Discrete diffraction managed solitons: Threshold phenomena and rapid decay for general nonlinearities

We prove a threshold phenomenon for the existence/non-existence of energy minimizing solitary solutions of the diffraction management equation for strictly positive and zero average diffraction. Our methods allow for a large class of nonlinearities, they are, for example, allowed to change sign, and the weakest possible condition, it only has to be locally integrable, on the local diffraction profile. The solutions are found as minimizers of a nonlinear and nonlocal variational problem which is translation invariant. There exists a critical threshold ?cr such that minimizers for this variational problem exist if their power is bigger than ?cr and no minimizers exist with power less than the critical threshold. We also give simple criteria for the finiteness and strict positivity of the critical threshold. Our proof of existence of minimizers is rather direct and avoids the use of Lions' concentration compactness argument. Furthermore, we give precise quantitative lower bounds on the exponential decay rate of the diffraction management solitons, which confirm the physical heuristic prediction for the asymptotic decay rate. Moreover, for ground state solutions, these bounds give a quantitative lower bound for the divergence of the exponential decay rate in the limit of vanishing average diffraction. For zero average diffraction, we prove quantitative bounds which show that the solitons decay much faster than exponentially. Our results considerably extend and strengthen the results of [15] and [16].Comment: 49 pages, no figure

Effect of TRB3 on Skeletal Muscle Mass Regulation and Exercise-Induced Adaptation

Skeletal muscle, which composes over 40% of body mass, is responsible for daily locomotion and energy metabolism. It is a malleable tissue that can adapt its structure and function in response to internal and external environmental stimuli. Changing muscle mass and energy substrate utilization is a common skeletal muscle adaptation in response to various pathological conditions, including type 2 diabetes and obesity. Failure to maintain skeletal muscle mass and function has been correlated with increasing morbidity and mortality, as well as poor quality of life. Hence, maintenance of skeletal muscle integrity is a recommended strategy in achieving better quality of life. TRB3 is a pseudokinase that is known to negatively regulate Akt phosphorylation, a key protein kinase in regulating protein turnover and energy metabolism in skeletal muscle. Although TRB3 is found in skeletal muscle and its expression is associated with Akt signaling, no previous studies have elucidated the effects of TRB3 on skeletal muscle mass regulation. The purpose of this dissertation was to determine the role of TRB3 in skeletal muscle mass regulation and exercise-induced adaptation. We hypothesized that TRB3 expression would regulate protein turnover by regulating Akt and its downstream proteins, mTOR and FOXO, at the basal state and under atrophic conditions. We also expected TRB3 expression to blunt exercise-induced skeletal muscle adaptation. In Aim 1, we tested whether TRB3 expression in mouse skeletal muscle regulated protein turnover through the Akt/mTOR/FOXO pathway at the basal state. We found that skeletal muscle protein turnover was regulated by TRB3 expression. In Aim 2, we examined whether TRB3 expression in mouse skeletal muscle affected food deprivation (FD)-induced skeletal muscle atrophy. We observed that muscle-specific TRB3 overexpression worsened FD- induced atrophy via increasing proteolysis systems, while TRB3 knockout prevented the atrophy by preserving protein synthesis. In Aim 3, we tested whether TRB3 expression was involved in exercise-induced skeletal muscle adaptation. Here, we found that muscle specific TRB3 overexpression blunted the benefits of exercise training in glucose uptake and mitochondrial adaptation. These findings provide evidence that TRB3 is a potential target to improve skeletal muscle integrity and quality in both healthy conditions and atrophic conditions

Discrete diffraction managed solitons: threshold phenomena and rapid decay for general nonlinearities

We prove a threshold phenomenon for the existence/non-existence of energy minimizing solitary solutions of the diffraction management equation for strictly positive and zero average diffraction. Our methods allow for a large class of nonlinearities, they are, for example, allowed to change sign, and the weakest possible condition, it only has to be locally integrable, on the local diffraction profile. The solutions are found as minimizers of a nonlinear and nonlocal variational problem which is translation invariant. There exists a critical threshold λcr such that minimizers for this variational problem exist if their power is bigger than λcr and no minimizers exist with power less than the critical threshold. We also give simple criteria for the finiteness and strict positivity of the critical threshold. Our proof of existence of minimizers is rather direct and avoids the use of Lions\u27 concentration compactness argument. Furthermore, we give precise quantitative lower bounds on the exponential decay rate of the diffraction management solitons, which confirm the physical heuristic prediction for the asymptotic decay rate. Moreover, for ground state solutions, these bounds give a quantitative lower bound for the divergence of the exponential decay rate in the limit of vanishing average diffraction. For zero average diffraction, we prove quantitative bounds which show that the solitons decay much faster than exponentially. Our results considerably extend and strengthen the results of [15] and [16]

Well-posedness of dispersion managed nonlinear Schr\"odinger equations

We prove local and global well-posedness results for the Gabitov-Turitsyn or dispersion managed nonlinear Schr\"odinger equation with a large class of nonlinearities and arbitrary average dispersion on $L^2(\mathbb{R})$ and $H^1(\mathbb{R})$. Moreover, when the average dispersion is non-negative, we show that the set of nonlinear ground states is orbitally stable.Comment: 29 page