Beatrice Pelloni
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View article: Jumps, cusps, and fractals in the solution of the periodic linear Benjamin–Ono equation
Jumps, cusps, and fractals in the solution of the periodic linear Benjamin–Ono equation Open
We establish two complementary results about the regularity of the solution of the periodic initial value problem for the linear Benjamin–Ono equation. We first give a new simple proof of the statement that, for a dense countable set of th…
View article: Semi-discrete optimal transport techniques for the compressible semi-geostrophic equations
Semi-discrete optimal transport techniques for the compressible semi-geostrophic equations Open
We prove existence of weak solutions of the 3D compressible semi-geostrophic (SG) equations with compactly supported measure-valued initial data. These equations model large-scale atmospheric flows. Our proof uses a particle discretisation…
View article: Jumps, cusps and fractals in the solution of the periodic linear Benjamin-Ono equation
Jumps, cusps and fractals in the solution of the periodic linear Benjamin-Ono equation Open
We establish two complementary results about the regularity of the solution of the periodic initial value problem for the linear Benjamin-Ono equation. We first give a new simple proof of the statement that, for a dense countable set of th…
View article: The phenomenon of revivals on complex potential Schrödinger’s equation
The phenomenon of revivals on complex potential Schrödinger’s equation Open
The mysterious phenomenon of revivals in linear dispersive periodic equations was discovered first experimentally in optics in the 19th century, then rediscovered several times by theoretical and experimental investigations. While the term…
View article: Revivals, or the Talbot effect, for the Airy equation
Revivals, or the Talbot effect, for the Airy equation Open
We study Dirichlet‐type problems for the simplest third‐order linear dispersive partial differential equations (PDE), often referred to as the Airy equation. Such problems have not been extensively studied, perhaps due to the complexity of…
View article: Jumps and cusps: a new revival effect in local dispersive PDEs
Jumps and cusps: a new revival effect in local dispersive PDEs Open
We study the presence of a non-trivial revival effect in the solution of linear dispersive boundary value problems for two benchmark models which arise in applications: the Airy equation and the dislocated Laplacian Schr{ö}dinger equation.…
View article: Revivals, or the Talbot effect, for the Airy equation
Revivals, or the Talbot effect, for the Airy equation Open
We study Dirichlet-type problems for the simplest third-order linear dispersive PDE, often referred to as the Airy equation. Such problems have not been extensively studied, perhaps due to the complexity of the spectral structure of the sp…
View article: EMYA: The European Mathematical Society Young Academy
EMYA: The European Mathematical Society Young Academy Open
Recently, EMS has initiated the instalment of the EMS Young Academy (European Mathematical Young Academy – EMYA). In this column we present the story behind the establishment of EMYA – presented by Beatrice Pelloni (vice-president of the E…
View article: The phenomenon of revivals on complex potential Schrödinger's equation
The phenomenon of revivals on complex potential Schrödinger's equation Open
The mysterious phenomena of revivals in linear dispersive periodic equations was discovered first experimentally in optics in the 19th century, then rediscovered several times by theoretical and experimental investigations. While the term …
View article: The role of periodicity in the solution of third order boundary value problems
The role of periodicity in the solution of third order boundary value problems Open
In this short paper, we elucidate how the solution of certain illustrative boundary value problems for the Airy equation $u_t+u_{xxx}=0$ on $[0,1]$ can be expressed as a perturbation of the solution of the purely periodic problem. The moti…
View article: A new implementation of the geometric method for solving the Eady slice equations
A new implementation of the geometric method for solving the Eady slice equations Open
We present a new implementation of the geometric method of Cullen & Purser (1984) for solving the semi-geostrophic Eady slice equations which model large scale atmospheric flows and frontogenesis. The geometric method is a Lagrangian discr…
View article: Time-periodic linear boundary value problems on a finite interval
Time-periodic linear boundary value problems on a finite interval Open
We study the large time behaviour of the solution of a linear dispersive PDEs posed on a finite interval, when the prescribed boundary conditions are time periodic. We use the approach pioneered in Fokas & Lenells 2012 for nonlinear integr…
View article: New revival phenomena for linear integro–differential equations
New revival phenomena for linear integro–differential equations Open
We present and analyze a novel manifestation of the revival phenomenon for linear spatially periodic evolution equations , in the concrete case of three nonlocal equations that arise in water wave theory and are defined by convolution kern…
View article: Existence for the Semi-Geostrophic Equations in Geostrophic Coordinates via Semi-Discrete Optimal Transport
Existence for the Semi-Geostrophic Equations in Geostrophic Coordinates via Semi-Discrete Optimal Transport Open
Using semi-discrete optimal transport, we construct global-in-time weak solutions of the 3-dimensional incompressible semi-geostrophic equations (SG) in geostrophic coordinates for arbitrary initial measures with compact support. This new …
View article: The Stability Principle and global weak solutions of the free surface semi-geostrophic equations in geostrophic coordinates
The Stability Principle and global weak solutions of the free surface semi-geostrophic equations in geostrophic coordinates Open
The semi-geostrophic equations are used widely in the modelling of large-scale atmospheric flows. In this note, we prove the global existence of weak solutions of the incompressible semi-geostrophic equations, in geostrophic coordinates, i…
View article: Evolution equations on time-dependent intervals
Evolution equations on time-dependent intervals Open
We study initial boundary value problems for linear evolution partial differential equations (PDEs) posed on a time-dependent interval $l_1(t)
View article: Cullen's Stability Principle and Weak Solutions of the Free-surface Semi-geostrophic Equations
Cullen's Stability Principle and Weak Solutions of the Free-surface Semi-geostrophic Equations Open
The semi-geostrophic equations are used widely in the modelling of large-scale atmospheric flows. In this note, we prove the global existence of weak solutions of the incompressible semi-geostrophic equations, in geostrophic coordinates, i…
View article: Nonlocal and Multipoint Boundary Value Problems for Linear Evolution Equations
Nonlocal and Multipoint Boundary Value Problems for Linear Evolution Equations Open
We derive the solution representation for a large class of nonlocal boundary value problems for linear evolution partial differential equations (PDE) with constant coefficients in one space variable. The prototypical example of such PDE is…
View article: A numerical implementation of the unified Fokas transform for evolution problems on a finite interval
A numerical implementation of the unified Fokas transform for evolution problems on a finite interval Open
We present the numerical solution of two-point boundary value problems for a third-order linear PDE, representing a linear evolution in one space dimension. To our knowledge, the numerical evaluation of the solution so far could only be ob…
View article: A numerical implementation of the unified Fokas transform for evolution problems on a finite interval
A numerical implementation of the unified Fokas transform for evolution problems on a finite interval Open
We present the numerical solution of two-point boundary value problems for a third order linear PDE, representing a linear evolution in one space dimension. The difficulty of this problem is in the numerical imposition of the boundary cond…
View article: Evolution PDEs and augmented eigenfunctions. Half-line
Evolution PDEs and augmented eigenfunctions. Half-line Open
The solution of an initial-boundary value problem for a linear evolution partial differential equation posed on the half-line can be represented in terms of an integral in the complex (spectral) plane. is representation is obtained by the …