CyberSec Research

We study the dynamics of polar core vortices in the easy plane phase of an atomic spin-1 Bose-Einstein condensate confined in a two-dimensional disc potential. A single vortex moves radially outward due to its interaction with background flows that arise from boundary effects. Pairs of opposite sign vortices, which tend to attract, move either radially inward or outward, depending on their strength of attraction relative to boundary effects. Pairs of same sign vortices repel. Spiral vortex dynamics are obtained for same-sign pairs in the presence of a finite axial magnetization. We quantify the dynamics for a range of realistic experimental parameters, finding that the vortex dynamics are accelerated with increasing quadratic Zeeman energy, consistent with existing studies in planar systems.

Supersolidity in a dipolar Bose-Einstein condensate (BEC), which is the coexistence of crystalline density modulation and global phase coherence, emerges from the interplay of contact interactions, long-range dipole-dipole forces, and quantum fluctuations. Although realized experimentally, stabilizing this phase at zero temperature often requires high peak densities. Here we chart the finite-temperature phase behavior of a harmonically trapped dipolar BEC using an extended mean-field framework that incorporates both quantum (Lee-Huang-Yang) and thermal fluctuation effects. We find that finite temperature can act constructively: it shifts the supersolid phase boundary toward larger scattering lengths, lowers the density threshold for the onset of supersolidity, and broadens the stability window of modulated phases. Real-time simulations reveal temperature-driven pathways (crystallization upon heating and melting upon cooling) demonstrating the dynamical accessibility and path dependence of supersolid order. Moreover, moderate thermal fluctuations stabilize single-droplet states that are unstable at zero temperature, expanding the experimentally accessible parameter space. These results identify temperature as a key control parameter for engineering and stabilizing supersolid phases, offering realistic routes for their observation and control in dipolar quantum gases.

In this paper, we propose an alternative approach to generate a new class of beating vector solitons. Unlike earlier procedures that use dark-bright or bright-dark soliton solutions to generate beating solitons, the method described here utilizes non-degenerate vector soliton solutions of the Manakov system. It involves linear superposition of such soliton solutions along with an intensity switching mechanism facilitated by cross-coupling between the optical modes. We find that the obtained beating solitons collide elastically with themselves and keep their beating feature unchanged after the collision. We also find that their beating nature can be controlled by allowing them to collide with degenerate beating solitons exhibiting energy-sharing collisions. The results presented in this work will provide new insights into beating solitons in Bose-Einstein condensates, nonlinear optics, and related areas of research.

Instantons, localised saddle points of the action, play an important role in describing non-perturbative aspects of quantum field theories, for example vacuum decay or violation of conservation laws associated with anomalous symmetries. However, there are theories in which no saddle point exists. In this paper, we revisit the idea of constrained instantons, proposed initially by Affleck in 1981, and develop it into a complete method for computing the vacuum decay rate in such cases. We apply this approach to the massive scalar field theory with a negative quartic self-interaction using two different constraints. We solve the field equations numerically and find a two-branch structure, with two distinct solutions for each value of the constraint. By counting the negative modes, we identify one branch of solutions as the constrained instantons and the other as the minima of the action subject to the constraint. We discuss their significance for the computation of the vacuum decay rate.

A simplified mathematical model is suggested to describe the dynamics of a quasi-monochromatic optical wave in the bulk of an effectively isotropic metamaterial with averaged dielectrical permittivity near zero (ENZ medium), in the presence of a weak spatial nonuniformity, Kerr nonlinearity as well as linear gain due to external pumping. The model is a vector Ginzburg-Landau equation of the general kind, with the dominating curl-curl term in the dispersive operator, and it resembles the equation for electromagnetic waves in plasma [E. A. Kuznetsov, 1974]. In the case of purely real Kerr coefficients, a split-step Fourier method is appropriate for numerical simulations. It makes possible to observe various variants of nontrivial evolution of both central-symmetric and toroidal vector wave structures trapped by a quadratic potential well, as well as nonlinear interaction between the longitudinal and transverse waves in the case of their combination.

Structured models, such as PDEs structured by age or phenotype, provide a setting to study pattern formation in heterogeneous populations. Classical tools to quantify the emergence of patterns, such as linear and weakly nonlinear analyses, pose significant mathematical challenges for these models due to sharply peaked or singular steady states. Here, we present a weakly nonlinear framework that extends classical tools to structured PDE models in settings where the base state is spatially uniform, but exponentially localized in the structured variable. Our approach utilizes WKBJ asymptotics and an analysis of the Stokes phenomenon to systematically resolve the solution structure in the limit where the steady state tends to a Dirac-delta function. To demonstrate our method, we consider a chemically structured (nonlocal) model of motile bacteria that interact through quorum sensing. For this example, our analysis yields an amplitude equation that governs the solution dynamics near a linear instability, and predicts a pitchfork bifurcation. From the amplitude equation, we deduce an effective parameter grouping whose sign determines whether the pitchfork bifurcation is subcritical or supercritical. Although we demonstrate our framework for a specific example, our techniques are broadly applicable.

Soliton dynamics in coupled Kerr microcavities is an important aspect of frequency comb technologies, with applications in optical communication and precision metrology. We investigate a minimal system consisting of two nearly identical coupled Kerr microresonators, each operating in the soliton regime and driven by a separate coherent beam, and analyze the mechanisms that govern their soliton interactions. In the weak-coupling regime, the system supports multiple soliton clusters characterized by distinct soliton separations and stability. Numerical simulations indicate that asymmetric perturbations can alter soliton separations or destroy these states, while the imposed pump phase difference plays a key role in cluster selection. Together, these findings highlight previously unexplored regimes of dissipative soliton organization and suggest new strategies for controlling soliton ensembles in integrated photonic platforms.

Dispersive shock waves (DSWs) are expanding nonlinear wave trains that arise when dispersion regularizes a steepening front, a phenomenon observed in fluids, plasmas, optics, and superfluids. Here we report the first experimental observation of DSWs in an intense electron beam, using the University of Maryland Electron Ring (UMER). A localized induction-cell perturbation produced a negative density pulse that evolved into a leading soliton-like peak followed by an expanding train of oscillations. The leading peak satisfied soliton scaling laws for width^2 vs inverse amplitude and velocity vs amplitude, while the total wave-train width increased linearly with time, consistent with Korteweg--de Vries (KdV) predictions. Successive peaks showed decreasing amplitude and velocity toward the trailing edge, in agreement with dispersive shock ordering. These results demonstrate that intense charged particle beams provide a new laboratory platform for studying dispersive hydrodynamics, extending nonlinear wave physics into the high-intensity beam regime.

A rescaled Manning potential is obtained in the analysis of scatterings of small- amplitude excitations with a kink defect. The generic model is a nonlinear Klein- Gordon Hamiltonian describing a one-dimensional chain of identical molecules, sub- jected to an hyperbolic single-particle substrate potential. To account for isotope effects that are likely to affect characteristic equilibrium parameters of the molec- ular chain, including the lattice spacing (i.e. the characteristic intermolecuar dis- tance) and/or the barrier height, the hyperbolic substrate potential is endowed with a real parameter whose variation makes it suitable for the description of molecu- lar excitations in a broad range of systems with inversion symmetry. These include hydrogen-bonded molecular crystals, {\alpha}-helix proteins, long polymer chains and two- state quantum-tunneling systems in general. Double-well models with deformable profiles are relevant in physical contexts where the equilibrium configurations are sensitive to atomic or molecular substitutions, dilution, solvation and so on.

A systematic analysis of the Eckhaus instability in the one-dimensional Ginzburg-Landau equation is presented. The analysis is based on numerical integration of the equation in a large (xt)-domain. The initial conditions correspond to a stationary, unstable spatially periodic solution perturbed by "noise." The latter consists of a set of spatially periodic modes with small amplitudes and random phases. The evolution of the solution is examined by analyzing and comparing the dynamics of three key characteristics: the solution itself, its spatial spectrum, and the value of the Lyapunov functional. All calculations exhibit four distinct, mutually agreed, well-defined regimes: (i) rapid decay of stable perturbations; (ii) latent changes, when the solution and the Lyapunov functional undergo minimal alterations while the Fourier spectrum concentrates around the most unstable perturbations; (iii) a phase-slip period, characterized by a sharp decrease in the Lyapunov functional; (iv) slow relaxation to a final stable state.

We study nonlinear bound states -- time-harmonic and spatially decaying ($L^2$) solutions -- of the nonlinear Schr\"odinger / Gross--Pitaevskii equations (NLS/GP) with a compactly supported linear potential. Such solutions are known to bifurcate from the $L^2$ bound states of an underlying Schr\"odinger operator $H_V=-\partial_x^2+V$. In this article we prove an extension of this result: for the 1D NLS/GP, nonlinear bound states also arise via bifurcation from the scattering resonance states and transmission resonance states of $H_V$, associated with the poles and zeros, respectively, of the reflection coefficients, $r_\pm(k)$, of $H_V$. The corresponding resonance states are non-decaying and only $L^2_{\rm loc}$. In contrast to nonlinear states arising from $L^2$ bound states of $H_V$, these resonance bifurcations initiate at a strictly positive $L^2$ threshold which is determined by the position of the complex scattering resonance pole or transmission resonance zero.

We present two new quantum algorithms for reaction-diffusion equations that employ the truncated Chebyshev polynomial approximation. This method is employed to numerically solve the ordinary differential equation emerging from the linearization of the associated nonlinear differential equation. In the first algorithm, we use the matrix exponentiation method (Patel et al., 2018), while in the second algorithm, we repurpose the quantum spectral method (Childs et al., 2020). Our main technical contribution is to derive the sufficient conditions for the diagonalization of the Carleman embedding matrix, which is indispensable for designing both quantum algorithms. We supplement this with an efficient iterative algorithm to diagonalize the Carleman matrix. Our first algorithm has gate complexity of O(d$\cdot$log(d)+T$\cdot$polylog(T/$\varepsilon$)). Here $d$ is the size of the Carleman matrix, $T$ is the simulation time, and $\varepsilon$ is the approximation error. The second algorithm is polynomial in $log(d)$, $T$, and $log(1/\varepsilon)$ - the gate complexity scales as O(polylog(d)$\cdot$T$\cdot$polylog(T/$\varepsilon$)). In terms of $T$ and $\varepsilon$, this is comparable to the speedup gained by the current best known quantum algorithm for this problem, the truncated Taylor series method (Costa et.al., 2025). Our approach has two shortcomings. First, we have not provided an upper bound, in terms of d, on the condition number of the Carleman matrix. Second, the success of the diagonalization is based on a conjecture that a specific trigonometric equation has no integral solution. However, we provide strategies to mitigate these shortcomings in most practical cases.

We all know that the first laser device was realised by Theodore Maiman at Hughes Labs in 1960. Less known is that the very first computer simulations of the relaxation oscillations displayed by Maiman's laser were also performed in 1960 on a digital IBM 704 computer. The reason is that lasers and almost all photonic devices are described by nonlinear equations that are more often than not impossible to be solved analytically, i.e. on a piece of paper. Since then the development and applications of lasers and photonic devices has progressed hand in hand with computer simulations and numerical programming. In this review we introduce and numerically solve the model equations for a variety of devices, lasers, lasers with modulated parameters, lasers with injection, Kerr resonators, saturable absorbers and optical parametric oscillators. By using computer simulations we demonstrate stability and instability of nonlinear solutions in these photonic devices via pitchfork, saddle-node, Hopf and Turing bifurcations; bistability, nonlinear oscillations, deterministic chaos, Turing patterns, conservative solitons; bright, dark and grey cavity solitons; frequency combs, spatial disorder, spatio-temporal chaos, defect mediated turbulence and even rogue waves. There has been a one-to-one correspondence between computer simulations of all these nonlinear features and laboratory experiments with applications in ultrafast optical communications, optical memories, neural networks, frequency standards, optical clocks, future GPS, astronomy and quantum technologies. All of this has been made possible by 'novel insights into spatio-temporal dynamics of lasers, nonlinear and quantum optical systems, achieved through the development and application of powerful techniques for small-scale computing' (2011 Occhialini Medal and Prize of the Institute of Physics and Societa' Italiana di Fisica).

In this work, we provide a full map of scattering scenarios between a Nielsen-Olesen vortex and antivortex. Importantly, in the deep type II regime, such a collision reveals a chaotic pattern in the final state formation with bounce windows immersed into annihilation regions. This structure is due to the energy transfer mechanism triggered by a quasinormal mode, specifically the Feshbach resonant mode, hosted by the vortex.

We investigate the adaptive Ambush strategy in cyclic models following the rules of the spatial rock-paper-scissors game. In our model, individuals of one species possess cognitive abilities to perceive environmental cues and assess the local density of the species they dominate in the spatial competition for natural resources. Based on this assessment, they either initiate a direct attack or, if the local concentration of target individuals does not justify the risk, reposition strategically to prepare an ambush. To quantify the evolutionary consequences of these behavioural strategies, we perform stochastic simulations, analysing emergent spatial patterns and the dependence of species densities on the threshold used by individuals to decide between immediate attack or anticipation. Our findings reveal that, despite being designed to enhance efficiency, cognitive strategies can reduce the abundance of the species due to the constraints of cyclic dominance. We identify an optimal decision threshold: attacking only when the local density of target individuals exceeds 15% provides the best balance between selection risk and long-term persistence. Furthermore, the Ambush strategy benefits low-mobility organisms, increasing coexistence probabilities by up to 53%. These results deepen the understanding of adaptive decision-making in spatial ecology, linking cognitive complexity to ecosystem resilience and extinction risk.

Controlling the behavior of nonlinear systems on networks is a paramount task in control theory, in particular the control of synchronization, given its vast applicability. In this work, we focus on pinning control and we examine two different approaches: the first, more common in engineering applications, where the control is implemented through an external input (additive pinning); the other, where the parameters of the pinned nodes are varied (parametric pinning). By means of the phase reduction technique, we show that the two pinning approaches are equivalent for weakly coupled systems exhibiting periodic oscillatory behaviors. Through numerical simulations, we validate the claim for a system of coupled Stuart--Landau oscillators. Our results pave the way for further applications of pinning control in real-world systems.

Electromagnetically induced transparency (EIT) is well known as a quantum optical phenomenon that permits a normally opaque medium to become transparent due to the quantum interference between transition pathways. This work addresses multi-soliton dynamics in an EIT system modeled by the integrable Maxwell-Bloch (MB) equations for a three-level $\Lambda $-type atomic configuration. By employing a generalized gauge transformation, we systematically construct explicit N-soliton solutions from the corresponding Lax pair. Explicit forms of one-, two-, three-, and four-soliton solutions are derived and analyzed. The resulting pulse structures reveal various nonlinear phenomena, such as temporal asymmetry, energy trapping, and soliton interactions. They also highlight coherent propagation, elastic collisions, and partial storage of pulses, which have potential implications for the design of quantum memory, slow light and photonic data transport in EIT media. In addition, the conservation of fundamental physical quantities, such as the excitation norm and Hamiltonian, is used to provide direct evidence of the integrability and stability of the constructed soliton solutions.

The dynamics of coupled Stuart-Landau oscillators play a central role in the study of synchronization phenomena. Previous works have focused on linearly coupled oscillators in different configurations, such as all-to-all or generic complex networks, allowing for both reciprocal or non-reciprocal links. The emergence of synchronization can be deduced by proving the linear stability of the limit cycle solution for the Stuart-Landau model; the linear coupling assumption allows for a complete analytical treatment of the problem, mostly because the linearized system turns out to be autonomous. In this work, we analyze Stuart-Landau oscillators coupled through nonlinear functions on both undirected and directed networks; synchronization now depends on the study of a non-autonomous linear system and thus novel tools are required to tackle the problem. We provide a complete analytical description of the system for some choices of the nonlinear coupling, e.g., in the resonant case. Otherwise, we develop a semi-analytical framework based on Jacobi-Anger expansion and Floquet theory, which allows us to derive precise conditions for the emergence of complete synchronization. The obtained results extend the classical theory of coupled oscillators and pave the way for future studies of nonlinear interactions in networks of oscillators and beyond.