Multi-breathers and high order rogue waves for the nonlinear Schr\"odinger equation on the elliptic function background

We construct the multi-breather solutions of the focusing nonlinear Schr\"odinger equation (NLSE) on the background of elliptic functions by the Darboux transformation, and express them in terms of the determinant of theta functions. The dynamics of the breathers in the presence of various kinds of backgrounds such as dn, cn, and non-trivial phase-modulating elliptic solutions are presented, and their behaviors dependent on the effect of backgrounds are elucidated. We also determine the asymptotic behaviors for the multi-breather solutions with different velocities in the limit $t\to\pm\infty$, where the solution in the neighborhood of each breather tends to the simple one-breather solution. Furthermore, we exactly solve the linearized NLSE using the squared eigenfunction and determine the unstable spectra for elliptic function background. By using them, the Akhmediev breathers arising from these modulational instabilities are plotted and their dynamics are revealed. Finally, we provide the rogue-wave and higher-order rogue-wave solutions by taking the special limit of the breather solutions at branch points and the generalized Darboux transformation. The resulting dynamics of the rogue waves involves rich phenomena: depending on the choice of the background and possessing different velocities relative to the background. We also provide an example of the multi- and higher-order rogue wave solution.Comment: 45 pages, 16 figure

Large Deviations for Stochastic Generalized Porous Media Equations

The large deviation principle is established for the distributions of a class of generalized stochastic porous media equations for both small noise and short time.Comment: 15 pages; BiBoS-Preprint No. 05-11-196; publication in preparatio

Darboux transformation and solitonic solution to the coupled complex short pulse equation

The Darboux transformation (DT) for the coupled complex short pulse (CCSP) equation is constructed through the loop group method. The DT is then utilized to construct various exact solutions including bright soliton, dark-soliton, breather and rogue wave solutions to the CCSP equation. In case of vanishing boundary condition (VBC), we perform the inverse scattering analysis to understand the soliton solution better. Breather and rogue wave solutions are constructed in case of non-vanishing boundary condition (NVBC). Moreover, we conduct a modulational instability (MI) analysis based on the method of squared eigenfunctions, whose result confirms the condition for the existence of rogue wave solution.Comment: 15 figure

Transportation-information inequalities for Markov processes (II) : relations with other functional inequalities

We continue our investigation on the transportation-information inequalities $W_pI$ for a symmetric markov process, introduced and studied in \cite{GLWY}. We prove that $W_pI$ implies the usual transportation inequalities $W_pH$, then the corresponding concentration inequalities for the invariant measure $\mu$. We give also a direct proof that the spectral gap in the space of Lipschitz functions for a diffusion process implies $W_1I$ (a result due to \cite{GLWY}) and a Cheeger type's isoperimetric inequality. Finally we exhibit relations between transportation-information inequalities and a family of functional inequalities (such as $\Phi$-log Sobolev or $\Phi$-Sobolev)

Effects of Empty Sites on Cooperation in the Prisoner’s Dilemma Game Based on Social Diversity

We study the effects of empty sites in the prisoner’s dilemma game based on social diversity by introducing some empty sites into a square lattice. The results reveal that the empty sites dramatically enhance the cooperation level for a wide range of temptation to defection values if two types of players coexist. By calculating the chances of different type-combinations of the players located on the square lattice, we find that there is an intermediate region where five social ranks are induced to satisfy the certain rank distributions and the cooperation level is significantly enhanced. Moreover, simulation results also show that the moderate gaps among the social ranks can favor cooperation for a larger occupation density

Enhanced multi-source data analysis for personalized sleep-wake pattern recognition and sleep parameter extraction

The file attached to this record is the author's final peer reviewed version.Sleep behavior is traditionally monitored with polysomnography, and sleep stage patterns are a key marker for sleep quality used to detect anomalies and diagnose diseases. With the growing demand for personalized healthcare and the prevalence of the Internet of Things, there is a trend to use everyday technologies for sleep behavior analysis at home, having the potential to eliminate expensive in-hospital monitoring. In this paper, we conceived a multi-source data mining approach to personalized sleep-wake pattern recog-nition which uses physiological data and personal information to facilitate fine-grained detection. Physiological data includes actigraphy and heart rate variability and personal data makes use of gender, health status and race infor-mation which are known influence factors. Moreover, we developed a personal-ized sleep parameter extraction technique fused with the sleep-wake approach, achieving personalized instead of static thresholds for decision-making. Results show that the proposed approach improves the accuracy of sleep and wake stage recognition, therefore, offers a new solution for personalized sleep-based health monitoring