Experimental Realization of the Peregrine Soliton in Repulsive Two-Component Bose-Einstein Condensates.

We experimentally realize the Peregrine soliton in a highly particle-imbalanced two-component repulsive Bose-Einstein condensate in the immiscible regime. The effective focusing dynamics and resulting modulational instability of the minority component provide the opportunity to dynamically create a Peregrine soliton with the aid of an attractive potential well that seeds the initial dynamics. The Peregrine soliton formation is highly reproducible, and our experiments allow us to separately monitor the minority and majority components, and to compare with the single component dynamics in the absence or presence of the well with varying depths. We showcase the centrality of each of the ingredients leveraged herein. Numerical corroborations and a theoretical basis for our findings are provided through three-dimensional simulations emulating the experimental setting and via a one-dimensional analysis further exploring its evolution dynamics.

Spatio-temporal modelling of phenotypic heterogeneity in tumour tissues and its impact on radiotherapy treatment.

We present a mathematical model that describes how tumour heterogeneity evolves in a tissue slice that is oxygenated by a single blood vessel. Phenotype is identified with the stemness level of a cell and determines its proliferative capacity, apoptosis propensity and response to treatment. Our study is based on numerical bifurcation analysis and dynamical simulations of a system of coupled, non-local (in phenotypic "space") partial differential equations that link the phenotypic evolution of the tumour cells to local tissue oxygen levels. In our formulation, we consider a 1D geometry where oxygen is supplied by a blood vessel located on the domain boundary and consumed by the tumour cells as it diffuses through the tissue. For biologically relevant parameter values, the system exhibits multiple steady states; in particular, depending on the initial conditions, the tumour is either eliminated ("tumour-extinction") or it persists ("tumour-invasion"). We conclude by using the model to investigate tumour responses to radiotherapy, and focus on identifying radiotherapy strategies which can eliminate the tumour. Numerical simulations reveal how phenotypic heterogeneity evolves during treatment and highlight the critical role of tissue oxygen levels on the efficacy of radiation protocols that are commonly used in the clinic.

The Role of Mobility in the Dynamics of the COVID-19 Epidemic in Andalusia.

Metapopulation models have been a popular tool for the study of epidemic spread over a network of highly populated nodes (cities, provinces, countries) and have been extensively used in the context of the ongoing COVID-19 pandemic. In the present work, we revisit such a model, bearing a particular case example in mind, namely that of the region of Andalusia in Spain during the period of the summer-fall of 2020 (i.e., between the first and second pandemic waves). Our aim is to consider the possibility of incorporation of mobility across the province nodes focusing on mobile-phone time-dependent data, but also discussing the comparison for our case example with a gravity model, as well as with the dynamics in the absence of mobility. Our main finding is that mobility is key toward a quantitative understanding of the emergence of the second wave of the pandemic and that the most accurate way to capture it involves dynamic (rather than static) inclusion of time-dependent mobility matrices based on cell-phone data. Alternatives bearing no mobility are unable to capture the trends revealed by the data in the context of the metapopulation model considered herein.

Discrete breathers in Klein-Gordon lattices: A deflation-based approach.

Deflation is an efficient numerical technique for identifying new branches of steady state solutions to nonlinear partial differential equations. Here, we demonstrate how to extend deflation to discover new periodic orbits in nonlinear dynamical lattices. We employ our extension to identify discrete breathers, which are generic exponentially localized, time-periodic solutions of such lattices. We compare different approaches to using deflation for periodic orbits, including ones based on Fourier decomposition of the solution, as well as ones based on the solution's energy density profile. We demonstrate the ability of the method to obtain a wide variety of multibreather solutions without prior knowledge about their spatial profile.

Dark solitons under higher-order dispersion.

We show theoretically that stable dark solitons can exist in the presence of pure quartic dispersion, and also in the presence of both quadratic and quartic dispersive effects, displaying a much greater variety of possible solutions and dynamics than for pure quadratic dispersion. The interplay of the two dispersion orders may lead to oscillatory non-vanishing tails, which enables the possibility of bound, potentially stable, multi-soliton states. Dark soliton-like states that connect to low-amplitude oscillations are also shown to be possible. Dynamical evolution results corroborate the stability picture obtained, and possible avenues for dark soliton generation are explored.

Spontaneous Formation of Star-Shaped Surface Patterns in a Driven Bose-Einstein Condensate.

We observe experimentally the spontaneous formation of star-shaped surface patterns in driven Bose-Einstein condensates. Two-dimensional star-shaped patterns with l-fold symmetry, ranging from quadrupole (l=2) to heptagon modes (l=7), are parametrically excited by modulating the scattering length near the Feshbach resonance. An effective Mathieu equation and Floquet analysis are utilized, relating the instability conditions to the dispersion of the surface modes in a trapped superfluid. Identifying the resonant frequencies of the patterns, we precisely measure the dispersion relation of the collective excitations. The oscillation amplitude of the surface excitations increases exponentially during the modulation. We find that only the l=6 mode is unstable due to its emergent coupling with the dipole motion of the cloud. Our experimental results are in excellent agreement with the mean-field framework. Our work opens a new pathway for generating higher-lying collective excitations with applications, such as the probing of exotic properties of quantum fluids and providing a generation mechanism of quantum turbulence.

Phenotypic variation modulates the growth dynamics and response to radiotherapy of solid tumours under normoxia and hypoxia.

In cancer, treatment failure and disease recurrence have been associated with small subpopulations of cancer cells with a stem-like phenotype. In this paper, we develop and investigate a phenotype-structured model of solid tumour growth in which cells are structured by a stemness level, which varies continuously between stem-like and terminally differentiated behaviours. Cell evolution is driven by proliferation and death, as well as advection and diffusion with respect to the stemness structure variable. Here, the magnitude and sign of the advective flux are allowed to vary with the oxygen level. We use the model to investigate how the environment, in particular oxygen levels, affects the tumour's population dynamics and composition, and its response to radiotherapy. We use a combination of numerical and analytical techniques to quantify how under physiological oxygen levels the cells evolve to a differentiated phenotype and under low oxygen level (i.e., hypoxia) they de-differentiate. Under normoxia, the proportion of cancer stem cells is typically negligible and the tumour may ultimately become extinct whereas under hypoxia cancer stem cells comprise a dominant proportion of the tumour volume, enhancing radio-resistance and favouring the tumour's long-term survival. We then investigate how such phenotypic heterogeneity impacts the tumour's response to treatment with radiotherapy under normoxia and hypoxia. Of particular interest is establishing how the presence of radio-resistant cancer stem cells can facilitate a tumour's regrowth following radiotherapy. We also use the model to show how radiation-induced changes in tumour oxygen levels can give rise to complex re-growth dynamics. For example, transient periods of hypoxia induced by damage to tumour blood vessels may rescue the cancer cell population from extinction and drive secondary regrowth.

A quantitative framework for exploring exit strategies from the COVID-19 lockdown.

Following the highly restrictive measures adopted by many countries for combating the current pandemic, the number of individuals infected by SARS-CoV-2 and the associated number of deaths steadily decreased. This fact, together with the impossibility of maintaining the lockdown indefinitely, raises the crucial question of whether it is possible to design an exit strategy based on quantitative analysis. Guided by rigorous mathematical results, we show that this is indeed possible: we present a robust numerical algorithm which can compute the cumulative number of deaths that will occur as a result of increasing the number of contacts by a given multiple, using as input only the most reliable of all data available during the lockdown, namely the cumulative number of deaths.

Kink-Kink and Kink-Antikink Interactions with Long-Range Tails.

In this Letter, we address the long-range interaction between kinks and antikinks, as well as kinks and kinks, in φ^{2n+4} field theories for n>1. The kink-antikink interaction is generically attractive, while the kink-kink interaction is generically repulsive. We find that the force of interaction decays with the 2n/(n-1)th power of their separation, and we identify the general prefactor for arbitrary n. Importantly, we test the resulting mathematical prediction with detailed numerical simulations of the dynamic field equation, and obtain good agreement between theory and numerics for the cases of n=2 (φ^{8} model), n=3 (φ^{10} model), and n=4 (φ^{12} model).

Demonstration of Dispersive Rarefaction Shocks in Hollow Elliptical Cylinder Chains.

We report an experimental and numerical demonstration of dispersive rarefaction shocks (DRS) in a 3D-printed soft chain of hollow elliptical cylinders. We find that, in contrast to conventional nonlinear waves, these DRS have their lower amplitude components travel faster, while the higher amplitude ones propagate slower. This results in the backward-tilted shape of the front of the wave (the rarefaction segment) and the breakage of wave tails into a modulated waveform (the dispersive shock segment). Examining the DRS under various impact conditions, we find the counterintuitive feature that the higher striker velocity causes the slower propagation of the DRS. These unique features can be useful for mitigating impact controllably and efficiently without relying on material damping or plasticity effects.

Three-Component Soliton States in Spinor F=1 Bose-Einstein Condensates.

Dilute-gas Bose-Einstein condensates are an exceptionally versatile test bed for the investigation of novel solitonic structures. While matter-wave solitons in one- and two-component systems have been the focus of intense research efforts, an extension to three components has never been attempted in experiments. Here, we experimentally demonstrate the existence of robust dark-bright-bright (DBB) and dark-dark-bright solitons in a multicomponent F=1 condensate. We observe lifetimes on the order of hundreds of milliseconds for these structures. Our theoretical analysis, based on a multiscale expansion method, shows that small-amplitude solitons of these types obey universal long-short wave resonant interaction models, namely, Yajima-Oikawa systems. Our experimental and analytical findings are corroborated by direct numerical simulations highlighting the persistence of, e.g., the DBB soliton states, as well as their robust oscillations in the trap.

Quasiperiodic granular chains and Hofstadter butterflies.

We study quasiperiodicity-induced localization of waves in strongly precompressed granular chains. We propose three different set-ups, inspired by the Aubry-André (AA) model, of quasiperiodic chains; and we use these models to compare the effects of on-site and off-site quasiperiodicity in nonlinear lattices. When there is purely on-site quasiperiodicity, which we implement in two different ways, we show for a chain of spherical particles that there is a localization transition (as in the original AA model). However, we observe no localization transition in a chain of cylindrical particles in which we incorporate quasiperiodicity in the distribution of contact angles between adjacent cylinders by making the angle periodicity incommensurate with that of the chain. For each of our three models, we compute the Hofstadter spectrum and the associated Minkowski-Bouligand fractal dimension, and we demonstrate that the fractal dimension decreases as one approaches the localization transition (when it exists). We also show, using the chain of cylinders as an example, how to recover the Hofstadter spectrum from the system dynamics. Finally, in a suite of numerical computations, we demonstrate localization and also that there exist regimes of ballistic, superdiffusive, diffusive and subdiffusive transport. Our models provide a flexible set of systems to study quasiperiodicity-induced analogues of Anderson phenomena in granular chains that one can tune controllably from weakly to strongly nonlinear regimes.This article is part of the theme issue 'Nonlinear energy transfer in dynamical and acoustical systems'.

Demonstrating an In Situ Topological Band Transition in Cylindrical Granular Chains.

We numerically investigate and experimentally demonstrate an in situ topological band transition in a highly tunable mechanical system made of cylindrical granular particles. This system allows us to tune its interparticle stiffness in a controllable way, simply by changing the contact angles between the cylinders. The spatial variation of particles' stiffness results in an in situ transition of the system's topology. This manifests as the emergence of a boundary mode in the finite system, which we observe experimentally via laser Doppler vibrometry. When two topologically different systems are placed adjacently, we analytically predict and computationally and experimentally demonstrate the existence of a finite-frequency topologically protected mode at their interface.

Adiabatic Invariant Approach to Transverse Instability: Landau Dynamics of Soliton Filaments.

Consider a lower-dimensional solitonic structure embedded in a higher-dimensional space, e.g., a 1D dark soliton embedded in 2D space, a ring dark soliton in 2D space, a spherical shell soliton in 3D space, etc. By extending the Landau dynamics approach [Phys. Rev. Lett. 93, 240403 (2004)PRLTAO0031-900710.1103/PhysRevLett.93.240403], we show that it is possible to capture the transverse dynamical modes (the "Kelvin modes") of the undulation of this "soliton filament" within the higher-dimensional space. These are the transverse stability or instability modes and are the ones potentially responsible for the breakup of the soliton into structures such as vortices, vortex rings, etc. We present the theory and case examples in 2D and 3D, corroborating the results by numerical stability and dynamical computations.

Analysis and observation of moving domain fronts in a ring of coupled electronic self-oscillators.

In this work, we consider a ring of coupled electronic (Wien-bridge) oscillators from a perspective combining modeling, simulation, and experimental observation. Following up on earlier work characterizing the pairwise interaction of Wien-bridge oscillators by Kuramoto-Sakaguchi phase dynamics, we develop a lattice model for a chain thereof, featuring an exponentially decaying spatial kernel. We find that for certain values of the Sakaguchi parameter α, states of traveling phase-domain fronts involving the coexistence of two clearly separated regions of distinct dynamical behavior, can establish themselves in the ring lattice. Experiments and simulations show that stationary coexistence domains of synchronization only manifest themselves with the introduction of a local impurity; here an incoherent cluster of oscillators can arise reminiscent of the chimera states in a range of systems with homogeneous oscillators and suitable nonlocal interactions between them.

When linear stability does not exclude nonlinear instability.

We describe a mechanism that results in the nonlinear instability of stationary states even in the case where the stationary states are linearly stable. This instability is due to the nonlinearity-induced coupling of the linearization's internal modes of negative energy with the continuous spectrum. In a broad class of nonlinear Schrödinger equations considered, the presence of such internal modes guarantees the nonlinear instability of the stationary states in the evolution dynamics. To corroborate this idea, we explore three prototypical case examples: (a) an antisymmetric soliton in a double-well potential, (b) a twisted localized mode in a one-dimensional lattice with cubic nonlinearity, and (c) a discrete vortex in a two-dimensional saturable lattice. In all cases, we observe a weak nonlinear instability, despite the linear stability of the respective states.

Highly nonlinear wave propagation in elastic woodpile periodic structures.

In the present work, we experimentally implement, numerically compute with, and theoretically analyze a configuration in the form of a single column woodpile periodic structure. Our main finding is that a Hertzian, locally resonant, woodpile lattice offers a test bed for the formation of genuinely traveling waves composed of a strongly localized solitary wave on top of a small amplitude oscillatory tail. This type of wave, called a nanopteron, is not only motivated theoretically and numerically, but is also visualized experimentally by means of a laser Doppler vibrometer. This system can also be useful for manipulating stress waves at will, for example, to achieve strong attenuation and modulation of high-amplitude impacts without relying on damping in the system.

Quasi-energies, parametric resonances, and stability limits in ac-driven PT-symmetric systems.

We introduce a simple model for implementing the concepts of quasi-energy and parametric resonances (PRs) in systems with the PT symmetry, i.e., a pair of coupled and mutually balanced gain and loss elements. The parametric (ac) forcing is applied through periodic modulation of the coefficient accounting for the coupling of the two degrees of freedom. The system may be realized in optics as a dual-core waveguide with the gain and loss applied to different cores, and the thickness of the gap between them subject to a periodic modulation. The onset and development of the parametric instability for a small forcing amplitude (V1) is studied in an analytical form. The full dynamical chart of the system is generated by systematic simulations. At sufficiently large values of the forcing frequency, ω, tongues of the parametric instability originate, with the increase of V1, as predicted by the analysis. However, the tongues following further increase of V1 feature a pattern drastically different from that in usual (non-PT) parametrically driven systems: instead of bending down to larger values of the dc coupling constant, V0, they maintain a direction parallel to the V1 axis. The system of the parallel tongues gets dense with the decrease of ω, merging into a complex small-scale structure of alternating regions of stability and instability. The cases of ω-->0 and ω-->∞ are studied analytically by means of the adiabatic and averaging approximation, respectively. The cubic nonlinearity, if added to the system, alters the picture, destabilizing many originally robust dynamical regimes, and stabilizing some which were unstable.

Vaccination compartmental epidemiological models for the delta and omicron SARS-CoV-2 variants.

We explore the inclusion of vaccination in compartmental epidemiological models concerning the delta and omicron variants of the SARS-CoV-2 virus that caused the COVID-19 pandemic. We expand on our earlier compartmental-model work by incorporating vaccinated populations. We present two classes of models that differ depending on the immunological properties of the variant. The first one is for the delta variant, where we do not follow the dynamics of the vaccinated individuals since infections of vaccinated individuals were rare. The second one for the far more contagious omicron variant incorporates the evolution of the infections within the vaccinated cohort. We explore comparisons with available data involving two possible classes of counts, fatalities and hospitalizations. We present our results for two regions, Andalusia and Switzerland (including the Principality of Liechtenstein), where the necessary data are available. In the majority of the considered cases, the models are found to yield good agreement with the data and have a reasonable predictive capability beyond their training window, rendering them potentially useful tools for the interpretation of the COVID-19 and further pandemic waves, and for the design of intervention strategies during these waves.

Matter-wave bright solitons in spin-orbit coupled Bose-Einstein condensates.

We study matter-wave bright solitons in spin-orbit coupled Bose-Einstein condensates with attractive interactions. We use a multiscale expansion method to identify solution families for chemical potentials in the semi-infinite gap of the linear energy spectrum. Depending on the linear and spin-orbit coupling strengths, the solitons may present either a sech2-shaped or a modulated density profile reminiscent of the stripe phase of spin-orbit coupled repulsive Bose-Einstein condensates. Our numerical results are in excellent agreement with our analytical findings and demonstrate the potential robustness of solitons for experimentally relevant conditions.