear Bäcklund Transformation

On the Lie algebras, generalized symmetries and darboux transformations of the fifth-order evolution equations in shallow water

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