Zlatko Jovanoski

Variational analysis of solitary waves in a homogeneous cubic-quintic nonlinear medium

Abstract Fundamental solitary waves in a cubic-quintic nonlinear medium are investigated both numerically and variationally. An equivalent particle model is used to understand the behaviour of the numerical solutions, and while a super-Gaussian trial function is found to have only a limited range of applicability in the variational analysis, a super-sech trial function is found to match the numerical solutions well for all input powers. Various limits are also investigated.

OPTICAL SOLITONS IN MULTI-DIMENSIONS WITH SPATIO-TEMPORAL DISPERSION AND NON-KERR LAW NONLINEARITY

This paper studies the dynamics of optical solitons in multi-dimensions with spatio-temporal dispersion and non-Kerr law nonlinearity. The integrability aspect is the main focus of this paper. Five different forms of nonlinearity are considered — Kerr law, power law, parabolic law, dual-power law and log law nonlinearity. The traveling wave hypothesis, ansatz approach and the semi-inverse variational principle are the integration tools that are adopted to retrieve the soliton solutions to the governing equation. As a result, several constraint conditions arise out of the integration process and represent necessary conditions for the existence of solitons.

Student interpretations of the terms in first-order ordinary differential equations in modelling contexts

A study of first-year undergraduate students′ interpretational difficulties with first-order ordinary differential equations (ODEs) in modelling contexts was conducted using a diagnostic quiz, exam questions and follow-up interviews. These investigations indicate that when thinking about such ODEs, many students muddle thinking about the function that gives the quantity to be determined and the equation for the quantitys rate of change, and at least some seem unaware of the need for unit consistency in the terms of an ODE. It appears that shifting from amount-type thinking to rates-of-change-type thinking is difficult for many students. Suggestions for pedagogical change based on our results are made.

Interaction of spatially separated oscillating solitons in biased two-photon photorefractive materials

This paper presents the dynamics of two spatially separated optical solitons in two-photon photorefractive materials. The variational formalism has been employed to derive evolution equations of different parameters which characterize the dynamics of two interacting solitons. This approach yields a system of coupled ordinary differential equations for evolution of different parameters characterizing solitons such as amplitude, spatial width, chirp, center of gravity, etc., which have been subsequently solved adopting numerical method to extract information on their dynamics. Depending on their initial separation and power, solitons are shown to either disperse or compresses individually and attract each other. Dragging and trapping of a probe soliton by another pump have been discussed.

Impacts of Biotic Resource Enrichment on a Predator–Prey Population

The environmental carrying capacity is usually assumed to be fixed quantity in the classical predator–prey population growth models. However, this assumption is not realistic as the environment generally varies with time. In a bid for greater realism, functional forms of carrying capacities have been widely applied to describe varying environments. Modelling carrying capacity as a state variable serves as another approach to capture the dynamical behavior between population and its environment. The proposed modified predator–prey model is based on the ratio-dependent models that have been utilized in the study of food chains. Using a simple non-linear system, the proposed model can be linked to an intra-guild predation model in which predator and prey share the same resource. Distinct from other models, we formulate the carrying capacity proportional to a biotic resource and both predator and prey species can directly alter the amount of resource available by interacting with it. Bifurcation and numerical analyses are presented to illustrate the system’s dynamical behavior. Taking the enrichment parameter of the resource as the bifurcation parameter, a Hopf bifurcation is found for some parameter ranges, which generate solutions that posses limit cycle behavior.

Cylindrically Symmetric Self-trapped Beams in a Higher-order Nonlinear Medium

Using a variational approach, we examine analytically the three-dimensional self-trapping of Gaussian beams in a medium with higher-order nonlinearity. We find exact analytic solutions for the spot width, and the conditions under which the Gaussian beam is stable to perturbation of its spot width. We also make a comparison with a Kerr-law medium.

Ship Stability and Parametric Rolling

Summary Ships, and therefore ship stability, is of vital importance for the transportation of humans and livestock, as well as providing the only means of transporting heavy cargoes between the continents. We present a simple model for ship stability based on the well known Mathieu equation, a second-order differential equation with periodic coefficients, which describes the phenomenon of parametric rolling. Using MATLAB, this model can be used in a classroom setting to introduce students to an important class of differential equations that are not ordinarily taught in the undergraduate engineering curriculum.

Two-color bright solitons in a three-level atomic system in the cascade configuration

We investigate two-component spatial optical solitons in a cascaded three-level atomic system. We derive an existence curve in the parameter space of power and spatial widths that reveals the existence of a plethora of coupled solitons. These solitons can exist with two different frequencies and also with two different widths. Our analytical results have been verified by direct numerical simulations. Stability analysis confirms that these solitons are stable.

GAUSSIAN BEAM PROPAGATION IN d-DIMENSIONAL CUBIC-QUINTIC NONLINEAR MEDIUM

The variational approach has been applied to study the propagation of Gaussian beams in a medium with d-transverse dimensions and a cubic-quintic nonlinearity. We find implicit conditions for the steady-state propagation and determine the stability of steady-state solutions to symmetric perturbations. In some parameter regime the stationary beam is stable and leads to the formation of light-bullets. We have also investigated the evolution of an initial Gaussian beam with finite radius of curvature into a steady-state beam. The results of the variational approach are congruous with recent numerical simulations.

Properties of combustion waves in a model with competitive exothermic reactions

We study the properties of travelling combustion waves in a diffusional thermal model with a two-step competitive exothermic-exothermic reaction mechanism. This investigation considers the system in one spatial dimension under adiabatic conditions. Based on the notion of the crossover temperature, the model is examined analytically using the activation energy asymptotic method to predict travelling combustion wave behaviour in the limit of large activation energies. The model is then studied numerically using a shooting-relaxation method over a wide range of parameter values, such as those describing the ratios of enthalpies, pre-exponential factors and activation energies. It is demonstrated that the flame speed as a function of these parameters is a single-valued monotonic function and there are two flame regimes identified—each region representing parameter values when one reaction dominates the other.