scholarly journals On the Conservation Laws and Exact Solutions of a Modified Hunter-Saxton Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Sait San ◽  
Emrullah Yaşar
Keyword(s):  
Liquid Crystals ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Nematic Liquid Crystals ◽  
Lie Point Symmetries ◽  
Local Conservation ◽  
The Relationship ◽  
Conservation Method

We study the modified Hunter-Saxton equation which arises in modelling of nematic liquid crystals. We obtain local conservation laws using the nonlocal conservation method and multiplier approach. In addition, using the relationship between conservation laws and Lie-point symmetries, some reductions and exact solutions are obtained.

Local conservation laws, symmetries, and exact solutions for a Kudryashov-Sinelshchikov equation

2017 ◽  
Vol 41 (4) ◽  
pp. 1631-1641 ◽  
Author(s):  
M. S. Bruzón ◽  
E. Recio ◽  
R. de la Rosa ◽  
M. L. Gandarias
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Local Conservation

scholarly journals On New Conservation Laws of Fin Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Gülden Gün Polat ◽  
Özlem Orhan ◽  
Teoman Özer
Keyword(s):  
Conservation Laws ◽  
Linear Form ◽  
Mathematical Physics ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Multiplier Method ◽  
Lie Point Symmetries ◽  
Invariant Solutions ◽  
Relationship Of ◽  
The Relationship

We study the new conservation forms of the nonlinear fin equation in mathematical physics. In this study, first, Lie point symmetries of the fin equation are identified and classified. Then by using the relationship of Lie symmetry andλ-symmetry, newλ-functions are investigated. In addition, the Jacobi Last Multiplier method and the approach, which is based on the factλ-functions are assumed to be of linear form, are considered as different procedures for lambda symmetry analysis. Finally, the corresponding new conservation laws and invariant solutions of the equation are presented.

scholarly journals Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 37-43 ◽  
Author(s):  
Emrullah Yaşar ◽  
Sait San ◽  
Yeşim Sağlam Özkan
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Shallow Water ◽  
Water Waves ◽  
Function Method ◽  
Boussinesq Equation ◽  
Lie Point Symmetries ◽  
Shallow Water Waves ◽  
Non Linear ◽  
Ill Posed

AbstractIn this work, we consider the ill-posed Boussinesq equation which arises in shallow water waves and non-linear lattices. We prove that the ill-posed Boussinesq equation is nonlinearly self-adjoint. Using this property and Lie point symmetries, we construct conservation laws for the underlying equation. In addition, the generalized solitonary, periodic and compact-like solutions are constructed by the exp-function method.

Conservation Laws and Exact Solutions with Symmetry Reduction of Nonlinear Reaction Diffusion Equations

2015 ◽  
Vol 16 (3-4) ◽  
pp. 191-196 ◽  
Author(s):  
Filiz Tascan ◽  
Arzu Yakut
Keyword(s):  
Differential Equations ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Lie Point Symmetries ◽  
Important Place ◽  
Physical Problems ◽  
Nonlinear Reaction

AbstractIn this work we study one of the most important applications of symmetries to physical problems, namely the construction of conservation laws. Conservation laws have important place for applications of differential equations and solutions, also in all physics applications. And so, this study deals conservation laws of first- and second-type nonlinear (NL) reaction diffusion equations. We used Ibragimov’s approach for finding conservation laws for these equations. And then, we found exact solutions of first- and second-type NL reaction diffusion equations with Lie-point symmetries.

The Logarithmic (1+1)$\boldsymbol{(1+1)}$-Dimensional KdV-Like and (2+1)$\boldsymbol{(2+1)}$-Dimensional KP-Like Equations: Lie Group Analysis, Conservation Laws and Double Reductions

2019 ◽  
Vol 20 (7-8) ◽  
pp. 747-755 ◽  
Author(s):  
İlker Burak Giresunlu ◽  
Emrullah Yaşar ◽  
Abdullahi Rashid Adem
Keyword(s):  
Conservation Laws ◽  
Group Analysis ◽  
Traveling Wave Solutions ◽  
Lie Point Symmetries ◽  
Double Reduction ◽  
Physical Processes ◽  
Soliton Theory ◽  
Simplest Equation Method ◽  
Invariance Theory ◽  
Conservation Method

AbstractWe investigate the logarithmic (1 + 1) dimensional KdV-like and (2 + 1) dimensional KP-like equations which model many physical processes in the field of soliton theory. In this paper, first, we get the classical Lie point symmetries using the invariance theory. Secondly, we obtain conservation laws of the underlying equations by incorporating the method of multiplier and nonlocal conservation method. A relationship between the obtained symmetries and conservation laws are shown. Then using the generalized double reduction theory for the associated symmetries, reductions are constructed. Finally traveling wave solutions are computed with the aid of the simplest equation method for the logarithmic (2 + 1) -dimensional KP-like equation.

scholarly journals On PT Symmetry Systems: Invariance, Conservation Laws, and Reductions

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
P. Masemola ◽  
A. H. Kara
Keyword(s):  
Schrödinger Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Schrodinger Equation ◽  
Lie Symmetries ◽  
Pt Symmetry ◽  
Noether Symmetries ◽  
Cubic Schrödinger Equation ◽  
The Relationship

An analysis of a PT symmetric coupler with “gain in one waveguide and loss in another” is made; a transformation in the PT system and some assumptions results in a scalar cubic Schrödinger equation. We investigate the relationship between the conservation laws and Lie symmetries and investigate a Lagrangian, corresponding Noether symmetries, conserved vectors, and exact solutions via “double reductions.”

Lie point symmetries exact solutions and conservation laws of perturbed Zakharov–Kuznetsov equation with higher-order dispersion term

2019 ◽  
Vol 34 (03) ◽  
pp. 1950027 ◽  
Author(s):  
Muhammad Nasir Ali ◽  
Aly R. Seadawy ◽  
Syed Muhammad Husnine
Keyword(s):  
Differential Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Nonlinear Partial Differential Equation ◽  
Higher Order ◽  
Wave Transformation ◽  
Lie Point Symmetries ◽  
Higher Order Dispersion ◽  
Order Dispersion ◽  
Dispersion Term

In this paper, Zakharov–Kuznetsov equation is investigated for exact solutions and conservation laws. The well-known Zakharov–Kuznetsov equation contains third-order dispersion, so its validity is restricted to the waves of small amplitudes only. When the amplitude of the wave increases, the velocity and the width of the soliton deviate from the prediction of this equation. To overcome this deficiency, higher-order dispersion term is added to the Zakharov–Kuznetsov equation. We obtain Lie point symmetries and conservation laws for this new model. Wave transformation is applied to convert the nonlinear partial differential equation into another nonlinear ordinary differential equation. Then exact solutions are computed for this with sine–cosine method and modified Kudryashov methods. Obtained solutions are new and are of significant importance in the field of plasma physics.

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