Papers

Noise-activated switching between coexisting stable states is a fundamental mechanism underlying stochastic dynamics in systems ranging from chemical reactions to neural networks. While this phenomenon is well understood for stationary attractors, it remains largely unexplored for limit cycles, whose periodic motion cannot be described by a static potential landscape. Here we experimentally demonstrate activated switching between two coexisting limit-cycle attractors in a driven nonlinear system of coupled resonators. Specifically, we introduce controlled fluctuations to directly observe the rare stochastic transitions between two limit cycles and measure their dependence on noise intensity and driving strength. The measured switching rates are well described by a large-deviation theory, which replaces the conventional activation barrier by the action along the most probable transition path. Our results extend the concept of activated dynamics from stationary to limit-cycle attractors and establish a framework for modeling stochastic transitions between limit cycles in driven-dissipative systems.

We study the finite-genus algebro-geometric solutions of the Wadati-Konno-Ichikawa (WKI) equation with the saturable nonlinearity and long-time asymptotic behaviors of their short-range perturbations. First, for both the focusing and defocusing reductions, we formulate the finite-genus Baker-Akhiezer functions as explicitly solvable the matrix Riemann-Hilbert (RH) problems on the complex spectral plane and obtain theta-function representations together with the reconstruction formulae for the WKI field and the reciprocal coordinate. We then consider the Cauchy problem of the defocusing WKI equation on a finite-genus algebro-geometric background. We construct the scattering data and RH problem, and perform a Deift-Zhou nonlinear steepest descent analysis. The space-time plane is divided into two transition regions, a Zakharov-Manakov (ZM) region, and a fast-decay region. The leading term is a phase-shifted finite-genus WKI solution. The transition corrections are governed by a Painlevé-XXXIV model, while the ZM radiation is described by parabolic-cylinder functions. The reciprocal-coordinate asymptotics are obtained simultaneously.

Chaotic itinerancy (CI), brought to attention, among others, by K. Ikeda, I. Tsuda and K. Kaneko in the early 1990s, is a phenomenon in which trajectories in a dynamical system experience periods of ordered motion near quasi-attractors interspersed with chaotic transitions between them. Possible maps in which CI was found include coupled map lattices (CML) and globally coupled one-dimensional chaotic maps (GCM). We study such maps using numerical methods and graph algorithms. Specifically, we partition the state space into a finite grid of compact subsets, and we represent the map using a multivalued mapping of grid elements. This mapping can be perceived as a directed graph, with grid elements as vertices and individual mappings between them as weighted edges. This setup provides a coarse view of global dynamics and opens the opportunity for using Markov chains and efficient graph algorithms to study dynamical features. In particular, invariant sets can be found by computing strongly connected components in the graph. Analysis of the transition matrix of the graph makes it possible to find its stationary distribution and to compute local entropy as a measure of expansion or instability in the system. Using these tools, we propose an algorithm for assessing whether a certain map possesses the CI property and show its application to dynamical systems: a globally coupled system of logistic maps and a variant of a CML system for which we conduct computations for a large range of parameters.

We show that the Coulomb branches of 5d $\mathcal{N}=1$ necklace quiver gauge theories on $\mathbb{R}^3 \times T^2$ are identified with the phase spaces of elliptic spin Ruijsenaars-Schneider models with dynamical inhomogeneities. This identification resolves the long-standing problem of determining a Poisson structure and a quantization of these models. We further show that the resulting quantum system is the integrable system governing supersymmetric indices of 4d $\mathcal{N}=1$ theories of class $\mathcal{S}_k$.

Synchronizing nonlinear dynamical networks typically requires solving matrix inequalities or detailed system models, which fail for large networks. This paper offers a simple fix : a purely graph-theoretic framework using only a single Lipschitz-like bound on the dynamics. Coupling strengths are computed directly from the digraph, bypassing inequality solvers entirely. The method succeeds where existing approaches encounter infeasibility due to connectivity patterns. It examines only $n-1$ directed paths per strongly connected component versus $\frac{n(n-1)}{2}$ undirected paths before, achieving $O(n^3)$ complexity. Results show network connectivity can be exploited to synchronize a large class of nonlinear dynamical networks.

We study integrable variable-mass sine-Gordon model (vmSG) with point defects. Using the Lax and Bäcklund-gauge formulations, we construct type-I and type-II defect matrices, derive the corresponding sewing conditions, and generate the bulk and defect contributions to an infinite hierarchy of conserved charges. The lowest members reduce to the standard sine-Gordon energy and momentum in the homogeneous limit, while for inhomogeneous backgrounds they define integrability-generated energy- and momentum-type quantities. Defect compatibility imposes matching conditions on the variable-mass functions across the defect. An analytical transmission factor $z$ characterizes soliton transmission, topological conversion, and absorption/emission processes. We then deform the type-I sewing conditions by parameters $\alpha$ and $\beta$ and derive the associated defect anomalies. Consistent one-soliton transmission is recovered for $\alpha \beta =1$, whereas for $\alpha \beta \neq 1$ parity-centered kink-kink and kink-antikink transmissions display vanishing lowest order integrated anomalies but no generic vanishing at higher order. Numerical simulations reproduce the integrable transmission/conversion regimes and show that non-integrable defects generate weak radiative tails and lowest order quasi-conserved charges, while quasi-conservation of the higher order charges remains unestablished. These results elucidate how exact defect integrability deforms into charge-dependent quasi-conservation and establish a framework for defect-controlled soliton transport in inhomogeneous media, with potential applications to nonuniform Josephson junctions, magnetic and nonlinear-optical systems, and effective molecular and DNA models.

The emergence of organized spatiotemporal patterns is ubiquitous in oscillatory systems, from neural populations to engineered networks. Identifying these patterns and tracking how they evolve over time remains challenging, particularly when systems exhibit transient dynamics. Here, we introduce a framework based on spatial ordinal patterns to characterize the spatiotemporal dynamics of oscillatory systems. Our approach acts directly on the phase rather than the amplitude, with additional patterns introduced to account for near-equal phases. This symbolic representation encodes local spatial ordering relations, capturing both phase gradients and synchronized clusters within a single framework. From this construction, we define a spatial permutation entropy that quantifies the diversity of spatiotemporal patterns at each point in time, enabling the detection of transient dynamics and regime transitions as they occur. We show that this approach distinguishes phase-locked states with identical levels of global synchronization but distinct spatial organization and also characterizes partially synchronized states. We demonstrate the method on synthetic oscillator networks across multiple spatiotemporal regimes, and on resting-state EEG recordings from human volunteers, where it distinguishes different conditions within individual volunteers.

A body floating atop a supersaturated fluid may accumulate bubbles along its underbelly, which can render the body rotationally unstable. But rotation can strip the surface of these bubbles when they make contact with the air above. To explore this coupling we perform experiments using cylinders floating on carbonated water. The cylinders exhibit an array of distinct dynamics, including constant rolling, periodic and aperiodic oscillation, chaos, and intermittent capsizing, which can be tuned by body size, mass distribution, and gas concentration. A continuum model for the bubble density dynamics, and Galerkin projection, tie the experiment to the generalized Lorenz system and its famed chaotic attractor. Even an extremely small center-of-mass offset can have a qualitative impact on the dynamics, and an offset of as little as a few percent can fully stabilize the system. The experiment represents a highly accessible physical realization of the Lorenz system, which, owing to continuous gas loss from the fluid to the air above, sweeps slowly without intervention across a classical bifurcation diagram in time.

We investigate vector Akhmediev breathers (ABs) in the degenerate and nondegenerate regimes for the coupled Sasa-Satsuma system describing two coupled ultrashort optical pulse envelopes. Based on the carrier-wavenumber relations of the two-component background, we examine three configurations: a single vanishing wavenumber, a pair of opposite wavenumbers, and two nonzero wavenumbers of unequal magnitudes. These configurations admit distinct maximal degrees of nondegeneracy. For each configuration, we derive the reduced spectral equations, construct the existence diagrams, and compare the resulting branches with the modulation-instability gain spectra. Owing to the distinct carrier-wavenumber symmetries, nondegenerate-type modes can occur in spectrally degenerate regions, whereas degenerate-type ones may also appear in spectrally nondegenerate regions. Moreover, all admissible branches in the degenerate regimes satisfy the state-transition condition and correspond to periodic-wave states, while such branches are also present in nondegenerate regimes. Direct numerical simulations of representative solutions further confirm the analytical predictions.

The solar wind is a turbulent, weakly collisional plasma characterized by non-Gaussian fluctuations, long-range correlations, and multifractal dynamics. We investigate the long-term evolution of these properties using hourly proton density measurements obtained directly by the Wind spacecraft over 1995–2025. The use of a single-spacecraft dataset provides an independent test of previous results derived from the multi-spacecraft OMNI database. The three components of the nonextensive $q$-triplet are estimated within one-year sliding windows shifted monthly: $q_{stat}$ from the distribution of density increments, $q_{rel}$ from the decay of the autocorrelation function, and $q_{sens}$ from the multifractal spectrum. Their mean values, $q_{stat}=1.71\pm0.07$, $q_{rel}=4.62\pm0.52$, and $q_{sens}=-0.45\pm0.27$, confirm the persistent presence of heavy-tailed statistics, slow relaxation, and weakly chaotic multifractal dynamics. The parameters nevertheless exhibit distinct temporal variability and relationships with solar activity. The Fourier spectrum of $q_{stat}$ contains a dominant period of approximately $10.1$ years, while its correlation with the sunspot number is positive and moderate ($r=0.611$). Correlation and mutual-information analyses show that $q_{stat}$ is primarily associated with solar proxies, whereas $q_{sens}$ displays stronger relationships with geomagnetic indices and $q_{rel}$ exhibits weaker dependencies. An anomalous enhancement of $q_{stat}$ around 2004 coincides with a sequence of intense interplanetary disturbances, although possible effects related to Wind's orbital transition must also be considered. These findings demonstrate the robustness of the nonextensive description and show that the statistical and dynamical properties of solar wind proton density show evidence of modulation by solar activity.

We present a systematic comparison between arbitrary precise arithmetic and integration, IEEE-754 compliant floating point arithmetic (fp16, bfp16, fp32, double precision fp64, and quadruple precision fp128), and two implementations of Posits (type III unum) for solving Newton's chaotic N-body problem. Each implementation is benchmarked with arbitrary precise calculations to objectively evaluate their performance in precision as well as speed. We rely on hardware and compiler implementations for fp64, and software implementations for arbitrary-precision arithmetic and Posits. Half precision arithmetic (fp16, bfp16, and Posits$<16,1>$) are insufficiently precise for solving Newton's equations of motion. Single precision (fp32, and Posits$<32,2>$) could be used for statistical ensemble calculations, but lead to relatively large errors in any individual strong encounter. All 64-bit implementations fp64 as well as Posits (Posits$<64,3>$) experience difficulty in our tests. One of the implementations of Posits (Universal) gives precision comparable to fp64 but is slow (by at least an orders of magnitude compared to fp64 after correcting for the more efficient hardware support for the latter). The other (CPPPosits) has a speed comparable to fp64 but has systematically larger errors (by about an order of magnitude compared to fp64 with excesses exceeding two orders of magnitude). As a consequence, this implementation leads to a systematic drift in the result space and has difficulty resolving close encounters. Posits and fp64 have difficulty when integrating a dynamical system in a moving reference frame; testing Galileo invariancy. In their current implementation, Posits do not seem to be the ideal alternative for fp64 when integrating chaotic or stiff ordinary differential equations, such as Newton's equations of motion.

We report new quasilinear and fully-nonlinear 5th-order Schwarzian evolution equations. These are symmetry-integrable evolution equations in 1+1 dimensions, i.e. equations that admit Lie-Bäcklund symmetries, whereby it is required that the equations are kept invariant under the Möbius transformation for their dependent variable.

In this paper, we study the emergence and dynamics of pure-quartic optical dispersive shock waves governed by the nonlinear Schrödinger (NLS) equation with self-defocusing Kerr nonlinearity and fourth-order dispersion. The corresponding dispersionless hydrodynamic system reveals a distinct mechanism for wave breaking, driven by the nonlinear self-steepening of the hydrodynamic velocity. We illustrate this mechanism through optical dam-break configurations described by Riemann problems and show that, in contrast to the classical quadratic NLS equation, wave breaking occurs prior to wave splitting even in the small amplitude regime. Numerical simulations of the pure-quartic NLS (PQNLS) equation and its hydrodynamic approximation confirm these qualitatively distinct dynamical regimes.

We construct novel higher-order lump solutions of the modified Kadomtsev-Petviashvili-I (mKP-I) equation by applying the generalized long-wave limit method with spectral perturbations. By tuning the phase parameters in the soliton solution, we obtain second- and third-order lump solutions, which, to the best of our knowledge, are reported here for the first time for the mKP-I equation. A detailed asymptotic analysis reveals that these degenerate lumps exhibit anomalous scattering analogous to that in the KP-I equation. However, in contrast to the KP-I case, no complete energy equipartition occurs after the collision, and the two lumps remain distinguishable in the subleading terms of their asymptotic amplitudes. This distinction highlights a difference in the interaction dynamics between the modified and standard KP-I equations.

Largest Lyapunov exponents (LLEs) quantify exponential sensitivity, but data-driven estimates are often obtained from finite-precision trajectories. We show that increasing the length of a single reduced-precision chaotic record can eventually degrade a forecast-error LLE estimate. Using the logistic map at r=4, an ESP32 single-precision trajectory is reproduced bit-for-bit by NumPy float32. Across 10,000 random float32 initial conditions, every trajectory reaches an exact recurrence before iteration 7612. For one long float32 record, the estimated LLE changes from 0.6853 at N=15,000 to 0.1827 at N=20,000 and approximately zero at N=30,000 as exact train-test histories saturate. At N=100,000, the long float32 record gives 0.0016, whereas independently restarted length-100 trajectories give 0.6917; matched float64 controls remain near ln(2)=0.6931. The collapse is reproduced for 28 representative initial conditions, and its onset is strongly correlated with the recurrence scale set by transient length and digital period (Pearson r=0.982). Thus, finite-state recurrence can turn additional samples into duplicate futures rather than new dynamical information, while independent restarts substantially delay this saturation.

Neural autoregressive models have rapidly emerged as powerful emulators of high-dimensional chaotic systems, yet their long-term instability and error growth remain poorly understood, leading to ad-hoc solutions. Here, we develop an eigenanalysis framework that reveals the dynamical origin of this error growth. By analyzing the Jacobian of the learned one-step update map with respect to the state, we show how inference-time error growth, and thus model stability, is governed by its spectral radius. Direct-step architectures (models that predict the next state from the previous one) generically admit unstable eigenvalues with magnitudes exceeding one, explaining the rapid divergence of these widely used models. In contrast, integration-constrained models (where the time derivative is estimated and integrated with a higher-order integrator) collapse their eigenspectrum onto the unit circle, yielding neutral stability and a universal linear error-scaling law. The largest eigenvalue of this Jacobian provides an architecture-agnostic, a priori diagnostic of short-term skill, long-term stability, and spectral bias, without requiring an expensive rollout. Leveraging this theory, we introduce a stability-promoting loss that explicitly regularizes Jacobian-driven error amplification, improving both forecast accuracy and dynamical robustness. Demonstrated across $29$ models spanning two architectures, several explicit and implicit integrators, and multiple loss functions on the Kuramoto-Sivashinsky system, our results establish a theoretical foundation for the design and evaluation of neural emulators of chaotic multi-scale dynamics. More broadly, our framework is a step toward the kind of a priori stability analysis that numerical analysis provides for discretizations of differential equations and that scientific machine learning currently lacks.

Adjoint methods provide a powerful route for gradient-based optimization, but their physical implementation is obstructed in generic nonlinear systems because the adjoint dynamics requires backward-time evolution, Jacobian transposition, and terminal-value constraints. Here we show that nonlinear parity-time ($\mathcal{PT}$)-symmetric systems overcome this obstruction. Using a class of nonlinear non-Hermitian resonator networks, we establish symmetry relations that map the formal adjoint dynamics onto experimentally accessible forward-time evolutions supplemented by controlled injections. This construction enables exact in-situ evaluation of adjoint gradients without requiring explicit backward propagation or matrix transposition. We demonstrate the approach in nonlinear $\mathcal{PT}$-symmetric resonator chains, where the resulting optimization protocol autonomously discovers parameter configurations that realize prescribed spatio-temporal functionalities, including uniform energy redistribution and targeted wave transport at predefined time windows. Our results identify $\mathcal{PT}$ symmetry as a resource for implementing computational sensitivities within physical systems and establish a route toward self-optimizing nonlinear machines.

We compute the dynamic critical exponent $z$ of the Yang–Lee edge singularity in relaxational dynamics using perturbative renormalization-group methods to the three-loop order. The calculation combines diagram reduction, analytic two-loop evaluation via parametric integration with hyperlogarithms, and numerical evaluation of three-loop integrals using the Sector Decomposition method. The perturbative series obtained is resummed using Padé and Padé-Borel-Leroy methods to produce estimates of $z$ in various spatial dimensions. The resulting values are consistent with previous perturbative and functional renormalization-group calculations.

In this article we consider the Drinfeld-Sokolov hierarchy of type $A$ and its similarity reduction. Then the similarity reduction is expressed as $2$ types of Hamiltonian systems, one is with a Poisson bracket and another a Poisson bracket of canonical coordinates. We also investigate a connection between the similarity reduction and the isomonodromy deformation system.

Spatial early warning signals (EWSs) seek evidence of an approaching tipping point from a single snapshot of many interacting elements. Existing theory largely assumes spatial homogeneity, whereas networks introduce systematic differences among nodes that may obscure fluctuation-based warning signals. We develop a mathematical framework for spatial EWSs in stochastic dynamical systems on networks. We find that the expected spatial variance, a popular spatial EWS, decomposes exactly into a structural contribution from heterogeneity in the equilibrium state and a fluctuation contribution determined by the stationary covariance. Near a simple steady-state bifurcation, the potentially divergent covariance concentrates along the critical eigendirection: the left eigenvector determines how strongly noise excites the critical fluctuation, while the right eigenvector determines its spatial pattern. Consequently, the spatial variance has a divergent fluctuation contribution when the limiting critical eigendirection is noise-excited and spatially nonuniform after centering. In contrast, the spatial coefficient of variation generally saturates, while skewness, kurtosis, and Moran's $I$ approach network-dependent limits without a universal warning direction. We also derive results for homogeneous networks, node-wise baseline subtraction as preprocessing, and Hopf bifurcations, for which the limiting distributions are qualitatively different. These results clarify when spatial EWSs provide reliable warnings and why their performance depends on network structure, noise, and preprocessing.

We investigate stochastic resonance in a hierarchy of delayed excitable systems, from a single FitzHugh–Nagumo neuron to single-layer and duplex networks with autaptic, pairwise, triadic higher-order, and interlayer interactions. We show that the order and organization of interactions generate distinct resonance regimes. Among the autapse-free single-layer architectures, pairwise and triadic coupling attain comparable maximal responses, but over different ranges of nominal coupling strength: the pairwise enhancement persists across the investigated coupling range, whereas the triadic enhancement is confined to weak coupling. Under the matched participation degrees used here, this difference is largely attributable to the doubled effective coupling generated by the hyperedge construction. Delayed autaptic feedback primarily shifts the optimal noise level and, within the configurations investigated, yields a genuine enhancement only when pairwise and triadic interactions coexist, revealing a non-additive coupling effect. In duplex networks, interlayer coupling acts predominantly as a response-equalization mechanism: it enhances the weaker layer, generally at the expense of the stronger one, with the redistribution controlled mainly by the uncoupled resonance-capacity gap. Across the sampled duplex configurations, increasing the interlayer delay largely suppresses the response within the investigated delay range. These results identify interaction order, resonance mismatch, and delay as key control parameters for noise-assisted signal processing in multilayer excitable systems.

In the late 1980s, Baulieu and Grossman demonstrated that the supersymmetric formulation of Langevin stochastic differential equations (SDEs), proposed earlier by Parisi and Sourlas, belongs to the family of Witten-type topological field theories (TFTs). From a certain angle, this finding may appear puzzling: TFTs have no local degrees of freedom, whereas SDEs do exhibit local fluctuations. In this paper, we address this apparent contradiction in the context of the supersymmetric theory of stochastic dynamics (STS), a generalization of the Parisi-Sourlas-Baulieu-Grossman approach to SDEs of arbitrary form. We further discuss how, within STS, dynamical chaos can be identified with the spontaneous breakdown of the corresponding topological supersymmetry, while 1/f noise can be understood as a consequence of the Goldstone theorem. The butterfly effect, in turn, calls for an effective field-theoretic description that may itself possess a hidden topological structure, as we speculate on the basis of the Ginzburg-Landau approach and insights from AdS/CFT duality.

We investigate integrability of Euler-Lagrange equations associated with 3D second-order Lagrangians of the form \begin{equation*} \int f(u_{xy},u_{xt},u_{yt})\ \text{d}x\text{d}y\text{d}t. \end{equation*} It is demonstrated that there are exactly four different types of such Lagrangian densities $f$: the first one is given by the formula $f=\sqrt{u_{xy}u_{xt}u_{yt}}$, the second and the third are more complicated (although still representable in elementary functions), whereas the most generic fourth one is expressible in terms of the Lobachevsky function, revealing unexpected links to spherical/hyperbolic trigonometry and Schläfly-type formulas. Dispersionless Lax pairs and integrable dispersive deformations of the corresponding Euler-Lagrange equations are also constructed.Remarkably, dispersive deformation of the first Lagrangian density coincides with the Lagrangian of the classical Darboux system arising in the theory of triply-orthogonal coordinate systems in $\mathbb{R}^3$. Dispersive deformations of the three other cases provide Lagrangian formulation of semi-discrete and fully discrete versions of the Darboux system, with one, two and three discrete variables, respectively.

A scuba diver holding constant depth balances on an unstable equilibrium: the gas carried in the suit and buoyancy compensator compresses with depth, so the buoyant force falls as the diver sinks and rises as the diver ascends. We represent the diver as a proportional-derivative controller that regulates this compressible-buoyancy saddle after a finite reaction delay, and we derive the governing delay differential equation from the vertical force balance and the isothermal gas law, reducing it to a damping ratio, two control gains, and a dimensionless delay. The characteristic spectrum, obtained by pseudospectral collocation of the semigroup generator and checked against a direct Newton solution of the characteristic equation, locates the Hopf boundary that separates stable hovering from sustained porpoising; for the baseline diver the critical reaction delay is 3.36 s and the onset period is 28.8 s. The bifurcation is supercritical, and because the saturating force is the quadratic hydrodynamic drag, the limit-cycle amplitude grows in proportion to the delay excess rather than as its square root. The safe-operating envelope shows that runaway ascent is triggered by saturation of the compensator, not by loss of linear stability, so a stable and an unstable diver can share the same escape threshold. As onset is approached, the lag-one autocorrelation and variance rise while the fitted recovery rate falls and matches the spectral abscissa, giving an eigenvalue-exact early warning of the transition.

Spectrally stable cascades can undergo enormous transient growth before eventually dying out. We show that this amplification is controlled by the directed architecture of the interactions, independently of the eigenvalues that determine asymptotic stability. More remarkably, when several critical subsystems are connected sequentially, their depth becomes a new control parameter for critical fluctuations: each additional critical stage generates a new cascade-size exponent, producing an infinite hierarchy of universality classes. This result follows because the fluctuating population produced at one stage becomes the random input to the next. The mechanism persists for both finite-variance and heavy-tailed reproduction and strongly enhances the probability of rare terminal events. Applied to multitype earthquake triggering, it provides a stationary mechanism for the anomalously large abundance of foreshock-mainshock sequences. Directed architecture can therefore control both transient amplification and critical statistics in cascade processes even when it leaves spectral stability unchanged.

New types of soliton and periodic waves are identified for a nonlinear dissipative medium where the pulse propagation is governed by the cubic complex Ginzburg-Landau equation. We find that the dynamical equation for the pulse amplitude supports two distinct types of kink and antikink solitons with different functional forms. It is found that the obtained kink and antikink soliton waveforms occur under the same fixed inverse velocity. The results also indicate that the periodic waves can propagate with variety of wave forms such as sn, cn, dn and their rational forms as well. It is also shown that in the long-wave limit, the derived periodic waves degenerate into different bright and dark soliton pulses. The stability analysis based on the theory of dispersive waves in nonlinear optics is developed. It is shown that some elliptic and soliton solutions are quasi-stable in the context of passive mode locking lasers described by complex Ginzburg-Landau equation.

For directed networks, the Laplacian need not be diagonalizable, so the standard master-stability variational equations cannot in general be fully decoupled into independent eigenmodes. We prove that this obstruction can always be removed by coupling-strength design: every weakly connected digraph admits a strictly positive weighting of its existing arcs for which the weighted in-degree Laplacian is diagonalizable. The construction uses a spanning directed acyclic subgraph with one source in each root strongly connected component, assigns distinct positive weighted indegrees to its non-source vertices, and then restores all remaining arcs with a common sufficiently small positive weight. The zero eigenvalue remains semisimple and all nonzero eigenvalues remain simple. We also give a discriminant criterion that computes an admissible interval of restoring weights. Thus, any fixed weakly connected directed topology can be positively weighted so that master-stability perturbations admit a complete modal decomposition.

We introduce a procedure to obtain lump solutions via the formation of a kink-antikink pair, consisting of the superposition of kinks whose distance from the origin is controlled by a single parameter $a$. For large values of $a$, a wide plateau appears in the solution, which we call a large lump. The procedure involves the use of a first-order equation that allows the construction of the potential associated with the lump solution. We then investigate several known scalar field models where the parent kinks are capable of giving rise to novel lumps. The lump inherits the tails of the parent kink, allowing for either short-range exponential profiles or long-range profiles characterized by distinct power-law decays. We also show how to verify if an arbitrary lump solution can be obtained via our method and illustrate this possibility with a novel vacuumless lump.

This work provides explicit expressions, in all sectors of the Fock space: soliton, antisoliton and breather, for the fundamental building blocks -the form factors- of the generalised integral kernels which represent the exponential of the field operator in the 1+1 dimensional Sine-Gordon quantum field theory. The obtained form factors reproduce the various results obtained earlier in the literature using, in particular, the free field approach.

We investigate the long-time behavior of optimal mixing in an advection-diffusion equation using a shell model framework. Our focus is on quantifying the decay of the scalar variance, measured by the negative Sobolev norm $H^{-1}$, under enstrophy-constrained stirring. We perform long-time computations using both local-in-time (maximizing the instantaneous mixing rate) and global-in-time (maximizing mixedness at a prescribed final time) optimization strategies. For mixing with diffusion ($\kappa>0$), the numerical results show that the scalar length scale eventually becomes limited by a generalized Batchelor scale, in close agreement with theoretical predictions. In this regime, the $H^{-1}$ mix-norm decays exponentially in time with a decay rate that is independent of the diffusivity $\kappa$. Compared with the purely advective case ($\kappa = 0$), diffusion significantly enhances the long-time mixing rate; moreover, increasing diffusivity further improves mixing efficiency by reducing the prefactor of the exponential decay. Guided by these numerical observations, we derive new conditional lower bounds on the $H^{-1}$ norm whose exponential decay rates are strictly independent of the diffusivity parameter $\kappa$, for all $\kappa > 0$. We further establish conditional upper bounds on the maximal rate of enhanced dissipation of the scalar variance, showing that the effective diffusion time scale is at least of the order $|\log\kappa|$.