scholarly journals Approximate symmetries and similarity solutions for wave equations on liquid films

2020 ◽  
pp. 18-18
Author(s):  
Sameerah Jamal ◽  
Andronikos Paliathanasis
Keyword(s):  
Conservation Laws ◽  
Small Amplitude ◽  
Wave Equations ◽  
Liquid Films ◽  
Similarity Solutions ◽  
Lie Symmetries ◽  
Long Waves ◽  
Approximate Symmetries ◽  
Approximate Similarity ◽  
Perturbation Parameters

We study the exact and approximate Lie symmetries for two equations which describe long waves with small amplitude on liquid films. Specifically, we study the 1+2 Benney-Luke and the 1+1 Benney-Lin equations, both from an exact and approximate perspective. To induce approximate symmetries, we show that terms involving derivatives higher than two are necessarily selected as the perturbation parameters. We construct conservation laws for both equations, and illustrate how the approximate point symmetries can be used to determine approximate similarity solutions.

Long waves on a beach

1967 ◽  
Vol 27 (4) ◽  
pp. 815-827 ◽  
Author(s):  
D. H. Peregrine
Keyword(s):  
Solitary Wave ◽  
Small Amplitude ◽  
Wave Equations ◽  
Equations Of Motion ◽  
Boussinesq Equations ◽  
Long Waves ◽  
Constant Depth ◽  
Reflected Wave ◽  
Long Wave ◽  
Non Linear

Equations of motion are derived for long waves in water of varying depth. The equations are for small amplitude waves, but do include non-linear terms. They correspond to the Boussinesq equations for water of constant depth. Solutions have been calculated numerically for a solitary wave on a beach of uniform slope. These solutions include a reflected wave, which is also derived analytically by using the linearized long-wave equations.

scholarly journals Some symmetries, similarity solutions and various conservation laws of a type of dispersive water waves

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yufeng Zhang ◽  
Na Bai ◽  
Hongyang Guan
Keyword(s):  
Conservation Laws ◽  
Water Waves ◽  
Water Wave ◽  
Wave Equations ◽  
Similarity Solutions ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Lie Transformation ◽  
Group Invariant Solutions ◽  
Special Solutions

Abstract We investigate the point symmetries, Lie–Bäcklund symmetries for a type of dispersive water waves. We obtain some Lie transformation groups, various group-invariant solutions, and some similarity solutions. Besides, we produce different formats of conservation laws of the dispersive water waves by using different schemes. Finally, we consider some special solutions of the stationary dispersive water-wave equations.

scholarly journals On the Symmetries and Conservation Laws of the Multidimensional Nonlinear Damped Wave Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-11
Author(s):  
Usamah S. Al-Ali ◽  
Ashfaque H. Bokhari ◽  
A. H. Kara ◽  
F. D. Zaman
Keyword(s):  
Wave Equation ◽  
Conservation Laws ◽  
Wave Equations ◽  
Lie Symmetries ◽  
Damped Wave Equation ◽  
Variable Damping ◽  
Nonlinear Damped Wave Equations ◽  
Nonlinear Damped Wave Equation ◽  
Similarity Reductions

We carry out a classification of Lie symmetries for the (2+1)-dimensional nonlinear damped wave equationutt+fuut=div(gugrad u)with variable damping. Similarity reductions of the equation are performed using the admitted Lie symmetries of the equation and some interesting solutions are presented. Employing the multiplier approach, admitted conservation laws of the equation are constructed for some new, interesting cases.

An analogy between electromagnetic and internal waves

2019 ◽  
Vol 485 (4) ◽  
pp. 428-433
Author(s):  
V. G. Baydulov ◽  
P. A. Lesovskiy
Keyword(s):  
Angular Momentum ◽  
Conservation Laws ◽  
Internal Wave ◽  
Special Relativity ◽  
Internal Waves ◽  
Maxwell Equations ◽  
Wave Equations ◽  
Lagrange Function ◽  
Lorentz Transformations ◽  
Symmetry Transformations

For the symmetry group of internal-wave equations, the mechanical content of invariants and symmetry transformations is determined. The performed comparison makes it possible to construct expressions for analogs of momentum, angular momentum, energy, Lorentz transformations, and other characteristics of special relativity and electro-dynamics. The expressions for the Lagrange function are defined, and the conservation laws are derived. An analogy is drawn both in the case of the absence of sources and currents in the Maxwell equations and in their presence.

Topological solitons, cnoidal waves and conservation laws of coupled wave equations

2013 ◽  
Vol 87 (12) ◽  
pp. 1233-1241 ◽  
Author(s):  
E. V. Krishnan ◽  
A. H. Kara ◽  
S. Kumar ◽  
A. Biswas
Keyword(s):  
Conservation Laws ◽  
Wave Equations ◽  
Cnoidal Waves ◽  
Topological Solitons ◽  
Coupled Wave Equations ◽  
Coupled Wave

Symmetry Reduction, Exact Solutions, and Conservation Laws of the (2+1)-Dimensional Dispersive Long Wave Equations

2009 ◽  
Vol 64 (9-10) ◽  
pp. 597-603 ◽  
Author(s):  
Zhong Zhou Dong ◽  
Yong Chen
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Symmetry Group ◽  
Infinite Number ◽  
Wave Equations ◽  
Direct Method ◽  
Discrete Groups ◽  
Point Symmetry Group ◽  
Long Wave ◽  
Full Symmetry

By means of the generalized direct method, we investigate the (2+1)-dimensional dispersive long wave equations. A relationship is constructed between the new solutions and the old ones and we obtain the full symmetry group of the (2+1)-dimensional dispersive long wave equations, which includes the Lie point symmetry group S and the discrete groups D. Some new forms of solutions are obtained by selecting the form of the arbitrary functions, based on their relationship. We also find an infinite number of conservation laws of the (2+1)-dimensional dispersive long wave equations.

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

2020 ◽  
Vol 13 (10) ◽  
pp. 2691-2701
Author(s):  
María-Santos Bruzón ◽  
◽  
Elena Recio ◽  
Tamara-María Garrido ◽  
Rafael de la Rosa
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Kdv Equation ◽  
Lie Symmetries

scholarly journals A Method for Generating Approximate Similarity Solutions of Nonlinear Partial Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Mazhar Iqbal ◽  
M. T. Mustafa ◽  
Azad A. Siddiqui
Keyword(s):  
Gas Flow ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Similarity Solutions ◽  
Standard Application ◽  
Approximate Similarity ◽  
Initial Boundary Value ◽  
Unsteady Gas Flow

Standard application of similarity method to find solutions of PDEs mostly results in reduction to ODEs which are not easily integrable in terms of elementary or tabulated functions. Such situations usually demand solving reduced ODEs numerically. However, there are no systematic procedures available to utilize these numerical solutions of reduced ODE to obtain the solution of original PDE. A practical and tractable approach is proposed to deal with such situations and is applied to obtain approximate similarity solutions to different cases of an initial-boundary value problem of unsteady gas flow through a semi-infinite porous medium.

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