Space-modulated Stability and Averaged Dynamics

In this brief note we give a brief overview of the comprehensive theory, recently obtained by the author jointly with Johnson, Noble and Zumbrun, that describes the nonlinear dynamics about spectrally stable periodic waves of parabolic systems and announce parallel results for the linearized dynamics near cnoidal waves of the Korteweg-de Vries equation. The latter are expected to contribute to the development of a dispersive theory, still to come.Comment: Proceedings of the "Journ\'ees \'Equations aux d\'eriv\'ees partielles", Roscoff 201

Linear Asymptotic Stability and Modulation Behavior near Periodic Waves of the Korteweg-de Vries Equation

We provide a detailed study of the dynamics obtained by linearizing the Korteweg-de Vries equation about one of its periodic traveling waves, a cnoidal wave. In a suitable sense, linearly analogous to space-modulated stability, we prove global-in-time bounded stability in any Sobolev space, and asymptotic stability of dispersive type. Furthermore, we provide both a leading-order description of the dynamics in terms of slow modulation of local parameters and asymptotic modulation systems and effective initial data for the evolution of those parameters. This requires a global-in-time study of the dynamics generated by a non normal operator with non constant coefficients. On the road we also prove estimates on oscillatory integrals particularly suitable to derive large-time asymptotic systems that could be of some general interest

Whitham Averaged Equations and Modulational Stability of Periodic Traveling Waves of a Hyperbolic-Parabolic Balance Law

In this note, we report on recent findings concerning the spectral and nonlinear stability of periodic traveling wave solutions of hyperbolic-parabolic systems of balance laws, as applied to the St. Venant equations of shallow water flow down an incline. We begin by introducing a natural set of spectral stability assumptions, motivated by considerations from the Whitham averaged equations, and outline the recent proof yielding nonlinear stability under these conditions. We then turn to an analytical and numerical investigation of the verification of these spectral stability assumptions. While spectral instability is shown analytically to hold in both the Hopf and homoclinic limits, our numerical studies indicates spectrally stable periodic solutions of intermediate period. A mechanism for this moderate-amplitude stabilization is proposed in terms of numerically observed "metastability" of the the limiting homoclinic orbits.Comment: 27 pages, 5 figures. Minor changes throughou

Linear asymptotic stability of small-amplitude periodic waves of the generalized Korteweg--de Vries equations

In this note, we extend the detailed study of the linearized dynamics obtained for cnoidal waves of the Korteweg--de Vries equation in \cite{JFA-R} to small-amplitude periodic traveling waves of the generalized Korteweg-de Vries equations that are not subject to Benjamin--Feir instability. With the adapted notion of stability, this provides for such waves, global-in-time bounded stability in any Sobolev space, and asymptotic stability of dispersive type. When doing so, we actually prove that such results also hold for waves of arbitrary amplitude satisfying a form of spectral stability designated here as dispersive spectral stability.Comment: 15 page

Modulated equations of Hamiltonian PDEs and dispersive shocks

Motivated by the ongoing study of dispersive shock waves in non integrable systems , we propose and analyze a set of wave parameters for periodic waves of a large class of Hamiltonian partial differential systems-including the generalized Korteweg-de Vries equations and the Euler-Korteweg systems-that are well-behaved in both the small amplitude and small wavelength limits. We use this parametrization to determine fine asymptotic properties of the associated modulation systems, including detailed descriptions of eigenmodes. As a consequence, in the solitary wave limit we prove that modulational instability is decided by the sign of the second derivative-with respect to speed, fixing the endstate-of the Boussinesq moment of instability; and, in the harmonic limit, we identify an explicit modulational instability index, of Benjamin-Feir type

Validation of percutaneous puncture trajectory during renal access using 4D ultrasound reconstruction

"Progress in Biomedical Optics and Imaging, vol. 16, nr. 43"Background: An accurate percutaneous puncture is essential for disintegration and removal of renal stones. Although this procedure has proven to be safe, some organs surrounding the renal target might be accidentally perforated. This work describes a new intraoperative framework where tracked surgical tools are superimposed within 4D ultrasound imaging for security assessment of the percutaneous puncture trajectory (PPT). Methods: A PPT is first generated from the skin puncture site towards an anatomical target, using the information retrieved by electromagnetic motion tracking sensors coupled to surgical tools. Then, 2D ultrasound images acquired with a tracked probe are used to reconstruct a 4D ultrasound around the PPT under GPU processing. Volume hole-filling was performed in different processing time intervals by a tri-linear interpolation method. At spaced time intervals, the volume of the anatomical structures was segmented to ascertain if any vital structure is in between PPT and might compromise the surgical success. To enhance the volume visualization of the reconstructed structures, different render transfer functions were used. Results: Real-time US volume reconstruction and rendering with more than 25 frames/s was only possible when rendering only three orthogonal slice views. When using the whole reconstructed volume one achieved 8-15 frames/s. 3 frames/s were reached when one introduce the segmentation and detection if some structure intersected the PPT. Conclusions: The proposed framework creates a virtual and intuitive platform that can be used to identify and validate a PPT to safely and accurately perform the puncture in percutaneous nephrolithotomy.The authors acknowledge to Foundation for Science and Technology (FCT) - Portugal for the fellowships with references: SFRH/BD/74276/2010.info:eu-repo/semantics/publishedVersio