Symmetries of differential equations. IV

By an application of the geometrical techniques of Lie, Cohen, and Dickson it is shown that a system of differential equations of the form [x^(r_i)]_i = F_i(where r_i > 1 for every i = 1 , ... ,n) cannot admit an infinite number of pointlike symmetry vectors. When r_i = r for every i = 1, ... ,n, upper bounds have been computed for the maximum number of independent symmetry vectors that these systems can possess: The upper bounds are given by 2n_ 2 + nr + 2 (when r> 2), and by 2n_2 + 4n + 2 (when r = 2). The group of symmetries of ͞x^r = ͞0 (r> 1) has also been computed, and the result obtained shows that when n > 1 and r> 2 the number of independent symmetries of these equations does not attain the upper bound 2n _2 + nr + 2, which is a common bound for all systems of differential equations of the form ͞x^r = F[t, ͞x, ... , ͞x^(r - 1 )] when r> 2. On the other hand, when r = 2 the first upper bound obtained has been reduced to the value n^2 + 4n + 3; this number is equal to the number of independent symmetry vectors of the system ͞x= ͞0, and is also a common bound for all systems of the form ͞x = ͞F (t ,͞x, ‾̇x)

Explicit differential characterization of the Newtonian free particle system in m > 1 dependent variables

In 1883, as an early result, Sophus Lie established an explicit necessary and sufficient condition for an analytic second order ordinary differential equation y_xx = F(x,y,y_x) to be equivalent, through a point transformation (x,y) --> (X(x,y), Y(x,y)), to the Newtonian free particle equation Y_XX = 0. This result, preliminary to the deep group-theoretic classification of second order analytic ordinary differential equations, was parachieved later in 1896 by Arthur Tresse, a French student of S. Lie. In the present paper, following closely the original strategy of proof of S. Lie, which we firstly expose and restitute in length, we generalize this explicit characterization to the case of several second order ordinary differential equations. Let K=R or C, or more generally any field of characteristic zero equipped with a valuation, so that K-analytic functions make sense. Let x in K, let m > 1, let y := (y^1, ..., y^m) in K^m and let y_xx^j = F^j(x,y,y_x^l), j = 1,...,m be a collection of m analytic second order ordinary differential equations, in general nonlinear. We provide an explicit necessary and sufficient condition in order that this system is equivalent, under a point transformation (x, y^1, ..., y^m) --> (X(x,y), Y^1(x,y),..., Y^m(x, y)), to the Newtonian free particle system Y_XX^1 = ... = Y_XX^m = 0. Strikingly, the (complicated) differential system that we obtain is of first order in the case m > 1, whereas it is of second order in S. Lie's original case m = 1.Comment: 76 pages, no figur

Submarine canyons and related features in the Alboran Sea: continental margins and major isolated reliefs

The analysis of a data set of multibeam bathymetry plus high resolution seismic and parametric profiles allow us to characterize the geomorphologic units on the Alboran Sea-floor as well as the evolution of morpho-sedimentary systems along the Pliocene and Quaternary, later than the main erosive Messinian event. Since the opening of the Gibraltar Straits, the sedimentary evolution of this basin has been controlled by the interchange of water masses between the Atlantic Ocean and the Mediterranean Sea. Basin physiography is also a consequence of the Pliocene-Quaternary compression which has progressively uplifted the sourrounding reliefs and deforms the interior and the margins of the basin. On this scenario, several submarine canyons and gullies have been developed in this basin which traverse especially the northern margin and the flanks of the Northern Alboran Ridge, without affecting the African margins. This fact must be related to the action of bottom contour currents which constitute the main morpho-sedimentary process. The influence of water masses distributed the sedimentary input carried by rivers and coming from the erosion of surrounding ranges. In the southern margin of this basin this influence is stronger and inhibits the development of transversal submarine canyons

The Guadiaro-Baños contourite drifts (SW Mediterranean). A geotechnical approach to stability analysis

Two Quaternary plastered contourite drifts, with terraced and low-mounded morphologies, make up the continental slope and base-of-slope in the northwestern Alboran Sea, respectively, between the Guadiaro and Baños turbidite systems, close to the Strait of Gibraltar. Considering their significant lateral extent, the link between the contourite drift deposits and landslides may be particularly important for hazard assessment. The physical properties, composition and geometry of contourite drifts have been proposed as key factors in slope stability, although this relationship still needs to be better constrained. In this work, new in-situ geotechnical data (cone penetration tests; CPTu) has been combined with morphostratigraphic, sedimentological, and (laboratory) geotechnical properties to determine the stability of the Guadiaro-Baños drifts. For the depositional domains of both drifts, the resulting sedimentary and geotechnical model describes low-plasticity granular and silty sands on the erosive terraced domain that evolve seawards to silty and silty-clay deposits with a higher plasticity and uniform geomechanical properties. For the shallower coarse-grained contourite sediments, the cohesion (c') and internal friction angle (ϕ') values are 0–9 kPa and 46–30°, respectively, whereas for the distal fine contourites the undrained shear strength gradient (∇Su) is 2 kPa/m. These properties allow us to establish high factors of safety for all the scenarios considered, including seismic loading. Slope failure may be triggered in the unlikely event that there is seismic acceleration of PGA > 0.19, although no potential glide planes have been observed within the first 20 m below the seafloor. This suggests that the contourite drifts studied tend to resist failure better than others with similar sedimentary characteristics. The interplay of several processes is proposed to explain the enhanced undrained shear strength: 1) the geometry of the drifts, defined by an upper contouritic terrace and lower low-mounded shapes; 2) recurrent low-intensity earthquakes with insufficient energy to trigger landslides, favouring increased strength due to dynamic compaction; and 3) cyclic loading induced by solitons/internal waves acting on the sediment.En prens

Understanding the complex geomorphology of a deep sea area affected by continental tectonic indentation: the case of the Gulf of Vera (Western Mediterranean)

We present a multidisciplinary study of morphology, stratigraphy, sedimentology, tectonic structure, and physical oceanography to report that the complex geomorphology of the Palomares continental margin and adjacent Algerian abyssal plain (i.e., Gulf of Vera, Western Mediterranean), is the result of the sedimentary response to the Aguilas Arc continental tectonic indentation in the Eurasian–Africa plate collision. The inden tation is imprinted on the basement of the margin with elongated metamorphic antiforms that are pierced by igneous bodies, and synforms that accommodate the deformation and create a complex physiography. The basement is partially covered by Upper Miocene deposits sealed by the regional Messinian Erosive Surface characterized by palaeocanyons that carve the modern margin. These deposits and outcropping basement highs are then covered and shaped by Plio-Quaternary contourites formed under the action of the Light Intermediate and Dense Deep Mediterranean bottom currents. Even though bottom currents are responsible for the primary sedimentation that shapes the margin, 97% of this region's seafloor is affected by mass-movements that modified contourite sediments by eroding, deforming, faulting, sliding, and depositing sediments. Mass-movement processes have resulted in the formation of recurrent mass-flow deposits, an enlargement of the submarine canyons and gully incisions, and basin-scale gravitational slides spreading above the Messinian Salinity Crisis salt layer. The Polopo, Aguilas and Gata slides are characterized by an extensional upslope domain that shapes the continental margin, and by a downslope contractional domain that shapes the abyssal plain with diapirs piercing (hemi)pelagites/sheet-like turbidites creating a seafloor dotted by numerous crests. The mass movements were mostly triggered by the interplay of the continental tectonic indentation of the Aguilas Arc with sedimentological factors over time. The indentation, which involves the progressively southeastward tectonic tilting of the whole land-sea region, likely generated a quasi-continuous oversteepening of the entire margin, thus reducing the stability of the contourites. In addition, tectonic tilting and subsidence of the abyssal plain favoured the flow of the underlying Messinian Salinity Crisis salt layer, contributing to the gravitational instability of the overlying sediments over large areas of the margin and abyssal plain