A. H. Kara

Relationship between Symmetries andConservation Laws

The fundamental relation between Lie-Bäcklund symmetry generators andconservation laws of an arbitrary differential equation is derived without regardto a Lagrangian formulation of the differential equation. This relation is used inthe construction of conservation laws for partial differential equations irrespectiveof the knowledge or existence of a Lagrangian. The relation enables one toassociate symmetries to a given conservation law of a differential equation.Applications of these results are illustrated for a range of examples.

A Basis of Conservation Laws for Partial Differential Equations

Abstract The classical generation theorem of conservation laws from known ones for a system of differential equations which uses the action of a canonical Lie–Bäcklund generator is extended to include any Lie–Bäcklund generator. Also, it is shown that the Lie algebra of Lie–Bäcklund symmetries of a conserved vector of a system is a subalgebra of the Lie–Bäcklund symmetries of the system. Moreover, we investigate a basis of conservation laws for a system and show that a generated conservation law via the action of a symmetry operator which satisfies a commutation rule is nontrivial if the system is derivable from a variational principle. We obtain the conservation laws of a class of nonlinear diffusion-convection and wave equations in (1 + 1)-dimensions. In fact we find a basis of conservation laws for the diffusion equations in the special case when it admits proper Lie–Bäcklund symmetries. Other examples are presented to illustrate the theory.

Lie–Bäcklund and Noether Symmetries with Applications

New identities relating the Euler–Lagrange, Lie–Bäcklund and Noether operators are obtained. Some important results are shown to be consequences of these fundamental identities. Furthermore, we generalise an interesting example presented by Noether in her celebrated paper and prove that any Noether symmetry is equivalent to a strict Noether symmetry, i.e. a Noether symmetry with zero divergence. We then use the symmetry based results deduced from the new identities to construct Lagrangians for partial differential equations. In particular, we show how the knowledge of a symmetry and its corresponding conservation law of a given partial differential equation can be utilised to construct a Lagrangian for the equation. Several examples are given.

Couette flow of a third-grade fluid with variable magnetic field

This investigation deals with the analytic solution for the time-dependent flow of an incompressible third-grade fluid which is under the influence of a magnetic field of variable strength. The fluid is in an annular region between two coaxial cylinders. The motion is induced due to an inner cylinder with arbitrary velocity. Group theoretic methods are employed to analyse the nonlinear problem and a solution for the velocity field is obtained analytically.

Exact flow of a third-grade fluid on a porous wall

The flow of a third-grade fluid occupying the space over a wall is studied. At the surface of the wall suction or blowing velocity is applied. By introducing a velocity field, the governing equations are reduced to a non-linear partial differential equation. The resulting equation is analysed analytically using Lie group methods.

Approximate Symmetries and Conservation Laws with Applications

The relationship between the approximateLie-Backlund symmetries and the approximate conservedforms of a perturbed equation is studied. It is shownthat a hierarchy of identities exists by which thecomponents of the approximate conserved vector or theassociated approximate Lie-Backlund symmetries aredetermined by recursive formulas. The results areapplied to certain classes of linear and nonlinear waveequations as well as a perturbed Korteweg-de Vriesequation. We construct approximate conservation laws forthese equations without regard to aLagrangian.

Nonlinear evolution-type equations and their exact solutions using inverse variational methods

We present the role of invariants in obtaining exact solutions of differential equations. Firstly, conserved vectors of a partial differential equation (p.d.e.) allow us to obtain reduced forms of the p.d.e. for which some of the Lie point symmetries (in vector field form) are easily concluded and, therefore, provide a mechanism for further reduction. Secondly, invariants of reduced forms of a p.d.e. are obtainable from a variational principle even though the p.d.e. itself does not admit a Lagrangian. In this latter case, the reductions carry all the usual advantages regarding Noether symmetries and double reductions. The examples we consider are nonlinear evolution-type equations such as the Korteweg–deVries equation, but a detailed analysis is made on the Fisher equation (which describes reaction–diffusion waves in biology, inter alia). Other diffusion-type equations lend themselves well to the method we describe (e.g., the Fitzhugh Nagumo equation, which is briefly discussed). Some aspects of Painleve properties are also suggested.

Approximate potential symmetries for partial differential equations

The method of approximate potential symmetries for partial differential equations with a small parameter is introduced. By writing a given perturbed partial differential equation R in a conserved form, an associated system S with potential variables as additional variables is obtained. Approximate Lie point symmetries admitted by S induce approximate potential symmetries of R. As applications of the theory, approximate potential symmetries for a perturbed wave equation with variable wave speed and a nonlinear diffusion equation with perturbed convection terms are obtained. The corresponding approximate group-invariant solutions are also derived.

Symmetry reduction, exact group-invariant solutions and conservation laws of the Benjamin–Bona–Mahoney equation

Abstract We show that the Benjamin–Bona–Mahoney (BBM) equation with power law nonlinearity can be transformed by a point transformation to the combined KdV–mKdV equation, that is also known as the Gardner equation. We then study the combined KdV–mKdV equation from the Lie group-theoretic point of view. The Lie point symmetry generators of the combined KdV–mKdV equation are derived. We obtain symmetry reduction and a number of exact group-invariant solutions for the underlying equation using the Lie point symmetries of the equation. The conserved densities are also calculated for the BBM equation with dual nonlinearity by using the multiplier approach. Finally, the conserved quantities are computed using the one-soliton solution.

A variational analysis of a non-Newtonian flow in a rotating system

This study deals with the flow of a third grade fluid past an infinite plate. Both fluid and plate are in a state of rigid body rotation. The analysis presented here deals with similarity solutions of the problem and is done in the following way. We consider a Lagrangian formulation which allows us to directly determine conserved quantities leading to a reduction of the resultant systems.