Papers

Skyrmions experience a sideways gyroscopic force when they move. Bound molecules can cancel this force; we call them gyroscopically neutral. We show that the quantum states available on each site of a magnet select the composition needed for cancellation. In two models of $SU(3)$ magnetism, this composition changes from a pair to a trimer. Both models assign exactly the same energy to every configuration of their common classical order parameter, isolating the effect of the local quantum states. Neutral molecules can also unwind when additional local states are allowed. Mean field calculations nevertheless find both compositions bound and locally stable, with the neutral trimer surviving sampled interaction changes. Comparing their motion reveals the changed cancellation rule, and prepared neutral molecules translate over short distances with little deformation.

We study nonlinear holes, bubbles, kink fronts, and quantum-droplet-like states in an elongated Bose mixture governed by an extended Gross-Pitaevskii equation with attractive cubic mean-field interactions and repulsive quartic Lee-Huang-Yang contribution. We classify the constant-amplitude backgrounds: the Bogoliubov-de Gennes spectrum shows that the upper branch is modulationally stable, whereas the lower branch is unstable to long-wavelength perturbations. Grand-potential and pressure determine the vacuum-finite-density coexistence point and the unique background supporting stationary kink and antikink fronts. Symmetric holes on the stable background have a density deficit and integral width that diverge near coexistence as the hole separates into two nonlinear fronts. A multiplicative kink-antikink construction connects bell-shaped states to broad flat-top droplets and approaches the stationary quantum-droplet family as the front separation increases. An antikink-kink sum with equal signs generates bubble-like density depletions without a phase jump, whereas their difference produces dark-hole-like profiles with a $\pi$ phase difference between the asymptotic backgrounds. Bubble depth increases with front separation, distinguishing gray-like and dark-like profiles. Real-time simulations show that deep bubbles persist under weak random perturbations, demonstrating finite-time robustness without establishing spectral stability. In the asymmetric two-component model, localized relative-density modes can induce spinor-driven breakup on a modulationally stable background, while robust holes persist near the kink limit. Collisions of kink-antikink droplet-like states display phase-dependent merging, asymmetric particle transfer, and effective reflection. These results connect nonlinear fronts with bubbles, phase-jump holes, and self-bound droplets in beyond-mean-field Bose mixtures.

We study pulse-number selection in a stochastic cubic–quintic complex Ginzburg–Landau model of a weakly seeded, normal-dispersion laser in the dissipative-soliton-resonance (DSR) regime. Without noise, a single pulse and pulse pairs persist at the same control parameters, and the total energy of an $N$-pulse state follows a ladder constructed from the single-pulse branch. The final energies of noisy trajectories lie close to the same ladder. Multipulsing therefore does not necessarily lose the single-pulse solution. It can reflect which coexisting state the noisy dynamics reaches. Optical seed injection suppresses energy-dependent multipulsing and, at larger seed power, produces a nonmonotonic DSR energy window. An energy–noise scan shows that energy dominates the multipulse probability, whereas additive noise produces only a modest trend common to all energies, without a noise optimum. The noise-induced formation statistics therefore do not establish canonical stochastic resonance or escape from a pre-existing soliton. Coherent control by a weak monochromatic seed extends predominantly single-pulse operation over a noticeably broader energy window, enhancing dissipative-soliton energy scalability. Within an adiabatic approximation, equal energy sharing among coexisting pulses is stable wherever the single-pulse energy grows less than proportionally with the control energy, as it does over the sampled DSR range. Energy exchange between the pulses then relaxes up to about 200 times more slowly than their total energy.

Influenza A persists by continually changing the antigens that its host population recognizes. How fast it evolves and how many hosts it infects are not independent, because the immunity that selects new variants is produced by the infections the pathogen has already caused. We analyze a model of Andreasen et.\ al.\ that captures this eco-evolutionary feedback, and we extend it to the case where infection may kill the host. In the model infected and recovered hosts are distributions over a one-dimensional antigenic space coupled by cross-immunity. The endemic state is a traveling pulse: the infected distribution keeps a fixed shape and advances through antigenic space at constant speed. In the limit of weak antigenic mutation we construct the pulse asymptotically and find that: (i) the speed of antigenic evolution decreases as prevalence rises, (ii) less standing antigenic variation accompanies either faster or slower antigenic evolution, depending on whether pathogens differ in immune evasion or host lifespan (iii) the total number of infected hosts is largest at an intermediate virulence when the host birth rate is sufficiently low.

Precise conditions are provided for the existence and criticality of Turing bifurcations in a general class of activator-inhibitor reaction-diffusion equations on a one-dimensional infinite domain. The class includes generalised Schnakenberg and Brusselator models, as well as other models with cubic autocatalytic nonlinear terms. Previous numerical work suggests the existence of a bifurcation structure containing localised patterns due to the so-called homoclinic snaking mechanism. This paper provides explicit calculations to justify those results. Two distinct scalings of parameters that lead to tractable normal-form coefficients are considered in the limit that the diffusion ratio $\delta \to 0$. First, under a small-parameter scaling, the Turing bifurcation is shown to be always subcritical. Second, a large-parameter scaling reveals the Turing bifurcation to be supercritical, leading, by continuity, to the existence of a codimension-two degenerate bifurcation. The sign of a 5th-order normal form coefficient is also computed, which is shown to have the correct sign for the local birth of homoclinic snaking. For the case of the Brusselator, two such codimension-two points can be found explicitly, as can the leading-order expression for the Maxwell point, in a parameter wedge about which localised patterns emerge. Numerical results are found to be consistent with the theory.

This data article provides the datasets, symbolic derivations, and scripts used to reproduce master diagrams, stationary-phase spectra, windowed first-order coherence functions, and quantum-noise stability maps for strongly chirped dissipative solitons of the cubic-quintic complex Ginzburg-Landau equation in normal and anomalous group-delay dispersion regimes. The repository includes node-regularized normal-dispersion spectra and energies; small-parameter expansions of the branch roots; cavity-map gain-loss update relations; Airy uniformization at the normal-dispersion spectral edge; anomalous-dispersion spectra and coherence calculations; and processed tables for plotting and stability analysis. OriginLab projects are accompanied by open-format .csv/.txt numerical tables to support reuse without proprietary plotting software. Data and code repository: https://doi.org/10.5281/zenodo.22690899.

We obtain a large number of exact hyperbolic, trigonometric, and periodic solutions in terms of Jacobi elliptic functions as well as algebraic solutions with a power law tail of the integrable local Fokas-Lennels equation and integrable nonlocal Fokas-Lennels equation. Further, we consider a one-parameter family of generalized Fokas-Lenells equations and obtain a few of their exact solutions.

General expressions are derived for the amplitude equation valid at a Turing bifurcation of a system of reaction-diffusion equations in one spatial dimension, with an arbitrary number of components. The normal form is computed up to fifth order, which enables the detection and analysis of codimension-two points where the criticality of the bifurcation changes. The expressions are implemented within a Python package, in which the user needs to specify only expressions for the reaction kinetics and the values of diffusion constants. The code is augmented with a Mathematica routine to compute curves of Turing bifurcations in a parameter plane and automatically detect codimension-two points. The software is illustrated with examples that show the versatility of the method including a case with cross-diffusion, a higher-order scalar equation, and a four-component system.

A wave bifurcation is the counterpart to a Turing instability in reaction-diffusion systems, but where the critical wavenumber corresponds to a pure imaginary pair rather than a zero temporal eigenvalue. Such bifurcations require at least three components and give rise to patterns that are periodic in both space and time. Depending on boundary conditions, these patterns can comprise either rotating or standing waves. Restricting to systems in one spatial dimension, complete formulae are derived for the evaluation of the coefficients of the weakly nonlinear normal form of the bifurcation up to order five, including those that determine the criticality of both rotating and standing waves. The formulae apply to arbitrary $n$-component systems ($n\geq 3$) and their evaluation is implemented in software which is made available as supplementary material. The theory is illustrated on two different versions of three-component reaction-diffusion models of excitable media that were previously shown to feature super- and subcritical wave instabilities and on a five-component model of two-layer chemical reaction. In each case, two-parameter bifurcation diagrams are produced to illustrate the connection between complex dispersion relations and different types of Hopf, Turing, and wave bifurcations, including the existence of several codimension-two bifurcations.

Necessary and sufficient conditions are provided for a diffusion-driven instability of a stable equilibrium of a reaction-diffusion system with $n$ components and a diagonal diffusion matrix. These can be either Turing or wave instabilities. Known necessary and sufficient conditions are reproduced for there to exist diffusion rates that cause a Turing bifurcation of a stable homogeneous state in the absence of diffusion. The method of proof here though, which is based on a study of dispersion relations in the contrasting limits in which the wavenumber tends to zero and to $\infty$, gives a constructive method for choosing diffusion constants. The results are illustrated on a model for the dispersion of malaria, a 3-component FitzHugh-Nagumo-like model proposed to study excitable wavetrains, and for two different coupled Brusselator systems with 4 components

We study the spectral stability of bright and dark solitons in the discrete nonlinear Schrödinger (DNLS) equation using multiprecision arithmetic. The eigenvalues governing stability are exponentially small in the lattice spacing and cannot be resolved with standard double precision. To address this, we develop a computational framework combining multiprecision arithmetic, an exact Jacobian for the stationary problem, and a squared-operator formulation for spectral analysis. This enables accurate resolution of exponentially small eigenvalues and direct comparison with exponential-asymptotic predictions. Our results show that onsite bright solitons are spectrally stable, whereas intersite bright solitons and both onsite and intersite dark solitons are unstable. Bright solitons require only a few eigenvalues and allow efficient large-scale computations, while dark solitons demand higher precision due to their proximity to the continuous spectrum. Simulations up to \(N=65{,}250\) grid points (31.7 GB RAM) highlight the necessity of multiprecision arithmetic for capturing beyond-all-orders spectral effects.

Self-organization of nonlinear waves underpins phenomena ranging from hydrodynamic pattern formation to optical frequency-comb generation. We introduce and numerically demonstrate a shift-register synchronization mechanism for pulse trains in quadratic nonlinear resonators. The synchronized state persists despite substantial group-velocity mismatch through continuous parametric energy exchange, resulting in deterministic discrete pulse shifts after each cavity round trip. Nonlinear parametric interaction modifies the effective walk-off, allowing the pulse trains to satisfy the shift-register synchronization conditions. The proposed mechanism reveals a previously unexplored synchronization regime in quadratic resonators and provides a route toward multicolour frequency combs, optical memories, and ultrafast photonic information processing.

We systematically investigate breather solutions in the modified Korteweg-de Vries (mKdV) equation via numerical simulations. We show that the breather solutions are stable under a variety of perturbations, including amplitude changes, modifications of internal parameters, and perturbations of the nonlinearity of the equation. Our results are not only consistent with the known analytical studies on the stability of breathers in integrable systems, extending them further, but also provide numerical evidence that such breather-type structures can persist and remain stable over the simulated time scales in some non-integrable settings with symmetric potentials.

Mass-conserving reaction-diffusion systems with two species correspond to a seemingly simple case where pattern formation occurs under the influence of a conservation law. Here, we first revisit their linear stability behavior and point out that generically two instabilities can occur: a stationary large-scale mass-conserving (Cahn-Hilliard) instability and an instability that combines features of a stationary large-scale non-mass-conserving (Allen-Cahn) instability and a oscillatory large-scale mass-conserving (conserved-Hopf) instability. We term it an Allen-Cahn-Hopf instability. Second, we investigate the nonlinear dynamics for a specific model related to the formation of cell polarization where only a Cahn-Hilliard instability can occur, i.e., all primary bifurcations are stationary. We analyze how secondary and further bifurcations subsequently give rise to various oscillatory states. The emerging rich spectrum of spatiotemporal behavior includes several period-doubling cascades related to different forms of spatial and temporal symmetry breaking. Beside regular states, three types of low-dimensional spatiotemporal chaos occur and involve transitions like fusion and an outer crises. Our results demonstrate the importance of nonlinear interactions in the dynamics of mass-conserving reaction-diffusion systems, and show that even a simple two-species system with primary bifurcations of Cahn-Hilliard type can show complex spatiotemporal behavior.

Probably not. For Airy-type initial data, although nearly all lobes accelerate, the principal lobe remains essentially stationary. Moreover, the overall wave packet gradually disperses.

A composite soliton can lose its binding through a change in the ordering of the vacua that fill its interior and its exterior, while the quantum correction to its mass stays small. We show this in the non-degenerate double sine-Gordon model, normal ordered at the meson mass of the primary vacuum. At one loop the central secondary vacuum becomes the true vacuum at a coupling of order the classical splitting of the vacua, because the quantum tension of a string of secondary vacuum carries the logarithm of the meson-mass ratio. Matrix-product-state calculations for $n=4$ confirm the inversion, extrapolate to the continuum, and are reproduced by the Gaussian effective potential. Finite-chain calculations at two lattice spacings show the string joining the two halves of the $Q=4$ multi-kink lengthening as its tension falls and, past the crossing, filling the box, with the slope of the energy in the box length equal to the vacuum-energy difference and the correction to the mass still under a tenth of the classical value.

We report on frequency bursting oscillations in a system of phase oscillators with adaptive and delayed coupling. Adaptation of the coupling strengths is considered slow and depends on the phase shift between the oscillators. We find due to the combined chain of adaptation, collective dynamics, and time delays, the system robustly achieves a state in which the oscillator's frequencies are nearly synchronized but detuned by an integer number of small adaptation frequencies. We demonstrate that this quantization of the detuning is caused by alternating slow and fast transitions. Moreover, the observed motions take the form of bursts of instantaneous frequency, and the number of spikes in each burst corresponds to the quantization level of the detuning. We provide a fast-slow analysis of this phenomenon and explain the mechanisms behind the emergence of bursts. Our findings indicate that these frequency bursting oscillations are robust and exist stably within finite parameter regions.

A two-component Bose–Einstein condensate can host a pseudospin Hopf texture coupled to the viscous normal flow of its thermal cloud. Motivated by OpenAI's reported finite-time Navier–Stokes singular solution, we evolve such a texture in a smooth flow reproducing the geometry and similarity scaling of the collapsing axisymmetric core. The evolution creates and annihilates preimage pairs while preserving the Hopf charge. The first one-to-three transition exhibits the square-root opening characteristic of a Thom $A_2$ fold. A subsequent three-to-five transition precedes a burst-like regime of recurrent fold catastrophes and near-caustic activity.

We study rolls and homoclinic snaking in a variation of the one-dimensional Swift-Hohenberg equation, whose standard forms are prototypical order-parameter models for pattern formation in the sciences. Motivated by classes of differential equation models that involve continuous non-smooth low order nonlinear terms, we replace the standard quadratic-cubic nonlinearity by $\nu |u|^\alpha-u^3$, $\alpha \in [1,2]$ with $\nu > 0$. In the vicinity of zero, for $\alpha<2$ this nonlinearity falls outside the scope of classical Taylor expansion and bifurcation analysis. Our partially analytical and partially numerical results highlight that the non-smooth term modifies the criticality of pattern-forming bifurcations and alters the associated branches of solutions. In particular, $\alpha\in(1,2)$ implies subcriticality of roll bifurcations for any $\nu>0$. At $\alpha=1$ differentiability is lost, which has a strong impact on the bifurcations of sign-changing rolls including the disappearance of homoclinic snaking. Homoclinic snaking thus emerges non-smoothly as $\alpha$ increases from $\alpha=1$, and persists when retaining an additional, e.g., quadratic term.

At an $O(2)$-equivariant Hopf bifurcation, the real parts $\zeta_R$ and $\xi_R$ of the two cubic normal-form coefficients govern selection between traveling and standing waves. In a fourth-order regularized $p$-system, $\xi_R$ was observed to vanish and the cubic nonlinearity to shift only frequencies. We show that both observations are structural: variational nonlinearities enforce them even when the linear part is dissipative and non-Hamiltonian. We consider a class of two-component PDEs on the circle whose quadratic and cubic nonlinearities are generated by reflection-invariant local Hamiltonians of the fields and finitely many of their spatial derivatives, through a common constant-coefficient Poisson operator. Then $\xi_R=0$ for every admissible linear part, while $\zeta_R$ depends only on the quadratic nonlinearity. For variational nonlinearities the direct cubic terms are purely imaginary. The mixed cascade term loses its real part because its second harmonic has zero temporal frequency and the resolvent keeps its parity, while the self-cascade at the doubled critical frequency keeps a real part. When $\zeta_R\ne0$, the bifurcating traveling waves are saddles, while the standing waves are supercritical and orbitally asymptotically stable for $\zeta_R<0$ and subcritical and unstable for $\zeta_R>0$. When the nonlinearities are the Poisson operator applied to polynomials in the fields alone, and its symbol does not vanish at the critical and second-harmonic wavenumbers, cancellation for every admissible linear part conversely characterizes variationality. In a dissipatively regularized Boussinesq family, as dissipation tends to zero, $\zeta_R$ vanishes linearly off resonance and diverges inversely at the 2:1 resonance. The Hamiltonian limit of the cubic coefficient is therefore singular.

Optical frequency combs (OFCs) are revolutionary light sources characterized by discrete and equally spaced spectral lines, and they have found widespread applications in metrology, spectroscopy, and communications. In the early stages, OFCs were realized using mode-locked lasers. With advancements in the fabrication of high-quality factor ($Q$) microresonators and the increasing demand for miniaturized and integrable photonic chips, microresonator-based OFCs, commonly referred to as microcombs, have been developed. Although early studies of microcombs primarily focused on single microresonators or a few resonators, a significant breakthrough occurred in 2021 when it was theoretically predicted that light propagating in the topological edge channel of an array of ring resonators could generate nested frequency combs known as topological OFCs. Since then, the field of topological OFCs has progressed rapidly, with experimental observations made in 2024. This Perspective will introduce the history of OFCs, placing particular emphasis on the emergence and development of topological OFCs, as well as exploring the research challenges and opportunities associated with them.

Dendritic silhouettes are cheap. Pond ice neurons, diffusion-limited aggregates, wiring-minimizing arbors, and biological neurons can share a branching statistic because they share a growth instability, not because they are functionally similar. We take that observation as a modeling constraint. The ice neuron only structurally resembles a biological neuron. A biological neuron carries an internal state and a field that it helps generate, letting geometry, dynamics, and field co-sculpt one another. In our construction, the ice neuron is the zero-gain limit of an interfacial system in which a FitzHugh–Nagumo state lives on the moving front and writes back into the field that drives it, with dimensionless gain $\gamma$. At $\gamma=0$, morphology and internal dynamics decouple into a Mullins–Sekerka growth band and a damped oscillator. Finite $\gamma$ deforms the spectrum, shifts the selected instability, and opens an oscillatory region below the isolated Hopf threshold. On a dendrite grown at $\gamma=0$ and then frozen, field closure increases distal persistence without changing the geometry; its first-order effect remains positive across repeated realizations of the same growth rule and localizes to a small set of non-cable field shortcuts. Nonlinear dynamics remain finite beyond the linear instability: below threshold, closure is almost the transmission; at spike amplitude, it is a correction to a regenerative cable event. A dendritic tree therefore is not intrinsically a structural spandrel. Shape alone is morphology without an internal state or a source term. That is the dendritic silhouette of the ice neuron. Excitability adds an active cable. Field closure is the biophysical mélange in which the neuron helps write the field that acts back on it, sculpting its dynamics and, during growth, its structure. That is a biological neuron.

Phase reduction is a powerful technique to obtain phase models from highly dimensional oscillatory systems. Starting from pairwise interactions, a first order approximation yields Kuramoto and Winfree phase models, while nonpairwise interactions emerge at the second order. Recently, phase models have been extended to account for nonpairwise interactions, notably, the higher-order Kuramoto model. Is there an intrinsic difference between physical and emergent nonpairwise interactions? And can we make use of the former to even out the effects of the latter? In this work, we exploit a recently developed parametrization method to compute the phase reduction and answer these questions. After a comparison of the new method with the classic Kuramoto-style phase reduction, we solve the network motifs of the emergent nonpairwise interactions and compare them with the physical once. Lastly, we exploit the motifs to adopt a coupling design approach. Our framework paves the way for further exploitations of physical nonpairwise interactions for applications in synchronization engineering.

In this article, we study the dipolar Bosonic quantum fluid. The fluid experiences mean-field, beyond mean-field, and three body interactions. We investigate their competition with dipolar interaction and fragmentation as a result of this competition. We further investigate the elementary excitations and note two distinct dispersion regimes, namely roton-mode and modulational instability. We support our observation by calculating the superfluid fraction and the condensate fraction.

The controlled dynamical generation of radial dam break flows (DBFs) is demonstrated in two-dimensional droplet environments. These ultracold mixtures feature competing mean-field and quantum fluctuation effects and are initialized on a disk-shaped uniform density. Depending on the value of the latter, reflecting also the dominance of attractive or repulsive interactions, three dynamical response regimes are identified. At small densities a moving wavetrain of bright ring solitons is observed (dubbed ring DBF), at intermediate densities a ring kink-dispersive shock wave, while at large densities the flows encompass composite ring kink-rarefaction waves moving towards the center. The emergence of these patterns, whose formation would be prohibited in two-dimensional Kerr media due to wave-collapse, is rooted in the competition between attraction and repulsion inherent to droplet environments, modeled by the extended Gross-Pitaevskii equation. The characterization of the ensuing nonlinear waveforms is further corroborated by reduced models based on the cubic-quintic nonlinear Schrödinger equation. The different wave patterns emanating from dam break flows can be experimentally realized using current state-of-the-art ultracold atom experiments.

Buckling of thin cylindrical shells under axial compression, a classical example of a subcritical instability, is highly sensitive to small imperfections, with minute geometric variations causing large changes in buckling threshold. To uncover the origin of this sensitivity, we use numerical continuation and bifurcation analysis while systematically varying the depth and size of a single localized Gaussian defect. We show that the instabilities of the imperfect shell originate from localized equilibria already present in the perfect shell. By breaking translation symmetry, the defect pins these equilibria and changes how they connect to the imperfect base state. Small changes in defect geometry can thereby switch the bifurcation that triggers buckling, producing non-monotonic and discontinuous changes in buckling threshold and abrupt changes in buckling mode. Imperfection sensitivity is therefore not simply sensitivity to imperfection magnitude, but sensitivity of the underlying bifurcation structure to imperfection geometry.

Phase-field trajectories governed by the Allen–Cahn equation determine the effective force $G=F'/\eps^2$, but not the free-energy potential $F$ and interfacial scale $\eps$ separately. We introduce surface-tension-calibrated Allen–Cahn identification (STAC), a two-step method that resolves this ambiguity using an independently supplied surface tension. First, space–time weak moments produce a linear inverse problem without differentiating the measured field. A Bernstein representation with positive coefficients enforces two stable pure phases and a single central energy maximum. Second, the surface tension fixes the remaining scale algebraically. We prove scale equivalence of the forward model, uniqueness of the calibrated pair within this class, and finite perturbation bounds that remain valid when the potential vanishes at the pure phases. Experiments with resolved synthetic phase fields separate the effects of the structural constraint, ridge regularization, and constitutive approximation error. Across 80 shared noisy records, unconstrained degree-five regression produces six potentials with an incorrect well structure. Positivity combined with ridge regularization eliminates these violations and reduces the mean force error from 8.70% to 5.66%. Lower-degree models are nevertheless more accurate in some tests, and an admissible law outside the positive Bernstein family shows how the constraint can introduce approximation bias. STAC thus recovers a structurally admissible, physically scaled potential while making explicit which information comes from the observed dynamics and which comes from the independent surface-tension datum.

We obtain exact solutions of the nonlinear Dirac equation in $1+1$ dimensions of the form $\psi(x,t) = e^{-i\omega t} \psi(x)$ for scalar-scalar (SS) plus vector-vector (VV) interaction with interaction Lagrangian given by $$L_{I} = \frac{g^2}{\kappa+1}[(\bar{\psi} \psi)^{\kappa+1} +\frac{1}{p} (\bar{\psi} \gamma_\mu \psi \bar{\psi} \gamma^{\mu} \psi)^{\kappa+1}]$$ where $p > 0$ but arbitrary otherwise. We look for solutions with $0 < \omega < m$ where $\omega, m$ are frequency and mass, respectively. We find solutions for all values of $\omega$ in this range. We compute the charge $Q$ and the energy $E$ for each of the solitary wave solutions and explore the region in the ($p, \kappa$) parameter space in which solitary wave bound states exist (i.e., for which $E/Q < m$). We show that for all the cases while both $E$ and $Q$ depend on the coupling constant $g$, their ratio $E/Q$ is independent of $g$. We further find that in case $p\kappa \le 1$, the charge density for all the solitary waves have only single hump while for $p \kappa > 1$ there is a transition from double to single hump and we determine it as a function of $\omega/m$. We notice that for all $p$ there is a transition at $\kappa=2$ in the behavior of $E/Q$ as a function of $\omega$ which we speculate is related to the onset of instability of the solutions at $\kappa=2$. We obtain the nonrelativistic reduction of the two-parameter family to a non-relativistic modified nonlinear Schrödinger equation (NLSE) and discuss the stability of the single hump olitary waves in the domain of validity of the modified NLSE.

We address three-dimensional (3D) solitons maintained by spin-orbit coupling (SOC) in the binary self-interacting Bose-Einstein condensate (BEC) held in the 3D optical lattice (OL). The analysis reveals that the SOC-OL interplay results in the formation of stable full-vortex (FV) soli-tons, built as sets of four density peaks residing in neighboring wells of the lattice potential, with the superimposed global vortical phase, and site-centered semi-vortices (SV), in which the vorticity is present in only one component. The full-vortex solitons, with their specific phase textures, do not exist in a uniform BEC with SOC. Full-vortex and semi-vortex states in the binary self-attractive BEC are stable despite the possibility of the supercritical collapse in the 3D system. Such states also exist, as gap solitons, in the self-repulsive 3D system. In terms of the chemical potential and number of particles, the stability regions of the 3D full-vortex solitons and semi-vortices expand with the increase of the SOC strength and OL depth. The results open the route to the creation of 3D complexes of vorticity-carrying condensates that can be realized with existing experimental techniques.

Recent experiments demonstrating superconducting-like characteristics in materials above their equilibrium transition temperatures following strong sub-picosecond optical excitation have stimulated theoretical interest in phase-disordered superconductors subject to a periodically modulated superfluid stiffness. Previous analyses of the single-layer optical conductivity have shown that such a drive can enhance the low-frequency $1/\omega$ response, even when the period-averaged stiffness remains unchanged. We extend these studies to bilayer systems, such as the YBa$_2$Cu$_3$O$_{6+x}$ (YBCO) cuprate, by considering drives with different symmetries. Specifically, we investigate a layer-symmetric drive, in which the two planes stiffen and soften simultaneously, and a layer-antisymmetric drive, in which one plane stiffens while the other softens. Both drives enhance correlations of the layer-averaged phase and generate comparable superconducting-like optical conductivities. Only the symmetric drive, however, strengthens relative-phase locking and the associated counterflow response to a magnetic field parallel to the planes. Thus, an enhanced $1/\omega$ optical conductivity does not by itself imply enhanced magnetic screening. In view of applications to driven YBCO, where superconducting-like in-plane optics and transient diamagnetism have both been reported, our result provides a symmetry diagnostic: within a phase-synchronization mechanism, the experimentally active low-frequency stiffness modulation must contain a substantial layer-symmetric component.