Complex singularities in KdV solutions

In the small dispersion regime, the KdV solution exhibits rapid oscillations in its spatio-temporal dependence. We show that these oscillations are caused by the presence of complex singularities that approach the real axis. We give a numerical estimate of the asymptotic dynamics of the poles.

Unsteady Separation and Navier-Stokes Solutions at High Reynolds Numbers

We compute the numerical solutions for Navier-Stokes and Prandtl’s equations in the case of a uniform bidimensional flow past an impulsively started disk. The numerical approx- imation is based on a spectral methods imple- mented in a Grid environment. We investigate the relationship between the phenomena of unsteady separation of the flow and the exponential decay of the Fourier spectrum of the solutions. We show that Prandtl’s solution develops a separation singularity in a finite time. Navier-Stokes solutions are computed over a range of Reynolds numbers from 3000 to 50000. We show that the appearance of large gradients of the pressure in the stream- wise direction, reveals that the visc…

Analysis of complex singularities in high-Reynolds-number Navier-Stokes solutions

AbstractNumerical solutions of the laminar Prandtl boundary-layer and Navier–Stokes equations are considered for the case of the two-dimensional uniform flow past an impulsively-started circular cylinder. The various viscous–inviscid interactions that occur during the unsteady separation process are investigated by applying complex singularity analysis to the wall shear and streamwise velocity component of the two solutions. This is carried out using two different methodologies, namely a singularity-tracking method and the Padé approximation. It is shown how the van Dommelen and Shen singularity that occurs in solutions of the Prandtl boundary-layer equations evolves in the complex plane be…

Complex singularities and PDEs

In this paper we give a review on the computational methods used to capture and characterize the complex singularities developed by some relevant PDEs. We begin by reviewing the classical singularity tracking method and give an example of application using the Burgers equation as a case study. This method is based on the analysis of the Fourier spectrum of the solution and it allows to determine and characterize the complex singularity closest to the real domain. We then introduce other methods generally used to detect the hidden singularities. In particular we show some applications of the Padé approximation, of the Kida method, and of Borel-Polya method. We apply these techniques to the s…

A note on the analytic solutions of the Camassa-Holm equation

Abstract In this Note we are concerned with the well-posedness of the Camassa–Holm equation in analytic function spaces. Using the Abstract Cauchy–Kowalewski Theorem we prove that the Camassa–Holm equation admits, locally in time, a unique analytic solution. Moreover, if the initial data is real analytic, belongs to H s ( R ) with s > 3 / 2 , ‖ u 0 ‖ L 1 ∞ and u 0 − u 0 x x does not change sign, we prove that the solution stays analytic globally in time. To cite this article: M.C. Lombardo et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).

Approccio bi-Hamiltoniano alle equazioni KP discrete

Singularity formation and separation phenomena in boundary layer theory

In this paper we review some results concerning the behaviour of the incompressible Navier–Stokes solutions in the zero viscosity limit. Most of the emphasis is put on the phenomena occurring in the boundary layer created when the no-slip condition is imposed. Numerical simulations are used to explore the limits of the theory. We also consider the case of 2D vortex layers, i.e. flows with internal layers in the form of a rapid variation, across a curve, of the tangential velocity.

Discrete KP Equation and Momentum Mapping of Toda System

Abstract A new approach to discrete KP equation is considered, starting from the Gelfand-Zakhharevich theory for the research of Casimir function for Toda Poisson pencil. The link between the usual approach through the use of discrete Lax operators, is emphasized. We show that these two different formulations of the discrete KP equation are equivalent and they are different representations of the same equations. The relation between the two approaches to the KP equation is obtained by a change of frame in the space of upper truncated Laurent series and translated into the space of shift operators.

Viscous-Inviscid Interactions in a Boundary-Layer Flow Induced by a Vortex Array

In this paper we investigate the asymptotic validity of boundary layer theory. For a flow induced by a periodic row of point-vortices, we compare Prandtl's solution to Navier-Stokes solutions at different $Re$ numbers. We show how Prandtl's solution develops a finite time separation singularity. On the other hand Navier-Stokes solution is characterized by the presence of two kinds of viscous-inviscid interactions between the boundary layer and the outer flow. These interactions can be detected by the analysis of the enstrophy and of the pressure gradient on the wall. Moreover we apply the complex singularity tracking method to Prandtl and Navier-Stokes solutions and analyze the previous int…

Regularized Euler-alpha motion of an infinite array of vortex sheets

We consider the Euler- $$\alpha $$ regularization of the Birkhoff–Rott equation and compare its solutions with the dynamics of the non regularized vortex-sheet. For a flow induced by an infinite array of planar vortex-sheets we analyze the complex singularities of the solutions.Through the singularity tracking method we show that the regularized solution has several complex singularities that approach the real axis. We relate their presence to the formation of two high-curvature points in the vortex sheet during the roll-up phenomenon.

LONG TIME BEHAVIOR OF A SHALLOW WATER MODEL FOR A BASIN WITH VARYING BOTTOM TOPOGRAPHY

We study the long time behavior of a shallow water model introduced by Levermore and Sammartino to describe the motion of a viscous incompressible fluid confined in a basin with topography. Here we prove the existence of a global attractor and give an estimate on its Hausdorff and fractal dimension.

On deformation of Poisson manifolds of hydrodynamic type

We study a class of deformations of infinite-dimensional Poisson manifolds of hydrodynamic type which are of interest in the theory of Frobenius manifolds. We prove two results. First, we show that the second cohomology group of these manifolds, in the Poisson-Lichnerowicz cohomology, is ``essentially'' trivial. Then, we prove a conjecture of B. Dubrovin about the triviality of homogeneous formal deformations of the above manifolds.

Route to chaos in the weakly stratified Kolmogorov flow

We consider a two-dimensional fluid exposed to Kolmogorov’s forcing cos(ny) and heated from above. The stabilizing effects of temperature are taken into account using the Boussinesq approximation. The fluid with no temperature stratification has been widely studied and, although relying on strong simplifications, it is considered an important tool for the theoretical and experimental study of transition to turbulence. In this paper, we are interested in the set of transitions leading the temperature stratified fluid from the laminar solution [U∝cos(ny),0, T ∝ y] to more complex states until the onset of chaotic states. We will consider Reynolds numbers 0 < Re ≤ 30, while the Richardson numb…

A computational method for the Helmholtz equation in unbounded domains based on the minimization of an integral functional

Abstract We study a new approach to the problem of transparent boundary conditions for the Helmholtz equation in unbounded domains. Our approach is based on the minimization of an integral functional arising from a volume integral formulation of the radiation condition. The index of refraction does not need to be constant at infinity and may have some angular dependency as well as perturbations. We prove analytical results on the convergence of the approximate solution. Numerical examples for different shapes of the artificial boundary and for non-constant indexes of refraction will be presented.

Numerical study of the primitive equations in the small viscosity regime

In this paper we study the flow dynamics governed by the primitive equations in the small viscosity regime. We consider an initial setup consisting on two dipolar structures interacting with a no slip boundary at the bottom of the domain. The generated boundary layer is analyzed in terms of the complex singularities of the horizontal pressure gradient and of the vorticity generated at the boundary. The presence of complex singularities is correlated with the appearance of secondary recirculation regions. Two viscosity regimes, with different qualitative properties, can be distinguished in the flow dynamics.

Singularities for Prandtl's equations.

We used a mixed spectral/finite-difference numerical method to investigate the possibility of a finite time blow-up of the solutions of Prandtl's equations for the case of the impulsively started cylinder. Our toll is the complex singularity tracking method. We show that a cubic root singularity seems to develop, in a time that can be made arbitrarily short, from a class of data uniformely bounded in H^1.

High Reynolds number Navier-Stokes solutions and boundary layer separation induced by a rectilinear vortex array

Numerical solutions of Prandtl’s equation and Navier Stokes equations are considered for the two dimensional flow induced by an array of periodic rec- tilinear vortices interacting with an infinite plane. We show how this initial datum develops a separation singularity for Prandtl equation. We investigate the asymptotic validity of boundary layer theory considering numerical solu- tions for the full Navier Stokes equations at high Reynolds numbers.

Transitions in a stratified Kolmogorov flow

We study the Kolmogorov flow with weak stratification. We consider a stabilizing uniform temperature gradient and examine the transitions leading the flow to chaotic states. By solving the equations numerically we construct the bifurcation diagram describing how the Kolmogorov flow, through a sequence of transitions, passes from its laminar solution toward weakly chaotic states. We consider the case when the Richardson number (measure of the intensity of the temperature gradient) is $$Ri=10^{-5}$$ , and restrict our analysis to the range $$0<Re<30$$ . The effect of the stabilizing temperature is to shift bifurcation points and to reduce the region of existence of stable drifting states. The…

Otitis media with effusion with or without atopy: audiological findings on primary schoolchildren

Objective: The objective of the study was to evaluate the role of atopy in otitis media with effusion (OME) in children attending primary school, focusing on the audiometric and tympanometric measurements among atopic and nonatopic subjects suffering from OME. Materials and Methods: Three hundred ten children (5-6 years old) were screened in Western Sicily by skin tests and divided into atopics (G1) and nonatopics (G2). The samples were evaluated for OME by pneumatic otoscopy, tympanogram, and acoustic reflex tests. The parameters considered were as follows: documented persistent middle ear effusion by otoscopic examination for a minimum of 3 months, presence of B or C tympanogram, absence …

Singular behavior of a vortex layer in the zero thickness limit

The aim of this paper is to study the Euler dynamics of a 2D periodic layer of non uniform vorticity. We consider the zero thickness limit and we compare the Euler solution with the vortex sheet evolution predicted by the Birkhoff-Rott equation. The well known process of singularity formation in shape of the vortex sheet correlates with the appearance of several complex singularities in the Euler solution with the vortex layer datum. These singularities approach the real axis and are responsible for the roll-up process in the layer motion.

Formation of Coherent Structures in Kolmogorov Flow with Stratification and Drag

We study a weakly stratified Kolmogorov flow under the effect of a small linear drag. We perform a linear stability analysis of the basic state. We construct the finite dimensional dynamical system deriving from the truncated Fourier mode approximation. Using the Reynolds number as bifurcation parameter we build the corresponding diagram up to Re=100. We observe the coexistence of three coherent structures.

Risk factors for otitis media with effusion: Case–control study in Sicilian schoolchildren

Objective To identify the prevalence and demographic, maternal and child risk factors for otitis media with effusion (OME) in Sicilian schoolchildren and analyse the results with reference to the review of the literature. Methods Associations of possible risk factors with prevalence of otitis media with effusion (OME) were studied in a cohort of 2097 children, aged 5–14 years. In order to determine OME, otoscopy and tympanometry were performed at 3-monthly intervals beginning at term date. Sixteen epidemiologically relevant features were inventoried by means of standardized questionnaires and skin tests were performed. Univariate analysis was performed to examine the association between det…

Turing pattern formation in the Brusselator system with nonlinear diffusion.

In this work we investigate the effect of density dependent nonlinear diffusion on pattern formation in the Brusselator system. Through linear stability analysis of the basic solution we determine the Turing and the oscillatory instability boundaries. A comparison with the classical linear diffusion shows how nonlinear diffusion favors the occurrence of Turing pattern formation. We study the process of pattern formation both in 1D and 2D spatial domains. Through a weakly nonlinear multiple scales analysis we derive the equations for the amplitude of the stationary patterns. The analysis of the amplitude equations shows the occurrence of a number of different phenomena, including stable supe…

The point prevalence of otitis media with eVusion among primary school children in Western Sicily

The objective of this study is to identify the prevalence of otitis media with effusion (OME) in primary school children and to value the possible predisposing factors focusing on relationship between allergy and OME in Western Sicily. 2,097 children attending primary school were screened from September 2006 to June 2007 in Sciacca. Children underwent pneumatic otoscopy, skin tests, tympanogram and acoustic reflex tests. Audiogram was performed if the child had a type B or a type C tympanogram. The criteria for diagnosis of OME were: documented persistent middle ear effusion by otoscopic examination for a minimum of 3 months, presence of B or C tympanogram, absence of ipsilateral acoustic r…

Analytic solutions and Singularity formation for the Peakon b--Family equations

This paper deals with the well-posedness of the b-family equation in analytic function spaces. Using the Abstract Cauchy-Kowalewski theorem we prove that the b-family equation admits, locally in time, a unique analytic solution. Moreover, if the initial data is real analytic and it belongs to H s with s>3/2, and the momentum density u 0-u 0, xx does not change sign, we prove that the solution stays analytic globally in time, for b≥1. Using pseudospectral numerical methods, we study, also, the singularity formation for the b-family equations with the singularity tracking method. This method allows us to follow the process of the singularity formation in the complex plane as the singularity a…

Characteristics of tinnitus with or without hearing loss: Clinical observations in Sicilian tinnitus patients

Objective: To analyze the clinical characteristics of tinnitus both in normal hearing subjects and in patients with hearing loss. Methods: The study considered 312 tinnitus sufferers, 176 males and 136 females, ranging from 21 to 83 years of age, who were referred to the Audiology Section of the Department of Bio-technology of Palermo University. The following parameters were considered: age, sex, hearing threshold, tinnitus laterality, tinnitus duration, tinnitus measurements and subjective disturbance caused by tinnitus. The sample was divided into two groups: Group 1 (G1) subjects with normal hearing; Group 2 (G2) subjects with hearing loss. Results: Among the patients considered, 115 ha…

Singularity formation for Prandtl’s equations

Abstract We consider Prandtl’s equations for an impulsively started disk and follow the process of the formation of the singularity in the complex plane using the singularity tracking method. We classify Van Dommelen and Shen’s singularity as a cubic root singularity. We introduce a class of initial data, uniformly bounded in H 1 , which have a dipole singularity in the complex plane. These data lead to a solution blow-up whose time can be made arbitrarily short within the class. This is numerical evidence of the ill-posedness of the Prandtl equations in H 1 . The presence of a small viscosity in the streamwise direction changes the behavior of the singularities. They stabilize at a distanc…

Unsteady Separation for High Reynolds Numbers Navier-Stokes Solutions

In this paper we compute the numerical solutions of Navier-Stokes equations in the case of the two dimensional disk impulsively started in a uniform back- ground flow. We shall solve the Navier-Stokes equations (for different Reynolds numbers ranging from 1.5 · 10^3 up to 10^5 ) with a fully spectral numerical scheme. We shall give a description of unsteady separation process in terms of large and small scale interactions acting over the flow. The beginning of these interactions will be linked to the topological change of the streamwise pressure gradient on the disk. Moreover we shall see how these stages of separation are related to the complex singularities of the solution. Infact the ana…

Singularity tracking for Camassa-Holm and Prandtl's equations

In this paper we consider the phenomenon of singularity formation for the Camassa-Holm equation and for Prandtl's equations. We solve these equations using spectral methods. Then we track the singularity in the complex plane estimating the rate of decay of the Fourier spectrum. This method allows us to follow the process of the singularity formation as the singularity approaches the real axis.

Long time behavior for a dissipative shallow water model

We consider the two-dimensional shallow water model derived by Levermore and Sammartino (Nonlinearity 14,2001), describing the motion of an incompressible fluid, confined in a shallow basin, with varying bottom topography. We construct the approximate inertial manifolds for the associated dynamical system and estimate its order. Finally, considering the whole domain R^2 and under suitable conditions on the time dependent forcing term, we prove the L^2 asymptotic decay of the weak solutions.

APPROXIMATE INERTIAL MANIFOLDS FOR THERMODIFFUSION EQUATIONS

In this paper, we consider the two dimensional equations of thermohydraulics, i.e. the coupled system of equations of fluid and temperature in the Boussinesq approximation. We construct a family of approximate Inertial Manifolds whose order decreases exponentially fast with respect to the dimension of the manifold. We give the explicit expression of the order of the constructed manifolds.

Intermittent and passivity based control strategies for a hyperchaotic system

In this paper a four-dimensional hyperchaotic system with only one equilibrium is consid- ered and it is shown how the control and the synchronization of this system can be realized via two different control techniques. Firstly, we propose a periodically intermittent con- troller to stabilize the system states to the equilibrium and to achieve the projective syn- chronization of the system both in its periodic and hyperchaotic regime. Then, based on the stability properties of a passive system, we design a linear passive controller, which only requires the knowledge of the system output, to drive the system trajectories asymptoti- cally to the origin. Using the same passivity-based method, …

The bi-Hamiltonian theory of the Harry Dym equation

We describe how the Harry Dym equation fits into the the bi-Hamiltonian formalism for the Korteweg-de Vries equation and other soliton equations. This is achieved using a certain Poisson pencil constructed from two compatible Poisson structures. We obtain an analogue of the Kadomtsev-Petviashivili hierarchy whose reduction leads to the Harry Dym hierarchy. We call such a system the HD-KP hierarchy. We then construct an infinite system of ordinary differential equations (in infinitely many variables) that is equivalent to the HD-KP hierarchy. Its role is analogous to the role of the Central System in the Kadomtsev-Petviashivili hierarchy.

Investigation of Tinnitus Patients in Italy: Clinical and Audiological Characteristics

Objective. 312 tinnitus sufferers were studied in order to analyze: the clinical characteristics of tinnitus; the presence of tinnitus-age correlation and tinnitus-hearing loss correlation; the impact of tinnitus on subjects' life and where possible the etiological/predisposing factors of tinnitus.Results. There is a slight predominance of males. The highest percentage of tinnitus results in the decades 61–70. Of the tinnitus sufferers, 197 (63.14%) have a hearing deficit (light hearing loss in 37.18% of cases). The hearing impairment results of sensorineural type in 74.62% and limited to the high frequencies in 58.50%. The tinnitus is referred as unilateral in 59.93%, a pure tone in 66.99%…

Prelingual sensorineural hearing loss and infants at risk: Western Sicily report.

Objective: To evaluate independent etiologic factor associated with sensorineural hearing loss (SNHL) in newborn at risk; to study the role of their interaction especially in NICU infants who present often multiple risk factors for SNHL. Methods: The main risk factors for SNHL reported by JCIH 2007 were evaluated on 508 infant at risk ranging from 4 to 20 weeks of life, transferred to the Audiology Department of Palermo from the main births centers of Western Sicily. After a global audiological assessment, performed with TEOAE, tympanometry and ABR, the prevalence and the effect of risk factors was statistically studied through univariate and multivariate analysis on the total population (n…

Transition to turbulence and Singularity in Boundary Layer Theory

We compute the solutions of Prandtl’s and Navier- Stokes equations for the two dimensional flow induced by an array of periodic rectilinear vortices interacting with a boundary in the halfplane. This initial datum develops, in a finite time, a separation singularity for Prandtl’s equation. We investigate the different stages of unsteady separation in Navier-Stokes solutions for various Reynolds numbers. We show the presence of a large- scale interaction between viscous boundary layer and inviscid outer flow in all Re regimes, while the presence of a small-scale interaction is visible only for moderate-high Re numbers. We also investigate the asymptotic validity of boundary layer theory in t…

A spectral approach to a constrained optimization problem for the Helmholtz equation in unbounded domains

We study some convergence issues for a recent approach to the problem of transparent boundary conditions for the Helmholtz equation in unbounded domains (Ciraolo et al. in J Comput Phys 246:78–95, 2013) where the index of refraction is not required to be constant at infinity. The approach is based on the minimization of an integral functional, which arises from an integral formulation of the radiation condition at infinity. In this paper, we implement a Fourier–Chebyshev collocation method to study some convergence properties of the numerical algorithm; in particular, we give numerical evidence of some convergence estimates available in the literature (Ciraolo in Helmholtz equation in unbou…

Well-posedness and singularity formation for the Camassa-Holm equation

We prove the well-posedness of Camassa--Holm equation in analytic function spaces both locally and globally in time, and we investigate numerically the phenomenon of singularity formation for particular initial data.

Assessing audiological, pathophysiological and psychological variables in tinnitus patients with or without hearing loss

The aim of this work is to study the characteristics of tinnitus both in normal hearing subjects and in patients with hearing loss. The study considered tinnitus sufferers, ranging from 21 to 83 years of age, who were referred to the Audiology Section of Palermo University in the years 2006–2008. The following parameters were considered: age, sex, hearing threshold, tinnitus laterality, tinnitus duration, tinnitus measurements and subjective disturbance caused by tinnitus. The sample was divided into Group1 (G1), 115 subjects with normal hearing, and Group2 (G2), 197 subjects with hearing loss. Especially for G2, there was a predominance of males compared to females (P = 0.011); the highest…

High Reynolds number Navier-Stokes solutions and boundary layer separation induced by a rectilinear vortex

Abstract We compute the solutions of Prandtl’s and Navier–Stokes equations for the two dimensional flow induced by a rectilinear vortex interacting with a boundary in the half plane. For this initial datum Prandtl’s equation develops, in a finite time, a separation singularity. We investigate the different stages of unsteady separation for Navier–Stokes solution at different Reynolds numbers Re = 103–105, and we show the presence of a large-scale interaction between the viscous boundary layer and the inviscid outer flow. We also see a subsequent stage, characterized by the presence of a small-scale interaction, which is visible only for moderate-high Re numbers Re = 104–105. We also investi…

Up-wind difference approximation and singularity formation for a slow erosion model

We consider a model for a granular flow in the slow erosion limit introduced in [31]. We propose an up-wind numerical scheme for this problem and show that the approximate solutions generated by the scheme converge to the unique entropy solution. Numerical examples are also presented showing the reliability of the scheme. We study also the finite time singularity formation for the model with the singularity tracking method, and we characterize the singularities as shocks in the solution.