The Generalized Dirichlet to Neumann map for the KdV equation on the half-line

For the two versions of the KdV equation on the positive half-line an initial-boundary value problem is well posed if one prescribes an initial condition plus either one boundary condition if $q_{t}$ and $q_{xxx}$ have the same sign (KdVI) or two boundary conditions if $q_{t}$ and $q_{xxx}$ have opposite sign (KdVII). Constructing the generalized Dirichlet to Neumann map for the above problems means characterizing the unknown boundary values in terms of the given initial and boundary conditions. For example, if $\{q(x,0),q(0,t) \}$ and $\{q(x,0),q(0,t),q_{x}(0,t) \}$ are given for the KdVI and KdVII equations, respectively, then one must construct the unknown boundary values $\{q_{x}(0,t),q_{xx}(0,t) \}$ and $\{q_{xx}(0,t) \}$, respectively. We show that this can be achieved without solving for $q(x,t)$ by analysing a certain ``global relation'' which couples the given initial and boundary conditions with the unknown boundary values, as well as with the function $\Phi^{(t)}(t,k)$, where $\Phi^{(t)}$ satisifies the $t$-part of the associated Lax pair evaluated at $x=0$. Indeed, by employing a Gelfand--Levitan--Marchenko triangular representation for $\Phi^{(t)}$, the global relation can be solved \emph{explicitly} for the unknown boundary values in terms of the given initial and boundary conditions and the function $\Phi^{(t)}$. This yields the unknown boundary values in terms of a nonlinear Volterra integral equation.Comment: 21 pages, 3 figure

Failure Analyses of Two Gas Turbine Wheels

AbstractThe results of the analysis of the catastrophic failures of two high pressure turbine wheels are discussed in this study. Fractographic and metallographic analyses on both the wheel and a set of failed blades of both wheels were performed to determine the possible events that led to failure. Both wheel materials had an austenitic microstructure, while blade materials were different for each case. One blade material is similar to INCONEL 738 nickel-based superalloy, while the other study is a single-crystal with dendritic growth microstructure. Facing two failures with apparently similar characteristics, once fractographic and metallographic analysis were performed, it was proved that failure modes respond to quite different origins in each case. This led to different corrective actions, according to each particular main contributing factor

Travelling waves for the Gross-Pitaevskii equation II

The purpose of this paper is to provide a rigorous mathematical proof of the existence of travelling wave solutions to the Gross-Pitaevskii equation in dimensions two and three. Our arguments, based on minimization under constraints, yield a full branch of solutions, and extend earlier results, where only a part of the branch was built. In dimension three, we also show that there are no travelling wave solutions of small energy.Comment: Final version accepted for publication in Communications in Mathematical Physics with a few minor corrections and added remark

Asymptotic models for the generation of internal waves by a moving ship, and the dead-water phenomenon

This paper deals with the dead-water phenomenon, which occurs when a ship sails in a stratified fluid, and experiences an important drag due to waves below the surface. More generally, we study the generation of internal waves by a disturbance moving at constant speed on top of two layers of fluids of different densities. Starting from the full Euler equations, we present several nonlinear asymptotic models, in the long wave regime. These models are rigorously justified by consistency or convergence results. A careful theoretical and numerical analysis is then provided, in order to predict the behavior of the flow and in which situations the dead-water effect appears.Comment: To appear in Nonlinearit

Nonlinear interfacial waves in a constant-vorticity planar flow over variable depth

Exact Lagrangian in compact form is derived for planar internal waves in a two-fluid system with a relatively small density jump (the Boussinesq limit taking place in real oceanic conditions), in the presence of a background shear current of constant vorticity, and over arbitrary bottom profile. Long-wave asymptotic approximations of higher orders are derived from the exact Hamiltonian functional in a remarkably simple way, for two different parametrizations of the interface shape.Comment: revtex, 4.5 pages, minor corrections, summary added, accepted to JETP Letter

Using 3D Stringy Gravity to Understand the Thurston Conjecture

We present a string inspired 3D Euclidean field theory as the starting point for a modified Ricci flow analysis of the Thurston conjecture. In addition to the metric, the theory contains a dilaton, an antisymmetric tensor field and a Maxwell-Chern Simons field. For constant dilaton, the theory appears to obey a Birkhoff theorem which allows only nine possible classes of solutions, depending on the signs of the parameters in the action. Eight of these correspond to the eight Thurston geometries, while the ninth describes the metric of a squashed three sphere. It therefore appears that one can construct modified Ricci flow equations in which the topology of the geometry is encoded in the parameters of an underlying field theory.Comment: 17 pages, Late

A microseismic study in a low seismicity area of Italy: the Città di Castello 2000-2001 experiment

Recent seismological studies contribute to better understand the first order characteristics of earthquake occurrence in Italy, identifying the potential sites for moderate to large size earthquakes. Ad hoc passive seismic experiments performed in these areas provide information to focus on the location and geometry of the active faults more closely. This information is relevant for assessing seismic hazard and for accurately constraining possible ground shaking scenarios. The area around the Città di Castello Basin, in the Northern Apennines (Central Italy), is characterized by the absence of instrumental seismicity (M > 2.5), it is adjacent to faults ruptured by recent and historical earthquakes. To better understand the tectonics of the area, we installed a dense network of seismic stations equipped with broadband and short period seismometers collecting data continuously for 8 months (October 2000-May 2001). The processing of ~ 900 Gbyte of data revealed a consistent background seismicity consisting of very low magnitude earthquakes (ML < 3.2). Preliminary locations of about 2200 local earthquakes show that the area can be divided into two regions with different seismic behaviour: an area to the NW, in between Sansepolcro and Città di Castello, where seismicity is not present. An area toward the SE, in between Città di Castello, Umbertide and Gubbio, where we detected a high microseismicity activity. These findings suggest a probable different mechanical behaviour of the two regions. In the latter area, the seismicity is confined between 0 and 8 km of depth revealing a rather well defined east-dipping, low angle fault 35 km wide that cuts through the entire upper crust down to 12-15 km depth. Beside an apparent structural complexity, fault plane solutions of background seismicity reveal a homogeneous pattern of deformation with a clear NE-SW extension