The K(m,n) equation with generalized evolution term studied by symmetry reductions and qualitative analysis

Bruzón, M. S.; Gandarias, M. L.; González, Graciela Adriana; Hansen, R. O.

doi:10.1016/j.amc.2012.03.084

Cited by 6 publications

(8 citation statements)

References 25 publications

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“…In the case of PDEs with two independent variables, the reduction procedure consists of obtaining a similarity variable that allows us to transform the PDE into an ordinary differential equation (ODE), which is, in general, easier to solve. Thus, symmetry groups can also be combined and applied with other methods to find exact and numerical solutions [20][21][22][23][24][25]. However, in this paper, we focus only on the application of solvable Lie groups and the reductions obtained from them.…”

Section: Introductionmentioning

confidence: 99%

Applications of Solvable Lie Algebras to a Class of Third Order Equations

et al. 2022

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A family of third-order partial differential equations (PDEs) is analyzed. This family broadens out well-known PDEs such as the Korteweg-de Vries equation, the Gardner equation, and the Burgers equation, which model many real-world phenomena. Furthermore, several macroscopic models for semiconductors considering quantum effects—for example, models for the transmission of electrical lines and quantum hydrodynamic models—are governed by third-order PDEs of this family. For this family, all point symmetries have been derived. These symmetries are used to determine group-invariant solutions from three-dimensional solvable subgroups of the complete symmetry group, which allow us to reduce the given PDE to a first-order nonlinear ordinary differential equation (ODE). Finally, exact solutions are obtained by solving the first-order nonlinear ODEs or by taking into account the Type-II hidden symmetries that appear in the reduced second-order ODEs.

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Section: Introductionmentioning

confidence: 99%

Applications of Solvable Lie Algebras to a Class of Third Order Equations

et al. 2022

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“…Substituting it into the original PDE, we obtain a traveling wave equation. Then, solving this equation by the bifurcation theory method of dynamical systems, [23][24][25][26][27][28] we can obtain the traveling wave solutions. Substituting Eq.…”

Section: Exact Traveling Wave Solutions Of the Shortwave Modelmentioning

confidence: 99%

Lie symmetries and exact solutions for a short-wave model

Chen¹,

Zhang²,

Wen³

2013

Chinese Phys. B

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“…and symmetry reductions and exact solutions were derived in [2,3]. The conservation laws for the compacton (2, 2) equation and the compacton (3, 3) equation were constructed in [4] by utilizing the multiplier approach.…”

Section: Introductionmentioning

confidence: 99%

“…For nonvariational problems there are different methods for the construction of conservation laws. In [7,8], Anco and Bluman gave a general algorithmic method to find all conservations laws for evolution equations like (2) and (3). In [9], a special method has been introduced by Ibragimov.…”

Section: Introductionmentioning

confidence: 99%