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By means of the Hamiltonian approach to two-dimensional wave motions in heterogeneous fluids proposed by Benjamin [1] we derive a natural Hamiltonian structure for ideal fluids, density stratified in four homogenous layers, constrained in a channel of fixed total height and infinite lateral length. We derive the Hamiltonian and the equations of motion in the dispersionless long-wave limit, restricting ourselves to the so-called Boussinesq approximation. The existence of special symmetric solutions, which generalise to the four-layer case the ones obtained in [11] for the three-layer case, is examined.

Nella tesi mostriamo che lo spazio delle orbite di B_2 può essere dotato di due struttre di varietà Frobenius associate rispettivamente alle equazione di Shrodinger non lineare defocusing e focusing. Motiviati da questo esempio, generalizziamo questa costruzione per ogni B_n, con n>2. Tale costruzione è basata su l'esistenza di un fascio piatto e omogeneo di cometriche definito sullo spazio delle orbite di B_n. La dimostrazione dell'esistenza di un fascio piatto e omogeneo si basa sulla procedura di Dubrovin-Saito modificata in modo opportuno. Partendo da questo fascio, si ricostruisce un'unica struttura di varietà di Frobenius M sullo spazio delle orbite di B_n, per ogni n>2, seguendo una procedura alternativa rispetto a quella standarda proposta da Dubrovin; questa complicazione tecnica è dovuta alla non regolarità del fascio di cometriche. Sorprendentemente, M è isomorfo a una struttura di varietà di Hurwitz-Frobenius. Questa è associata alla gerarchia constrained KP. Tale identificazione rende possibile calcolare esplicitamente le costanti di struttura del prodotto duale di M., In the thesis, we show that the orbit space of B_2 can be equipped with two Frobenius manifolds structures related to the defocusing and focusing NLS (nonlinear Shrodinger) equation respectively. Motivated by this example, we generalize this construction for any B_n, with n>2. Such a construction is based on the existence of a homogeneous flat pencil of cometrics defined on the orbit space of B_n. The proof of the existence of a homogeneous flat pencil relies on the Dubrovin-Saito procedure suitably modified. Starting from this pencil, one can reconstruct a unique Frobenius manifold structure M on the orbit space of B_n, for any n > 2, by following an alternative procedure with respect to the standard one presented by Dubrovin; this technical obstacle is due to the non-regularity of the pencil of cometrics. Remarkably, M is isomorphic to a Hurwitz-Frobenius manifold structure. This is related to the constrained KP hierarchy. Such an identification makes it possible to explicitly compute the structure constants corresponding to the dual product of M.

Originatesi nel contesto delle teorie di campo bidimensionali, le varietà di Frobenius sono punto di incontro tra diverse aree della matematica, includendo la teoria dei sistemi integrabili, la geometria algebrica, la teoria delle singolarità e molte altre. Mentre la maggior parte della teoria è stata sviluppata sotto una certa ipotesi di semi semplicità, la nostra attenzione si focalizza sul più ampio scenario governato dall’ipotesi meno restringente di una certa nozione di regolarità. Studiamo varietà di Frobenius regolari non semisemplici e altre strutture geometriche che le generalizzano progressivamente: F-varietà, F-varietà di tipo flat e F-varietà di tipo bi-flat. In particolare, il risultato principale di questa tesi è la costruzione di una classe di F-varietà regolari non segni semplici di tipo bi-flat legate a sistemi integrabili, alla geometria di Nijenhuis e alla teoria delle funzioni di Lauricella., Originated in the context of two-dimensional topological field theories, Frobenius manifolds lie at the intersection of several areas of mathematics, embracing integrable systems, algebraic geometry, singularity theory and many more. While most of the theory has been developed under some semisimplicity assumption, we pay our attention to the broader case governed by a milder assumption of regularity. We study regular non-semisimple Frobenius manifolds and other geometric structures progressively generalizing them: F-manifolds, flat F-manifolds and bi-flat F-manifolds. In particular, the main result of this thesis is the construction of a class of regular non-semisimple bi-flat F-manifolds related to integrable systems of hydrodynamic type, Nijenhuis geometry and the theory of Lauricella functions.

This thesis presents some recently published results regarding two classical models of fluid dynamics: the 1D shallow water equations (SWE) and the 3D incompressible semigeostrophic equations., This thesis presents some recently published results regarding two classical models of fluid dynamics: the 1D shallow water equations (SWE) and the 3D incompressible semigeostrophic equations.

In this work we aim to give a first attempt at describing spectral properties and resonances of the Laplacian perturbed by point interaction on unbounded domains. In the first chapter, which is preparatory but still contains not previously published material, we define the N-point interactions in an unbounded domain, with Dirichlet, Neumann or Robin boundary conditions using the theory of self-adjoint extensions. We also give a different treatment through a different road and, following a work Teta, where he defines quadratic forms corresponding to the point interaction Laplacians in R^n, we construct quadratic forms corresponding to delta-interactions in unbounded domains. The second chapter is devoted to the study of the spectrum of 1-point interaction in unbounded domains. The point spectrum in these domains is constituted at most of an eigenvalue, due to Krein formula. The existence and behavior of the eigenvalue is determined by two elements: the value of the parameter alpha and the position of the point interaction with respect to the boundary. General conditions are found for exterior domains in R^3 and R^2 with Dirichlet boundary conditions and for domains \Omega, star-shaped with respect to the point in which the "delta intersction" is located, for Neumann boundary conditions, that include the half-space and other domains with unbounded boundary. Then the critical alpha for which an eigenvalue exists is determined, in some particular cases in which the Green's function is known explicitly such as the half-space, the half-plane, the exterior of the disk and the exterior of a sphere. The behavior of alpha_ textup c changes in a noticeable way when compared to the one in R^ n for a single interaction. In the third chapter we define resonances for point interactions in domains (according to a definition agreeing with the Sjostrand-Zworski theory for regular perturbation of the laplacian and of their more general "black-box" definition) and study their distribut, In this work we aim to give a first attempt at describing spectral properties and resonances of the Laplacian perturbed by point interaction on unbounded domains. In the first chapter, which is preparatory but still contains not previously published material, we define the N-point interactions in an unbounded domain, with Dirichlet, Neumann or Robin boundary conditions using the theory of self-adjoint extensions. We also give a different treatment through a different road and, following a work Teta, where he defines quadratic forms corresponding to the point interaction Laplacians in R^n, we construct quadratic forms corresponding to delta-interactions in unbounded domains. The second chapter is devoted to the study of the spectrum of 1-point interaction in unbounded domains. The point spectrum in these domains is constituted at most of an eigenvalue, due to Krein formula. The existence and behavior of the eigenvalue is determined by two elements: the value of the parameter alpha and the position of the point interaction with respect to the boundary. General conditions are found for exterior domains in R^3 and R^2 with Dirichlet boundary conditions and for domains \Omega, star-shaped with respect to the point in which the "delta intersction" is located, for Neumann boundary conditions, that include the half-space and other domains with unbounded boundary. Then the critical alpha for which an eigenvalue exists is determined, in some particular cases in which the Green's function is known explicitly such as the half-space, the half-plane, the exterior of the disk and the exterior of a sphere. The behavior of alpha_ textup c changes in a noticeable way when compared to the one in R^ n for a single interaction. In the third chapter we define resonances for point interactions in domains (according to a definition agreeing with the Sjostrand-Zworski theory for regular perturbation of the laplacian and of their more general "black-box" definition) and study their distribut

A Hamiltonian reduction approach is defined, studied, and finally used to derive asymptotic models of internal wave propagation in density stratified fluids in two-dimensional domains. Beginning with the general Hamiltonian formalism of Benjamin (1986 J. Fluid Mech. 165 445-74) for an ideal, stably stratified Euler fluid, the corresponding structure is systematically reduced to the setup of two homogeneous fluids under gravity, separated by an interface and confined between two infinite horizontal plates. A long-wave, small-amplitude asymptotics is then used to obtain a simplified model that encapsulates most of the known properties of the dynamics of such systems, such as bidirectional wave propagation and maximal amplitude travelling waves in the form of fronts. Further reductions, and in particular devising an asymptotic extension of Dirac's theory of Hamiltonian constraints, lead to the completely integrable evolution equations previously considered in the literature for limiting forms of the dynamics of stratified fluids. To assess the performance of the asymptotic models, special solutions are studied and compared with those of the parent equations

The universal typical-signal estimators of entropy and cross-entropy based on the asymptotics of recurrence and waiting times play an important role in information theory. Building on their construction, we introduce and study universal typical-signal estimators of entropy production in the context of nonequilibrium statistical mechanics of one-sided shifts over finite alphabets.

In this note we provide an analytical proof of the zero helicity condition for systems governed by the Gross-Pitaevskii equation (GPE). The proof is based on the hydrodynamic interpretation of the GPE, and the direct use of Noether's theorem by applying Kleinert's multi-valued gauge theory. As a by-product we also demonstrate the conservation and quantization of the circulation for the GPE.

We present a result which allows us to deform a Poisson-Nijenhuis manifold into a Poisson quasi-Nijenhuis manifold by means of a closed 2-form. Under an additional assumption, the deformed structure is also Poisson-Nijenhuis. We apply this result to show that the canonical Poisson-Nijenhuis structure on R2n gives rise to both the Poisson-Nijenhuis structure of the open (or non periodic) n-particle Toda lattice, introduced by Das and Okubo [7], and the Poisson quasi-Nijenhuis structure of the closed (or periodic) n-particle Toda lattice, described in our recent work [8].

Il sistema linfatico è una componente importante del sistema circolatorio, responsabile del trasporto del fluido interstiziale in tutto il corpo. Il fluido interstiziale si chiama linfa quando si trova all'interno del sistema linfatico. Il movimento della linfa svolge un ruolo chiave nel trasporto di cellule immunitarie, proteine, metastasi tumorali, farmaci e così via. Inoltre, un trasporto linfatico compromesso può causare linfedema, una patologia che consiste in un accumulo di fluido interstiziale nei tessuti. L'obiettivo di questa tesi è quello di sviluppare modelli matematici che descrivono il comportamento della linfa nel sistema linfatico, in particolare il suo flusso attraverso linfangioni e linfonodi. Analizziamo questi modelli, li risolviamo esplicitamente in geometrie semplificate e proponiamo simulazioni numeriche per affrontare scenari più generali. Innanzitutto proponiamo un modello del flusso linfatico in un linfangione, cioè la parte del vaso linfatico compresa tra due valvole. Grazie alla sua geometria, siamo in grado di sviluppare un modello 1D utilizzando la teoria dei sistemi iperbolici quasilineari. Diversi modelli tentano di descrivere il flusso linfatico in un linfangione; abbiamo proposto un modello con una descrizione fenomenologica del comportamento della parete, tenendo conto dei fenomeni di inibizione e regolazione dello shear stress legati alle contrazioni. Successivamente, ci concentriamo nel modellare il flusso linfatico attraverso un linfonodo perché, per quanto ci risulta, non è stato ampiamente esplorato da un punto di vista meccanico. Inoltre, alcune evidenze suggeriscono che i linfonodi sono essenziali per la regolazione del trasporto della linfa a causa dello scambio di fluidi con i vasi sanguigni al loro interno. In effetti, il linfedema secondario spesso deriva dal danneggiamento o dalla rimozione dei linfonodi. A causa della struttura e del funzionamento complessi del linfonodo, abbiamo elaborato alcuni modelli matemat, The lymphatic system is an important component of the circulatory network, responsible for the transportation of interstitial fluid throughout the body. The interstitial fluid is called lymph when inside the lymphatics. The movement of lymph plays a key role in transporting immune cells, proteins, cancer metastasis, drugs, and so on. Moreover, impaired lymph transport can result in lymphoedema, a condition that consists of an accumulation of interstitial fluid in the tissues. The goal of this thesis is to develop mathematical models that describe the lymph behavior in the lymphatic system, in particular its flow through lymphangions and lymph nodes. We analyze these models, solve them explicitly in simplified geometries, and propose numerical simulations to handle more general scenarios. First of all, we propose a model of the lymph flow in a lymphangion, that is the part of the lymphatic vessel between two valves. Due to its geometry, we are able to develop a 1D model using the theory of quasilinear hyperbolic systems. Several models try to describe the lymph flow in a lymphangion; we proposed a model with a phenomenological description of the wall behavior, taking into account the shear stress inhibition and regulation phenomena related to the contractions. After that, we focus most of our attention to model the lymph flow through a lymph node because, to our knowledge, it has not been extensively explored from a mechanical perspective. Moreover, there is evidence that the lymph nodes are essential for regulating lymph transport due to the fluid exchange with the blood vessels within them. Indeed, secondary lymphoedema often results from damage to or removal of lymph nodes. Due to the complex structure and functioning of the lymph node, the attempt to describe the motion of the lymph within it has inspired us with some very interesting mathematical models which have been worth studying in detail.

Consideriamo dapprima l'impostazione della meccanica lagrangiana proposta da Darryl Holm ei suoi collaboratori, in cui possono essere le equazioni evolutive delle onde interne ottenuto mediante la tecnica di riduzione di Eulero-Poincare, sotto il cosiddetto colonnare ́ ipotesi di movimento ansatz. Proseguiamo quindi con la spiegazione del modello classico proposto da T. Wu e Camassa-Choi per formulare equazioni evolutive dell'interno onde, mediante espansione di Taylor in direzione verticale. Infine, in base a il classico formalismo hamiltoniano per moti ondulatori 2D in fluidi eterogenei da Benjamin, deriviamo la struttura hamiltoniana di fluidi stratificati a 2 strati, con e senza dispersione, sotto Debolmente Nonlineare e Lievemente Nonlineare assunzione., We first consider the Lagrangian mechanics setting proposed by Darryl Holm and his collaborators, in which the evolutionary equations of internal waves can be obtained via the Euler-Poincare reduction technique, under the so-called columnar ́ motion ansatz assumption. Then, we go on explaining the classical model proposed by T. Wu and Camassa-Choi to formulate evolutionary equations of the internal waves, by means of Taylor-expansion in the vertical direction. Finally, based on the classical Hamiltonian formalism for 2D wave motions in heterogeneous fluids by Benjamin, we derive the Hamiltonian structure of 2-layer stratified fluids, with and without dispersion, under the Weakly Nonlinear and Mildly Nonlinear assumption.

In questa tesi ci si propone di analizzare le implicazioni fisiche della geometria dello spazio ambiente nel contesto dei condensati di Bose-Einstein (BEC) e le possibili applicazioni nell’ambito dei modelli analogici della cosmologia dei buchi neri. A tal fine si deriva la formulazione idrodinamica dell'equazione di Gross-Pitaevskii (GPE) nel caso di una generica varietà Riemanniana e si osserva la comparsa di una nuova forza, che dipende essenzialmente da due parametri: la geometria della varietà e le derivate prime del profilo di densità. Si studiano le condizioni stazionarie in relazione alla presenza di varietà a curvatura scalare negativa. Analizzando tali varietà si stabilisce una relazione esplicita tra le superfici a curvatura negativa e l'equazione di seno-Gordon, che risulta un'approssimazione della GPE nel caso di accoppiamento di fasi. Assumendo condizioni stazionarie, si ottiene un nuovo tipo di equazioni di Einstein e si è spinti a ricercare altri legami tra le equazioni che governano i condensati e la cosmologia. A tal fine si considerano i BEC relativistici, che vengono utilizzati nello studio del comportamento dell'universo primordiale e della sua espansione. Facendo uso delle conoscenze ottenute nel caso di varietà Riemanniane generiche, otteniamo nuove equazioni di Einstein nel caso multi-dimensionale. Successivamente, si considerano i modelli analogici utilizzati per lo studio della formazione di buchi neri e per il calcolo della radiazione di Hawking. Attraverso un processo di linearizzazione si nota come sia possibile far emergere una metrica acustica Lorentziana che governi il moto delle fluttuazioni della fase; a questo scopo si considera il caso di un vortice dritto che presenta un profilo di densità in cui le derivate prime assumono un valore massimo all’interno del tubo vorticoso e la geometria dello spazio ambiente diventa rilevante. In questa situazione si scopre che è effettivamente possibile far emergere una metrica Lorentzia, In this thesis we analyze the physical implications of the geometry of the ambient space in the context of Bose-Einstein condensates (BECs) and possible applications to the field of analogue models in the cosmology of black holes. To this end we derive the hydrodynamic formulation of the Gross-Pitaevskii equation (GPE) in the case of a generic Riemannian manifold. We observe the appearance of a new force, which essentially depends on two parameters: the geometry of the manifold and the first derivatives of the density profile. The stationary conditions are studied in relation to the presence of manifolds with negative scalar curvature. By analyzing these manifolds, an explicit relationship is established between the negatively curved surfaces and the sine-Gordon equation, which results in an approximation of the GPE in the presence of phase coupling. By assuming stationary conditions, we obtain a new type of Einstein field equations and we look for other possible connections between the equations governing condensates and cosmology. For this purpose, we consider relativistic BECs, that are used in the study of the early universe and its expansion, and we obtain Einstein equation in the multi-dimensional case. Then we consider the analogue models used for the study of the formation of black holes and for the calculation of Hawking radiation. Through a linearization process it is possible to derive a Lorentzian acoustic metric for the phase fluctuations; for this purpose, we consider the case of a straight vortex defect with a density profile where the first derivatives have maximum value inside the vortex tube and the geometry of the ambient space becomes relevant. In this situation it turns out that it is possible to determine a Lorentzian metric, and some useful approximations are proposed for its explicit computation. Finally, some concluding remarks are presented on possible future research directions, given by the study of the evolution of isophase surfaces in r

We call internal-wave billiard the dynamical system of a point particle that moves freely inside a planar domain (the table) and is reflected by its boundary according to this nonstandard rule: the angles that the incident and reflected velocities form with a fixed direction (representing gravity) are the same. These systems are point particle approximations for the motion of internal gravity waves in closed containers, hence the name. For a class of tables similar to rectangular trapezoids, but with the slanted leg replaced by a general curve with downward concavity, we prove that the dynamics has only three asymptotic regimes: (1) there exist a global attractor and a global repellor, which are periodic and might coincide; (2) there exists a beam of periodic trajectories, whose boundary (if any) comprises an attractor and a repellor for all the other trajectories; (3) all trajectories are dense (that is, the system is minimal). Furthermore, in the prominent case where the table is an actual trapezoid, we study the sets in parameter space relative to the three regimes. We prove in particular that the set for (1) has positive measure (giving a rigorous proof of the existence of Arnol’d tongues for internal-wave billiards), whereas the sets for (2) and (3) are non-empty but have measure zero.

Questa tesi si occupa di una classe particolare di liotropici: i cristalli liquidi cromatici (CLCs). Essi in fase nematica godono della simmetria testa-coda, cioè sono materiali non chirali con una tendenza dei loro elementi costitutivi a raggrupparsi in modo da poter definire un direttore su scala mesoscopica che non ha polarità. Lo stato fondamentale dei cristalli liquidi nematici ordinari è raggiunto quando il direttore è uniforme nello spazio. Quando i CLCs sono confinati in capillari cilindrici con condizioni al contorno degeneri, si osserva che acquisiscono una configurazione non uniforme. In particolare, il loro stato fondamentale nel capillare cilindrico, spesso indicato come 'escaped twist' (ET), consiste in due configurazioni simmetriche di opposta chiralità, ognuna che si verifica con la stessa probabilità come ci si aspetta dalla mancanza di chiralità negli aggregati molecolari che costituiscono questi materiali. Nonostante la chiara indicazione che i CLCs nello spazio tridimensionale mostrano un comportamento diverso dai comuni cristalli liquidi nematici, la teoria di Oseen-Frank per i nematici è stata applicata per razionalizzare gli esperimenti con i capillari e quindi per determinare la configurazione dello stato fondamentale ET. Si tratta di una teoria variazionale che postula una densità di energia libera quadratica nel gradiente del direttore che penalizza tutte le distorsioni dall'allineamento uniforme. Classicamente vengono identificate quattro distorsioni, che corrispondono a quattro costanti elastiche indipendenti; si tratta delle costanti splay K_{11}, twist K_{22}, bend K_{33}, e saddle-splay K_{24}. Le disuguaglianze di Ericksen assicurano che la densità di energia di Frank sia positiva definita, e l'emergere spontaneo della chiralità non è concepibile quando sono valide. Così, la teoria elastica di Frank giustifica le configurazioni osservate dei CLCs sotto confinamento cilindrico solo se la disuguaglianza di Ericksen pertinen, This dissertation is concerned with a peculiar class of lyotropics: chromatic liquid crystals (CLCs). Chromonics in the nematic phase enjoy the head-tail symmetry, that is, they are non-chiral materials with a tendency for their constitutive to bundle together so that a director can be defined at a mesoscopic scale which lacks polarity. The ground state of ordinary nematic liquid crystals is attained when the director is uniform in space. When CLCs are confined in capillary cylinders with degenerate boundary conditions that would be compatible with the uniform alignment of the director along the cylinder's axis, they are observed to acquire a nonuniform arrangement instead. Expecially, their ground state in cylindrical capillary, often referred to as escaped twist (ET) ground state, is two-fold; it consists of two symmetric twisted configurations (left- and right-handed), each variant occurring with the same likelihood, as was to be expected from the lack of chirality in the molecular aggregates that constitute these materials. Despite the clear indication that CLCs in three-dimensional space exhibit a different behavior from common nematic liquid crystals, the Oseen-Frank theory for nematics has been applied to rationalize the experiments with capillary tubes and so to determine the configuration of the ET ground state. This is a variational theory which posits a free energy density quadratic in the director gradient that penalizes all distortions of the director away from the unifom alignment. Four basic distortions are classically identified, which correspond to four independent elastic constants; these are the splay K_{11}, twist K_{22}, bend K_{33}, and saddle-splay K_{24} constants, which have recently been re-interpreted in a new light by Selinger. Ericksen's inequalities ensure that Frank's energy density is positive definite, and the spontaneous emergence of chirality in the nematic texture is not conceivable when they hold. Thus, Frank's elastic theory jus

We review evolutionary models on quantum graphs expressed by linear and nonlinear partial differential equations. Existence and stability of the standing waves trapped on quantum graphs are studied by using methods of the variational theory, dynamical systems on a phase plane, and the Dirichlet-to-Neumann mappings.

We consider the shift transformation on the space of infinite sequences over a finite alphabet endowed with the invariant product measure, and examine the presence of a hole on the space. The holes we study are specified by the sequences that do not contain a given finite word as initial sub-string. The measure of the set of sequences that do not fall into the hole in the first n iterates of the shift is known to decay exponentially with n, and its exponential rate is called escape rate. In this paper we provide a complete characterization of the holes with maximal escape rate. In particular we show that, contrary to the case of equiprobable symbols, ordering the holes by their escape rate corresponds to neither the order by their measure nor by the length of the shortest periodic orbit they contain. Finally, we adapt our technique to the case of shifts endowed with Markov measures, where preliminary results show that a more intricate situation is to be expected.

The paper is devoted to the analysis of the blow-ups of derivatives, gradient catastrophes (GCs) and dynamics of mappings of Rn → Rn associated with the n-dimensional homogeneous Euler equation. Several characteristic features of the multi-dimensional case (n > 1) are described. Existence or nonexistence of blow-ups in different dimensions, boundness of certain linear combinations of blow-up derivatives and the first occurrence of the GC are among of them. It is shown that the potential solutions of the Euler equations exhibit blow-up derivatives in any dimension n. Several concrete examples in two- and three-dimensional cases are analysed. Properties of Rnu → Rnx mappings defined by the hodograph equations are studied, including appearance and disappearance of their singularities.

An extensive experimental study of the shapes of two-dimensional bipolar droplets of the chromonic nematic phase of disodium cromoglycate (DSCG) sandwiched between glass plates, by Kim et al was published in (2013 J. Phys.: Condens. Matter 25 404202). The paper includes a mathematical model of this system. We have extended this study by further theoretical modelling. Our results are in good, quantitative agreement with the experimental data. The model has produced what promises to be a more accurate estimate for the isotropic surface tension at the nematic/isotropic solution interface-and predicts a regime of shape bistability (which has not yet been observed) for larger droplets, where tactoids (pointed, zeppelin-shaped droplets) and smooth-edged discoids can coexist in equilibrium. The general method presented in this paper is also applied to the tactoids formed by F-actin filaments in solution, for which an estimate is given for the value of the isotropic surface tension at the nematic/isotropic interface.

Line defects are one-dimensional phase singularities (forming knots and links) that arise in a variety of physical systems. In these systems, isophase surfaces (Seifert surfaces) have the phase defects as boundary, and these Seifert surfaces define a framing of the normal bundle of each link component. We define the individual helicity for each component of a link singularity, and prove that each individual helicity is zero if and only if there exists a Seifert framing for the link. We extend these results to multi-armed defects. We prove that under anti-parallel reconnection of defect strands total twist is conserved.

We study the well posedness of the nonlinear Schrödinger (NLS) equation with a point interaction and power nonlinearity in dimension two and three. Behind the autonomous interest of the problem, this is a model of the evolution of so called singular solutions that are well known in the analysis of semilinear elliptic equations. We show that the Cauchy problem for the NLS considered enjoys local existence and uniqueness of strong (operator domain) solutions, and that the solutions depend continuously from initial data. In dimension two well posedness holds for any power nonlinearity and global existence is proved for powers below the cubic. In dimension three local and global well posedness are restricted to low powers.

We consider the problem of modifying a network topology in such a way as to delay the propagation of a disease with minimal disruption of the network capacity to reroute goods/items/passengers. We find an approximate solution to the Susceptible-Infected-Susceptible (SIS) model, which constitutes an upper bound to its exact solution. This upper bound allows direct structure-epidemic dynamic relations via the total communicability function. Using this approach we propose a strategy to remove edges in a network that significantly delays the propagation of a disease across the network with minimal disruption of its capacity to deliver goods/items/passengers. We apply this strategy to the analysis of the UK airport transportation network weighted by the number of passengers transported in 2003. We find that the removal of all flights connecting four origin-destination pairs in the UK delays the propagation of a disease by more than 300%, with a minimal deterioration of the transportation capacity of this network. These time delays in the propagation of a disease represent an important non-pharmaceutical intervention to confront an epidemic, allowing for better preparations of the health systems, while keeping the economy moving with minimal disruptions.

We study the ODE/IM correspondence for all the states of the quantum Boussinesq model. We consider a particular class of third order linear ordinary differential operators and show that the generalised monodromy data of such operators provide solutions to the Bethe Ansatz equations of the Quantum Boussinesq model.

We study the ODE/IM correspondence for all states of the quantum g^ -KdV model, where g^ is the affinization of a simply-laced simple Lie algebra g. We construct quantum g^ -KdV opers as an explicit realization of the class of opers introduced by Feigin and Frenkel (Exploring new structures and natural constructions in mathematical physics, Math. Soc. Japan, Tokyo, 2011), which are defined by fixing the singularity structure at 0 and ∞, and by allowing a finite number of additional singular terms with trivial monodromy. We prove that the generalized monodromy data of the quantum g^ -KdV opers satisfy the Bethe Ansatz equations of the quantum g^ -KdV model. The trivial monodromy conditions are equivalent to a complete system of algebraic equations for the additional singularities.

The classical dam-break problem for the shallow water system with a dry/vacuum downstream state is revisited in the context of exact solutions which generalize the Riemann setup of a Heaviside jump between constant states to continuous initial data. Two main setups are considered, chosen to illustrate how local properties of the dependent variables at the vacuum point influence the evolution over different time scales. For the first case, the elevation (density) variable is initially continuous but not differentiable at the dry (vacuum) point: no gradient catastrophe develops in this variable, with the time evolution eventually merging into a single (shifted) Stoker wave. Conversely, for the second case, the elevation joins the dry state with vanishing first derivative and a curvature jump: for an instant in time a global gradient catastrophe at the fixed contact point forms and immediately evolves as a Stoker parabolic simple wave, allowing the contact point to split into two moving points, one at the dry bed and one at a fixed elevation, where curvature singularities persist at all times. Although in both cases shocks develop for the velocity field, these are nongeneric, in that, in contrast to the usual case, infinitely many conservation laws are satisfied at all times. Long time evolution is further analyzed with the help of new stretched ``unfolding"" variables to extract the details of the asymptotic approach to a rarefaction Stoker-like wave.

Physical knots observed in various contexts – from DNA biology to vortex dynamics and condensed matter physics – are found to undergo topological simplification through iterated recombination of knot strands following a common, qualitative pattern that bears remarkable similarities across fields. Here, by interpreting evolutionary processes as geodesic flows in a suitably defined knot polynomial space, we show that a new measure of topological complexity allows accurate quantification of the probability of decay pathways by selecting the optimal unlinking pathways. We also show that these optimal pathways are captured by a logarithmic best-fit curve related to the distribution of minimum energy states of tight knots. This preliminary approach shows great potential for establishing new relations between topological simplification pathways and energy cascade processes in nature.