Papers

The ability to precisely shape the properties of light over multiple degrees of freedom constitutes the foundation of modern photonic architectures. Broadband sources with addressable spectro-temporal content are for instance critical for applications spanning biomedical imaging, material processing, lidar, and ultrafast spectroscopy, but also key enabling technologies in quantum information processing, high-capacity communications, and optical computing. Yet, flexible and adjustable spectro-temporal shaping so far remains a significant challenge, as it requires the simultaneous manipulation of both frequency and temporal domains across broad spectral bandwidths and extended timescales, from femtoseconds to nanoseconds. Importantly, conventional approaches to spectro-temporal processing typically rely on bulky and complex optical systems that lack scalability and flexibility, ultimately hampering their utility for applications requiring addressable multiphotonic processes. Here, we present a framework for spectro-temporal wavepacket control over a broad bandwidth, merging programmable integrated photonics with high-speed optical characterization. Leveraging machine leaning for tailoring nonlinear pulse propagation, we experimentally report scalable and on-demand shaping of broadband ultrafast pulse patterns. We demonstrate the efficacy of our method through reconfigurable power across 400 nm bandwidth with picosecond resolution, with promises for applications where precise control over light is paramount, in particular those needing versatile and controllable multiphoton excitations.

This thesis investigates the impact of resonant mode-particle interactions on transport and confinement in toroidal fusion plasmas. Using analytical and numerical approaches, we study how intrinsic and externally applied magnetic perturbations affect plasma particles with different kinetic characteristics, focusing on resonant interactions with guiding-center motion. We develop a novel, computationally efficient method based on Action-Angle variables to identify resonance locations and characterize resonances in guiding-center phase space, including the number of islands in resonance chains and the formation of transport barriers. Using the drift center (DC) approximation, we derive analytical expressions for orbital frequencies and the kinetic $q$ factor for large-aspect-ratio (LAR) equilibria. These results provide the conditions for mode-particle resonances, which can strongly influence particle, momentum, and energy transport and, consequently, plasma confinement. The analytical results are validated against numerical simulations, demonstrating an efficient tool for predicting transport barriers and energetic-particle behavior under non-axisymmetric perturbations. We further extend the orbital-frequency analysis from LAR equilibria to numerically reconstructed, realistic equilibria using a computationally efficient, semi-analytical geometrical method applicable to arbitrary unperturbed equilibria. Finally, for the LAR equilibrium, we extend the analysis to time-dependent perturbations and demonstrate the emergence of Arnold diffusion in guiding-center phase space, highlighting the role of the Arnold web in particle transport. Overall, this work advances the understanding of mode-particle resonant interactions and provides computational tools for predicting transport and confinement in realistic fusion configurations.

We investigate novel vector localized wave solutions in the coupled Lakshmanan-Porsezian-Daniel equa?tions, which describe the dynamics of the Heisenberg ferromagnetic spin chain. We present Tajiri-Watanabe breathers, rogue waves and resonant modes in both the degenerate and non-degenerate regions, together with the degenerate beating solitons. The fourth-order effect induces state transitions in both regions. In particular, the degenerate breathers can be transformed into solitons, whereas such transitions are absent in the coupled Hirota equations. Moreover, beating solitons can be converted into stable solitons only in the degenerate region, the phenomena not found in the Manakov system. We further uncover the state transitions of the resonant modes and derive the corresponding transition conditions for each branch. We derive the physical spectra and subsequently identify the state transition conditions in the spectral domain for both the degenerate and non-degenerate cases. These spectra provide an additional characterization of the transition dynamics. Finally, direct numerical simulations are performed to verify the validity of the exact solutions.

The Julia set is defined by the closure of repelling periodic points in chaotic systems. Why does this structure not appear in integrable systems? In this paper, we address this question by demonstrating the existence of the closure of divergences of the periodic equations, which we designate as the "anti-Julia set." We also call the sets before taking the closures the pre-Julia and pre-anti-Julia sets, respectively. We illustrate the transition mechanism by considering a complex map that interpolates between integrable and non-integrable dynamics, by introducing a real deformation parameter $a$. For $0<a\le 1/2$, we show that the Julia set and the anti-Julia set coincide in the complex plane, although the pre-Julia and pre-anti-Julia sets remain completely disjoint. At the integrable limit $a\to 0$, these two dual structures undergo a critical collision and subsequent annihilation, reminiscent of a cosmological Big Crunch. When $1/2<a<1$, on the other hand, the boundary of the pre-anti-Julia set includes the Julia set. We analytically characterize these phenomena, focusing in particular on the asymptotic behavior of the pre-anti-Julia set as it approaches the integrable limit, and provide numerical visualizations that elucidate the underlying mechanisms of this Big Crunch phenomenon.

Condensates can accelerate reactions by concentrating reactants, yet product accumulation can eliminate this advantage. We show that product-induced suppression of condensation generates rotating, deforming, and propagating condensate domains through spontaneous symmetry breaking. Their reorganization shifts reaction sites away from accumulated products while retaining reactant enrichment, yielding rates above those of static condensates and homogeneous states. Reaction flows can thus self-organize condensates into dynamic reactors, with implications for biological and synthetic systems.

Infectious diseases may interact by competing for the same hosts or by facilitating subsequent infections. Understanding the dynamics of such multi-pathogen systems, particularly those subject to endemic seasonality and mitigation, is essential for designing robust public health interventions. We propose a three-stage modeling framework to disentangle the interplay between seasonality and behavioral feedback as a way of mitigation. First, we analyze a coupled susceptible-infectious-recovered-susceptible (SIRS) system without external forcing and show that the abrupt transition between the disease-free and endemic equilibria arises from a backward bifurcation-induced first-order phase transition. Second, we independently examine seasonality and behavioral feedback, characterizing where and when oscillatory behavior is induced near critical tipping points. Third, we demonstrate that their combination generates complex multi-annual wave patterns, with high-incidence cycles driven by seasonality and low-incidence intervals driven by behavioral feedback. By mapping stability as a function of seasonal forcing, mitigation strength, and cooperativity, we identify distinct period-doubling cascades with chaotic signatures arising from different mechanisms: the interplay between seasonality and behavior, and inter-pathogen cooperativity with the backward bifurcation it induces. We then analyze how these mechanisms interact across parameter ranges. Altogether, we show that cooperation fundamentally expands the spectrum of possible epidemic patterns, highlighting the importance of considering multi-pathogen interactions in epidemic modeling and control strategies.

We test whether the Kuramoto model quantitatively describes spatial synchronization in regional epidemic waves, using daily COVID-19 incidence for 400 German Kreise (2021–2023) and weekly ILI rates for 12 European countries (ECDC, 2021–2026). Bandpass filtering and Hilbert-transform phase extraction yield high global order parameters ($\langle r\rangle\approx0.87$ for Germany; $\langle r\rangle\approx0.97$ for Europe), yet both are dominated by a common-mode driver rather than pairwise local coupling. Removing the common mode reveals genuine short-range synchronization: an exponentially decaying residual local order with correlation length $\xi=152\pm6$~km in Germany, and a significant neighbor-vs.-non-neighbor contrast (Mann–Whitney $p=0.005$) in the European panel. We interpret this convergent pattern across two diseases and two spatial scales as a universal two-scale structure: a long-range common-mode field superimposed on shorter-range diffusive coupling. Cross-validation against a companion gauge-mediated Doi–Peliti framework shows that district-level effective screening mass $m_R$ is uncorrelated with local~$r$ (null result), the national $(m_R,r)$ phase portrait independently reproduces a previously reported hysteresis loop, and pairwise phase lags quantitatively match fractional-calculus predictions. A frequency–degree correlation provides the structural signature of explosive synchronization; excess synchronization between districts sharing the same Bundesland implicates shared school-holiday calendars and state-level policy as local coupling drivers.

Programmable control of individual pulses in a high-repetition-rate (typically MHz) ultrafast pulse train is a long-standing goal for optical arbitrary waveform synthesis. Here, we report a programmable pulse-by-pulse shaper that enables deterministic spectral-temporal control of ultrafast pulses at a repetition rate of $\sim$ 20~MHz. By synchronizing an FPGA-driven electro-optic modulation on the stretched waveform in a temporal 4$f$ shaping system, the regime writes pulse-index-dependent spectral phase profiles onto individual pulses. We demonstrate three levels of programmable ability: zero-order phase coding that maps pulse-by-pulse phase sequences into double-slit-like spectral-temporal interference; first-order phase programming that produces arbitrary temporal trajectories with deterministic delay; and fractional-order phase engineering that generates programmable temporal breathing. By launching the shaped pulse train into a nonlinear fiber stage, the programmed temporal breathing is converted into one spectral breathing. We further construct a phase-level-dependent regime map of nonlinear spectral breathing, revealing transitions from weak single-envelope breathing to multi-peak spectral splitting and strongly breathing merged-spectrum dynamics, in agreement with numerical simulations. This pulse-by-pulse spectral-temporal synthesis platform establishes pulse index as a programmable degree of freedom for ultrafast pulse shaping and provides a route toward real-time and pulse resolved optical arbitary waveform synthesis.

We investigate shortcut-to-adiabaticity protocols for the rapid compression of optical solitons in nonlocal media. Using a variational approximation, we derive an effective Ermakov-like equation for the soliton width and employ inverse engineering to design propagation-dependent control parameters. Three control strategies are compared, based respectively on modulation of the characteristic nonlocal length, the Kerr nonlinearity, and the external parabolic confinement. We show that all three protocols enable high-fidelity compression over strongly reduced propagation distances. By comparing control smoothness and final fidelity, we find that the relative performance of the protocols in the very short-distance regime depends on the chosen criterion, revealing a trade-off between implementation smoothness and target-profile accuracy. These results establish a practical and general route to fast soliton manipulation in nonlocal media.

A Poincaré section replaces a flow by a return map, but for a chaotic system this map is usually known only from sampled crossings. We show that coarse transport can be read directly from Koopman spectral data, without fitting the map. Measure-preserving EDMD retains the isometric structure; riggedDMD then approximates spectral measures and constructs finite regularized wave packets. Packet phase supplies a finite-resolution transport coordinate; low modulus marks a singular skeleton where the phase becomes ill-conditioned. We demonstrate the idea on the Rössler system, a 32-mode Kuramoto–Sivashinsky Galerkin system, and the forced Duffing oscillator. The packets yield coarse symbolic models on sections ranging from an almost one-dimensional curve to a visibly thick set. Their graphs organize observed low-period orbits and guide targeted searches for others. In Duffing Regime~II, a seven-region rule accounts for $91\%$–$94\%$ of filtered one-step transitions, while failures in the lowest retained modulus decile occur at $5.08$–$5.20$ times the overall rate. The packets are not Koopman eigenfunctions, nor are the regions exact Markov partitions. Together these computations show how spectral information beyond isolated eigenpairs can expose chaotic transport directly from trajectories.

Physical reservoir computing (PRC) with neuromorphic networks offers a promising approach to brain-inspired information processing, exploiting emergent nonlinear dynamics of physical neural networks as a computational resource. This study demonstrates fully autonomous closed-loop prediction of the Mackey–Glass (MG) chaotic time series using a simulated neuromorphic nanowire network as the physical reservoir. Two strategies are evaluated: the virtual node (VN) method, which expands the feature space by temporal multiplexing of reservoir states, and a non-VN approach that uses all physical node readouts directly without temporal multiplexing. Results are reported for two values of the MG time delay parameter, $\tau = 18$ and $\tau = 21$, the latter representing a more complex chaotic regime not previously evaluated for this class of physical reservoir. Over a short prediction horizon of $T = 100$ timesteps, the VN approach achieves autonomous prediction accuracies of $90.4$% and $89.7$% at $\tau = 18$ and $\tau = 21$, respectively, while the non-VN approach achieves $81.5$% and $76.2$%. Long-horizon analysis over $T = 500$ timesteps shows that both approaches reproduce the qualitative attractor structure and dominant spectral content of the true MG signal, with trajectories remaining bounded throughout. These results suggest that the intrinsic dynamics of neuromorphic nanowire networks are sufficient to support meaningful autonomous chaotic time series prediction without virtual node augmentation, and that performance may improve further as physical network sizes scale to the millions of nodes achievable in hardware. As this study uses simulated networks, extrapolation to physically fabricated large-scale arrays remains to be validated experimentally.

We analyze an inverse problem for determining the piston speed and the associated flow field from a prescribed leading shock and the initial data in a shock tube. The gas flow is described by the isentropic Euler equations (i.e., the $p$-system), while the trajectory of the leading shock is prescribed as a given Lipschitz curve. Under an Oleĭnik-type entropy condition on the leading shock, we develop a modified wavefront tracking scheme to construct the flow field behind the shock. This construction enables us to determine the corresponding piston speed and the associated flow field.

In a mechanistic model of the dawn chorus, Kaye showed that heterogeneous activation thresholds and a shared feedback signal determined by the population's active fraction can produce abrupt collective activation and hysteresis. We extend this mechanism to a network of interacting agents. Each node has a continuous activation level, and a nonnegative row-stochastic matrix determines how node activities contribute to individual feedback. We prove that sufficiently weak feedback yields a unique globally attracting equilibrium. For all feedback strengths, the homogeneous dynamics exactly reproduce Kaye's scalar equation; consequently, network topology does not alter the folds or cusp of the homogeneous branch, and no heterogeneous mode becomes unstable before the homogeneous mode. For equitable partitions, the network admits an exact quotient system in which nodes within a block receive the same aggregate input from every block. When blocks are uncoupled, the quotient reduces to independent copies of Kaye's scalar equation. We show that every assignment of stable scalar equilibria to blocks persists under sufficiently weak interblock coupling, producing a combinatorial family of stable quotient equilibria that lift to stable full-network equilibria and remain under small perturbations that break exact equitability. For two symmetrically coupled blocks, branches with unequal block activities terminate at a pair of symmetry-related cusp bifurcations. Near the onset of bistability, we derive scaling laws for the interblock coupling at which these bifurcations occur, the activity difference between the blocks at bifurcation, and the corresponding shift of the external stimulus from the scalar cusp. Numerical continuation confirms the scaling laws for gamma, logistic, and normal threshold distributions.

This paper deals with two kinds of Bäcklund transformations for scalar hyperbolic partial differential equations. We prove that both these types of transformations map solutions of a Darboux integrable equation into solutions of, generally speaking, another but also Darboux integrable equation. The latter fact can be used to roughly check the completeness of a list of Darboux integrable equations. To illustrate this, we apply the above transformations to several equations from a well-known list of Darboux integrable equations and, as a result, obtain a Darboux integrable equation which is absent in this list, but is already known at present. As a generalization of the last equation, we construct a family of Darboux integrable equations that is parametrized by three arbitrary functions, each of which depends on two arguments. This family is probably new.

Synchronization can be observed in various systems in physics and biology. The choruses of male frogs are known as an example of biological synchronization in which the well-organized temporal structure, i.e., anti-phase synchronization between neighbors, is realized. Given that male frogs produce sounds to advertise their territories to competitors, the dynamics of phases and spatial coordinates should be mutually coupled in the frog choruses. In this study, we examined the spatio-temporal dynamics in the choruses consisting of male Japanese tree frogs and other acoustic animals. First, we carried out playback experiments using actual frogs and observed that male Japanese tree frog synchronized in anti-phase with the stimuli of a similar frequency but did not synchronize with the stimuli of a much different frequency. Second, we modeled the choruses with two species as a system of coupled mobile oscillators and numerically evaluated how the spatio-temporal structure depends on the similarity of call frequencies. Numerical simulations of the model showed that (1) the two-cluster antisynchronization is established in the same species when the distributions of call frequencies are much different between two species and (2) the two-cluster antisynchronization is disturbed when the distributions of call frequencies are similar. These results highlight the occurrence of various spatio-temporal patterns in the proposed model, indicating the importance of repulsive effects with different weights on the variation of the spatio-temporal patterns.

Multiscale simulation requires coupling physics models that operate at different characteristic length and time scales, because no single method spans the range needed for most problems of interest. Moving to a coarser-grained representation unlocks longer length and time scales, but it discards the microscopic interactions that build morphology. A fine-grained model can be initialized from an arbitrary packing and left to self-assemble into a physically meaningful structure; a lower-resolution model cannot, and must inherit its starting configuration from a higher-fidelity simulation. The length scales accessible to the coarse-grained model are therefore set not by the coarse-grained method itself, but by the largest fine-grained configuration that can be affordably equilibrated. A representative example is the scale-up from particle-based molecular dynamics (MD) to a lattice-based representation such as kinetic Monte Carlo (kMC). In this work, we present a cellular automata (CA) approach for generating arbitrarily large lattice starting configurations. CA is a natural fit for this task: short-ranged local rules drive the evolution of a lattice, and their repeated application gives rise to emergent long-range order, thematically mirroring how short-ranged interactions in MD produce self-assembled morphology. We use a configuration from a higher-fidelity simulation as a training set and learn the CA rules from it via logistic regression. As a proof of principle, we develop these rules for a hydrated anion exchange membrane (AEM), generate new starting configurations, and benchmark their performance in mesoscale kMC simulations against an MD-derived "ground truth." We then demonstrate the ability to generate substantially larger lattices and show that their behavior in kMC is consistent with that of the smaller CA benchmark configurations.

Framewise variance normalization removes fluctuation amplitude from an observation, but not from the dynamics that generate later patterns. The current normalized spectrum therefore need not determine its own future. We study this effect in three two-dimensional phase-separation models that change transport, add active driving, or suppress large-scale separation. For paired initial fields, their normalized spectra evolve identically in the linear regime. The restoring model provides an exact example: damping does not directly change how the normalized spectrum evolves when both equations are applied to the same field. It nevertheless changes the hidden amplitude, which alters the later nonlinear shape. An exact comparison separates the immediate effect of changing the equation from the effect of reaching a different state through earlier evolution. It also reveals strong cancellation between these effects in the mobility and active models. As a practical test, spectral histories improve classification by about 4-6 percentage points at several later observation windows, although the gain depends on quench depth and spectral construction. Temporal spectra can therefore reveal dynamical consequences of information absent from every individual normalized frame.

Massive jellyfish swarms observed at sea can extend over tens of kilometres and contain millions of individuals, yet the mechanisms governing their formation and large-scale dynamics remain poorly understood. Agent-based models provide a framework for describing this dynamics based on jellyfish responses to ocean currents and environmental cues, but become computationally prohibitive when extended to large populations and spatial scales relevant to ocean circulation. Here we derive a continuous kinetic theory from an active-particle model of jellyfish motion. The resulting Fokker-Planck framework incorporates transport by prescribed currents, stochastic reorientation, direct interactions and stimulated steering, allowing chemical signalling to be represented through a coupled field. We further derive a hydrodynamic closure for large swarms by exploiting the separation between fast orientational and slow spatial dynamics, yielding a reduced density equation suitable for implementation in ocean-current models. This framework provides a route from individual behavioural mechanisms to continuum descriptions of jellyfish populations and establishes a basis for constraining model parameters using observations and in-situ measurements. This approach offers a theoretical foundation for future numerical prediction of large jellyfish swarm formation and evolution in realistic ocean flows.

We present a Hessian-based approach to numerically continue periodic orbits in dynamical systems. A loop (periodic orbit candidate) is parametrized as a Fourier series; a loss function is defined based on the deviation of the loop from the physical differential equations. Unlike previous work relying on hand-derived Jacobians, our method automates the process by leveraging automatic differentiation, a common machine learning technique. The continuation direction can be determined by the flat directions of the loss landscapes (directions with zero eigenvalues), making the search of periodic orbits efficient and guided. Our method is integrator-free, precisely initializes oscillations around unstable fixed points, and efficiently detects orbit family intersections and subharmonic bifurcations. As a demonstration, we present full continuations of periodic double pendulum oscillations from fixed points, showing bifurcations along orbit families and categorizing branches of periodic orbits. In particular, we find periodic orbits where both pendulum masses are never simultaneously at rest, which to our knowledge has been missing in the literature.

Chaotic evolution measures how rapidly nearby configurations separate, but the existence of such rapid evolution does not by itself determine what physical information the evolution actually carries. For fixed-coupling JIMWLK evolution with fresh transverse-local, color-isotropic noise, we show that this distinction becomes exact - leading to a no go theorem. At finite rapidity step and finite transverse regulator, the exact linearized update depends on the Wilson-line background only through local adjoint rotations of fresh color-isotropic noise. Once the past is conditioned upon, these rotations leave the complete local noise measure invariant, so the tangent process has the same probability law for any target ensemble evolved with the same regulated fixed-coupling dynamics, including dilute and saturated ensembles. After the exact global color-rotation mode is removed, the reduced tangent process found to have a positive leading Lyapunov exponent which indicate chaotic evolution. Fixed-coupling JIMWLK therefor can be dynamically chaotic while its tangent-only statistics carry no information about the target ensemble. Spatially nonlocal noise however evades the argument because correlations between distinct sites leave relative Wilson lines in the conditional covariance.

We present a simple algebraic proof of meromorphic non-integrability of a 'mysterious' case of a quadratic Hamiltonian studied by Sokolov and Wolf. The result is obtained by applying the Morales-Ramis method, based on differential Galois theory.

We study the operator entanglement of local operators in one-dimensional brickwork circuits whose two-site gate satisfies the braid relation; throughout this work, we call such a gate a Yang–Baxter gate. We establish upper bounds for several structured, overlapping classes of Yang–Baxter gates. We show that the operator Schmidt rank remains uniformly bounded in time for all qubit Yang–Baxter gates and, in arbitrary local dimension, for permutation gates obtained from non-degenerate Yang–Baxter maps. We also show that it grows at most polynomially for involutive dual-unitary Yang–Baxter gates and for arbitrary phase dressings of permutation gates obtained from non-degenerate Yang–Baxter maps. These results imply, respectively, constant and logarithmic upper bounds on the operator entanglement. Conversely, we construct a seven-state involutive Yang–Baxter gate without dual unitarity and a one-site operator whose exact operator Schmidt rank grows exponentially, although the corresponding operator entropies remain undetermined. Entanglement growth in the general Yang–Baxter case remains open. All proofs and selected examples were constructed by ChatGPT 5.6 Sol.

Detecting weak transient signals in noise is a persistent challenge in sensing, communication, and electrophysiology. Here, we demonstrate a physical weak-signal detector based on a semiconductor negative differential resistance (NDR) device operated near a Hopf bifurcation. A coherent input that persists over the response time of the dynamical system can drive a transition between quiescent and oscillatory states, whereas faster stochastic fluctuations are largely suppressed. This nonlinear transformation converts a weak analog threshold crossing into all-or-none voltage spikes and therefore provides asynchronous signal detection without a reference clock. Using a modulated photovoltaic signal, we detect a weak frequency component of 100 Hz as a demonstration, at an input signal-to-noise amplitude ratio as low as 1/500, and reliably recover it in the output spectrum. We then apply the same principle to neuronal multisite extracellular recordings. After standard band-pass filtering, the raw microelectrode signal is transformed by the NDR dynamics, enhancing the distinction between neuronal spikes and background fluctuations. The resulting detected spike times agree closely with those obtained using a traditional spike-detection pipeline. These results establish bifurcation-engineered NDR dynamics as a compact hardware approach to noise-resilient signal discrimination and event-based analog-to-digital conversion that will be useful for neuroprosthetic devices.

Solutions of the Whitham modulation equations for one-phase periodic waves obeying the sine-Gordon equation are found that describe the evolution of an oscillatory region behind an instability front propagating into the instability region. A simple self-similar solution describes the whole region between two fronts of instability resulting from a localized initial disturbance in the unstable state. Another hodograph solution represents typical waves close to the instability fronts. This theory generalizes the approach used previously for systems obeying the nonlinear Schroedinger equation.

Higher-order networks exhibit rich critical phenomena that cannot be captured by traditional pairwise models. Here, we develop an analytical dimension-reduction framework that maps higher-order networked dynamics onto an effective low-dimensional system, allowing accurate prediction of tipping boundaries, bistability regions, and the nature of phase transitions. We demonstrate the power of this framework across a range of dynamical processes, revealing distinct effects of higher-order interactions on transition continuity and hysteresis. Furthermore, we find that system resilience exhibits a profound dependence on the alignment between pairwise and higher-order connectivity, with assortative mixing enhancing tipping toward active states. Our findings establish a general theory for understanding the critical transitions in higher-order networks, offering new insights for anticipating and managing systemic risk in complex systems.

There is a body of research devoted to understanding how complex singularities of solutions of nonlinear partial differential equations (pdes) spontaneously emerge at $t=0^+$ and propagate for $t>0$, and how their behaviour affects the solution on the real axis. Despite the importance of the small-time limit in these studies, there is still a lack of understanding of how complex singularities are born at $t=0^+$, including for initial conditions that are not analytic functions of the spatial variable. In this paper, we use Burgers' equation as a prototype nonlinear pde and study the complex-plane singularities for initial conditions that are piecewise smooth. Using matched asymptotic expansions, we show how infinitely many singularities emerge from points of discontinuity in a pattern that can be described using branches of the Lambert-$W$ function. For various initial conditions, we observe how these singularities rearrange themselves to align with the appropriate exactly-described long-time behaviour, including sigmoid-shaped travelling waves, constant-area (triangular wave) similarity solutions and $N$-wave solutions. In terms of Burgers' equation, our small-time asymptotic analysis of the singularity propagation for piecewise-continuous initial conditions illustrates the types of generic behaviours that arise for inner regions when diffusion dominates advection. More generally, this work is a step towards understanding complex-plane behaviour of solutions of nonlinear partial differential equations with non-analytic initial conditions.

Unstable periodic orbits and steady states are believed to form the backbone of spatiotemporal chaos and turbulence, yet their computation in thermally driven flows remains scarce for transitional regimes. In this work we compute and characterize a steady state and three families of periodic orbits in two-dimensional Rayleigh-Bénard at $\mathrm{Pr}=1$, near the transition to chaos. We find that in its route to chaos, the flow hops between several sets of orbits after becoming quasiperiodic. We use Floquet analysis to study the stability of the orbits obtained and characterize their bifurcations, showing how the appearance of primary and secondary frequencies, as well as phase-locking mechanisms, are all related to the dynamics of the orbits. We study in detail how the flow shadows the orbits found and determine in which regimes each orbit is dynamically relevant or not. Our analysis also reveals two important insights: (1) all symmetries are broken before the onset of chaos, and (2) this onset does not alter the behavior of the heat transport.

We study the stability of collective neutrino oscillations using a distributional approach motivated by the statistical mechanics of Kuramoto synchronization. Treating the ensemble of neutrino flavor polarization vectors in the thermodynamic limit $N\to\infty$, we derive an exact nonlinear Fokker–Planck (continuity) equation for the one-body distribution $F(\vec{\mathbf{S}},\omega,t)$ on the flavor sphere. This equation admits a two-parameter family of azimuthally symmetric stationary solutions, whose stability we analyze by linearizing around them. The resulting eigenvalue condition determines the growth or decay rate of small perturbations from any initial distribution – not merely from a state close to full flavor coherence – thereby going significantly beyond the conventional linear stability analysis of collective modes. In special limits the condition reproduces known synchronization thresholds in the two-beam model, providing a non-trivial check of the framework. We present analytical results for the eigenvalue equation and explore stability phase diagrams for physically relevant frequency distributions.

This paper describes classical soliton scattering in the 2+1d integrable chiral model, related to the Bogomolny monopole equation and self-dual Yang-Mills, as a case study for higher-dimensional integrable scattering. Extended line solitons exhibit nontrivial scattering with lump solitons and with other line solitons, and $N \to N$ interactions can be factorised into a series of $2 \to 2$ interactions. Scattering between parallel line solitons in 3d is related to lump soliton scattering in 2d, allowing for direct comparison with results in 1+1d integrable field theories.

We report the measurement of the critical velocity of a paraxial superfluid of light flowing past a mobile impurity, realized using a dual-wavelength optical scheme in a warm vapor of $^{87}\text{Rb}$. The superflow is two-dimensional by nature, while the impurity features a highly repulsive potential, a large tunable radius, and a finite mass. The critical velocity is evaluated by simultaneously monitoring three complementary observables: the number of vortices nucleated in the wake of the impurity, the hydrodynamic force exerted on it, and its displacement velocity relative to the flow. These independent methods yield consistent values for the critical velocity, which remain systematically below the Landau criterion and decrease with the impurity radius, in qualitative agreement with theoretical expectations. Our results demonstrate the reliability of these combined diagnostics, establishing paraxial superfluids of light as a versatile platform for exploring the impurity problem in quantum hydrodynamics.