Nauman Raza

Sobolev gradient approach for the time evolution related to energy minimization of Ginzburg-Landau functionals

The Sobolev gradient technique has been discussed previously in this journal as an efficient method for finding energy minima of certain Ginzburg-Landau type functionals [S. Sial, J. Neuberger, T. Lookman, A. Saxena, Energy minimization using Sobolev gradients: application to phase separation and ordering, J. Comput. Phys. 189 (2003) 88-97]. In this article a Sobolev gradient method for the related time evolution is discussed.

Approximating time evolution related to Ginzburg-Landau functionals via Sobolev gradient methods in a finite-element setting

Sobolev gradients have previously [1] been used to approximate time evolution related to a model A functional in a finite-difference setting in this journal. Here a related approach in a finite-element setting is discussed.

Energy minimization related to the nonlinear Schrödinger equation

In this the window of the Sobolev gradient technique to the problem of minimizing a Schrodinger functional associated with a nonlinear Schrodinger equation. We show that gradients act in a suitably chosen Sobolev space (Sobolev gradients) can be used in finite-difference and finite-element settings in a computationally efficient way to find minimum energy states of Schrodinger functionals.

On solitons: the biomolecular nonlinear transmission line models with constant and time variable coefficients

Abstract The article studies the dynamics of solitons in electrical microtubule model, which describes the propagation of waves in nonlinear dynamical system. Microtubules are not only a passive support of a cell but also they have highly dynamic structures involved in cell motility, intracellular transport and signaling. The underlying model has been considered with constant and variable coefficients of time function. The solitary wave ansatz has been applied successfully to extract these solitons. The corresponding integrability criteria, also known as constraint conditions, naturally emerge from the analysis of these models.

NUMERICAL SOLUTIONS OF INTEGRO-DIFFERENTIAL EQUATIONS USING SOBOLEV GRADIENT METHODS

A numerical procedure for solving a class of integro-differential equations of Volterra type using the Sobolev gradient method is presented. Results are compared with those from the variational iteration method (VIM) and Adomian decomposition method (ADM) (Batiha, B., Noorani, M. S. M. and Hashmi, I. [2008] Numerical solutions of the nonlinear integro-differential equations, Int. J. Open Probl. Compt. Math.1, 34–42). The capabilities of our codes are briefly described and test results from some examples are presented.

Numerical solution of Burgers’ equation by the Sobolev gradient method

Abstract Burgers’ equation is solved numerically with Sobolev gradient methods. A comparison is shown with other numerical schemes presented in this journal, such as modified Adomian method (MAM) [1] and by a variational method (VM) which is based on the method of discretization in time [2] . It is shown that the Sobolev gradient methods are highly efficient while at the same time retaining the simplicity of steepest descent algorithms.

Optical dark and dark-singular soliton solutions of (1+2)-dimensional chiral nonlinear Schrodinger’s equation

ABSTRACT In this study, the dynamical analysis of optical dark and singular solitons is carried out for chiral (1+2)-dimensional nonlinear Schrödinger’s equation with the implementation of extended direct algebraic and extended trial equation method independently. The constraint conditions guarantee the perseverance of these soliton solutions. Along with optical dark and singular solitons, these integration techniques yield other wave solutions such as Jacobi elliptic function, rational function, and hyperbolic function solutions as outgrowth.

Optical soliton solutions of the cubic-quintic non-linear Schrödinger’s equation including an anti-cubic term

Abstract A wide range of problems in different fields of the applied sciences especially non-linear optics is described by non-linear Schrödinger’s equations (NLSEs). In the present paper, a specific type of NLSEs known as the cubic-quintic non-linear Schrödinger’s equation including an anti-cubic term has been studied. The generalized Kudryashov method along with symbolic computation package has been exerted to carry out this objective. As a consequence, a series of optical soliton solutions have formally been retrieved. It is corroborated that the generalized form of Kudryashov method is a direct, effectual, and reliable technique to deal with various types of non-linear Schrödinger’s equations.

Numerical approximation of time evolution related to Ginzburg-Landau functionals using weighted Sobolev gradients

Sobolev gradients have been discussed in Sial et al. (2003) as a method for energy minimization related to Ginzburg-Landau functionals. In this article, a weighted Sobolev gradient approach for the time evolution of a Ginzburg-Landau functional is presented for different values of @k. A comparison is given between the weighted and unweighted Sobolev gradients in a finite element setting. It is seen that for small values of @k, the weighted Sobolev gradient method becomes more and more efficient compared to using the unweighted Sobolev gradient. A comparison with Newtons method is given where the failure of Newtons method is demonstrated for a test problem.

Bright, dark and dark-singular soliton solutions of nonlinear Schrödinger's equation with spatio-temporal dispersion

ABSTRACT This article presents an analytical study of the propagation of solitons through optical fibres. We consider a nonlinear Schrödinger equation with spatio-temporal dispersion and quadratic–cubic nonlinearity. Jacobi elliptic functions are used as an ansatz to extract optical dark and bright solitons as well as Jacobi elliptic solutions. The extended direct algebraic method gives dark and dark-singular soliton solutions. The constraint conditions which guarantee the existence of soliton solutions are listed.