Abdul H. Kara

A note on a symmetry analysis and exact solutions of a nonlinear fin equation

A similarity analysis of a nonlinear fin equation has been carried out by M. Pakdemirli and A.Z. Sahin [Similarity analysis of a nonlinear fin equation, Appl. Math. Lett. (2005) (in press)]. Here, we consider a further group theoretic analysis that leads to an alternative set of exact solutions or reduced equations with an emphasis on travelling wave solutions, steady state type solutions and solutions not appearing elsewhere.

Conservation laws for optical solitons with spatio-temporal dispersion

The conservation laws for optical solitons with spatio-temporal dispersion are obtained in this paper. There are three conserved quantities that are reported in this paper. They are the power, linear momentum, and the Hamiltonian. The conserved quantities, from their respective densities, are obtained from 1-soliton solution that was reported earlier. Five types of nonlinear media are taken into account.

Wave Equations in Bianchi Space-Times

We investigate the wave equation in Bianchi type III space-time. We construct a Lagrangian of the model, calculate and classify the Noether symmetry generators, and construct corresponding conserved forms. A reduction of the underlying equations is performed to obtain invariant solutions.

Double reductions/analysis of the Drinfeld-Sokolov-Wilson equation

In this paper, we show that the recently developed formulation of the association between symmetries and conservation laws lead to double reductions of the third-order Drinfeld-Sokolov-Wilson system of partial differential equations (PDEs). For illustrative purposes only, we carry out the procedure for a version of the well-known third-order scalar Korteweg-de Vries (KdV) equation via a scaling symmetry of the equation.

SOLITONS AND CONSERVATION LAWS IN NEUROSCIENCES

This paper obtains the 1-soliton solution by the ansatz method for the proposed model that governs the propagation of solitons through the neurons. This model is an improved one that describes the solitons in neurosciences more accurately. The ansatz method is applied to obtain the 1-soliton solution to the model. The Lie symmetry analysis is subsequently applied to obtain the conservation laws for the model.

Invariant solutions of certain nonlinear evolution type equations with small parameters

The Fisher equation, which arises in the study of reaction diffusion waves in biology, does not display a high level of symmetry properties. Consequently, only travelling wave solutions are obtainable using the method of invariants. This has a direct bearing on studying perturbed forms of the equation which may arise from considering, e.g., damping or dissipative factors. We show, here, how one can get around this limitation by appending some unknown function to the perturbation and obtain interesting practical results using invariants. The ideas have significant consequences for equations which do not admit large class of symmetry properties. The method used in this analysis is then extended to other classes of evolution type equations that involve perturbations, for, e.g., the KdV type equations.

Potential symmetry generators and associated conservation laws of perturbed nonlinear equations

Some recent results on approximate Lie group methods and previously developed concepts on potential symmetries are extended and applied to (mainly) nonlinear systems perturbed by a small parameter. The potential (or auxilliary) form of the perturbed system necessarily requires a knowledge of an `approximate conservation law of the system. We then also use knowledge of `approximate potential symmetries to calculate new approximate potential conservation laws.

Invariance, Conservation Laws, and Exact Solutions of the Nonlinear Cylindrical Fin Equation

Fins are heat exchange surfaces which are used widely in industry. The partial differential equation arising from heat transfer in a fin of cylindrical shape with temperature dependent thermal diffusivity are studied. The method of multipliers and invariance of the differential equations is employed to obtain conservation laws and perform double reduction.

Symmetries, conservation laws, and ‘integrability’ of difference equations

A number of nontrivial conservation laws of some difference equations, viz., the discrete Liouville equation and the discrete sine-Gordon equation, are constructed using first principles. Symmetries and the more recent ideas and notions of characteristics (multipliers) for difference equations are also discussed.

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

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.