Nectarios C. Papanicolaou

Numerical solutions of boundary value problems for variable coefficient generalized KdV equations using Lie symmetries

Abstract The exhaustive group classification of a class of variable coefficient generalized KdV equations is presented, which completes and enhances results existing in the literature. Lie symmetries are used for solving an initial and boundary value problem for certain subclasses of the above class. Namely, the found Lie symmetries are applied in order to reduce the initial and boundary value problem for the generalized KdV equations (which are PDEs) to an initial value problem for nonlinear third-order ODEs. The latter problem is solved numerically using the finite difference method. Numerical solutions are computed and the vast parameter space is studied.

Kawahara solitons in Boussinesq equations using a robust Christov-Galerkin spectral method

We develop a robust Christov-Galerkin spectral technique for computing interacting localized wave solutions of and fourth and sixth-order generalized wave equations. To this end, a special complete orthonormal system of functions in L^2(-~,~) is used whose rate of convergence is shown to be exponential for the cases under consideration. For the time-stepping, an implicit algorithm is chosen which makes use of the banded structure of the matrices representing the different spatial derivatives. As featuring examples, the head-on collision of solitary waves is investigated for a sixth-order generalized Boussinesq equation and a fourth-order Boussinesq type equation with a linear term. Its solutions comprise monotone shapes (sech-es) and damped oscillatory shapes (Kawahara solitons). The numerical results are validated against published data in the literature using the method of variational imbedding.

A Mode-Matching Approach to Electromagnetic Wave Propagation in Nematic Liquid Crystals

In this paper, we present a computationally efficient and highly accurate numerical method for the analysis of electromagnetic wave propagation in nematic liquid crystal (N-LC) cells. An iterative procedure is employed where the mode-matching technique (MMT) is used to solve the time-harmonic Maxwell equations inside the N-LC cell, whereas a finite-difference method (FDM) with relaxation is utilized to treat the nonlinear stationary Ginzburg-Landau equation for the director field. The angular distortion of the directors in the N-LC cell depends on the applied electric field which, in turn, affects the anisotropic dielectric properties of the medium. Numerical results are obtained for various values of the governing parameters. These simulations provide further insight into the Fréedericksz transition with special emphasis on resonances, bi-stability, hysteresis, phase shift between ordinary and extraordinary waves (birefringence), and soft anchoring effects. Obtained results are compared and validated against measurements and data published in the literature.

Soft Polarization Diffraction Coefficient for a Conducting Cylinder-Tipped Wedge

TM, electromagnetic scattering from a perfectly conducting wedge with a cylindrical tip is formulated using a mode-matching technique. The eigenmode expansion is then written in a convenient form, where the total field is expressed as a superposition of a wedge-diffraction term based on the uniform theory of diffraction, a geometrical optics term, and a Correction Field term due to the presence of the cylindrical tip. The obtained diffraction coefficient can be easily incorporated into existing ray-tracing, high-frequency codes for the prediction of scattered fields from electrically large structures. The underlined formulation and obtained expressions are verified by comparing numerical results with the finite element method.

The Christov-Galerkin spectral method in complex arithmetics

We apply the Christov-Galerkin spectral method for the numerical investigation of the interaction of solitons in the Cubic Nonlinear Schrodinger Equation. The issues of convergence are addressed and an algorithm is devised for the application of the method. Results are obtained for the interaction of solitons with different phase velocities and different carrier frequencies. The interactions are shown to be elastic, save for the phase shifts. The latter are extracted from the numerical solution and discussed.

Numerical modeling of electromagnetic wave propagation in a liquid crystal cell at oblique incidence

This paper presents a robust numerical method for the analysis of wave propagation in nematic liquid crystals. The structure is excited by a plane wave incident at an oblique angle with respect to the normal to the liquid-crystal cell. The underlined formulation is based on an eigenvalue problem which is solved analytically in order to obtain the governing field expressions inside a homogeneous, thin crystal layer. The liquid-crystal cell is comprised of N such layers. Enforcing the continuity of the tangential electric and magnetic fields at the interfaces formed by the various layers, a matrix system is generated. Solution of the linear system of equations results in the light intensity inside the liquid crystal, which is coupled to a non-linear differential equation for the director tilt angle. This equation is solved using either an explicit or implicit finite-difference scheme. An iteration process continues until convergence is reached for the coupled problem. The proposed numerical method was validated against published results that were generated by approximate analytical methods. Further simulations and studies were conducted emphasizing on the physics of the problem and related interesting phenomena.

Modal analysis and solution of electromagnetic wave propagation in cholesteric liquid crystal cells

Cholesteric Liquid Crystals (Ch-LC) are anisotropic and inhomogeneous materials with intriguing and useful properties in the visible region of the electromagnetic spectrum. In this paper, we use eigenmode analysis to obtain the supported field expressions in a thin, homogeneous sub-layer of the Ch-LC cell; the liquid crystal is made of multiple sub-layers. The cell is sandwiched between layers of dielectric and is excited by an elliptically polarized plane wave at an oblique incidence. The governing field expressions in the dielectric layers are also obtained using eigenmode analysis. A mode-matching technique (MMT) is then employed to enforce the continuity of the tangential electric and magnetic fields at the interface of two neighboring layers, thus, resulting in a matrix system representative of the problem at hand. Solution of the linear system of equations yields the reflection and transmission coefficients on the two principal planes as well as the expansion coefficients for the modal fields inside the Ch-LC and the dielectric layers. The underlined formulation is verified by comparing results to the open literature.