CyberSec Research

We propose a framework for topological soliton dynamics in trapped spinor superfluids, decomposing the force acting on the soliton by the surrounding fluid into the buoyancy force and spin-corrections arising from the density depletion at soliton core and the coupling between the orbital motion and the spin mixing, respectively. For ferrodark solitons (FDSs) in spin-1 Bose-Einstein Condensates (BECs), the spin correction enables mapping the FDS motion in a harmonic trap to the atomic-mass particle dynamics in an emergent quartic potential. Initially placing a type-I FDS near the trap center, a single-sided oscillation happens, which maps to the particle moving around a local minimum of the emergent double-well potential. As the initial distance of a type-II FDS from the trap center increases, the motion exhibits three regimes: trap-centered harmonic and anharmonic, followed by single-sided oscillations. Correspondingly the emergent quartic potential undergoes symmetry breaking, transitioning from a single minimum to a double-well shape, where particle motion shifts from oscillating around the single minimum to crossing between two minima via the local maximum, then the motion around one of the two minima. In a hard-wall trap with linear potential, the FDS motion maps to a harmonic oscillator.

Enhancement of the predictive power and robustness of nonlinear population dynamics models allows ecologists to make more reliable forecasts about species' long term survival. However, the limited availability of detailed ecological data, especially for complex ecological interactions creates uncertainty in model predictions, often requiring adjustments to the mathematical formulation of these interactions. Modifying the mathematical representation of components responsible for complex behaviors, such as predation, can further contribute to this uncertainty, a phenomenon known as structural sensitivity. Structural sensitivity has been explored primarily in non-spatial systems governed by ordinary differential equations (ODEs), and in a limited number of simple, spatially extended systems modeled by nonhomogeneous parabolic partial differential equations (PDEs), where self-diffusion alone cannot produce spatial patterns. In this study, we broaden the scope of structural sensitivity analysis to include spatio-temporal ecological systems in which spatial patterns can emerge due to diffusive instability. Through a combination of analytical techniques and supporting numerical simulations, we show that pattern formation can be highly sensitive to how the system and its associated ecological interactions are mathematically parameterized. In fact, some patterns observed in one version of the model may completely disappear in another with a different parameterization, even though the underlying properties remain unchanged.

We look for traveling waves of the semi-discrete conservation law $4\dot u_j +u_{j+1}^2-u_{j-1}^2 = 0$, using variational principles related to concepts of ``hidden convexity'' appearing in recent studies of various PDE (partial differential equations). We analyze and numerically compute with two variational formulations related to dual convex optimization problems constrained by either the differential-difference equation (DDE) or nonlinear integral equation (NIE) that wave profiles should satisfy. We prove existence theorems conditional on the existence of extrema that satisfy a strict convexity criterion, and numerically exhibit a variety of localized, periodic and non-periodic wave phenomena.

We reconsider the dynamics of localized states in the deterministic and stochastic discrete nonlinear Schr\"odinger equation. Localized initial conditions disperse if the strength of the nonlinear part drops below a threshold. Localized states are unstable in a noisy environment. As expected, an infinite temperature state emerges when multiplicative noise is applied, while additive noise yields unbounded dynamics since conservation of normalization is violated.

Topological edge states typically arise at the boundaries of topologically nontrivial structures or at interfaces between regions with differing topological invariants. When topological systems are extended into the nonlinear regime, linear topological edge states bifurcate into nonlinear counterparts, and topological gap solitons emerge in the bulk of the structures. Despite extensive studies of these two types of nonlinear states, self-induced topological edge states localized at the physical boundaries of originally nontopological structures remain underexplored. Unlike the previously reported self-induced topological transitions driven by nonlinear couplings, which are conceptually straightforward but less common in realistic interacting systems, here we experimentally realize self-induced topological edge states in a lattice with onsite nonlinearity. Leveraging the strong and tunable nonlinearity of electrical circuits, we systematically investigate the localized states in a nonlinear Su-Schrieffer-Heeger model. Besides revisiting the nonlinear topological edge states and topological gap solitons, we uncover a novel type of self-induced topological edge states which exhibit the hallmark features of linear topological edge states, including sublattice polarization, phase jumps, and decaying tails that approach zero. A distinctive feature of these states is the boundary-induced power threshold for existence. Our results are broadly applicable and can be readily extended to photonic and cold atomic systems, where onsite nonlinearities naturally arise from interparticle interactions. Our work unveils new opportunities for exploring novel correlated topological states of light and matter, and paves the way for the development of robust photonic devices and topological quantum computation.

{The problem of laser beam concentration in a focal spot via wavefront variations is formulated as a maximization of the $beam$ $propagation$ $functional$ defined as the light power passing through aperture of an arbitrary shape located in the far field. Variational principle provides the necessary and sufficient conditions for at least the $local$ $maximum$ of the $beam$ $propagation$ $functional$. The wavefront shape is obtained as exact solution of nonlinear integral equation.}

We report the experimental observation of discrete bright matter-wave solitons with attractive interaction in an optical lattice. Using an accordion lattice with adjustable spacing, we prepare a Bose-Einstein condensate of cesium atoms across a defined number of lattice sites. By quenching the interaction strength and the trapping potential, we generate both single-site and multi-site solitons. Our results reveal the existence and characteristics of these solitons across a range of lattice depths and spacings. We identify stable regions of the solitons, based on interaction strength and lattice properties, and compare these findings with theoretical predictions. Our results provide insights into the quench dynamics and collapse mechanisms, paving the way for further studies on transport and dynamical properties of matter-wave solitons in lattices.

In the present work we examine multi-hump solutions of the nonlinear Schr{\"o}dinger equation in the blowup regime of the one-dimensional model with power law nonlinearity, bearing a suitable exponent of $\sigma>2$. We find that families of such solutions exist for arbitrary pulse numbers, with all of them bifurcating from the critical case of $\sigma=2$. Remarkably, all of them involve ``bifurcations from infinity'', i.e., the pulses come inward from an infinite distance as the exponent $\sigma$ increases past the critical point. The position of the pulses is quantified and the stability of the waveforms is also systematically examined in the so-called ``co-exploding frame''. Both the equilibrium distance between the pulse peaks and the point spectrum eigenvalues associated with the multi-hump configurations are obtained as a function of the blowup rate $G$ theoretically, and these findings are supported by detailed numerical computations. Finally, some prototypical dynamical scenarios are explored, and an outlook towards such multi-hump solutions in higher dimensions is provided.

Nonlinear stage of higher-order modulation instability (MI) phenomena in the frame of multicomponent nonlinear Schr\"odinger equations (NLSEs) are studied analytically and numerically. Our analysis shows that the $N$-component NLSEs can reduce to $N-m+1$ components, when $m(\leq N)$ wavenumbers of the plane wave are equal. As an example, we study systematically the case of three-component NLSEs which cannot reduce to the one- or two-component NLSEs. We demonstrate in both focusing and defocusing regimes, the excitation and existence diagram of a class of nondegenerate Akhmediev breathers formed by nonlinear superposition between several fundamental breathers with the same unstable frequency but corresponding to different eigenvalues. The role of such excitation in higher-order MI is revealed by considering the nonlinear evolution starting with a pair of unstable frequency sidebands. It is shown that the spectrum evolution expands over several higher harmonics and contains several spectral expansion-contraction cycles. In particular, abnormal unstable frequency jumping over the stable gaps between the instability bands are observed in both defocusing and focusing regimes. We outline the initial excitation diagram of abnormal frequency jumping in the frequency-wavenumber plane. We confirm the numerical results by exact solutions of multi-Akhmediev breathers of the multi-component NLSEs.

We present a framework for controlling the collective phase of a system of coupled oscillators described by the Kuramoto model under the influence of a periodic external input by combining the methods of dynamical reduction and optimal control. We employ the Ott-Antonsen ansatz and phase-amplitude reduction theory to derive a pair of one-dimensional equations for the collective phase and amplitude of mutually synchronized oscillators. We then use optimal control theory to derive the optimal input for controlling the collective phase based on the phase equation and evaluate the effect of the control input on the degree of mutual synchrony using the amplitude equation. We set up an optimal control problem for the system to quickly resynchronize with the periodic input after a sudden phase shift in the periodic input, a situation similar to jet lag, and demonstrate the validity of the framework through numerical simulations.

We study the propagation of narrow solitons through various profiles of dispersive shock waves (DSW) for the generalized Korteweg-de Vries equation. We consider situations in which the soliton passes through the DSW region quickly enough and does not get trapped in it. The idea is to consider the motion of such solitons through DSW as motion along some smooth effective profile. In the case of KdV and modified KdV, based on the law of conservation of momentum and the equation of motion, this idea is proven rigorously; for other cases of generalized KdV, we take this as a natural generalization. In specific cases of self-similar decays for KdV and modified KdV, a method for selecting an effective field is demonstrated. For the case of generalized KdV, a hypothesis is proposed for selecting an effective field for any wave pulse that is not very large compared to the soliton. All proposed suggestions are numerically tested and demonstrate a high accuracy of reliability.

Motile eukaryotic cells display distinct modes of migration that often occur within the same cell type. It remains unclear, however, whether transitions between the migratory modes require changes in external conditions, or whether the different modes are coexisting states that emerge from the underlying signaling network. Using a mass-conserved reaction-diffusion model of small GTPase signaling with F-actin mediated feedback, we uncover a bistable regime where a polarized mode of migration coexists with spatiotemporal oscillations. Indeed, experimental observations of D. discoideum show that, upon collision with a rigid boundary, cells may switch from polarized to disordered motion.

We investigate existence, stability, and instability of anchored rotating spiral waves in a model for geometric curve evolution. We find existence in a parameter regime limiting on a purely eikonal curve evolution. We study stability and instability both theoretically in this limiting regime and numerically, finding both oscillatory, at first convective instability, and saddle-node bifurcations. Our results in particular shed light onto instability of spiral waves in reaction-diffusion systems caused by an instability of wave trains against transverse modulations.

We are concerned with the low regularity of self-similar solutions of two-dimensional Riemann problems for the isentropic Euler system. We establish a general framework for the analysis of the local regularity of such solutions for a class of two-dimensional Riemann problems for the isentropic Euler system, which includes the regular shock reflection problem, the Prandtl reflection problem, the Lighthill diffraction problem, and the four-shock Riemann problem. We prove that it is not possible that both the density and the velocity are in $H^1$ in the subsonic domain for the self-similar solutions of these problems in general. This indicates that the self-similar solutions of the Riemann problems with shocks for the isentropic Euler system are of much more complicated structure than those for the Euler system for potential flow; in particular, the density and the velocity are not necessarily continuous in the subsonic domain. The proof is based on a regularization of the isentropic Euler system to derive the transport equation for the vorticity, a renormalization argument extended to the case of domains with boundary, and DiPerna-Lions-type commutator estimates.

We examine the effect of fractionality on the bloch oscillations (BO) of a 1D tight-binding lattice when the discrete Laplacian is replaced by its fractional form. We obtain the eigenmodes and the dynamic propagation of an initially localized excitation in closed form as a function of the fractional exponent and the strength of the external potential. We find an oscillation period equal to that of the non-fractional case. The participation ratio is computed in closed form and it reveals that localization of the modes increases with a deviation from the standard case, and with an increase of the external constant field. When nonlinear effects are included, a competition between the tendency to Bloch oscillate, and the trapping tendency typical of the Kerr effect is observed, which ultimately obliterates the BO in the limit of large nonlinearity.

Cavity polaritons, hybrid half-light half-matter excitations in quantum microcavities in the strong-coupling regime demonstrate clear signatures of quantum collective behavior, such as analogues of BEC and superfluidity at surprisingly high temperatures. The analysis of the formation of these states demands an account of the relaxation processes in the system. Although there are well-established approaches for the description of some of them, such as finite lifetime polariton, an external optical pump, and coupling with an incoherent excitonic reservoir, the treatment of pure energy relaxation in a polariton fluid still remains a puzzle. Here, based on the quantum hydrodynamics approach, we derive the corresponding equations where the energy relaxation term appears naturally. We analyze in detail how it affects the dynamics of polariton droplets and the dispersion of elementary excitations of a uniform polariton condensate. Although we focus on the case of cavity polaritons, our approach can be applied to other cases of bosonic condensates, where the processes of energy relaxation play an important role.

Vertical thermal convection system exhibits weak turbulence and spatio-temporally chaotic behavior. In this system, we report seven equilibria and 26 periodic orbits, all new and linearly unstable. These orbits, together with four previously studied in Zheng et al. (2024) bring the number of periodic orbit branches computed so far to 30, all solutions to the fully non-linear three-dimensional Navier-Stokes equations. These new invariant solutions capture intricate flow patterns including straight, oblique, wavy, skewed and distorted convection rolls, as well as bursts and defects in rolls. Most of the solution branches show rich spatial and/or spatio-temporal symmetries. The bifurcation-theoretic organisation of these solutions are discussed; the bifurcation scenarios include Hopf, pitchfork, saddle-node, period-doubling, period-halving, global homoclinic and heteroclinic bifurcations, as well as isolas. Given this large number of unstable orbits, our results may pave the way to quantitatively describing transitional fluid turbulence using periodic orbit theory.

In this work, we explore the robustness of a bit-flip operation against thermal and quantum noise for bits represented by the symmetry-broken pairs of the period-doubled (PD) states in a classical parametric oscillator and discrete time crystal (DTC) states in a fully-connected open spin-cavity system, respectively. The bit-flip operation corresponds to switching between the two PD and DTC states induced by a defect in a periodic drive, introduced in a controlled manner by linearly ramping the phase of the modulation of the drive. In the absence of stochastic noise, strong dissipation results in a more robust bit-flip operation in which slight changes to the defect parameters do not significantly lower the success rate of bit-flips. The operation remains robust even in the presence of stochastic noise when the defect duration is sufficiently large. The fluctuations also enhance the success rate of the bit-flip below the critical defect duration needed to induce a switch. By considering parameter regimes in which the DTC states in the spin-cavity system do not directly map to the PD states, we reveal that this robustness is due to the system being quenched by the defect towards a new phase that has enough excitation to suppress the effects of the stochastic noise. This allows for precise control of the bit-flip operations by tuning into the preferred intermediate state that the system will enter during a bit-flip operation. We demonstrate this in a modified protocol based on precise quenches of the driving frequency.