Traveling Wave Solutions and Chaotic Motions for a Perturbed Nonlinear Schrödinger Equation with Power-Law Nonlinearity and Higher-Order Dispersions

This chapter aims to study and solve the perturbed nonlinear Schrödinger (NLS) equation with the power-law nonlinearity in a nano-optical fiber, based upon different methods such as the auxiliary equation method, the Stuart and DiPrima’s stability analysis method, and the bifurcation theory. The existence of the traveling wave solutions is discussed, and their stability properties are investigated through the modulational stability gain spectra. Moreover, the development of the chaotic motions for the systems is pointed out via the bifurcation theory. Taking into account an external periodic perturbation, we have analyzed the chaotic behavior of traveling waves through quasiperiodic route to chaos.

Periodicity disruption of a model quasi-biennial oscillation of equatorial winds

<p>In the Earth's atmosphere, fast propagating equatorial waves generate slow reversals of the large scale stratospheric winds with a  period of about 28 months. This quasi-biennial oscillation is a spectacular manifestation of wave-mean flow interactions in stratified fluids, with analogues in other planetary atmospheres and laboratory experiments. Recent observations of a disruption of this periodic behavior have been attributed to external perturbations, but the mechanism explaining the disrupted response has remained elusive. We show the existence of secondary bifurcations and a quasiperiodic route to chaos in simplified models of the equatorial atmosphere ranging from the classical Holton-Lindzen-Plumb model to fully nonlinear simulations of stratified fluids. Perturbations of the slow oscillations are widely amplified in the proximity of the secondary bifurcation point. This suggests that intrinsic dynamics may be equally influential as external variability in explaining disruptions of regular wind reversals  [1].</p><p>[1] Renaud, A., Nadeau, L. P., & Venaille, A. (2019). Periodicity Disruption of a Model Quasibiennial Oscillation of Equatorial Winds. Physical Review Letters, 122(21), 214504.<br> </p>