Symmetric invariant manifolds in the Fermi-Pasta-Ulam lattice

The Fermi-Pasta-Ulam (FPU) lattice with periodic boundary conditions and $n$ particles admits a large group of discrete symmetries. The fixed point sets of these symmetries naturally form invariant symplectic manifolds that are investigated in this short note. For each $k$ dividing $n$ we find $k$ degree of freedom invariant manifolds. They represent short wavelength solutions composed of $k$ Fourier-modes and can be interpreted as embedded lattices with periodic boundary conditions and only $k$ particles. Inside these invariant manifolds other invariant structures and exact solutions are found which represent for instance periodic and quasi-periodic solutions and standing and traveling waves. Some of these results have been found previously by other authors via a study of mode coupling coefficients and recently also by investigating `bushes of normal modes'. The method of this paper is similar to the latter method and much more systematic than the former. We arrive at previously unknown results without any difficult computations. It is shown moreover that similar invariant manifolds exist also in the Klein-Gordon lattice and in the thermodynamic and continuum limits.Comment: 14 pages, 1 figure, accepted for publication in Physica

Influence of the Geometry Alteration of the Landslide Slope on its Stability: A Case Study in the Carnian Alps (Italy)

Abstract The paper presents the principles of the slope reprofiling and proves the effectiveness of this stabilization measure. The case study of two adjacent landslides in the National Road 52 "Carnica" in the Tagliamento River valley, the Carnian Alps (46°23′49″N, 12°42′51″E) are the example allowing for illustration of this approach. The phenomena have been studied for more than a decade, making it possible to carry out a detailed geological and geomorphological reconstruction. That was done on the basis of a large amount of monitoring data collected during that period. Since the landslides are threatening an important road, countermeasure works to ameliorate the stability conditions of the slides need to be designed. The paper focuses on the creation of a numerical model consistent with monitoring data and capable of reconstructing the dynamics of both landslides. Two cross-sections, one for each landslide, were selected for the analysis. The geometry of the slip surface was determined on the basis of control points, such as slip surface readings from inclinometers, and geomorphological evidence for the contour. The FLAC2D code was used to evaluate the current stability of these landslides and to determine the effectiveness of changing the slope geometry by removing material from the upper part of the slope and putting it to the lowest part as reinforcement

STABILITY ANALYSIS OF THE SLOPE SUBJECTED TO THE DYNAMIC LOADING

This paper presents the results of a stability analysis of a slope located in the immediate vicinity of a railway line. The plans for the extension of this railway track include the construction of another line, which would run parallel to the existing one, within a few metres distance. It is expected that intensive goods train traffic will generate both static and dynamic forces in the underlying subsoil. Consequently, seismic vibrations will be generated in the subsoil, propagating mainly not only in the horizontal direction but also in the vertical direction. The method of seismic coefficient of the earthquake intensity determined by a pseudo-static method and horizontal component of acceleration is appropriate and recommended because it is simple, and the safety factor of the slope is calculated in the same way as in conventional stability calculations

Site investigation and modelling at "La Maina" landslide (Carnian Alps, Italy)

International audienceThe Sauris reservoir is a hydroelectric basin closed downstream by a 136 m high, double arc concrete dam. The dam is firmly anchored to a consistent rock (Dolomia dello Schlern), but the Lower Triassic clayey formations, cropping out especially in the lower part of the slopes, have made the whole catchment basin increasingly prone to landslides. In recent years, the "La Maina landslide" has opened up several joints over a surface of about 100 000 m2, displacing about 1 500 000 m3 of material. Particular attention is now being given to the evolution of the instability area, as the reservoir is located at the foot of the landslide. Under the commission of the Regional Authority for Civil Protection a numerical modelling simulation in a pseudo-time condition of the slope was developed, in order to understand the risk for transport infrastructures, for some houses and for the reservoir and to take urgent mesaures to stabilize the slope. A monitoring system consisting of four inclinometers, three wire extensometers and ten GPS bench-mark pillars was immediately set up to check on surface and deep displacements. The data collected and the geological and geomorphological evidences was used to carry out a numerical simulation. The reliability of the results was checked by comparing the model with the morphological evidence of the movement. The mitigation measures were designed and realised following the indications provided by the model

Evaluation of seismic effects on the landslide deposits of Monte Salta (Eastern Italian Alps) using distinct element method

International audienceThe aim of the paper is to present the modelling of the ground effects of seismic waves on a large debris deposit lying on a steep mountain slope, with particular attention paid to the potential triggering of slope movements. The study site is a mass of 2.5 million m3 rock fall deposit, named "Monte Salta Landslide", located on the northern slope of the Vajont valley, at the border between Veneto and Friuli Venezia Giulia regions in north-eastern Italy. Several historical landslide events were reported in the area in the past, first one dating back to the 17th century. The landslide deposit completely mantles the slope with a thick cover of rock blocks. The Mt. Salta landslide is conditioned by the presence of Mt. Borgà regional thrust, which uplifts Jurassic limestone on the top of Cretaceous rock units. Above the thrust zone, folded and highly fractured rock mass dips steeply towards the slope free face, producing highly unstable setting. The study area has been classified as high seismic hazard and different vulnerable elements can be affected by the remobilisation of debris, among which a village, a national road and a big quarry that was opened, with the intent to exploit the part of the landslide deposit for construction purposes. In this study, numerical analysis was performed, to simulate the slope behaviour using distinct element method and applying UDEC code. The 2-D models were built on three cross-sections and elasto-plastic behaviour was assumed, both for rock matrix and discontinuities. The earthquake effect was modelled in pseudo-dynamic way, i.e. by magnifying the acceleration and applying also its horizontal component. The expected seismic acceleration in the study area was calculated on the basis of previous studies as equal to 0.28 g. The results proved that the increase of the vertical component alone has a small influence on the deformational behaviour of the system. Hence, the acceleration vector was deviated at 5° and then at 10° from the vertical. A small increment of the displacement was observed in the first case, whereas very large movements occurred in the second. Therefore, it can be concluded that, besides the magnitude of the earthquake, even small seismic waves in horizontal direction could trigger significant movements and therefore hazardous conditions. The modelled scenario should be helpful for planning of the functional countermeasure works and civil defence evacuation plan

On Completely Integrability Systems of Differential Equations

In this note we discuss the approach which was given by Wazwaz for the proof of the complete integrability to the system of nonlinear differential equations. We show that his method presented in [Wazwaz A.M. Completely integrable coupled KdV and coupled KP systems, Commun Nonlinear Sci Simulat 15 (2010) 2828-2835] is incorrect.Comment: 14 pages. This paper was sent to the Communications in Nonlinear Science and Numerical Simulatio

A study on relativistic lagrangian field theories with non-topological soliton solutions

We perform a general analysis of the dynamic structure of two classes of relativistic lagrangian field theories exhibiting static spherically symmetric non-topological soliton solutions. The analysis is concerned with (multi-) scalar fields and generalized gauge fields of compact semi-simple Lie groups. The lagrangian densities governing the dynamics of the (multi-) scalar fields are assumed to be general functions of the kinetic terms, whereas the gauge-invariant lagrangians are general functions of the field invariants. These functions are constrained by requirements of regularity, positivity of the energy and vanishing of the vacuum energy, defining what we call "admissible" models. In the scalar case we establish the general conditions which determine exhaustively the families of admissible lagrangian models supporting this kind of finite-energy solutions. We analyze some explicit examples of these different families, which are defined by the asymptotic and central behaviour of the fields of the corresponding particle-like solutions. From the variational analysis of the energy functional, we show that the admissibility constraints and the finiteness of the energy of the scalar solitons are necessary and sufficient conditions for their linear static stability against small charge-preserving perturbations. Furthermore we perform a general spectral analysis of the dynamic evolution of the small perturbations around the statically stable solitons, establishing their dynamic stability. Next, we consider the case of many-components scalar fields, showing that the resolution of the particle-like field problem in this case reduces to that of the one-component case. The study of these scalar models is a necessary step in the analysis of the gauge fields.Comment: LaTeX (uses elsart.cls), 57 pages, 9 Postscript figures, final version to appear in Annals of Physics, some minor corrections adde

Construction of Pressure Tunnels

The paper focuses on two pressure tunnels in the design of “Katy-Myscowa” water reservoir. One of them serves as a discharge conduit, whereas the other plays an energetic role. Their depths range between 0 and 75 metres and their diameters equal 5 m. Tunnels are located in the rock mass of Carpathian flysch which is anisotropic and heterogeneous, composed of layers of sandstone and clay shales and intersected with interbedding fissures and numerous joints. The paper is divided in two parts. The first part focuses on methods of excavating and supporting, as well as injecting and sealing (i.e. waterproofing) the tunnel. In the second part, a numerical analysis using the FLAC2D code based on the finite difference method was carried for calculating displacements and internal forces in the preliminary support and in permanent lining. Results of the analysis allow for the assessment of conditions in the tunnel during its excavation and exploitation stages

Appraisal of landslide processes based on the results of inclinometric measurements

The paper presents the principles of the inclinometric method in the measurements of underground displacements of slopes are described and some example results. The advantages of the method and the practical examples are shown: results of the measurements on the slopes in the flysch Carpathians and Polish Lowland - i.e. in the Beskid Niski Mts., Beskid Ma³y Mts. and in Koronowo near Bydgoszcz. The capabilities of the method are described in detail. These include determination of the depth of the landslide slip surface or zone, description of the slide mechanism (rotational or translational), estimation of displacement rate and its connection with the external conditions, and determination of the movement direction

Construction of Pressure Tunnels

The paper focuses on two pressure tunnels in the design of “Katy-Myscowa” water reservoir. One of them serves as a discharge conduit, whereas the other plays an energetic role. Their depths range between 0 and 75 metres and their diameters equal 5 m. Tunnels are located in the rock mass of Carpathian flysch which is anisotropic and heterogeneous, composed of layers of sandstone and clay shales and intersected with interbedding fissures and numerous joints. The paper is divided in two parts. The first part focuses on methods of excavating and supporting, as well as injecting and sealing (i.e. waterproofing) the tunnel. In the second part, a numerical analysis using the FLAC2D code based on the finite difference method was carried for calculating displacements and internal forces in the preliminary support and in permanent lining. Results of the analysis allow for the assessment of conditions in the tunnel during its excavation and exploitation stages