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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{\"o}dinger equation. In both cases, we consider boundary conditions of Dirichlet-type. We prove that, at suitable times, jump discontinuities in the initial profile are revived in the solution not only as jump discontinuities but also as logarithmic cusp singularities. We explicitly describe these singularities and show that their formation is due to interactions between the symmetries of the underlying spatial operators with the periodic Hilbert transform., Comment: 35 pages, 3 figures, 2 appendices

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 spatial operator. Our specific interest is to determine whether the peculiar phenomenon of revivals, also known as Talbot effect, is supported by these boundary conditions, which for third order problems are not reducible to periodic ones. We prove that this is the case only for a very special choice of the boundary conditions, for which a new type of weak cusp revival phenomenon has been recently discovered. We also give some new results on the functional class of the solution for other cases., Comment: 20 pages, 4 figures, 1 appendix

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 has been used systematically and consistently by many authors, there is no consensus on a rigorous definition. In this paper, we describe revivals modulo a regularity condition in a large class of Schr\"odinger's equations with complex bounded potentials. As we show, at rational times the solution is given explicitly by finite linear combinations of translations and dilations of the initial datum, plus an additional continuous term., Comment: 17 pages, 2 figures. This paper is published in https://ems.press/journals/zaa/articles/14297854

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 discretisation, where the PDE is approximated by a particle system. An important property of the discretisation is that it is energy conserving. We restate the geometric method in the language of semi-discrete optimal transport theory and exploit this to develop a fast implementation that combines the latest results from numerical optimal transport theory with a novel adaptive time-stepping scheme. Our results enable a controlled comparison between the Eady-Boussinesq vertical slice equations and their semi-geostrophic approximation. We provide further evidence that weak solutions of the Eady-Boussinesq vertical slice equations converge to weak solutions of the semi-geostrophic Eady slice equations as the Rossby number tends to zero., Comment: 43 pages, 9 figures

We study the phenomenon of revivals for the linear Schr\"odinger and Airy equations over a finite interval, by considering several types of non-periodic boundary conditions. In contrast with the case of the linear Schr\"odinger equation examined recently (which we develop further), we prove that, remarkably, the Airy equation does not generally exhibit revivals even for boundary conditions very close to periodic. We also describe a new, weaker form of revival phenomena, present in the case of certain Robin-type boundary conditions for the linear Schr\"odinger equation. In this weak revival, the dichotomy between the behaviour of the solution at rational and irrational times persists, but in contrast with the classical periodic case, the solution is not given by a finite superposition of copies of the initial condition., Comment: 26 pages, 8 figures, typos corrected, main results moved to the introduction, reduced size of the figures

We present and analyse 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 kernels. Revival in these cases is manifested in the form of dispersively quantised cusped solutions at rational times. We give an analytic description of this phenomenon, and present illustrative numerical simulations.

We give a new and constructive proof of the existence of 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 proof, based on semi-discrete optimal transport techniques, works by characterising discrete solutions of SG in geostrophic coordinates in terms of trajectories satisfying an ordinary differential equation. It is advantageous in its simplicity and its explicit relation to Eulerian coordinates through the use of Laguerre tessellations. Using our method, we obtain improved time-regularity for a large class of discrete initial measures, and we compute explicitly two discrete solutions. The method naturally gives rise to an efficient numerical method, which we illustrate by presenting simulations of a 2-dimensional semi-geostrophic flow in geostrophic coordinates generated using a numerical solver for the semi-discrete optimal transport problem coupled with an ordinary differential equation solver., Comment: 35 pages, 2 figures

We study initial boundary value problems for linear evolution partial differential equations (PDEs) posed on a time-dependent interval $l_1(t)

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, in a three-dimensional domain with a free upper boundary. The proof, based on an energy minimisation argument originally inspired by Cullen's Stability Principle, uses optimal transport results as well as the analysis of Hamiltonian ODEs in spaces of probability measures as studied by Ambrosio and Gangbo. We also give a general formulation of Cullen's Stability Principle in a rigorous mathematical framework.

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 conditions, and to our knowledge, no such computations exist. Instead of computing the evolution numerically, we evaluate the solution representation formula obtained by the unified transform of Fokas. This representation involves complex line integrals, but in order to evaluate these integrals numerically, it is necessary to deform the integration contours using appropriate deformation mappings. We formulate a strategy to implement effectively this deformation, which allows us to obtain accurate numerical results., Comment: 20 pages, 14 figures

We derive the solution representation for a large class of nonlocal boundary value problems for linear evolution PDEs with constant coefficients in one space variable. The prototypical such PDE is the heat equation, for which problems of this form model physical phenomena in chemistry and for which we formulate and prove a full result. We also consider the third order case, which is much less studied and has been shown by the authors to have very different structural properties in general. The nonlocal conditions we consider can be reformulated as \emph{multipoint conditions}, and then an explicit representation for the solution of the problem is obtained by an application of the Fokas transform method. The analysis is carried out under the assumption that the problem being solved is well posed, i.e.\ that it admits a unique solution. For the second order case, we also give criteria that guarantee well-posedness., Comment: 28 pages, 4 figures

Sir, We write as senior scientists about a problem vital to the scientific enterprise and prosperity. Nowadays, funding is a lengthy and complex business. First, universities themselves must approve all proposals for submission. Funding agencies then subject those that survive to peer review, a process by which a few researchers, usually acting anonymously, assess a proposal's chances that it will achieve its goals, is the best value for money, is relevant to a national priority and will impact on a socio-economic problem. Only 25% of proposals received by the funding agencies are funded. These protracted processes force researchers to exploit existing knowledge, severely discourage open-ended studies and are hugely time-consuming. They are also new: before 1970, few researchers wrote proposals. Now they are virtually mandatory.

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. This representation is obtained by the {\em unified transform} introduced by Fokas in the 90's. On the other hand, it is known that many initial-boundary value problems can be solved via a classical transform pair, constructed via the spectral analysis of the associated spatial operator. For example, the Dirichlet problem for the heat equation can be solved by applying the Fourier sine transform pair. However, for many other initial-boundary value problems there is {\em no} suitable transform pair in the classical literature. Here we pose and answer two related questions: Given any well-posed initial-boundary value problem, does there exist a (non-classical) transform pair suitable for solving that problem? If so, can this transform pair be constructed via the spectral analysis of a differential operator? The answer to both of these questions is positive and given in terms of {\em augmented eigenfunctions}, a novel class of spectral functionals. These are eigenfunctions of a suitable differential operator in a certain generalised sense, they provide an effective spectral representation of the operator, and are associated with a transform pair suitable to solve the given initial-boundary value problem., Comment: 2 figures. arXiv admin note: text overlap with arXiv:1303.2205

The problem of heat conduction in one-dimensional piecewise homogeneous composite materials is examined by providing an explicit solution of the one-dimensional heat equation in each domain. The location of the interfaces is known, but neither temperature nor heat flux are prescribed there. Instead, the physical assumptions of their continuity at the interfaces are the only conditions imposed. The problem of two semi-infinite domains and that of two finite-sized domains are examined in detail. We indicate also how to extend the solution method to the setting of one finite-sized domain surrounded on both sides by semi-infinite domains, and on that of three finite-sized domains., Comment: 25 pages + 7 page appendix (electronic supplementary material)

We give a characterisation of the spectral properties of linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and determined by general linear boundary conditions. The boundary conditions may be such that the resulting operator is not selfadjoint. We associate the spectral properties of such an operator $S$ with the properties of the solution of a corresponding boundary value problem for the partial differential equation $\partial_t q \pm iSq=0$. Namely, we are able to establish an explicit correspondence between the properties of the family of eigenfunctions of the operator, and in particular whether this family is a basis, and the existence and properties of the unique solution of the associated boundary value problem. When such a unique solution exists, we consider its representation as a complex contour integral that is obtained using a transform method recently proposed by Fokas and one of the authors. The analyticity properties of the integrand in this representation are crucial for studying the spectral theory of the associated operator., Comment: 1 figure