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S. Pasquali

Publications and source records attributed to S. Pasquali.

12 recordsLinked to original sources
arXiv2026

Controllability for 2D water waves: effects of bottom topography and constant vorticity

In this paper we consider two-dimensional water waves, under the action of gravity and surface tension. We prove a controllability result for irrotational waves in a fluid domain with finite depth and general bottom topography. The result holds for an open and dense set of bottom topographies in $H^{s+1/2}(\mathbb{T})$ (where $s$ is sufficiently large) that do not touch the free surface. We point out that the bottom topographies that we allow for are not necessarily small perturbations of the flat bottom case: this leads to many technical difficulties, since the eigenvalues of the Dirichlet-Neumann operator at a still free surface with general bottom topography are not explicit, nor are they necessarily close (for low frequencies) to the eigenvalues of the corresponding operator for the flat bottom case. In turn, this leads to a more involved argument to prove Ingham-type estimates, which are needed to prove observability, and motivates the restriction mentioned above on the admissible bottom topographies. We also prove a controllability result for waves with constant vorticity in a fluid domain with flat bottom topography.

math.AP↗
arXiv2025

Instabilities of internal gravity waves in the two-dimensional Boussinesq system

We consider a two-dimensional, incompressible, inviscid fluid with variable density, subject to the action of gravity. Assuming a stable equilibrium density profile, we adopt the so-called Boussinesq approximation, which neglects density variations in all terms except those involving gravity. This model is widely used in the physical literature to describe internal gravity waves. In this work, we prove a modulational instability result for this system. Specifically, we show that the Floquet operator associated with the linearization around a small-amplitude traveling wave admits at least one eigenvalue with positive real part, bifurcating from a double eigenvalue of the unperturbed linear operator. This can be regarded as the first rigorous justification of the Parametric Subharmonic Instability (PSI) of inviscid internal waves, wherein energy is transferred from an initially excited primary wave to two secondary waves with different frequencies. Our approach uses Floquet-Bloch decomposition and Kato's similarity transformations to compute rigorously the perturbed eigenvalues without requiring boundedness of the perturbed operator - differing fundamentally from prior analyses involving viscosity. Notably, the inviscid setting is especially relevant in oceanographic applications, where viscous effects are often negligible.

math.AP↗
arXiv2025

Analytical study of a generalized Dirichlet-Neumann operator for three-dimensional water waves with vorticity

In this paper we consider three-dimensional water waves with vorticity, under the action of gravity. We discuss a generalized Zakharov-Craig-Sulem formulation of the problem introduced by Castro and Lannes, which involves a generalized Dirichlet-Neumann operator. We study this operator in detail, extending some well-known results about the classical Dirichlet-Neumann operator for irrotational water waves, such as the Taylor expansion in homogeneous powers of the wave profile, the computation of its differential and a paralinearization result. We stress the fact that no geometric condition on either the velocity field or the vorticity is assumed.

math.AP↗
arXiv2025

Two-dimensional water waves with constant vorticity and general bottom topography

In this paper we consider two-dimensional water waves with constant vorticity, under the action of gravity and surface tension, in a fluid domain with finite depth and general bottom topography. We present a formulation which generalizes the one by Zakharov-Craig-Sulem for irrotational water waves, and the one by Constantin-Ivanov-Prodanov for water waves with constant vorticity and flat bottom topography. We study in detail an operator which appears in such formulation, extending well-known results for the classical Dirichlet-Neumann operator, such as an analiticity result, the Taylor expansion in homogeneous powers of the wave profile, and a paralinearization formula. As an application, we prove a local well-posedness result.

math.AP↗
arXiv2018

The Fermi-Pasta-Ulam problem and its underlying integrable dynamics: an approach through Lyapunov Exponents

FPU models, in dimension one, are perturbations either of the linear model or of the Toda model; perturbations of the linear model include the usual $β$-model, perturbations of Toda include the usual $α+β$ model. In this paper we explore and compare two families, or hierarchies, of FPU models, closer and closer to either the linear or the Toda model, by computing numerically, for each model, the maximal Lyapunov exponent $χ$. We study the asymptotics of $χ$ for large $N$ (the number of particles) and small $ε$ (the specific energy $E/N$), and find, for all models, asymptotic power laws $χ\simeq Cε^a$, $C$ and $a$ depending on the model. The asymptotics turns out to be, in general, rather slow, and producing accurate results requires a great computational effort. We also revisit and extend the analytic computation of $χ$ introduced by Casetti, Livi and Pettini, originally formulated for the $β$-model. With great evidence the theory extends successfully to all models of the linear hierarchy, but not to models close to Toda.

math.DS↗
arXiv2008

Mean Field and the Single Homopolymer

We develop a statistical model for a confined chain molecule based on a monomer grand canonical ensemble. The molecule is subject to an external chemical potential, a backbone interaction, and an attractive interaction between all monomers. Using a Gaussian variable formalism and a mean field approximation, we analytically derive a minimum principle from which we can obtain relevant physical quantities, such as the monomer density, and we explore the limit in which the chain is subject to a tight confinement. Through a numerical implementation of the minimization process we show how we can obtain density profiles in three dimensions for arbitraty potentials, and we test the limits of validity of the theory.

cond-mat.stat-mech↗
arXiv2008

Numerical studies of Casimir interactions

We study numerically the Casimir interaction between dielectrics in both two and three dimensions. We demonstrate how sparse matrix factorizations enable one to study torsional interactions in three dimensions. In two dimensions we study the full cross-over between non-retarded and retarded interactions as a function of separation. We use constrained factorizations in order to measure the interaction of a particle with a rough dielectric surface and compare with a scaling argument.

quant-ph↗
arXiv2007

Fluctuation-induced interactions between dielectrics in general geometries

We study thermal Casimir and quantum non-retarded Lifshitz interactions between dielectrics in general geometries. We map the calculation of the classical partition function onto a determinant which we discretize and evaluate with the help of Cholesky factorization. The quantum partition function is treated by path integral quantization of a set of interacting dipoles and reduces to a product of determinants. We compare the approximations of pairwise additivity and proximity force with our numerical methods. We propose a ``factorization approximation'' which gives rather good numerical results in the geometries that we study.

cond-mat.stat-mech↗
arXiv2007

Numerical methods for fluctuation driven interactions between dielectrics

We develop a discretized theory of thermal Casimir interactions to numerically calculate the interactions between fluctuating dielectrics. From a constrained partition function we derive a surface free energy, while handling divergences that depend on system size and discretization. We derive analytic results for parallel plate geometry in order to check the convergence of the numerical methods. We use the method to calculate vertical and lateral Casimir forces for a set of grooves.

cond-mat.stat-mech↗
arXiv2006

Mapping a Homopolymer onto a Model Fluid

We describe a linear homopolymer using a Grand Canonical ensemble formalism, a statistical representation that is very convenient for formal manipulations. We investigate the properties of a system where only next neighbor interactions and an external, confining, field are present, and then show how a general pair interaction can be introduced perturbatively, making use of a Mayer expansion. Through a diagrammatic analysis, we shall show how constitutive equations derived for the polymeric system are equivalent to the Ornstein-Zernike and P.Y. equations for a simple fluid, and find the implications of such a mapping for the simple situation of Van der Waals mean field model for the fluid.

cond-mat.stat-mech↗
arXiv1998

Folding and Aggregation of Designed Proteins

Studies of how protein fold have shown that the way protein clumps form in the test tube is similar to how proteins form the so-called ``amyloid'' deposits that are the pathological signal of a variety of diseases, among them the memory disorder Alzheimer's. Protein aggregation have traditionally been connected to either unfolded or native states. Inclusion body formation (disordered aggregation) has been assumed to arise from hydrophobic aggregation of the unfolded or denaturated states, while the amyloid fibrils (ordered aggregation) have been assumed to arise from native-like conformations in a process analogous to the polymerization of hemoglobin S. Making use of lattice-model simulations we find that both ordered and disordered aggregation arise from elementary structures which eventually build the folding nucleus of the heteropolymers, and takes place when some of the most strongly interacting amino acids establish their contacts leading to the formation of a specific subset of the native structure. These elementary structures can be viewed as the partially folded intermediates suggested to be involved in the aggregation of a number of proteins. These results have evolutionary implications, as the elementary structures forming the folding core of designed proteins contain the residues which are conserved among the members of homologous sequences.

cond-mat↗