Bifurcation of gap solitons through catastrophe theory

In the theory of optical gap solitons, slowly-moving finite-amplitude Lorentzian solutions are found to mediate the transition from bright to coexistent dark-antidark solitary wave pairs when the laser frequency is detuned out of the proper edge of a dynamical photonic bandgap. Catastrophe theory is applied to give a geometrical description of this strongly asymmetrical 'morphing' process.Comment: 6 pages, 3 figures, submitted to Phys. Rev.

Dimensions of design space: a decision-theoretic approach to optimal research design

Bayesian decision theory can be used not only to establish the optimal sample size and its allocation in a single clinical study, but also to identify an optimal portfolio of research combining different types of study design. Within a single study, the highest societal pay-off to proposed research is achieved when its sample sizes, and allocation between available treatment options, are chosen to maximise the Expected Net Benefit of Sampling (ENBS). Where a number of different types of study informing different parameters in the decision problem could be conducted, the simultaneous estimation of ENBS across all dimensions of the design space is required to identify the optimal sample sizes and allocations within such a research portfolio. This is illustrated through a simple example of a decision model of zanamivir for the treatment of influenza. The possible study designs include: i) a single trial of all the parameters; ii) a clinical trial providing evidence only on clinical endpoints; iii) an epidemiological study of natural history of disease and iv) a survey of quality of life. The possible combinations, samples sizes and allocation between trial arms are evaluated over a range of costeffectiveness thresholds. The computational challenges are addressed by implementing optimisation algorithms to search the ENBS surface more efficiently over such large dimensions.Bayesian decision theory; expected value of information; research design; costeffectiveness analysis

The TTˉ\textrm{T}\bar{\textrm{T}} perturbation and its geometric interpretation

Starting from the recently-discovered $\textrm{T}\bar{\textrm{T}}$-perturbed Lagrangians, we prove that the deformed solutions to the classical EoMs for bosonic field theories are equivalent to the unperturbed ones but for a specific field-dependent local change of coordinates. This surprising geometric outcome is fully consistent with the identification of $\textrm{T}\bar{\textrm{T}}$-deformed 2D quantum field theories as topological JT gravity coupled to generic matter fields. Although our conclusion is valid for generic interacting potentials, it first emerged from a detailed study of the sine-Gordon model and in particular from the fact that solitonic pseudo-spherical surfaces embedded in $\mathbb R^3$ are left invariant by the deformation. Analytic and numerical results concerning the perturbation of specific sine-Gordon soliton solutions are presented.Comment: v2 : Expanded version with new comments, numerical results and 16 figures added. Minor typos corrected. Extra references added. 25 pages. 4 figures v3 : JHEP version. Section 6 added. Minor typos corrected. Extra reference added. 30 pages. 5 figure

A look at the inner structure of the 2-adic ring C*-algebra and its automorphism groups

We undertake a systematic study of the so-called 2-adic ring C\u87-algebra Q2. This is the universal C\u87-algebra generated by a unitary U and an isometry S2 such that S2U = U2S2 and S2S\u87 2+US2S\u87 2U\u87 = 1. Notably, it contains a copy of the Cuntz algebra O2 = C\u87(S1;S2) through the injective homomorphism mapping S1 to US2. Among the main results, the relative commutant C\u87(S2)\u9c 9 Q2 is shown to be trivial. This in turn leads to a rigidity property enjoyed by the inclusion O2 ` Q2, namely the endomorphisms of Q2 that restrict to the identity on O2 are actually the identity on the whole Q2. Moreover, there is no conditional expectation from Q2 onto O2. As for the inner structure of Q2, the diagonal subalgebra D2 and C\u87(U) are both proved to be maximal abelian in Q2. The maximality of the latter allows a thorough investigation of several classes of endomorphisms and automorphisms of Q2. In particular, the semigroup of the endomorphisms xing U turns out to be a maximal abelian subgroup of Aut(Q2) topologically isomorphic with C(T;T). Finally, it is shown by an explicit construction that Out(Q2) is uncountable and non- abelian

Diagonal automorphisms of the 22-adic ring C∗C^*-algebra

The $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$ naturally contains a copy of the Cuntz algebra $\mathcal{O}_2$ and, a fortiori, also of its diagonal subalgebra $\mathcal{D}_2$ with Cantor spectrum. This paper is aimed at studying the group ${\rm Aut}_{\mathcal{D}_2}(\mathcal{Q}_2)$ of the automorphisms of $\mathcal{Q}_2$ fixing $\mathcal{D}_2$ pointwise. It turns out that any such automorphism leaves $\mathcal{O}_2$ globally invariant. Furthermore, the subgroup ${\rm Aut}_{\mathcal{D}_2}(\mathcal{Q}_2)$ is shown to be maximal abelian in ${\rm Aut}(\mathcal{Q}_2)$. Saying exactly what the group is amounts to understanding when an automorphism of $\mathcal{O}_2$ that fixes $\mathcal{D}_2$ pointwise extends to $\mathcal{Q}_2$. A complete answer is given for all localized automorphisms: these will extend if and only if they are the composition of a localized inner automorphism with a gauge automorphism.Comment: Improved exposition and corrected some typos and inaccuracie

Conserved currents and TTˉs\text{T}\bar{\text{T}}_s irrelevant deformations of 2D integrable field theories

It has been recently discovered that the $\text{T}\bar{\text{T}}$ deformation is closely-related to Jackiw-Teitelboim gravity. At classical level, the introduction of this perturbation induces an interaction between the stress-energy tensor and space-time and the deformed EoMs can be mapped, through a field-dependent change of coordinates, onto the corresponding undeformed ones. The effect of this perturbation on the quantum spectrum is non-perturbatively described by an inhomogeneous Burgers equation. In this paper, we point out that there exist infinite families of models where the geometry couples instead to generic combinations of local conserved currents labelled by the Lorentz spin. In spirit, these generalisations are similar to the $\text{J}\bar{\text{T}}$ model as the resulting theories and the corresponding scattering phase factors are not Lorentz invariant. The link with the $\text{J}\bar{\text{T}}$ model is discussed in detail. While the classical setup described here is very general, we shall use the sine-Gordon model and its CFT limit as explanatory quantum examples. Most of the final equations and considerations are, however, of broader validity or easily generalisable to more complicated systems.Comment: 39 pages, 3 figures. v2: typos corrected, extended version with more results on the link between the classical and the quantum analysi

On the Cahn-Hilliard-Brinkman system

We consider a diffuse interface model for phase separation of an isothermal incompressible binary fluid in a Brinkman porous medium. The coupled system consists of a convective Cahn-Hilliard equation for the phase field $\phi$, i.e., the difference of the (relative) concentrations of the two phases, coupled with a modified Darcy equation proposed by H.C. Brinkman in 1947 for the fluid velocity $\mathbf{u}$. This equation incorporates a diffuse interface surface force proportional to $\phi \nabla \mu$, where $\mu$ is the so-called chemical potential. We analyze the well-posedness of the resulting Cahn-Hilliard-Brinkman (CHB) system for $(\phi,\mathbf{u})$. Then we establish the existence of a global attractor and the convergence of a given (weak) solution to a single equilibrium via {\L}ojasiewicz-Simon inequality. Furthermore, we study the behavior of the solutions as the viscosity goes to zero, that is, when the CHB system approaches the Cahn-Hilliard-Hele-Shaw (CHHS) system. We first prove the existence of a weak solution to the CHHS system as limit of CHB solutions. Then, in dimension two, we estimate the difference of the solutions to CHB and CHHS systems in terms of the viscosity constant appearing in CHB

Shocks in nonlocal media

We investigate the formation of collisionless shocks along the spatial profile of a gaussian laser beam propagating in nonlocal nonlinear media. For defocusing nonlinearity the shock survives the smoothing effect of the nonlocal response, though its dynamics is qualitatively affected by the latter, whereas for focusing nonlinearity it dominates over filamentation. The patterns observed in a thermal defocusing medium are interpreted in the framework of our theory.Comment: 5 pages, 5 figure

Generalised Born-Infeld models, Lax operators and the TTˉ\textrm{T} \bar{\textrm{T}} perturbation

Surprising links between the deformation of 2D quantum field theories induced by the composite $\textrm{T} \bar{\textrm{T}}$ operator, effective string models and the $AdS/$CFT correspondence, have recently emerged. The purpose of this article is to discuss various classical aspects related to the deformation of 2D interacting field theories. Special attention is given to the sin(h)-Gordon model, for which we were able to construct the $\textrm{T} \bar{\textrm{T}}$-deformed Lax pair. We consider the Lax pair formulation to be the first essential step toward a more satisfactory geometrical interpretation of this deformation within the integrable model framework. Furthermore, it is shown that the 4D Maxwell-Born-Infeld theory, possibly with the addition of a mass term or a derivative-independent potential, corresponds to a natural extension of the 2D examples. Finally, we briefly comment on 2D Yang-Mills theory and propose a modification of the heat kernel, for a generic surface with genus $p$ and $n$ boundaries, which fully accounts for the $\textrm{T} \bar{\textrm{T}}$ contribution.Comment: 22 pages, 2 figures, v2: new comments, hyperlinks and minor typos correcte