scholarly journals Symmetry Reductions, Exact Solutions, and Conservation Laws of a Modified Hunter-Saxton Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Andrew Gratien Johnpillai ◽  
Chaudry Masood Khalique
Keyword(s):  
Differential Equations ◽  
Conservation Laws ◽  
Ordinary Differential Equations ◽  
Lie Algebra ◽  
Point Of View ◽  
Third Order ◽  
Theoretic Point ◽  
Group Invariant Solutions ◽  
Nonlocal Conservation Laws ◽  
Symmetry Generators

We study a modified Hunter-Saxton equation from the Lie group-theoretic point of view. The Lie point symmetry generators of the underlying equation are derived. We utilize the Lie algebra admitted by the equation to obtain the optimal system of one-dimensional subalgebras of the Lie algebra of the equation. These subalgebras are then used to reduce the underlying equation to nonlinear third-order ordinary differential equations. Exact traveling wave group-invariant solutions for the equation are constructed by integrating the reduced ordinary differential equations. Moreover, using the variational method, we construct infinite number of nonlocal conservation laws by the transformation of the dependent variable of the underlying equation.

scholarly journals Symmetries and Conservation Laws for Some Compacton Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
M. L. Gandarias ◽  
M. S. Bruzón ◽  
M. Rosa
Keyword(s):  
Differential Equations ◽  
Qualitative Analysis ◽  
Conservation Laws ◽  
Ordinary Differential Equations ◽  
Dynamical Behaviour ◽  
Point Of View ◽  
Nonlinear Dispersion ◽  
Multipliers Method

We consider some equations with compacton solutions and nonlinear dispersion from the point of view of Lie classical reductions. The reduced ordinary differential equations are suitable for qualitative analysis and their dynamical behaviour is described. We derive by using the multipliers method some nontrivial conservation laws for these equations.

Exact Group Invariant Solutions and Conservation Laws of the Complex Modified Korteweg–de Vries Equation

2013 ◽  
Vol 68 (8-9) ◽  
pp. 510-514 ◽  
Author(s):  
Andrew G. Johnpillai ◽  
Abdul H. Kara ◽  
Anjan Biswas
Keyword(s):  
Conservation Laws ◽  
Symmetry Algebra ◽  
Lie Symmetry ◽  
Point Of View ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Symmetry Approach ◽  
De Vries ◽  
Group Invariant Solutions ◽  
Symmetry Generators

We study the scalar complex modified Korteweg-de Vries (cmKdV) equation by analyzing a system of partial differential equations (PDEs) from the Lie symmetry point of view. These systems of PDEs are obtained by decomposing the underlying cmKdV equation into real and imaginary components. We derive the Lie point symmetry generators of the system of PDEs and classify them to get the optimal system of one-dimensional subalgebras of the Lie symmetry algebra of the system of PDEs. These subalgebras are then used to construct a number of symmetry reductions and exact group invariant solutions to the system of PDEs. Finally, using the Lie symmetry approach, a couple of new conservation laws are constructed. Subsequently, respective conserved quantities from their respective conserved densities are computed.

scholarly journals Discrete Symmetry Transformations of Third Order Ordinary Differential Equations and Applications

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Discrete Symmetries ◽  
Discrete Symmetry ◽  
Third Order ◽  
Symmetry Transformations ◽  
Paper List

Third order ordinary differential equations have already been classified by the Lie algebra they admit. Invariant equations corresponding to these Lie algebras are also available in the literature [17]. In this paper, list of all discrete symmetries corresponding to these invariant ordinary differential equations, are obtained. Some particular examples are given to show the significance of the work.

Boundary-Value Problems for Third-Order Lipschitz Ordinary Differential Equations

2014 ◽  
Vol 58 (1) ◽  
pp. 183-197 ◽  
Author(s):  
John R. Graef ◽  
Johnny Henderson ◽  
Rodrica Luca ◽  
Yu Tian
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Optimal Length ◽  
The Third

AbstractFor the third-order differential equationy′″ = ƒ(t, y, y′, y″), where, questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

Sign in / Sign up

Export Citation Format

Share Document