A Rigorous Justification of the Modulation Approximation to the 2D Full Water Wave Problem

We consider the 2D inviscid incompressible irrotational infinite depth water wave problem neglecting surface tension. Given wave packet initial data, we show that the modulation of the solution is a profile traveling at group velocity and governed by a focusing cubic nonlinear Schrodinger equation, with rigorous error estimates in Sobolev spaces. As a consequence, we establish existence of solutions of the water wave problem in Sobolev spaces for times in the NLS regime provided the initial data is suitably close to a wave packet of sufficiently small amplitude in Sobolev spaces

Modulational Instability in Equations of KdV Type

It is a matter of experience that nonlinear waves in dispersive media, propagating primarily in one direction, may appear periodic in small space and time scales, but their characteristics --- amplitude, phase, wave number, etc. --- slowly vary in large space and time scales. In the 1970's, Whitham developed an asymptotic (WKB) method to study the effects of small "modulations" on nonlinear periodic wave trains. Since then, there has been a great deal of work aiming at rigorously justifying the predictions from Whitham's formal theory. We discuss recent advances in the mathematical understanding of the dynamics, in particular, the instability of slowly modulated wave trains for nonlinear dispersive equations of KdV type.Comment: 40 pages. To appear in upcoming title in Lecture Notes in Physic

Comprehensive security assessment of combined MTD techniques for the cloud

Moving Target Defense (MTD) is a proactive security solution, which can be utilized by cloud computing in order to thwart cyber attacks. Many MTD techniques have been proposed, but there is still a lack of systematic evaluation methods for assessing the effectiveness of the proposed MTD techniques, especially when multiple MTD techniques are to be used in combinations. In this paper, we aim to address the aforementioned issue by proposing an approach for modeling and analysis of MTD techniques. We consider four security metrics: system risk, attack cost, return on attack, and availability to quantify the security of the cloud before and after deploying MTD techniques. Moreover, we propose a Diversity MTD technique to deploy OS diversification with various variants on multiple VMs and also combined Shuffle, Diversity, and Redundancy MTD techniques to improve the security of the cloud. We analyze the security metrics before and after deploying the proposed techniques to show the effectiveness of them. We also utilize importance measures based on network centrality measures into security analysis phase to improve the scalability of the MTD evaluation