Thresholds for breather solutions on the Discrete Nonlinear Schr\"odinger Equation with saturable and power nonlinearity

We consider the question of existence of periodic solutions (called breather solutions or discrete solitons) for the Discrete Nonlinear Schr\"odinger Equation with saturable and power nonlinearity. Theoretical and numerical results are proved concerning the existence and nonexistence of periodic solutions by a variational approach and a fixed point argument. In the variational approach we are restricted to DNLS lattices with Dirichlet boundary conditions. It is proved that there exists parameters (frequency or nonlinearity parameters) for which the corresponding minimizers satisfy explicit upper and lower bounds on the power. The numerical studies performed indicate that these bounds behave as thresholds for the existence of periodic solutions. The fixed point method considers the case of infinite lattices. Through this method, the existence of a threshold is proved in the case of saturable nonlinearity and an explicit theoretical estimate which is independent on the dimension is given. The numerical studies, testing the efficiency of the bounds derived by both methods, demonstrate that these thresholds are quite sharp estimates of a threshold value on the power needed for the the existence of a breather solution. This it justified by the consideration of limiting cases with respect to the size of the nonlinearity parameters and nonlinearity exponents.Comment: 26 pages, 10 figure

An Evaluation Of The Relationships Between Resilient Safety Culture, Safety Risk Parameters, And Mindfulness In The International Air Show Community

A convergent mixed-methods approach with data triangulation was utilized to assess the strength of relationships between operational risk factors, hazardous attitude, and resilient safety culture when mediated by mindfulness in the international air show community. An anonymous online survey of respondents’ perceptions, semi-structured interviews of air show experts, focus-group on air show performers, field observation at an air show, and a documentary analysis of air show safety event data was used to collect data. The quantitative findings suggest a good fit of a hypothesized structural model showing the relationships between study variables using structural equation modeling (SEM). Mindfulness (MF) significantly mediates the predictive relationship between hazardous attitudes (HA), risk perception (RP), risk tolerance (RT), and resilient safety culture (RSC) with a high effect size. There was significant predictive relationship between MF and RSC with medium effect size. Demographically, married respondents had significantly lower mean scores on MF compared to single and divorced while single respondents had higher mean scores on RT than married or divorced. The qualitative findings indicate that the RSC of air show performers has a negative correlation with RT and HA. The triangulation suggests military air show background was strongly correlated with RSC, MF, and a negative correlation to HA. This study provides a validated measurement model to assess the relationships between the study variables and fills a gap in the literature related to resilient safety culture in the airshow community. Theoretical and practical implications of this study provide a framework for continuous improvement of safety in the air show community

Self trapping transition for a nonlinear impurity within a linear chain

In the present work we revisit the issue of the self-trapping dynamical transition at a nonlinear impurity embedded in an otherwise linear lattice. For our Schr\"odinger chain example, we present rigorous arguments that establish necessary conditions and corresponding parametric bounds for the transition between linear decay and nonlinear persistence of a defect mode. The proofs combine a contraction mapping approach applied in the fully dynamical problem in the case of a 3D-lattice, together with variational arguments for the derivation of parametric bounds for the creation of stationary states associated with the expected fate of the self-trapping dynamical transition. The results are relevant for both power law nonlinearities and saturable ones. The analytical results are corroborated by numerical computations.Comment: 16 pages, 7 figures. To be published in Journal of Mathematical Physic

Επίλυση στοχαστικής Μη γραμμικής εξίσωσης με τη μέθοδο του Πολυωνυμικού Χάους

Εθνικό Μετσόβιο Πολυτεχνείο--Μεταπτυχιακή Εργασία. Διεπιστημονικό-Διατμηματικό Πρόγραμμα Μεταπτυχιακών Σπουδών (Δ.Π.Μ.Σ.) “Υπολογιστική Μηχανική