Search

We solve the fifth-order Korteweg-de Vries (fKdV) equation which is a modified KdV equation perturbed by a fifth-order derivative term multiplied by a small parameter $\epsilon^2$, with $0< \epsilon \ll 1$. Unlike the KdV equation, the stationary fKdV equation does not exhibit exactly localized 1-soliton solution, instead it allows a solution which has a well defined central core similar to that of the KdV 1-soliton solution, accompanied by extremely small oscillatory standing wave tails on both sides of the core. The amplitude of the standing wave tail oscillations is $\mathcal{O}(\exp(-1/\epsilon))$, i.e. it is beyond all orders small in perturbation theory. The analytical computation of the amplitude of these transcendentally small tail oscillations has been carried out up to $\mathcal{O}(\epsilon^5)$ order corrections by using the complex method of matched asymptotics. Also the long-standing discrepancy between the $\mathcal{O}(\epsilon^2)$ perturbative result of Grimshaw and Joshi (1995) and the numerical results of Boyd (1995) has been resolved. In addition to the stationary symmetric weakly localized solitary wave-like solutions, we analyzed the stationary asymmetric solutions of the fKdV equation which decay exponentially to zero on one side of the (slightly asymmetric) core and blows up to large negative values on other side of the core. The asymmetry is quantified by computing the third derivative of the solution at the origin which also turns out to be beyond all orders small in perturbation theory. The analytical computation of the third derivative of a function at the origin has also been carried out up to $\mathcal{O}(\epsilon^5)$ order corrections. We use the exponentially convergent pseudo-spectral method to solve the fKdV equation numerically. The analytical and the numerical results show remarkable agreement., Comment: A thesis submitted to the institute of Physics (E\"otv\"os Lor\'and University, Budapest, Hungary) in partial fulfillment of the requirements for the degree of Masters of Science Physics

Nonlinear two-dimensional internal gravity waves (IGWs) in the atmospheres of the Earth and the Sun are studied. The resulting two-dimensional nonlinear equation has the form of a generalized nonlinear Schr\"{o}dinger equation with nonlocal nonlinearity, that is when the nonlinear response depends on the wave intensity at some spatial domain. The modulation instability of IGWs is predicted, and specific cases for the Earth's atmosphere are considered. In a number of particular cases, the instability thresholds and instability growth rates are analytically found. Despite the nonlocal nonlinearity, we demonstrate the possibility of critical collapse of IGWs due to the scale homogeneity of the nonlinear term in spatial variables., Comment: to be published in Phys. Rev. E

Davydova-Lashkin-Fokas-Lenells equation (DLFLE) is a gauged equivalent form of Fokas-Lenells equation (FLE) that addresses both spatio-temporal dispersion (STD) and nonlinear dispersion (ND) effects. The balance between those effects results a soliton which has always been an interesting topic in research due to its potential applicability as signal carrier in information technology. We have induced a variation to the dispersion effects and apply Hirota bilinear method to realise soliton solution of the proposed DLFLE and explore how the soliton dynamic behaves in accordance to the variation of the dispersion effects. The proposed equation is applicable for number of systems like ultrashort optical pulse, ioncyclotron plasma wave, Bose-Einstein condensate (BEC) matter-wave soliton under certain external fields, etc. The study on such systems under varying effects is very limited and we hope our work can benefit the researchers to understand soliton dynamics more and work on various other nonlinear fields under varying effects., Comment: Under review in journal

Synchronization of activity among myocytes constituting vital organs, e.g., the heart, is crucial for physiological functions. Self-organized coordination in such heterogeneous ensemble of excitable and oscillatory cells is therefore of clinical importance. We show by varying the strength of intercellular coupling and the electrophysiological diversity, a wide range of collective behavior emerges including clusters of synchronized activity. Strikingly, stretch-activated currents allow waves of mechanical deformation to alter the activity of neighboring cells, promoting robust global coherence., Comment: 6 pages, 3 figures + 1 page Supplementary Information for 3 supplementary videos

The cubic nonlinear Schrodinger equation (NLS) in one dimension is considered in the presence of an intensity-dependent dispersion term. We study bright solitary waves with smooth profiles which extend from the limit where the dependence of the dispersion coefficient on the wave intensity is negligible to the limit where the solitary wave becomes singular due to vanishing dispersion coefficient. We analyze and numerically explore the stability for such smooth solitary waves, showing with the help of numerical approximations that the family of solitary waves becomes unstable in the intermediate region between the two limits, while being stable in both limits. This bistability, that has also been observed in other NLS equations with the generalized nonlinearity, brings about interesting dynamical transitions from one stable branch to another stable branch, that are explored in direct numerical simulations of the NLS equation with the intensity-dependent dispersion term., Comment: 18 pages, 9 figures

The generation of second and third harmonics by an acoustic wave propagating along one dimension in a weakly nonlinear elastic medium that is loaded harmonically in time with frequency $\omega_0$ at a single point in space, is analyzed by successive approximations starting with the linear case. It is noted that nonlinear waves have a speed of propagation that depends on their amplitude. It is also noted that both a free medium as well as a loaded medium generate higher harmonics, but that although the second harmonic of the free medium scales like the square of the linear wave, this is no longer the case when the medium is externally loaded. The shift in speed of propagation due to the nonlinearities is determined imposing that there be no resonant terms in a successive approximation solution scheme to the homogeneous problem. The result is then used to solve the inhomogeneous case also by successive approximations, up to the third order. At second order, the result is a second harmonic wave whose amplitude is modulated by a long wave, whose wavelength is inversely proportional to the shift in the speed of propagation of the linear wave due to nonlinearities. The amplitude of the long modulating wave scales like the amplitude of the linear wave to the four thirds. At short distances from the source a scaling proportional to the amplitude of the linear wave squared is recovered, as is a second harmonic amplitude that grows linearly with distance from the source and depends on the third-order elastic constant only. The third order solution is the sum of four amplitude-modulated waves, two of them oscillate with frequency $\omega_0$ and the other two, third harmonics, with $3\omega_0$. In each pair, one term scales like the amplitude of the linear wave to the five-thirds, and the other to the seven-thirds.

We study the formation of breathers in multi-dimensional lattices with nonlocal coupling that decays algebraically. By variational methods, the exact relationship between various parameters (dimension, nonlinearity, nonlocal parameter $\alpha$) that defines positive excitation thresholds is characterized. At the anti-continuum regime, there exists a family of unique ground states that determines excitation thresholds. We not only characterize the sharp spatial decay of ground states, which varies continuously in $\alpha$, but also identify the time decay of dispersive waves, which undergoes a discontinuous transition in $\alpha$.

Buildup and switching mechanisms of solitons in complex nonlinear systems are fundamentally important dynamical regimes. Using a novel strongly nonlinear optical system,the work reveals a new buildup scenario for soliton molecules , which includes a long-duration stage dominated by the emergence of transient NLPs modes to withstand strong disturbances arising from turbulence and extreme nonlinearity in the optical cavity. Systematic simulations reveal effects of the PC rotation angle and intra-cavity nonlinearity on the periodic phase transitions between the different soliton states, and accurately reproduce the experimentally observed buildup and switching mechanisms. These findings could enhance our fundamental study and points to potential uses in designing information encoding systems., Comment: To be published in LASER & PHOTONICS REVIEWS

Spontaneous pattern formation in homogeneous systems is ubiquitous in nature. Although Turing demonstrated that spatial patterns can emerge in reaction-diffusion (RD) systems when the homogeneous state becomes linearly unstable, it remains unclear whether the Turing mechanism is the only route for pattern formation. Here, we develop an efficient algorithm to systematically map the solution landscape to find all steady-state solutions. By applying our method to generic RD models, we find that stable spatial patterns can emerge via saddle-node bifurcations before the onset of Turing instability. Furthermore, by using a generalized action in functional space based on large deviation theory, our method is extended to evaluate stability of spatial patterns against noise. Applying this general approach in a three-species RD model, we show that though formation of Turing patterns only requires two chemical species, the third species is critical for stabilizing patterns against strong intrinsic noise in small biochemical systems.

We study quantum droplets emerging in a quasi-one-dimensional asymmetric mixture of two atomic species with different intra-component coupling constants. We find that such mixtures support a rich variety of multipole quantum droplets, where the macroscopic wavefunction of one component changes its sign and features distinctive multipole structure, while the wavefunction of another component does not have zeros. Such multipole droplets have no counterparts in the reduced single-component model frequently used to describe symmetric one-dimensional mixtures. We study transformations of multipole states upon variation of the chemical potential of each component and demonstrate that quantum droplets can split into separated fundamental states, transform into flat-top multipoles, or into multipole component coupled to flat-top state with several humps on it, akin to anti-dark solitons. Multipole quantum droplets described here are stable in large part of their existence domain. Our findings essentially broaden the family of quantum droplet states emerging in the beyond-meanfield regime and open the way for observation of such heterostructured states in Bose-Bose mixtures., Comment: 7 pages, 7 figures; accepted for Physical Review A as a Letter

The control over the geometry and topology of quantum systems is crucial for advancing novel quantum technologies. This work provides a synthesis of recent insights into the behaviour of quantum vortices within atomic Bose-Einstein condensates (BECs) subject to curved geometric constraints. We highlight the significant impact of the curvature on the condensate density and phase distribution, particularly in quasi-one-dimensional waveguides for different angular momentum states. An engineered periodic transport of the quantized vorticity between density-coupled ring-shaped condensates is discussed. The significant role of curved geometry in shaping the dynamics of rotational Josephson vortices in long atomic Josephson junctions is illustrated for the system of vertically stacked toroidal condensates. Different methods for the controlled creation of rotational Josephson vortices in coupled ring systems are described in the context of the formation of long-lived vortex configurations in shell-shaped BECs with cylindrical geometry. Future directions of explorations of vortices in curved geometries with implications for quantum information processing and sensing technologies are discussed., Comment: 13 pages, 6 figures

We present an exact solution to the problem of a self-consistent equilibrium force-free magnetic flux rope. Unlike other approaches, we use magnetostatic equations and assume only a relatively rapid decrease in the axial magnetic field at infinity. For the first time we obtain a new nonlinear equation for the axial current density, the derivation of which does not require any phenomenological assumptions. From the resulting nonlinear equation, we analytically find the radial profiles of the components of the magnetic field strength and current density.

$\pi$ modes are unique topological edge states appearing in Floquet systems with periodic modulations of the underlying lattice structure in evolution variable, such as dynamically modulated Su-Schrieffer-Heeger (SSH) lattices. These edge states are anomalous states usually appearing between Floquet replicas of the same band, even if standard topological index remains zero for this band. While linear and nonlinear $\pi$ modes were observed in conservative systems, they have never been studied in nonlinear regime in the non-Hermitian systems with structured gain and losses. Here we show that SSH waveguide array with periodically oscillating waveguide positions in propagation direction and with parity-time symmetric refractive index landscape, can support $\pi$ modes that are damped or amplified at different ends of the array. By including nonlinearity and nonlinear absorption into our continuous system, we achieve stable lasing in $\pi$ mode at one end of the array. The representative feature of this system is that lasing in it is thresholdless and it occurs even at low gain-loss amplitudes. The degree of localization of lasing $\pi$ modes can be flexibly controlled by the amplitude of transverse waveguide oscillations. This work therefore introduces a new type of topological Floquet laser and a route to manipulation of $\pi$ modes by structured gain and losses., Comment: 9 pages, 3 figures, to appear in APL Photonics

We theoretically demonstrate a new regime of the formation of ultrashort optical solitons in spherical silica microresonators with whispering gallery modes.The solitons are driven by a coherent CW pump at the frequency in the range of normal dispersion, and the energy is transferred from this pump to the solitons via two channels: Raman amplification and the inverse Cherenkov effect. We discuss three different regimes of soliton propagation and we also show that these Raman solitons can be controlled by weak coherent CW signals.

The Burgers hierarchy consists of nonlinear evolutionary PDEs with increasingly higher-order dispersive and nonlinear terms, resulting in a rich variety of soliton solutions. Notable members include the Burgers equation and the Sharma-Tasso-Olver equation, which are applied in plasma physics, fluid mechanics, optics, and biophysics to describe nonlinear waves in inhomogeneous media. Various soliton and multi-soliton solutions have been identified, and the fission and fusion of solitons have been explored through analytical and numerical techniques. Recently, deep learning methods, particularly Physics Informed Neural Networks (PINNs), have emerged as powerful tools for solving PDEs. These methods use deep neural networks to minimize PDE residuals and satisfy boundary and initial conditions while fitting relevant data. Although PINNs have been applied to equations like Burgers' and Korteweg-de Vries, higher-order members of the Burgers hierarchy remain unexplored. In this study, we employ a PINN algorithm to approximate multi-soliton solutions of linear combinations of equations within the Burgers hierarchy. This semi-supervised approach encodes the PDE and relevant data, determining PDE parameters and resolving the linear combination to discover the PDE that best describes the data. Additionally, we employ gradient-enhanced PINNs (gPINNs) and a conservation law specific to the generic Burgers' hierarchy to improve training accuracy. The results demonstrate the effectiveness of PINNs in describing multi-soliton solutions within the generic Burgers' hierarchy. They also verify the potential for training refinement and accuracy improvement using enhanced approaches in certain cases, while enabling the inference of the PDE model that best describes the observed solitary structures.

Homogeneous planar superfluids exhibit a range of low-energy excitations that also appear in highly excited states like superfluid turbulence. In dilute gas Bose-Einstein condensates, the Jones- Roberts soliton family includes vortex dipoles and rarefaction pulses in the low and high velocity regimes, respectively. These excitations carry both energy and linear momentum, making their decay characteristics crucial for understanding superfluid dynamics. In this work, we develop the theory of planar soliton decay due to thermal effects, as described by the stochastic projected Gross-Pitaevskii theory of reservoir interactions. We analyze two distinct damping terms involving transfer between the condensate and the non-condensate reservoir: particle transfer that also involves energy and usually drives condensate growth, and number-conserving energy transfer. We provide analytical treatments for both the low and high velocity regimes and identify conditions under which either mechanism dominates. Our findings indicate that energy damping prevails at high phase space density. These theoretical results are supported by numerical studies covering the entire velocity range from vortex dipole to rarefaction pulse. We use interaction energy to characterize rarefaction pulses, analogous to the distance between vortices in vortex dipoles, offering an experimentally accessible test for finite temperature theory in Bose-Einstein condensates., Comment: 13 pages + appendices, 15 figures. Comments welcome!

We study two-dimensional stationary soliton gas in the framework of the time-independent reduction of the Kadomtsev-Petviashvili (KPII) equation, which coincides with the integrable two-way ``good'' Boussinesq equation in the xy-plane. This (2+0)D reduction enables the construction of the kinetic equation for the stationary gas of KP solitons by invoking recent results on (1+1)D bidirectional soliton gases and generalised hydrodynamics of the Boussinesq equation. We then use the kinetic theory to analytically describe two basic types of 2D soliton gas interactions: (i) refraction of a line soliton by a stationary soliton gas, and (ii) oblique interference of two soliton gases. We verify the analytical predictions by numerically implementing the corresponding KPII soliton gases via exact N-soliton solutions with N-large and appropriately chosen random distributions for the soliton parameters. We also explicitly evaluate the long-distance correlations for the two-component interference configurations. The results can be applied to a variety of physical systems, from shallow water waves to Bose-Einstein condensates., Comment: 11 pages, 4 figures

We hereby develop the theory of Turing instability for reaction-diffusion systems defined on m-directed hypergraphs, the latter being generalization of hypergraphs where nodes forming hyperedges can be shared into two disjoint sets, the head nodes and the tail nodes. This framework encodes thus for a privileged direction for the reaction to occur: the joint action of tail nodes is a driver for the reaction involving head nodes. It thus results a natural generalization of directed networks. Based on a linear stability analysis we have shown the existence of two Laplace matrices, allowing to analytically prove that Turing patterns, stationary or wave-like, emerges for a much broader set of parameters in the m-directed setting. In particular directionality promotes Turing instability, otherwise absent in the symmetric case. Analytical results are compared to simulations performed by using the Brusselator model defined on a m-directed d-hyperring as well as on a m-directed random hypergraph.

We study the dynamics of a gas bubble in a fluid with surface tension, initially near a spherical equilibrium. While there are many studies and applications of radial bubble dynamics, the theory of general deformations from a spherical equilibrium is less developed. We aim to understand how asymmetrically perturbed equilibrium bubbles evolve toward spherical equilibrium due to thermal or viscous dissipation in an incompressible liquid. We focus on the isobaric approximation [Prosperetti, JFM, 1991], under which the gas pressure within the bubble is spatially uniform and obeys the ideal gas law. The liquid outside the bubble is incompressible, irrotational, and has surface tension. We prove that any equilibrium gas bubble must be spherical by showing that the bubble boundary is a closed surface of constant mean curvature. We then study the initial value problem (IVP) for the coupled PDEs, constitutive laws and interface conditions of the isobaric approximation for general (asymmetric) small initial perturbations of the spherical bubble in the linearized approximation. Our first result, considering thermal damping without viscosity, proves that the linearized IVP is globally well-posed. The monopole (radial) component of the perturbation decays exponentially over time, while the multipole (non-radial) components undergo undamped oscillations. This indicates a limitation of the isobaric model for non-spherical dynamics. Our second result, incorporating viscous dissipation, shows that the IVP is linear and nonlinearly ill-posed due to an incompatibility of normal stress boundary conditions, for non-spherical solutions, and the irrotationality assumption. Our study concludes that to accurately capture the dynamics of general deformations of a gas bubble, the model must account for either vorticity generated at the bubble-fluid boundary, spatial non-uniformities in the gas pressure, or both.

We generalize the Kuramoto model by interpreting the $N$ variables on the unit circle as eigenvalues of a $N$-dimensional unitary matrix $U$, in three versions: general unitary, symmetric unitary and special orthogonal. The time evolution is generated by $N^2$ coupled differential equations for the matrix elements of $U$, and synchronization happens when $U$ evolves into a multiple of the identity. The Ott-Antonsen ansatz is related to the Poisson kernels that are so useful in quantum transport, and we prove it in the case of identical natural frequencies. When the coupling constant is a matrix, we find some surprising new dynamical behaviors.

We investigate the emergence of sustained spatio-temporal behaviors in reaction-phase separation systems. We focus on binary systems, in which either one or both species can phase separate, and we discuss the stability of the homogeneous state determining the conditions for the emergence of a Hopf-type bifurcation. We then examine the effects of a specific autocatalytic chemical reaction, and computationally determine the full solutions to the partial differential equations. We find that when both species phase separate, sustained pulsed dynamics arise in one dimension. When considered in two dimensions, the system generates persistent, complex dynamic droplets, which do not generally appear if only one of the species can phase separate. We finally discuss the emergence of dynamics with complex features, which can be understood using the framework of a cellular automata.

We develop fs laser-fabricated asymmetric couplers and zig-zag arrays consisting of single and two-mode waveguides with bipartite nonlinearity. The fundamental mode ($s$ orbital) is near resonance with the neighboring higher-order $p$ orbital, causing efficient light transfer at low power. Due to Kerr nonlinearity, the coupler works as an all-optical switch between $s$ and $p$ orbitals. Single- and double-peak spatial solitons of $s$-$p$ orbitals form in the lattice due to the bipartite nature of the on-site nonlinearity. We probe highly localized bulk and edge solitons, peaked at the $s$ and $p$ orbitals, spectrally residing in the photonic band gap. Our work will be important for exploring inter-orbital couplings and nonlinear interactions in intricate photonic devices., Comment: 4 pages, 7 figures and supplementary information

We report that conventional saturable periodic structures, in sharp contrast to the conventional systems with different nonlinearities which exhibit the typical S- shaped optical bi- and multi-stable states, reveal some unusual and unique nonlinear dynamics. These include the onset of ramp-like optical bistability (OB) and optical multistability (OM) curves which further transit into mixed OM states combining both ramp-like states followed by the S-shaped multistable curves. We also extend this study to another domain of physics, namely parity-time ($\mathcal{PT}$)- symmetry, by including equal amount of gain and loss into the system which then establishes additional degree of freedom by enabling the investigation into additional two domains which are the unbroken and broken $\mathcal{PT}$- symmetric regimes. Although these bi- and multi-stable states are unusual and unique, when the frequency detuning is introduced, the revival of S-shaped stable states is possible but only in the presence of unbroken $\mathcal{PT}$- symmetry. Conversely, the broken $\mathcal{PT}$- symmetry which usually generates ramp-like multistable states, gives rise to the birth of novel multistable states with a vortex like envelope, (the curve that features simultaneous increase in the critical switch-up and switch-down powers with an increase in the input power) causing a novel structure which has not been reported in the existing literature of different physical systems manifesting multi-stable states., Comment: 17 pages, 10 figures. arXiv admin note: substantial text overlap with arXiv:2404.13724, arXiv:2404.13828

In the framework of the model of a spatially non-uniform Bose-Einstein condensate with helicoidal spin-orbit (SO) coupling, we find abnormal Peregrine solitons (PSs) on top of flat and periodic backgrounds, with ultra-high amplitudes. We explore the roles of the SO coupling strength and helicity pitch in the creation of these anomalously tall PSs and find that their amplitude, normalized to the background height, attains indefinitely large values. The investigation of the modulation instability (MI) in the same system demonstrates that these PSs exist in a range of relatively weak MI, maintaining the feasibility of their experimental observation., Comment: To be published as a Letter in Phys. Rev. Research

This study deals with a piecewise $\phi^2$ scalar field theory in $(1+1)$ dimensions. The scalar field potential is designed with a triple-well shape, engendering kink solutions with asymmetric square-well linearized potentials. Thus, the localized and delocalized modes in this model can be obtained analytically in terms of transcendental equations. This allows us to explore kink-antikink and antikink-kink collisions with any desired number of localized and delocalized modes. We obtain new scenarios of resonance windows suppression, shedding light on the role of higher excited modes in kink scattering.

Creating crystal bilayers twisted with respect to each other would lead to large periodic supercell structures, which can support a wide range of novel electron correlated phenomena, where the full understanding is still under debate. Here, we propose a new scheme to realize a nonlinearity-induced dynamical self-organized twisted-bilayer lattice in an atomic Bose-Einstein condensate (BEC). The key idea here is to utilize the nonlinear effect from the intrinsic atomic interactions to couple different layers and induce a dynamical self-organized supercell structure, dramatically distinct from the conventional wisdom to achieve the static twisted-bilayer lattices. To illustrate that, we study the dynamics of a two-component BEC and show that the nonlinear interaction effect naturally emerged in the Gross-Pitaevskii equation of interacting bosonic ultracold atoms can dynamically induce both periodic (commensurable) and aperiodic (incommensurable) moir\'{e} structures. One of the interesting moir\'{e} phenomena, i.e., the flat-band physics, is shown through investigating the dynamics of the wave packet of BEC. Our proposal can be implemented using available state-of-the-art experimental techniques and reveal a profound connection between the nonlinearity and twistronics in cold atom quantum simulators., Comment: 6 pages, 5 figures

It was recently established that self-healing slip pulses under uniform prestress $\tau_b$ are unstable frictional rupture modes, i.e., they either slowly expand/decay with time t. Furthermore, their dynamics were shown to follow a reduced-dimensionality description corresponding to a special $L(c)$ line in a plane defined by the pulse propagation velocity $c(t)$ and size $L(t)$. Yet, uniform prestress is rather the exception than the rule in natural faults. We study the effects of a spatially-varying prestress $\tau_b(x)$ on 2D slip pulses, initially generated under a uniform $\tau_b$ along a rate-and-state friction fault. We consider periodic and constant-gradient prestress $\tau_b(x)$ around the reference uniform $\tau_b$. For a periodic $\tau_b(x)$, pulses either sustain and form quasi-limit cycles in the $L-c$ plane or decay predominantly monotonically along the $L(c)$ line, depending on the instability index of the initial pulse and the properties of the periodic $\tau_b(x)$. For a constant-gradient $\tau_b(x)$, expanding/decaying pulses closely follow the $L(c)$ line, with systematic shifts determined by the sign and magnitude of the gradient. We also find that a spatially-varying $\tau_b(x)$ can revert the expanding/decaying nature of the initial reference pulse. Finally, we show that a constant-gradient $\tau_b(x)$, of sufficient magnitude and specific sign, can lead to the nucleation of a back-propagating rupture at the healing tail of the initial pulse, generating a bilateral crack-like rupture. This pulse-to-crack transition, along with the above-described effects, demonstrate that rich rupture dynamics merge from a simple, nonuniform prestress. Furthermore, we show that as long as pulses exist, their dynamics are related to the special $L(c)$ line, providing an effective, reduced-dimensionality description of unsteady slip pulses under spatially-varying prestress., Comment: 7 pages, 5 figures + Supplementary Materials & References

Early warning signals (EWSs) forewarn a sudden transition (or tipping) from a desirable state to an undesirable state. However, we observe that EWSs detect an impending tipping past bifurcation points when control parameters are varied fast; this questions the applicability of EWSs in real-world systems. When a control parameter is changed at a finite rate, the tipping is also delayed, providing a borrowed stability (in the parameter space) before the system tips. In this study, we use the Hurst exponent as EWS in a thermoacoustic system - a horizontal Rijke tube. We find that upon receiving an EWS alert, a quick reversal of the control parameter within the region of borrowed stability cannot always prevent tipping in real-world systems. We show this failure is due to the (i) delay in receiving the EWS alert and (ii) dispersion observed in the warning points received. For fast variation of parameters, where preventive measures fall short, we demonstrate EWS-based control actions to rescue the system after tipping. Our results in a real-world system for a fast variation of parameter highlight the limits of applicability of EWSs in preventing tipping., Comment: 13 pages, 7 figures

Like Life, Lenia CA support a range of patterns that move, interact with their environment, and/or are modified by said interactions. These patterns maintain a cohesive, self-organizing morphology, i.e. they exemplify autopoiesis, the self-organization principle of a network of components and processes maintaining themselves. Recent work implementing Asymptotic Lenia as a reaction-diffusion system reported that non-Platonic behavior in standard Lenia may depend on the clipping function, and that ALenia gliders are likely not subject to non-Platonic instability. In this work I show an example of a glider in ALenia that depends on a certain simulation coarseness for autopoietic competence: when simulated with too fine spatial or temporal resolution the glider no longer maintains its morphology or dynamics. I also show that instability maps of the asymptotic Lenia glider, and others in different CA framworks, show fractal retention of fine boundary detail down to the limit of floating point precision., Comment: Published at the 2024 Conference on Artificial Life

Canard cascading (CC) is observed in dynamical networks with global adaptive coupling. It is a fast-slow phenomenon characterized by a recurrent sequence of fast transitions between distinct and slowly evolving quasi-stationary states. In this letter, we uncover the dynamical mechanisms behind CC, using an illustrative example of globally and adaptively coupled semiconductor lasers, where CC represents sequential switching on and off the lasers. Firstly, we show that CC is a robust and truly adaptive network effect that is scalable with network size and does not occur without adaptation. Secondly, we uncover multiple saddle slow manifolds (unstable quasi-stationary states) linked by heteroclinic orbits (fast transitions) in the phase space of the system. This allows us to identify CC with a novel heteroclinic canard orbit that organises different unstable quasi-stationary states into an intricate fast-slow limit cycle. Although individual quasi-stationary states are unstable (saddles), the CC cycle as a whole is attractive and robust to parameter changes.

We predict the existence of a novel type of temporal localized structure in injected Kerr-Gires-Tournois interferometers. These bright pulses exist in the normal dispersion regime, yet they do not correspond to the usual scenario of domain wall locking that induces complex shape multi-stability, weak stability as well as a reduced domain of existence. The new states are observed beyond the mean-field limit and out of the bistable region. Their shape is uniquely defined, with peak intensities beyond that of the upper steady state, and they are stable over a broad range of the injection field, highlighting their potential for optical frequency comb generation., Comment: 5 pages 4 figures

We consider interaction between the small amplitude travelling waves ("sound") and the shock waves in the transmission line containing both nonlinear capacitors and nonlinear inductors. We calculate for the "sound" wave the coefficient of reflection from (the coefficient of transmission through) the shock wave. These coefficients are expressed in terms of the wave speeds and the wave impedances. When only the capacitors or only the inductors are nonlinear, the coefficients are expressed in terms of the wave speeds only. We explicitly include into consideration of the shocks the dissipation, introducing ohmic resistors shunting the inductors and also in series with the capacitors. This allows us to describe the shocks as physical objects of finite width and study their profiles. In some particular cases the profiles were obtained in terms of elementary functions., Comment: pdfLaTeX, 8 pages, 4 figures. The paper was edited substantially

Nonlinear elastic metamaterials are known to support a variety of dynamic phenomena that enhance our capacity to manipulate elastic waves. Since these properties stem from complex, subwavelength geometry, full-scale dynamic simulations are often prohibitively expensive at scales of interest. Prior studies have therefore utilized low-order effective medium models, such as discrete mass-spring lattices, to capture essential properties in the long-wavelength limit. While models of this type have been successfully implemented for a wide variety of nonlinear elastic systems, they have predominantly considered dynamics depending only on the instantaneous kinematics of the lattice, neglecting history-dependent effects, such as wear and plasticity. To address this limitation, the present study develops a lattice-based modeling framework for nonlinear elastic metamaterials undergoing plastic deformation. Due to the history- and rate-dependent nature of plasticity, the framework generally yields a system of differential-algebraic equations whose computational cost is significantly greater than an elastic system of comparable size. We demonstrate the method using several models inspired by classical lattice dynamics and continuum plasticity theory, and explore means to obtain empirical plasticity models for general geometries., Comment: 14 pages, 12 figures

We elucidate the generic bifurcation behavior of local and global order in short-range interacting nonreciprocal lattice systems without conservation laws. We show that a critical magnitude of nearest-neighbor correlations within the respective lattices controls the emergence of coherent oscillations of global order as a result of frustration. Local order is maintained during these oscillations, implying non-trivial spatio-temporal correlations. Ghost states emerge in the strong-interaction regime. The residence time in either of these states eventually diverges, giving rise to ordered non-equilibrium steady states via a saddle-node-infinite-period bifurcation. Our work provides a comprehensive microscopic understanding of collective phenomena in nonreciprocal matter without conservation laws, which may be relevant e.g. for understanding enzyme clustering.

A quasi-one-dimensional Bose-Einstein condensate loaded into a quasi-periodic potential created by two sub-lattices of comparable amplitudes and incommensurate periods is considered. Although the conventional tight-binding approximation is not applicable in this setting, the description can still be reduced to a discrete model that accounts for the modes below the mobility edge. In the respective discrete lattice, where no linear hopping exists, solutions and their dynamics are governed solely by nonlinear interactions. Families of nonlinear modes, including those with no linear limit, are described with a special focus on dimers, which correspond to breather solutions of the Gross-Pitaevskii equation with a quasi-periodic potential. The breathers are found to be stable for negative scattering lengths. Localization and stable propagation of breathers are also observed for positive scattering lengths at relatively weak and moderate nonlinearities., Comment: 7 pages, 5 figures

We propose a scheme to create monopoles with half-integer magnetic charges in a spinful cold atom system. With a minimal monopole in the center, we derive the ground-state single-vortex wave function on the sphere and develop the vortex's kinematic equation in the presence of an external electromagnetic field. The vortex's trajectory is generally depicted by the precession of the system. We further formulate the inter-vortex interaction and build up a theory of multi-vortex dynamics in high-charge monopole systems. We predict the vortices'trajectory in the bi-vortex system and figure out stable vortex (line) patterns in multi-vortex systems. Our study provides deep insights into properties of magnetic monopoles and vortices and paves the way for experimental verification., Comment: 6+2+3 pages, 4+1 figures, 1 table

Excited states (ESs) of two- and three-dimensional (2D and 3D) solitons of the semivortex (SV) and mixed-mode (MM) types, supported by the interplay of the spin-orbit coupling (SOC) and local nonlinearity in binary Bose-Einstein condensates, are unstable, on the contrary to the stability of the SV and MM solitons in their fundamental states. We propose a stabilization strategy for these states in 3D, combining SOC and long-range Rydberg-Rydberg interactions (RRI), in the presence of a spatially-periodic potential, that may include a parity-time (PT)-symmetric component. ESs of the SV solitons, which carry integer vorticities S and S+1 in their two components, exhibit robustness up to S= 4. ESs of MM solitons feature an interwoven necklace-like structure, with the components carrying opposite fractional values of the orbital angular momentum. Regions of the effective stability of the 3D solitons of the SV and MM types (both fundamental ones and ESs), are identified as functions of the imaginary component of the PT-symmetric potential and strengths of the SOC and RRI terms., Comment: to be published in Chaos, Solitons & Fractals

We revisit here the dynamics of an engineered dimer granular crystal under an external periodic drive in the presence of dissipation. Earlier findings included a saddle-node bifurcation, whose terminal point initiated the observation of chaos; the system was found to exhibit bistability and potential quasiperiodicity. We now complement these findings by the identification of unstable manifolds of saddle periodic solutions (saddle points of the stroboscopic map) within the system dynamics. We unravel how homoclinic tangles of these manifolds lead to the appearance of a chaotic attractor, upon the apparent period-doubling bifurcations that destroy invariant tori associated with quasiperiodicity. These findings complement the earlier ones, offering more concrete insights into the emergence of chaos within this high-dimensional, experimentally accessible system., Comment: 10 pages, 8 figures

Soliton pulsations are ubiquitous feature of non-stationary soliton dynamics in mode-locked lasers and many other physical systems. To overcome difficulties related to huge amount of necessary computations and low efficiency of traditional numerical methods in modeling the evolution of non-stationary solitons, we propose a two-parallel bidirectional long short-term memory recurrent neural network, with the main objective to predict dynamics of vector-soliton pulsations in various complex states, whose real-time dynamics is verified by experiments. Besides, the scheme of coded information storage based on the TP-Bi_LSTM RNN, instead of actual pulse signals, is realized too. The findings offer new applications of deep learning to ultrafast optics and information storage., Comment: To be published in Laser & Photonics Reviews;https://doi.org/10.1002/lpor.202400097

The compacton, peakon, and Burgers-Hopf equations regularized by the Galerkin truncation preserving finite Fourier modes are found to support new travelling waves and interacting solitonic structures amidst weaker less-ordered components (`longons'). Different perspectives focusing on the zero-Hamiltonian solitonic, chaotic-looking, and stationary longons are also offered., Comment: arXiv admin note: text overlap with arXiv:2404.08583

We unveil a new scenario for the formation of dissipative localised structures in nonlinear systems. Commonly, the formation of such structures arises from the connection of a homogeneous steady state with either another homogeneous solution or a pattern. Both scenarios, typically found in cavities with normal and anomalous dispersion, respectively, exhibit unique fingerprints and particular features that characterise their behaviour. However, we show that the introduction of a periodic non-Hermitian modulation in Kerr cavities hybridises the two established soliton formation mechanisms, embodying the particular fingerprints of both. In the resulting novel scenario, the stationary states acquire a dual behaviour, playing the role that was unambiguously attributed to either homogeneous states or patterns. These fundamental findings have profound practical implications for frequency comb generation, introducing unprecedented reversible mechanisms for real-time manipulation., Comment: 6 pages, 4 figures, to appear in Physical Review Letters. Comments welcomed

Motivated by the recent introduction of an integrable coupled massive Thirring model by Basu-Mallick et al, we introduce a new coupled Soler model. Further we generalize both the coupled massive Thirring and the coupled Soler model to arbitrary nonlinear parameter $\kappa$ and obtain exact solitary wave solutions in both cases. Remarkably, it turns out that in both the models, because of the conservation laws of charge and energy, the exact solutions we find seem to not depend on how we parameterize them, and the charge density of these solutions is related to the charge density of the single field solutions found earlier by a subset of the present authors. In both the models, a nonrelativistic reduction of the equations leads to the same conclusion that the solutions are proportional to those found in the one component field case., Comment: 9 pages, 2 figures

We present a computational study of the pairwise interactions between defects in the recently introduced non-reciprocal Cahn-Hilliard model. The evolution of a defect pair exhibits dependence upon their corresponding topological charges, initial separation, and the non-reciprocity coupling constant $\alpha$. We find that the stability of isolated topologically neutral targets significantly affects the pairwise defect interactions. At large separations, defect interactions are negligible and a defect pair is stable. When positioned in relatively close proximity, a pair of oppositely charged spirals or targets merge to form a single target. At low $\alpha$, like-charged spirals form rotating bound pairs, which are however torn apart by spontaneously formed targets at high $\alpha$. Similar preference for charged or neutral solutions is also seen for a spiral target pair where the spiral dominates at low $\alpha$, but concedes to the target at large $\alpha$. Our work sheds light on the complex phenomenology of non-reciprocal active matter systems when their collective dynamics involves topological defects.

This study investigates the onset of linear instabilities and their later nonlinear interactions in the shear layer of an initially-laminar jet using a combination of stability analysis and data from high-fidelity flow simulations. We provide a complete picture of the vortex-pairing process. Hydrodynamic instabilities initiate the transition to turbulence, causing the shear layer to spread rapidly. In this process, the shear layer rolls up to form vortices, accompanied by the exponential growth of the fundamental frequency. As the fundamental frequency grows, it gains energy from the mean flow. Subsequently, as it saturates and begins to decay, the fundamental vortices start to pair. During this vortex pairing process, the subharmonic vortex acquires energy both linearly from the mean flow and nonlinearly through a reverse cascade from the fundamental. The process concludes when the subharmonic vortex eventually saturates. Similarly, two subharmonic vortices merge to form a second subharmonic vortex. Our results confirm Kelly's (1967) hypothesis of a resonance mechanism between the fundamental and subharmonic, which supplies energy to the subharmonic. In this multi-tonal, convective-dominated flow, we clarify the ambiguity surrounding the fundamental frequency by demonstrating that the spatially most amplified frequency should be considered fundamental, rather than the structure associated with the spectral energy peak. For the initially-laminar jet considered here, the fundamental frequency corresponds to the fourth largest spectral peak, highlighting the important distinction between the energetically and dynamical significance of a tone. Despite its low energy, the fundamental frequency is dynamically dominant as it determines all other spectral peaks and supplies energy to the subharmonics through a reverse energy cascade.

We present an analytical model of integrable turbulence in the focusing nonlinear Schr\"odinger (fNLS) equation, generated by a one-parameter family of finite-band elliptic potentials in the semiclassical limit. We show that the spectrum of these potentials exhibits a thermodynamic band/gap scaling compatible with that of soliton and breather gases depending on the value of the elliptic parameter m of the potential. We then demonstrate that, upon augmenting the potential by a small random noise (which is inevitably present in real physical systems), the solution of the fNLS equation evolves into a fully randomized, spatially homogeneous breather gas, a phenomenon we call breather gas fission. We show that the statistical properties of the breather gas at large times are determined by the spectral density of states generated by the unperturbed initial potential. We analytically compute the kurtosis of the breather gas as a function of the elliptic parameter m, and we show that it is greater than 2 for all non-zero m, implying non-Gaussian statistics. Finally, we verify the theoretical predictions by comparison with direct numerical simulations of the fNLS equation. These results establish a link between semiclassical limits of integrable systems and the statistical characterization of their soliton and breather gases., Comment: 11 pages, 11 figures

We investigate the behavior of minimizers of perturbed Dirichlet energies supported on a wire generated by a regular simple curve $\gamma$ and defined in the space of $\mathbb{S}^2$-valued functions. The perturbation $K$ is represented by a matrix-valued function defined on $\mathbb{S}^2$ with values in $\mathbb{R}^{3 \times 3}$. Under natural regularity conditions on $K$, we show that the family of perturbed Dirichlet energies converges, in the sense of $\Gamma$-convergence, to a simplified energy functional on $\gamma$. The reduced energy unveils how part of the antisymmetric exchange interactions contribute to an anisotropic term whose specific shape depends on the curvature of $\gamma$. We also discuss the significant implications of our results for studies of ferromagnetic nanowires when Dzyaloshinskii-Moriya interaction (DMI) is present.

We study the radiation in kink collision via a model that varies between $\phi^6$ theory and $\phi^2$ theory with some discontinuities. Both numerical and analytical methods were used to investigate The kink-antikink(KAK) and antikink-kink(AKK) collision. In the numerical analysis, we found the critical velocities in both collisions increased with $n$. We also found a finite lifetime oscillon window in KAK collision for $n=2$. In the analytical part, we found a family of shock wave solutions that describes radiation in the kink collision perfectly. Moreover, an analytical AKK solution at $n\rightarrow\infty$ and $v=1$ was found by considering a certain limit of these solutions., Comment: 27 pages, 68 figures

We report on the theoretical analysis as well as the numerical simulations about the nonlinear dynamics of cavity solitons in a passive Kerr resonator operating in the multistable regime under the condition of a sufficiently strong pump. In this regime, the adjacent tilted cavity resonances might overlap, thus leading to the co-existence of combinatory states of temporal cavity solitons and the extended modulation instability patterns. Very interestingly, the cavity in the regime of multistablity may sustain distinct families of cavity solitons, vividly termed as super cavity solitons with much higher intensity and broader spectra if compared with those in the conventional bi-stable regime. The description of such complex cavity dynamics in the multstable regime requires either the infinite-dimensional Ikeda map, or the derived mean-field coupled Lugiato-Lefever equations by involving the contributing cavity resonances. With the latter model, for the first time, we revealed the existence of different orders of super cavity solitons, whose stationary solutions were obtained by using the Newton-Raphson algorithm. Along this line, with the continuation calculation, we have plotted the Hopf / saddle-node bifurcation curves, thus identifying the existing map of the stable and breathing (super) cavity solitons. With this defined parameter space, we have proposed an efficient method to excite and switch the super cavity solitons by adding an appropriate intensity (or phase) perturbation on the pump. Such deterministic cavity soliton manipulation technique is demonstrated to underpin the multi-level coding, which may enable the large capacity all-optical buffering based on the passive fiber ring cavities., Comment: 11 pages,5 figures

In this study, we investigate the existence and stability of in-phase and out-of-phase dissipative solitons in a dual-waveguide lattice with linear localized gain and nonlinear losses under both focusing and defocusing nonlinearities. Numerical results reveal that both types of dissipative solitons bifurcate from the linear amplified modes, and their nonlinear propagation constant changes to a real value when nonlinearity, linear localized gain, and nonlinear losses coexist. We find that increasing the linear gain coefficient leads to an increase in the power and propagation constant of both types of dissipative solitons. For defocusing nonlinearity, in-phase solitons are stable across their entire existence region, while focusing nonlinearity confines them to a small stable region near the lower cutoff value in the propagation constant. In contrast, out-of-phase solitons have a significantly larger stable region under focusing nonlinearity compared to defocusing nonlinearity. The stability regions of both types of dissipative solitons increase with increasing nonlinear losses coefficient. Additionally, we validate the results of linear stability analysis for dissipative solitons using propagation simulations, showing perfect agreement between the two methods., Comment: 6 pages, 6 figures, to be appear on Chinese Journal of Physics

We elaborate a fractional discrete nonlinear Schr\"{o}dinger (FDNLS) equation based on an appropriately modified definition of the Riesz fractional derivative, which is characterized by its L\'{e}vy index (LI). This FDNLS equation represents a novel discrete system, in which the nearest-neighbor coupling is combined with long-range interactions, that decay as the inverse square of the separation between lattice sites. The system may be realized as an array of parallel quasi-one-dimensional Bose-Einstein condensates composed of atoms or small molecules carrying, respectively, a permanent magnetic or electric dipole moment. The dispersion relation (DR) for lattice waves and the corresponding propagation band in the system's linear spectrum are found in an exact form for all values of LI. The DR is consistent with the continuum limit, differing in the range of wavenumbers. Formation of single-site and two-site discrete solitons is explored, starting from the anti-continuum limit and continuing the analysis in the numerical form up to the existence boundary of the discrete solitons. Stability of the solitons is identified in terms of eigenvalues for small perturbations, and verified in direct simulations. Mobility of the discrete solitons is considered too, by means of an estimate of the system's Peierls-Nabarro potential barrier, and with the help of direct simulations. Collisions between persistently moving discrete solitons are also studied., Comment: 15 pages, 8 figures (to be published in Phys. Rev. E, 2024)