Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion

Bruzón, M. S.; Recio, Elena; Garrido, Tamara-María; Rosa, Rafael de la

doi:10.3934/dcdss.2020222

Cited by 4 publications

(5 citation statements)

References 22 publications

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“…In [39], Lie symmetries, conservation laws and exact solutions for a generalized quasilinear KdV equation with degenerate dispersion were found.…”

Section: Contributionsmentioning

confidence: 99%

Roadmap of the Multiplier Method for Partial Differential Equations

Alvarez-Valdez,

Fernandez-Anaya

2023

Mathematics

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This review paper gives an overview of the method of multipliers for partial differential equations (PDEs). This method has made possible a lot of solutions to PDEs that are of interest in many areas such as applied mathematics, mathematical physics, engineering, etc. Looking at the history of the method and synthesizing the newest developments, we hope to give it the attention that it deserves to help develop the vast amount of work still needed to understand it and make the best use of it. It is also an interesting and a relevant method in itself that could possibly give interesting results in areas of mathematics such as modern algebra, group theory, topology, etc. The paper will be structured in such a manner that the last review known for this method will be presented to understand the theoretical framework of the method and then later work done will be presented. The information of four recent papers further developing the method will be synthesized and presented in such a manner that anyone interested in learning this method will have the most relevant information available and have all details cited for checking.

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“…In [39], Lie symmetries, conservation laws and exact solutions for a generalized quasilinear KdV equation with degenerate dispersion were found.…”

Section: Contributionsmentioning

confidence: 99%

Roadmap of the Multiplier Method for Partial Differential Equations

Alvarez-Valdez,

Fernandez-Anaya

2023

Mathematics

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“…Each low-order multiplier Q corresponds to a conserved density T and flux X through the characteristic Equation (13). All low-order local conservation laws have the general form T(x, t, u), X(x, t, u, u x , u t , u xx , u tx , u xxx ).…”

Section: Low-order Conservation Lawsmentioning

confidence: 99%

“…For instance, in [10][11][12] the authors apply the multiplier method to different types of wave equations in order to find conservation laws. In [13] a dispersive equation based on the well-known KdV equation is studied obtaining as well conservation laws by the application of this method. In the same way, conservation laws are determined in [14] for a general nonlinear diffusion-reaction equation.…”

Section: Introductionmentioning

confidence: 99%

Lie Symmetries and Conservation Laws for the Viscous Cahn-Hilliard Equation

2022

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In this paper, we study a viscous Cahn–Hilliard equation from the point of view of Lie symmetries in partial differential equations. The analysis of this equation is motivated by its applications since it serves as a model for many problems in physical chemistry, developmental biology, and population movement. Firstly, a classification of the Lie symmetries admitted by the equation is presented. In addition, the symmetry transformation groups are calculated. Afterwards, the partial differential equation is transformed into ordinary differential equations through symmetry reductions. Secondly, all low-order local conservation laws are obtained by using the multiplier method. Furthermore, we use these conservation laws to determine their associated potential systems and we use them to investigate nonlocal symmetries and nonlocal conservation laws. Finally, we apply the multi-reduction method to reduce the equation and find a soliton solution.

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“…In [28], the authors obtained the conservation laws and discussed the physical meaning of the corresponding conserved quantities. A classification of conservation laws of a generalized quasilinear KdV equation was provided in [29] too. Moreover, a (1 + 1)-dimensional coupled modified KdV-type system was studied in [30], constructing its conservation laws also using the multiplier method.…”

Section: Introductionmentioning

confidence: 99%

Lie Point Symmetries, Traveling Wave Solutions and Conservation Laws of a Non-linear Viscoelastic Wave Equation

Márquez

Bruzón

2021

Mathematics

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This paper studies a non-linear viscoelastic wave equation, with non-linear damping and source terms, from the point of view of the Lie groups theory. Firstly, we apply Lie’s symmetries method to the partial differential equation to classify the Lie point symmetries. Afterwards, we reduce the partial differential equation to some ordinary differential equations, by using the symmetries. Therefore, new analytical solutions are found from the ordinary differential equations. Finally, we derive low-order conservation laws, depending on the form of the damping and source terms, and discuss their physical meaning.

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Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion

Cited by 4 publications

References 22 publications

Roadmap of the Multiplier Method for Partial Differential Equations

Roadmap of the Multiplier Method for Partial Differential Equations

Lie Symmetries and Conservation Laws for the Viscous Cahn-Hilliard Equation

Lie Point Symmetries, Traveling Wave Solutions and Conservation Laws of a Non-linear Viscoelastic Wave Equation

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