Papers

A chemical transition state is a phase-space bottleneck: trajectories approach along stable directions and leave along unstable directions. Lagrangian betweenness (LB), introduced in finite-time transport theory for fluid flows, combines backward and forward deformation in a way suggestive of this gather-and-disperse geometry. We ask what LB measures when the transition-state geometry is known. For a linear rank-one saddle, LB is spatially constant even though the stable and unstable manifolds are present. Relative nonlinear corrections remain small on shrinking initial neighborhoods, including observation times on the logarithmic local escape scale. At fixed spatial resolution, however, increasing observation time can concentrate the spatial variation of LB near a stable or unstable manifold. A separable quartic Hamiltonian makes these two limits explicit. A nonseparable Hamiltonian shows local flattening near a hyperbolic periodic orbit. We then use HCN/CNH isomerization, where dividing surfaces and gap times are established by phase-space transition-state theory. Before any exit, larger LB generally accompanies longer eventual gap times. The incoming reactive action provides a dynamical interpretation: initial conditions closer to the normally hyperbolic invariant manifold have longer local passages and greater accumulated stretching during passage. Among trajectories still inside at 0.5 ps that exit before 5 ps, early LB does not usefully rank exit times. LB therefore characterizes transition-state-organized transport in this example; a large value is neither an intrinsic manifold marker nor a general measure of molecular residence time.

Synchronization is a collective phenomenon in which interacting nonlinear oscillators develop a persistent relation between their phases and frequencies. In quantum systems, its description requires connecting the classical dynamics of limit-cycle oscillators with quantum fluctuations, dissipation, and the notion of phase. In this tutorial review, we offer a systematic framework for analyzing synchronization in classical and quantum dynamical systems. Starting from amplitude-phase dynamics, we introduce successively topics of relevance, such as phase locking, the Adler equation, Arnold tongues, and the Kuramoto model, and use these concepts to analyze coupled oscillators and their stability. We then formulate the corresponding description for quantum limit-cycle oscillators using Lindblad master equations and phase-space methods, with explicit examples of two and many coupled oscillators. We further discuss noise, synchronization measures, and the extreme quantum regime, including synchronization in spin systems. The emphasis throughout is on analytical derivations and a practical methodology for identifying synchronized states, determining their stability, and characterizing synchronization in quantum systems. The review is designed as a self-contained guide from which we hope that students can learn the essential concepts and techniques, while researchers can readily find the analytical tools and methodological insights needed to approach new synchronization problems.

We present a classification of supersymmetric bilinear KdV-type equations admitting unconstrained three-super-soliton solutions. Extending Hirota's classical three-soliton criterion to the supersymmetric setting, we derive the complete bosonic and fermionic compatibility conditions governing the existence of three-super-soliton solutions. We prove that every supersymmetric bilinear KdV-type equation possesses unconstrained one- and two-super-soliton solutions, whereas three-super-soliton solutions exist only when eight integrability conditions are satisfied. These conditions provide a supersymmetric analogue of Hirota's classical integrability criterion and naturally recover the fermionic relations previously introduced by Carstea, revealing their structural origin. As a consequence, we classify the supersymmetric extensions of Hirota bilinear KdV-type equations and show that only a subset of Hietarinta's classical classification remains valid in the unrestricted supersymmetric framework.

We observe the emergence of extreme events in an active fluid involving two distinct chemical species that regulate active stress. One species is slow diffusing and the other is fast diffusing, and the growth of the fast-diffusing species is modelled using a nonlinear logistic term. We demonstrate the presence of extreme events in the temporal evolution of the concentrations, as well as in the spatial profile of the system,in regimes of merging-emerging soliton-like dynamics and spatio-temporal chaos, through analysis of the time-series, bifurcation diagrams, probability distribution functions of the concentration of the two species, return maps and distributions of inter-event intervals. Interestingly, we also find evidence of pronounced bunching of extreme events and super-extreme events in the slow chemical species in the soliton-like regime. We go on to systematically explore the dependence of the extreme event occurrences on the Péclet number and the strength of the nonlinear growth term, and find that the probability of extreme events increases after a critical Péclet number, while increasing the nonlinearity suppresses extreme events. Lastly, in order to gain further insight, we investigate a modified mode-truncated reduced order model comprising of coupled differential equations mimicking this active fluid system. We find that this reduced order model also exhibits extreme events whose emergence is correlated with a sudden expansion in attractor size due to a crisis arising from attractor collision.So these results demonstrate the existence of extreme events in an active fluid system, and are of potential relevance to biological phenomena where active transport plays an important role.

Classical theories of random neural networks typically assume independent connectivity, overlooking the local motif structures prevalent in biological circuits. Here, we investigate how four second-order synaptic motifs (chain, reciprocal, convergent, and divergent) shape the dynamics of nonlinear firing-rate networks. While previous studies have established that chain correlations generate outlier eigenvalues, we demonstrate that these motifs also jointly reshape the Jacobian eigenvalue bulk. Using the path-integral formalism, we derive a dynamic mean-field theory which reveals that the chain motif acts as a retarded feedback of the ensemble-mean activity through the response kernel, producing a rich repertoire of dynamical regimes, including ferromagnetic states and limit cycles. At sufficiently large magnitude, negative chain correlations produce a glassy, multistable regime that was previously mainly associated with partially symmetric networks. Our theory also distinguishes convergent from divergent motifs: divergent correlations primarily rescale temporal noise, while convergent correlations suppress temporal chaos by converting nonzero mean activity into quenched heterogeneity. Finally, analyses of the Lyapunov spectrum and participation-ratio dimension show that motif structure changes the geometry of chaotic activity, reducing entropy production and attractor dimensionality even when the effective spectral edge is held fixed. Together, these findings establish second-order motifs as a fundamental structural mechanism governing the dynamical regimes of local cortical circuits.

I consider vortex solitons in two-dimensional star networks created by several one-dimensional line waveguide arrays (or rays) brought in close proximity. First waveguide in each array belongs to the ring of certain radius and its modal field overlaps with modal fields of waveguides from other rays laying on this ring, as well as with modal fields of waveguides in this line array. This creates a system with discrete rotational symmetry that is capable of supporting various localized linear vortex states due to local coupling between waveguides on the central ring, where line arrays approach each other. Localized vortex states emerge from the continuum of the states extended along the rays of the structure upon variation of the radius of the central ring. I show how different vortex modes with progressively increasing topological charges emerge when the number of rays in structure and, therefore, its discrete rotational symmetry increases. Each such linear localized mode gives rise to a family of thresholdless vortex solitons in nonlinear optical medium, whose topological charges are limited by the discrete rotational symmetry of the system. Vortex solitons in star networks exhibit qualitatively different behavior with increasing power depending on whether the propagation constant of linear vortex that gives rise to a family of solitons is located above or below the band of extended states. Stability properties of vortex solitons nontrivially depend on their topological charge and discrete rotational symmetry of the system, showing very different picture from usual ring-like waveguide arrays.

How can a neural circuit rhythmically burst when none of its constituent neurons can endogenously do so? We address this question through a bottom-up reconstruction of a 4-cell neural circuit modeled after the swim central pattern generator (CPG) of the sea slug Dendronotus iris. We first map the intrinsic regimes of a swim interneuron (SiN) model neuron and show that slow mutual inhibition can generate anti-phase bursting in a half-center oscillator (HCO) assembled from tonic-spiking or quiescent cells. A slow–fast phase-space deconstruction explains this pairwise rhythm as a release mechanism. We then move on to CPG parametrization, in which cells 1 and 2 are quiescent, while cells 3 and 4 are tonic spikers but their isolated HCO can only exhibit tonic spiking or suppression. Bursting therefore does not arise at either the cellular or HCO level. It appears only after the two modules are assembled into the complete 4-cell network, where cross excitation, cross inhibition, and rectified electrical coupling act together to support a robust network rhythm-generation. Event-based symbolic encoding and GPU-parallel parameter sweeps show that this higher-order collective state occupies extended parameter domains rather than a single tuned point, and they quantify how the network interactions reshape those domains and the spike counts per burst. Symbolic sweeps identify the healthy range of activity regimes (periodic sequences) and transition boundaries; Lempel–Ziv complexity is used as a descriptor of aperiodic symbolic output rather than as proof of chaos or dynamical instability. Our results establish a hierarchy of rhythm generation—from intrinsic cell dynamics, through conditional HCO bursting, to bursting that emerges only in the fully assembled 4-cell CPG—and provide experimentally accessible predictions for perturbing its chemical and electrical couplings.

We examine a 4-dimensional generalization of predator-prey models describing interactions among four species. We show that these systems exhibit a rich variety of asymptotic behaviors, with solutions that are either integrable, leading to non-sustainable dynamics characterized by unbounded population growth or collapse, or chaotic, where all species coexist over time. Moreover, we review the integrable cases, which are classified into two-parametric families, and examine three-parametric families that behave similarly to the known two-parametric integrable cases. We also analyze chaotic Lotka-Volterra models in terms of their Lyapunov exponents and Poincare' surfaces of section, which turn out to reveal underlying structures that are compatible with what is expected from sustainable dynamics.

Integrable partial differential equations, such as the Korteweg–De Vries (KdV) equation, admit infinite hierarchies of commuting higher symmetries. Their symmetry reductions give rise to integrable finite-dimensional dynamical systems solvable in terms of Abelian functions. This underlies the finite-gap integration method for the KdV equation, introduced by S.P. Novikov and subsequently extended to many other integrable systems. In this paper, I introduce the concept of reduction ideals and a new class of reductions for integrable differential–difference equations, leading to integrable finite-dimensional systems in both commutative and noncommutative settings. This class includes periodic reductions, symmetry reductions, and a broad family of other reductions defined in terms of the associated Lax operator. The reduction constraints define integrable maps that enable solutions of the reduced systems to be extended to solutions of the corresponding differential-difference equations. The construction is illustrated using the Volterra hierarchy.

We investigate a discontinuous route from integrability to chaos using a confined stochastic random walk and a deterministic stadium-like billiard. In both systems, the stationary diffusive observable exhibits a finite jump at the transition: it vanishes at the unperturbed limit but approaches a finite, geometry-controlled value for arbitrarily small nonzero perturbations, providing the characteristic order-parameter signature of a first-order transition. Despite this discontinuity, the relaxation timescale diverges as the transition is approached, revealing critical slowing down. Both models exhibit normal diffusion with $\beta=1/2$, a perturbation-independent stationary state with $\alpha=0$, and a crossover iteration scaling as $n_x\propto\lambda^{-2}$, yielding $z=-2$, where $\lambda$ denotes the corresponding perturbation parameter. The common exponent set $(\alpha,\beta,z)=(0,1/2,-2)$ originates from the same coarse-grained mechanism: normal diffusion within a finite accessible domain with a diffusion coefficient that vanishes quadratically at the transition. The agreement between stochastic transport and deterministic chaotic scattering provides strong evidence for a common class of discontinuous dynamical transitions and extends the statistical-mechanics description of phase transitions to integrability-breaking dynamics.

Can the transition from integrability to chaos be discontinuous? We show that it can, and that the resulting first-order dynamical transition coexists with critical slowing down. Using an analytically tractable confined random walk and a deterministic stadium-like billiard, we find a finite jump of the stationary diffusive observable at the transition while the relaxation time diverges. Both systems display normal diffusion and the same exponent set $(\alpha,\beta,z)=(0,1/2,-2)$. We trace this agreement to a common coarse-grained mechanism: diffusion in a finite accessible domain with a diffusion coefficient that vanishes quadratically with the perturbation. The results identify a discontinuous route from integrability to chaos and provide evidence for a broader universality class of first-order dynamical transitions.

For a model of a thin spherical layer of an incompressible rotating fluid subjected to the action of Coriolis and buoyancy forces, both analytical solutions in the form of stationary thermal spots moving along the equator and self-similar solutions are found. The influence of the parameters determining the shape of the spots on their propagation speed and direction is studied. It is found that the speed and direction of motion of the spots depend not only on the sign of their thermal contrast with the background flow, but also on the ratio of the semi-axes determining their shape. In particular, if the ratio of the equatorial semi-axis to the meridional one $b/a$ is equal to $\pi/2$, the spots are at rest. If the spots are elongated along the equatorial axis to such an extent that $b/a>\pi/2$, then “cold” spots move westward and “hot” spots move eastward. Under the condition $0.747\leq b/a<\pi/2$, the situation is reversed: “hot” spots move westward and “cold” spots move eastward. Self-similar solutions in the model are realized strictly in the form of circular thermal spots, and their radius varies according to a power law with a scaling exponent $k$, which is determined by the behavior of the cross-frontal buoyancy gradient on the spot contour. We note that the regime $k=1/4$ is realized under the assumption of a constant buoyancy gradient, while the regime $k=1/6$ corresponds to the conservation of total buoyancy.

We construct a class of integrable chiral fermionic quantum field theories with interactions that depend on both space and time. The construction starts from a unitary, regular, difference-form two-body scattering matrix satisfying braiding unitarity and the Yang–Baxter equation. Each freely moving right- or left-moving particle carries a characteristic coordinate that remains constant along its trajectory, and the scattering matrix depends on differences of these coordinates. The many-body amplitudes form a discrete connection over the coordinate-ordering sectors. The Yang–Baxter equation gives local path independence, while periodic winding on a spatial circle produces quantum Knizhnik–Zamolodchikov (qKZ) difference equations expressing global flatness. Once a reference amplitude satisfying the exchange and qKZ constraints is specified, these transport equations determine the exact wavefunction in every fixed-particle-number sector. Independent reparametrizations of the right- and left-moving characteristics give couplings that depend on both space and time. For the chiral $SU(2)$ Gross–Neveu realization, we solve the qKZ equations for this full characteristic family using off-shell Bethe vectors. The resulting amplitudes are written in the characteristic labels, and their pullbacks give exact many-body wavefunctions for the corresponding space-time-dependent couplings. The two-body amplitude is a ratio of Gamma functions that resums the phase shifts from successive scattering processes. Equal-slope affine maps give a spatially homogeneous specialization in which the coupling depends only on time, and the Bethe ansatz with constant coupling is recovered when the qKZ step vanishes.

We present an extended study of the gravitationally localized soliton-like solutions to the minimally-coupled Einstein-Dirac-Higgs equations, embedded in asymptotically Minkowski spacetimes. The equations of motion describing these Yukawa-coupled Dirac stars are generalized to any even number of constituent fermions. We first expand the discussion of two-fermion ground-state solutions initially analyzed in Leith et al. [Phys. Rev. D. 107, 106020 (2023)]. Upon extending our analysis to excited states and many-fermion states, we find that both exhibit different behaviors to their two-fermion ground-state counterparts. In these excited states and many-fermion states, the Higgs field may exhibit stepwise increases and at times decrease within the soliton, which has not been seen for the two-fermion ground states. We also propose the potential cause of a significant degree of ADM-to-fermion mass disparity, which is present in states with strong Yukawa-coupling.

We describe how strong vorticity fluctuations appear in three-dimensional incompressible turbulence driven by a large-scale random force. By combining analytical and numerical methods, we construct a vorticity instanton: the most probable force history driving the flow to a prescribed large vorticity at a given point and time. We show how the smooth, large-scale force produces a long vortex filament with a thin, viscous-scale core. Our main finding is that the standard mechanism of amplifying a vortex tube through radial compression and axial stretching is only effective up to a threshold value. Producing vorticity above the threshold requires an additional ingredient: vorticity waves propagating along the tube. The final vorticity is the result of two colliding waves. This mechanism has striking similarities to the one recently suggested by OpenAI for creating infinite vorticity in finite-time Navier–Stokes blow-up.

We develop a matrix pseudodifference operator framework that places theAdler-Pfaff and Pfaff-Toda hierarchies in a common algebraic setting. The bi-infinite Adler-Pfaff hierarchy then becomes a $2\times2$ matrix pseudodifference Lax hierarchy. We introduce the Matrix Pfaff-Toda hierarchy by dressing the matrix Laurent algebra $M_{2N}\bigl(\mathbb{C}[\mathcal{S},\mathcal{S}^{-1}]\bigr)$ using a Pfaff-Toda splitting of the matrix pseudodifference algebra. In the one-component case, its diagonal sector recovers the continuous Pfaff-Toda hierarchy, while the off-diagonal directions supply the missing odd flows, and the powers of a single bare operator reconstruct the full Adler-Pfaff hierarchy. For general $N$, on the regular factorization locus, the multicomponent Pfaff-Toda tau-functions of Savchenko and Zabrodin realize the commuting diagonal sector, with its dressing equations derived directly from the fermionic bilinear identity. Reductions to the block-diagonal and scalar cases recover multicomponent and scalar $2D$ Toda, identifying the even Pfaff hierarchy with an anti-diagonal Toda subhierarchy, and yielding the Krichever-Zabrodin C-Toda hierarchy.

We formulate the linearization of the fourth Painlevé equation as a linearized Hamiltonian system and transform the scalar variational equation to normal form, with the first-derivative term eliminated. From the canonical fundamental solutions of the Jimbo–Miwa Lax pair, we construct quadratic expressions $Q_{IV}$ and $R_{IV}$ that satisfy the identity $\partial_x^2Q_{IV}-U_{IV}Q_{IV}=\partial_\lambda R_{IV}$. Here $U_{IV}$ is the coefficient in the normal-form equation. A rapid-decay cycle with a vanishing boundary term yields an integral solution of the linearized equation. Every nontrivial such cycle has asymptotic branches in at least two distinct Stokes sectors; otherwise the integral vanishes by Cauchy's theorem. A second rapid-decay homology class gives another integral solution. The Wronskian of the two integral solutions depends holomorphically on the monodromy data, so a single nonzero value implies generic linear independence. For one set of complex data, direct quadrature gives numerical evidence of a nonzero Wronskian. We also derive integral formulas for variations of the Stokes multipliers, the connection matrix, and the local monodromy, and verify the contour representation by substitution into the linearized equation.

We consider the application of Sundman transformations to the first four Painlevé equations $P_I$-$P_{IV}$, with $P_V$ and $P_{VI}$ being left for future work. Since the Painlevé equations are nonautonomous, this requires first of all the construction of an equivalent autonomous system, a process which we discuss in detail, and which we characterize as being inverse to the use of Lie symmetries to obtain a reduction of order of an autonomous system. Secondly, we apply a Sundman transformation to this autonomous system, and identify the resulting equation with a corresponding nonautonomous equation. We give an explicit mapping between solutions of this nonautonomous equation and the original Painlevé equation. Interesting special cases of the nonautonomous equation found by Sundman transformation include, for $P_{II}$, an equation with coefficients expressed via Airy functions, and for $P_{IV}$, an equation with coefficients expressed via Weber-Hermite functions. We also discuss an alternative autonomisation process. Finally, a variety of possible directions for future work are identified, along with interesting open problems.

Reducing the complexity of ride-pooling paths is a central challenge in systems with distributed demand. Here we show that such dynamics admit an exact coarse-grained representation: for a broad class of routing algorithms whose decisions depend only on path lengths, the full network can be reduced to an effective network of active nodes weighted by shortest-path distances without altering the resulting trajectories, up to stochastic degeneracy breaking. The reduction therefore defines an equivalence class of network representations generating identical path dynamics. We further demonstrate that for globally optimizing dispatchers this equivalence is systematically violated through degeneracy amplification, yet remains quantitatively accurate beyond the exactly solvable regime. Our results identify when spatial structure can be integrated out without loss of dynamical fidelity, providing a general framework for the analysis of interacting path processes.

The deployment of a geoengineering program to cool Earth's climate may be imminent. It is crucial that tools be developed to ensure that such a program would achieve its objectives while minimizing disruption. Here we exploit recently developed differentiable atmospheric models to demonstrate a novel geoengineering control strategy. In the differentiable primitive-equation atmospheric model JAX-GCM we impose a uniform $+4$\,K ocean warming and ask what pattern of sea-surface temperature cooling – in five ocean-masked zonal bands of prescribed SST forcings whose amplitudes are free – returns land near-surface air temperature closest to the model's own unwarmed climatology. This idealized set-up represents a cooling pattern that could be delivered physically either by marine cloud brightening or stratospheric aerosol injection. Gradients through chaotic dynamics decorrelate from the true sensitivity beyond the Lyapunov horizon, so we optimize greedily over segments of 8 to 14 days, following receding-horizon control. The learned strategy removes $92.3 \pm 0.4\%$ of the realized land warming across a ten-member ensemble of two-year rollouts, and a three-year run sustains it. If we use the spatial pattern of land temperature as the optimization objective, the distributions of precipitation, evaporation, and specific humidity over land are restored as well, even though they are not included in the objective function. The learned strategy from JAX-GCM replayed in the AI emulators LUCIE and NeuralGCM without re-optimization is successful, suggesting robustness. These promising results demonstrate a strategy for designing optimal climate interventions that can be applied broadly for geoengineering scenarios under consideration.

A resonator with one localized cubic nonlinearity can acquire additional cusp pairs when it is coupled to an otherwise linear network. At first-harmonic order, the cusp locus is a phase contour of the driving-point receptance $G(\Omega)$ at the nonlinear coordinate. We resolve that contour into the contribution of an active pole and its background. For a lightly damped, isolated modal pole on a slowly varying, nearly real background, cusp-pair birth occurs once the participation of that pole at the nonlinear coordinate exceeds a closed-form threshold set by its damping half-width and that background. On the weak-mixing hardening branch generated by a pole above the primary resonance, the birth coupling then grows as the square root of the damping at fixed mode separation, and of the mode separation at fixed damping. The curvature of the full cusp contour determines whether the local fold set opens a gap (beaks type) or creates an isolated loop (lips type). The isolated-pole approximation reduces this classifier to a background-receptance rule whose range of validity is quantified on 100 random networks. The same ensemble tests the pole ladder numerically, while multi-harmonic continuation follows the whole reconnection sequence—connected band, gap, isolated lens, none—in the original equations at the fold of the birth locus, where the classifier is least protected. The construction also extends to coupled-mode Kerr networks, where the driving-point response sets the intrinsic fold geometry and the drive-port transfer response sets its accessibility.

We study the rational quadratic differential equation \[ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{ay^2+by+cx}{y^2}, \qquad a,b,c\in\C, \] under the non-degeneracy assumptions \[ c\neq 0,\qquad 2ay+b\not\equiv 0. \] Equivalently, after clearing the denominator, we consider the polynomial vector field \[ \dot{x}=y^2,\qquad \dot{y}=ay^2+by+cx. \] We give a complete, directly checkable classification of its Darboux curves. If $a=0$, no non-constant Darboux polynomial exists. If $a\neq 0$, a non-constant Darboux polynomial exists if and only if \[ c=-ab\qquad\text{or}\qquad c=-2ab. \] In these two cases the unique irreducible Darboux polynomials, up to non-zero constant multiples, are respectively \[ y-ax,\qquad y^2-2bx. \] Consequently every non-constant Darboux polynomial is a non-zero constant multiple of a positive integral power of the corresponding irreducible factor. We then place this classification in the framework of Riccati (R-)integrability and rational potentials. Excluding the exceptional branch $c=-2ab$, the analytic integrability theorem identifies $c=-ab$ as the branch admitting a global Riccati-type analytic first integral. For $c=-2ab$, we compute the first four transverse variational groups along the transformed Darboux divisor over the rational function field. Their dimensions are $1,2,3,4$; the fourth is the full group of invertible fourth-order transverse jets. We prove that this exceptional branch admits no Riccati first integral and hence is not R-integrable. Finally, we relate the branch $c=-ab$ to the Quartic Inverse Riccati (QIR) class and discuss an invariant-based classification problem for quartic Abel equations.

We survey a geometric approach to the Navier-Stokes equations developed by the author over the past fifteen years. The central object is a Riemannian space of fourteen dimensions whose metric is Ricci-flat precisely on the solutions of the Navier-Stokes system, so that the behaviour of a viscous incompressible fluid becomes a property of curvature. The space decomposes into a flat six-dimensional part and two dual quadruples of coordinates, Eulerian and Lagrangian, which exhibits the two classical descriptions of a fluid as two halves of a single geometry. We also recall the associated six-dimensional metric, whose integrability condition is the incompressibility of the fluid; the link with the projective geometry of second-order ordinary differential equations in the sense of E. Cartan; and the use of Cartan's invariants to construct a three-dimensional analogue of the Taylor-Green vortex. An application of the same construction to the equations of rotation of a rigid body is indicated.

We study the two-field nonlinear dispersive system \[ u_t=u_{xxx}+6uu_x,\qquad v_t=v_{xxx}+6(uv)_x, \] viewed simultaneously as a coupled wave equation, as the tangent covering of Korteweg–de Vries (KdV), and as a Hamiltonian flow on a tangent Poisson manifold. The main purpose is to make these viewpoints interact at theorem level. First, we prove that the Magri Poisson pencil of KdV admits a complete tangent lift to an explicit compatible pair of matrix Hamiltonian operators. The corresponding recursion operator has triangular tangent form and generates the lifted KdV hierarchy. Second, we identify a five-dimensional point-symmetry algebra together with the infinite hierarchy of tangent generalized symmetries. Third, reduction by the traveling-wave subgroup produces a four-dimensional Hamiltonian system that is the complete tangent lift of the scalar KdV profile dynamics. On the nonsingular elliptic locus, the base profile is written in Weierstrass form and every tangent traveling wave is classified explicitly by derivatives with respect to the energy and integration constants.

Fast-flowing glaciers, known as ice streams, control the majority of ice sheet mass loss due to thermomechanical coupling at their base, which can induce basal lubrication and highly variable flow. Even quite simple models can show `binge-purge' oscillations. Recent work has highlighted that coupling between divergent ice streams can also give rise to chaotic variability. In this paper, we return to a model of ice streams with a divergent coupling to identify and correct some physical inconsistencies, most notably a global volume conservation error. We propose and validate a revised Divergent Coupling (DC) model that ensures rigorous mass balance. Bifurcation analysis and Poincaré sections reveal previously unidentified complex frequency-locking (Arnold tongues), a period-adding cascade and chaos associated with torus breakdown. We also consider an equivalent model with a Convergent Coupling (CC), where we find no evidence of chaotic variability. These findings suggest the importance of specific topologies on ice stream variability.

Oscillators defined on curved spaces provide a natural framework for exploring geometry-induced nonlinear dynamics. The two-dimensional Mathews-Lakshmanan oscillator and Higgs oscillator are prominent examples of geometrically induced nonlinear oscillators. The two-dimensional Higgs oscillator in flat space arises from a conformal (stereographic) projection of the harmonic oscillator on the sphere S2 onto the two-dimensional plane. Its one-dimensional counterpart, defined by a non-Euclidean geometry with curvature parameter, exhibits chaotic dynamics and extreme events (EEs) when subjected to damping and external driving forces [1]. In this work, we introduce coupling through mass interaction among multiple one-dimensional damped and forced Higgs oscillators and investigate the resulting collective nonlinear dynamics within a many-particle framework. More importantly, we establish the connection between the conformal mass m(x) and the extrema of velocity through a power-law relation, demonstrating how the conformal mass regulates the turning points leading to the emergence of extreme events (EEs). Furthermore, to provide additional insight into the underlying mechanism, we quantify the energy synchronization error among the oscillators, thereby revealing how energy synchronization evolves during the onset of EEs.

We study the polynomial generalized Korteweg–de Vries family \[ u_t+P_m(u)u_x+\kappa u_{xxx}=0,\qquad \kappa\neq0, \] where $P_m$ is a real polynomial of degree $m\ge1$. The purpose is not to generate isolated closed-form waves, but to identify the structural threshold at which pole-type Painlevé behavior and elliptic traveling-wave geometry are simultaneously lost. A Weiss–Tabor–Carnevale dominant-balance calculation gives the universal principal exponent $p=-2/m$. Hence only $m=1$ and $m=2$ can possess a principal Laurent branch with integer pole order; their resonance sets are respectively $\{-1,4,6\}$ and $\{-1,3,4\}$. These two sectors are, after affine and Galilean transformations, the KdV and modified KdV equations, so their compatibility conditions and complete-integrability structures are inherited from the classical hierarchies. In contrast, every degree $m\ge3$ has a fractional leading exponent and therefore fails the strong WTC pole criterion at the first step. The results provide a concise bridge between singularity analysis, Hamiltonian form, algebraic-curve genus and exact nonlinear waves.

We propose a learning procedure for system identification in interacting particle systems from single-snapshot observations of collective behaviors, unlike existing approaches that rely on observations of trajectories. This setting leads to a fundamentally ill-posed inverse problem, which we solve by using a regularization strategy based on the empirical distribution of observed configurations, drawn from different, unobserved initial conditions. We test our learning procedure on a variety of representative models with steady-state and quasi-stationary patterns, where collective behaviors encode implicit information about the interaction mechanisms, demonstrating that our approach enables stable and accurate recovery of the underlying interaction laws, leading to faithful reproduction of the collective behavior, and in many cases even of the dynamics leading up to it.

Nonlinearities in multimode mechanical systems can give rise to rich dynamical phenomena with great potential for sensing applications and for future quantum experiments. We demonstrate that the coupled center-of-mass and rotational motion of a Meissner-levitated micromagnet offer a promising platform for nonlinear dynamics, combining low dissipation, magnetic tunability, strong intrinsic Duffing nonlinearities, and nonlinear intermodal couplings. We use this tunability to demonstrate the generation of a mechanical frequency comb in the micromagnet dynamics, realized by parametric excitation of two translational modes followed by cascaded nonlinear frequency mixing, which produces a phononic comb with tunable spacing. At large amplitudes, nonlinear coupling to a low-frequency librational mode of the magnet leads to parametric excitation and phase locking of that mode, generating a dense spectral fine structure at subharmonics of the drive. These results establish levitated micromagnets as a platform for nonlinear multimode mechanics, with potential applications in precision sensing and quantum-limited metrology.

Linear response is cheap to measure; the bistability boundaries it organizes are not. For a passive network with one localized cubic, the driving-point receptance $G$ fixes the period-one cusp set at fundamental-harmonic order: cusps lie on a fixed phase contour of $G$, a tangency of that contour under parameter variation creates a pair, and its curvature separates a gap opening from an isolated loop. For a two-mode absorber the linear prediction locates a benchmark birth coupling to $0.3\%$, and to $0.03\%$ once a third-harmonic correction of scale $|G(3\Omega)/G(\Omega)|$ is included.