CyberSec Research

We consider an $N$-soliton solution of the focusing nonlinear Schr\"{o}dinger equations. We give conditions for the synchronous collision of these $N$ solitons. When the solitons velocities are well separated and the solitons have equal amplitude, we show that the local wave profile at the collision point scales as the $\operatorname{sinc}(x)$ function. We show that this behaviour persists when the amplitudes of the solitons are i.i.d. sub-exponential random variables. Namely the central collision peak exhibits universality: its spatial profile converges to the $\operatorname{sinc}(x)$ function, independently of the distribution. We derive Central Limit Theorems for the fluctuations of the profile in the near-field regime (near the collision point) and in the far-regime.

We investigate the nonlinear dynamics of dark solitons in a one-dimensional Bose-Einstein condensate confined to a curved geometry. Using the Gross-Pitaevskii equation in curvilinear coordinates and a perturbative expansion in the local curvature, we derive a set of coupled evolution equations for the soliton velocity and the curvature. For the case of constant curvature, such as circular geometries, the soliton dynamics is governed solely by the initial velocity and curvature. Remarkably, the soliton travels a nearly constant angular trajectory across two orders of magnitude in curvature, suggesting an emergent conserved quantity, independent of its initial velocity. We extend our analysis to elliptical trajectories with spatially varying curvature and show that soliton dynamics remain determined by the local curvature profile. In these cases, the model of effective constant curvature describes accurately the dynamics given the local curvature has smooth variation. When the soliton crosses regions of rapid curvature variation and/or non-monotonic behavior, the model fails to describe to soliton dynamics, although the overall behavior can still be fully mapped to the curvature profile. Our results provide a quantitative framework for understanding the role of geometry in soliton dynamics and pave the way for future studies of nonlinear excitations in curved quantum systems.

We construct the dynamic models governing two nonreciprocally coupled fields for cases with zero, one, and two conservation laws. Starting from two microscopic nonreciprocally coupled Ising models, and using the mean-field approximation, we obtain closed-form evolution equations for the spatially resolved magnetization in each lattice. For single spin-flip dynamics, the macroscopic equations in the thermodynamic limit are closely related to the nonreciprocal Allen-Cahn equations, i.e. conservation laws are absent. Likewise, for spin-exchange dynamics within each lattice, the thermodynamic limit yields equations similar to the nonreciprocal Cahn-Hilliard model, i.e. with two conservation laws. In the case of spin-exchange dynamics within and between the two lattices, we obtain two nonreciprocally coupled equations that add up to one conservation law. For each of these cases, we systematically map out the linear instabilities that can arise. Our results provide a microscopic foundation for a broad class of nonreciprocal field theories, establishing a direct link between non-equilibrium statistical mechanics and macroscopic continuum descriptions.

We report stable composite vortex solitons in the model of a three-dimensional photonic crystal with the third-harmonic (TH) generation provided by the quasi-phase-matched quadratic nonlinearity. The photonic crystal is designed with a checkerboard structure in the $\left( x\text{,}% y\right) $ plane, while the second-order nonlinear susceptibility, $d(z)$, is modulated along the propagation direction as a chains of rectangles with two different periods. This structure can be fabricated by means of available technologies. The composite vortex solitons are built of fundamental-frequency (FF), second-harmonic (SH), and TH components, exhibiting spatial patterns which correspond to vortex with topological charges $s=1$, a quadrupole with $s=2$, and an anti-vortex structure with $s = -1$, respectively. The soliton profiles feature rhombic or square patterns, corresponding to phase-matching conditions $\varphi =0$ or $\pi $, respectively, the rhombic solitons possessing a broader stability region. From the perspective of the experimental feasibility, we show that both the rhombic and square-shaped composite vortex solitons may readily propagate in the photonic crystals over distances up to $\sim 1$ m. The TH component of the soliton with $s=\mp 1$ is produced by the cascaded nonlinear interactions, starting from the FF vortex component with $s=\pm 1$ and proceeding through the quadrupole SH one with $s=2$. These findings offer a novel approach for the creation and control of stable vortex solitons in nonlinear optics.

We study the properties of non-topological solitons in two-dimensional conformal field theory. The spectrum of linear perturbations on these solutions is found to be trivial, containing only symmetry-related zero modes. The interpretation of this feature is given by considering the relativistic generalization of our theory in which the conformal symmetry is violated. It is explicitly seen that the restoration of this symmetry leads to the absence of decay/vibrational modes.

We studied the long-term nonequilibrium dynamics of q-state Potts models with q = 4, 5, 6, and 8 using Monte Carlo simulations on a two-dimensional square lattice. When the contact energies between the nearest neighbors for the standard Potts models are used, cyclic changes in the q homogeneous phases and q-state coexisting wave mode appear at low and high flipping energies, respectively, for all values of q. However, for a factorizable q value, dynamic modes with skipping states emerge, depending on the contact energies. For q = 6, a spiral wave mode with three domain types (one state dominant or two states mixed) and cyclic changes in three homogeneous phases are found. Although three states can coexist spatially under thermal equilibrium, the scaling exponents of the transitions to the wave modes are modified from the equilibrium values.

We propose and theoretically characterize three-dimensional spatio-temporal thermalization of a continuous-wave classical light beam propagating along a multi-mode optical waveguide. By combining a non-equilibrium kinetic approach based on the wave turbulence theory and numerical simulations of the field equations, we anticipate that thermalizing scattering events are dramatically accelerated by the combination of strong transverse confinement with the continuous nature of the temporal degrees of freedom. In connection with the blackbody catastrophe, the thermalization of the classical field in the continuous temporal direction provides an intrinsic mechanism for adiabatic cooling and, then, spatial beam condensation. Our results open new avenues in the direction of a simultaneous spatial and temporal beam cleaning.

We investigate the dynamics of multi-lump waves in a generalized spatial symmetric higher-dimensional dispersive water wave model using an analytical approach. This involves the construction of explicit solutions using Hirota's bilinear method and generalized polynomial expansions. The dynamical study shows that the multi-lump waves are non-interacting and reveal different geometrical patterns.

In this paper, we systematically investigate the intricate dynamics of the breather-to-soliton transitions and nonlinear wave interactions for the higher-order modified Gerdjikov-Ivanov equation. We discuss the transition conditions of the breather-to-soliton and obtain different types of nonlinear converted waves, including the W-shaped soliton, M-shaped soliton, multi-peak soliton, anti-dark soliton and periodic wave solution. Meanwhile, the interactions among the above nonlinear converted waves are explored by choosing appropriate parameters. Furthermore, we derive the double-pole breather-to-soliton transitions and apply the asymptotic analysis method to analyze the dynamics of the asymptotic solitons for the double-pole anti-dark soliton.

We study the Swift-Hohenberg equation - a paradigm model for pattern formation - with "large" spatially periodic coefficients and find a Turing bifurcation that generates patterns whose leading order form is a Bloch wave modulated by solutions of a Ginzburg-Landau type equation. Since the interplay between forcing wavenumber and intrinsic wavenumber crucially shapes the spectrum and emerging patterns, we distinguish between resonant and non-resonant regimes. Extending earlier work that assumed asymptotically small coefficients, we tackle the more involved onset analysis produced by O(1) forcing and work directly in Bloch space, where the richer structure of the bifurcating solutions becomes apparent. This abstract framework is readily transferable to more complex systems, such as reaction-diffusion equations arising as dryland vegetation models, where topography induces spatial heterogeneity.

Tree-grass coexistence is a defining feature of savanna ecosystems, which play an important role in supporting biodiversity and human populations worldwide. While recent advances have clarified many of the underlying processes, how these mechanisms interact to shape ecosystem dynamics under environmental stress is not yet understood. Here, we present and analyze a minimalistic spatially extended model of tree-grass dynamics in dry savannas. We incorporate tree facilitation of grasses through shading and grass competing with trees for water, both varying with tree life stage. Our model shows that these mechanisms lead to grass-tree coexistence and bistability between savanna and grassland states. Moreover, the model predicts vegetation patterns consisting of trees and grasses, particularly under harsh environmental conditions, which can persist in situations where a non-spatial version of the model predicts ecosystem collapse from savanna to grassland instead (a phenomenon called ''Turing-evades-tipping''). Additionally, we identify a novel ''Turing-triggers-tipping'' mechanism, where unstable pattern formation drives tipping events that are overlooked when spatial dynamics are not included. These transient patterns act as early warning signals for ecosystem transitions, offering a critical window for intervention. Further theoretical and empirical research is needed to determine when spatial patterns prevent tipping or drive collapse.

We observe entanglement between collective excitations of a Bose-Einstein condensate in a configuration analogous to particle production during the preheating phase of the early universe. In our setup, the oscillation of the inflaton field is mimicked by the transverse breathing mode of a cigar-shaped condensate, which parametrically excites longitudinal quasiparticles with opposite momenta. After a short modulation period, we observe entanglement of these pairs which demonstrates that vacuum fluctuations seeded the parametric growth, confirming the quantum origin of the excitations. As the system continues to evolve, we observe a decrease in correlations and a disappearance of non-classical features, pointing towards future experimental probes of the less understood interaction-dominated regime.

Recent research has demonstrated Reservoir Computing's capability to model various chaotic dynamical systems, yet its application to Hamiltonian systems remains relatively unexplored. This paper investigates the effectiveness of Reservoir Computing in capturing rogue wave dynamics from the nonlinear Schr\"{o}dinger equation, a challenging Hamiltonian system with modulation instability. The model-free approach learns from breather simulations with five unstable modes. A properly tuned parallel Echo State Network can predict dynamics from two distinct testing datasets. The first set is a continuation of the training data, whereas the second set involves a higher-order breather. An investigation of the one-step prediction capability shows remarkable agreement between the testing data and the models. Furthermore, we show that the trained reservoir can predict the propagation of rogue waves over a relatively long prediction horizon, despite facing unseen dynamics. Finally, we introduce a method to significantly improve the Reservoir Computing prediction in autonomous mode, enhancing its long-term forecasting ability. These results advance the application of Reservoir Computing to spatio-temporal Hamiltonian systems and highlight the critical importance of phase space coverage in the design of training data.

In the present work we study discrete nonlinear Schr{\"o}dinger models combining nearest (NN) and next-nearest (NNN) neighbor interactions, motivated by experiments in waveguide arrays. While we consider the more experimentally accessible case of positive ratio $\mu$ of NNN to NN interactions, we focus on the intriguing case of competing such interactions $(\mu<0)$, where stationary states can exist only for $-1/4 < \mu < 0$. We analyze the key eigenvalues for the stability of the pulse-like stationary (ground) states, and find that such modes depend exponentially on the coupling parameter $\eps$, with suitable polynomial prefactors and corrections that we analyze in detail. Very good agreement of the resulting predictions is found with systematic numerical computations of the associated eigenvalues. This analysis uses Borel-Pad\'{e} exponential asymptotics to determine Stokes multipliers in the solution; these multipliers cannot be obtained using standard matched asymptotic expansion approaches as they are hidden beyond all asymptotic orders, even near singular points. By using Borel-Pad\'{e} methods near the singularity, we construct a general asymptotic template for studying parametric problems which require the calculation of subdominant Stokes multipliers.

We introduce rd-spiral, an open-source Python library for simulating 2D reaction-diffusion systems using pseudo-spectral methods. The framework combines FFT-based spatial discretization with adaptive Dormand-Prince time integration, achieving exponential convergence while maintaining pedagogical clarity. We analyze three dynamical regimes: stable spirals, spatiotemporal chaos, and pattern decay, revealing extreme non-Gaussian statistics (kurtosis $>96$) in stable states. Information-theoretic metrics show $10.7\%$ reduction in activator-inhibitor coupling during turbulence versus $6.5\%$ in stable regimes. The solver handles stiffness ratios $>6:1$ with features including automated equilibrium classification and checkpointing. Effect sizes ($\delta=0.37$--$0.78$) distinguish regimes, with asymmetric field sensitivities to perturbations. By balancing computational rigor with educational transparency, rd-spiral bridges theoretical and practical nonlinear dynamics.

In dynamical systems on networks, one assigns the dynamics to nodes, which are then coupled via links. This approach does not account for group interactions and dynamics on links and other higher dimensional structures. Higher-order network theory addresses this by considering variables defined on nodes, links, triangles, and higher-order simplices, called topological signals (or cochains). Moreover, topological signals of different dimensions can interact through the Dirac-Bianconi operator, which allows coupling between topological signals defined, for example, on nodes and links. Such interactions can induce various dynamical behaviors, for example, periodic oscillations. The oscillating system consists of topological signals on nodes and links whose dynamics are driven by the Dirac-Bianconi coupling, hence, which we call it Dirac-Bianconi driven oscillator. Using the phase reduction method, we obtain a phase description of this system and apply it to the study of synchronization between two such oscillators. This approach offers a way to analyze oscillatory behaviors in higher-order networks beyond the node-based paradigm, while providing a ductile modeling tool for node- and edge-signals.

We study two-dimensional, stationary square and hexagonal patterns in the thermocapillary deformational thin-film model for the fluid height $h$\begin{equation*} \partial_t h+\nabla\cdot\left(h^3\left(\nabla\Delta h-g\nabla h\right)+M\frac{h^2}{(1+h)^2}\nabla h\right)=0,\quad t>0,\quad x\in\mathbb{R}^2, \end{equation*} that can be formally derived from the B\'enard-Marangoni problem via a long-wave approximation. Using a linear stability analysis, we show that the flat surface profile corresponding to the pure conduction state destabilises at a critical Marangoni number $M^*$ via a conserved long-wave instability. For any fixed absolute wave number $k_0$, we find that square and hexagonal patterns bifurcate from the flat surface profile at $M=M^* + 4k_0^2$. Using analytic global bifurcation theory, we show that the local bifurcation curves can be extended to global curves of square and hexagonal patterns with constant absolute wave number and mass. We exclude that the global bifurcation curves are closed loops through a global bifurcation in cones argument, which also establishes nodal properties for the solutions. Furthermore, assuming that the Marangoni number is uniformly bounded on the bifurcation branch, we prove that solutions exhibit film rupture, that is, their minimal height tends to zero. This assumption is substantiated by numerical experiments.

We note that the non-orthogonality of states and their coincidence at the degeneracy point are both admitted by nonlinear Hermitian systems and linear non-Hermitian systems. These striking characteristics motivate us to re-investigate the localized waves of nonlinear Hermitian systems and the eigenvalue degeneracies of linear non-Hermitian models, based on several well-known Lax integrable systems that have wide applications in nonlinear optics. We choose nonlinear Schrodinger equation integrability hierarchy to demonstrate the quantitative relations between dynamics of nonlinear Hermitian systems and eigenvalue degeneracies of linear non-Hermitian models. Specifically, the degeneracies of the real or imaginary spectrum of the linear non-Hermitian matrices are uncovered to clarify several essential characteristics of nonlinear localized waves, such as breathers, rogue waves, and solitons. We find that the exceptional points generally correspond to rogue waves for modulational instability cases and dark solitons with maximum velocity for the modulational stability cases. These insights provide another interesting perspective for understanding nonlinear localized waves, and hint that there are closer relations between nonlinear Hermitian systems and linear non-Hermitian systems.