Lie symmetries and invariant solutions of the shallow-water equation

Velan, M. Senthil; Lakshmanan, M.

doi:10.1016/0020-7462(95)00063-1

Cited by 19 publications

(5 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…L X u = 0 (10) then we shall say that u (t, x, y) is invariant under the action of the one-parameter point transformation with generator the vector field X.…”

Section: Whenmentioning

confidence: 99%

See 1 more Smart Citation

Lie symmetry analysis and similarity solutions for the Camassa-Choi equations

Paliathanasis

2021

Preprint

View full text Add to dashboard Cite

The method of Lie symmetry analysis of differential equations is applied to determine exact solutions for the Camassa-Choi equation and its generalization. We prove that the Camassa-Choi equation is invariant under an infinite-dimensional Lie algebra, with an essential five-dimensional Lie algebra. The application of the Lie point symmetries leads to the construction of exact similarity solutions.

show abstract

“…L X u = 0 (10) then we shall say that u (t, x, y) is invariant under the action of the one-parameter point transformation with generator the vector field X.…”

Section: Whenmentioning

confidence: 99%

“…The Lie symmetry analysis of the Camassa-Holm equation has been previously performed in [10]. Similarity solutions to the shallow water equations with a variable bottom were found in [11], while the Lie symmetry analysis of the rotating shallow-water equation was performed in [12].…”

Section: Introductionmentioning

confidence: 99%

Lie symmetry analysis and similarity solutions for the Camassa-Choi equations

Paliathanasis

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The Lie symmetries for the one-dimensional shallow-water system were determined in [23]. The complete symmetry classification of shallow-water equations for a twodimensional flow was performed in [24], while the same problem in a rotating frame with non-zero Coriolis component was investigated in [25], while a varying bottom topography was considered in [26]; see also [27][28][29].…”

Section: Introductionmentioning

confidence: 99%

Symmetry Analysis for the 2D Aw-Rascle Traffic-Flow Model of Multi-Lane Motorways in the Euler and Lagrange Variables

Paliathanasis

2023

Symmetry

View full text Add to dashboard Cite

A detailed symmetry analysis is performed for a microscopic model used to describe traffic flow in two-lane motorways. The traffic flow theory employed in this model is a two-dimensional extension of the Aw-Rascle theory. The flow parameters, including vehicle density, and vertical and horizontal velocities, are described by a system of first-order partial differential equations belonging to the family of hydrodynamic systems. This fluid-dynamics model is expressed in terms of the Euler and Lagrange variables. The admitted Lie point symmetries and the one-dimensional optimal system are determined for both sets of variables. It is found that the admitted symmetries for the two sets of variables form different Lie algebras, leading to distinct one-dimensional optimal systems. Finally, the Lie symmetries are utilized to derive new similarity closed-form solutions.

show abstract

“…Applications of Lie symmetries cover a wide range of physical and natural systems, from classical mechanics [38,39], fluid dynamics [40,41], plasma physics [42], optics [43], gravitational physics [44,45], financial mathematics [46,47] and other areas of applied mathematics.…”

Section: Introductionmentioning

confidence: 99%

Lie symmetries and similarity solutions for the generalized Zakharov equations

Krishnakumar,

Devi,

Paliathanasis

2020

Preprint

View full text Add to dashboard Cite

The theory of Lie point symmetries is applied to study the generalized Zakharov system with two unknown parameters. The system reduces into a three-dimensional real value functions system, where we find that admits five Lie point symmetries. From the resulting point, we focus on these which provide travel-wave similarity transformation. The reduced system can be integrated while we remain with a system of two second-order nonlinear ordinary differential equations. The parameters of the latter system are classified in order the equations to admit Lie point symmetries.Exact travel-wave solutions are found, while the generalized Zakharov system can be described by the one-dimensional Ermakov-Pinney equation.

show abstract

Lie symmetries and invariant solutions of the shallow-water equation

Cited by 19 publications

References 9 publications

Lie symmetry analysis and similarity solutions for the Camassa-Choi equations

Lie symmetry analysis and similarity solutions for the Camassa-Choi equations

Symmetry Analysis for the 2D Aw-Rascle Traffic-Flow Model of Multi-Lane Motorways in the Euler and Lagrange Variables

Lie symmetries and similarity solutions for the generalized Zakharov equations

Contact Info

Product

Resources

About