Optical soliton propagation in metamaterials; evolutionary pattern formation for competing populations under seasonal forcing
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Optical metamaterials is a cutting edge technology that is being studied. These Optical metamaterials possess both negative permittivity and negative permeability that cannot be found in nature; but can be engineered by using advanced processing technology. The dynamics of soliton propagation through these optical metamaterials is governed by the nonlinear Schr\ddot{o}dinger's equation(NLSE) with a few perturbation terms. There are a couple of ways this study can be done. One of them is to get specific values of the Super-Gaussian pulse parameters by the aid of collective variables; to recover bright 1-soliton solution by the aid of travelling wave hypothesis for bright soliton solutions of power law and dual-power law media; to obtain doubly periodic functions to the model with mapping method; to retrieve exact soliton solutions by the method of undetremined coefficients which is known as Ansatz scheme; To extract bright and exotic soliton solutions by semi-inverse variational principle; to illustrate the controllability of the Raman soliton self-frequency shift in non-linear metamaterials by numerical results.
Population models can be used to understand the Honey Bee Population Dynamics and other species at interest and also be used to understand the spread of parasites, viruses, and disease. For example, explore the impact of different death rates of forager bees on colony growth and development, evaluate the effects of artificial feeding on bee colony population dynamics, recognize the importance of pollination to our food systems and economics. In our model we describe the effects of seasonal variations on competing population dynamics. We begin with the well understood fisher's equation applied to competing species. Competition is modeled as a non-local effect through a convolution integral with an asymmetric kernel function. Seasonal variations are added by perturbing the competition term with a sinusoidal term, psin(ft). The extent of the non-local coupling is determined by a parameter delta, with delta = 0 corresponding to localization; the degree of asymmetry is characterized by alpha, so that when alpha goes to 0, the problem becomes symmetric non-local coupling, and psin(ft), describing the severity of disturbance, with p=0 corresponding to static interactions. We study the case where the model admits a stable coexistence equilibrium solution. We access stability conditions of this critical point as a function of alpha and p and determine unstable wave number bands with delta beyond the stability boundary. We show the evolutionary nonlinear patterns under varying seasonal forcing with sufficiently non-localization.