Nonlinear Sciences

We propose a method to controllably generate six kinds of nonlinear waves on continuous waves, including the one- and multi-peak solitons, the Akhmediev, Kuznetsov-Ma, and Taijiri-Watanabe breathers, and stable periodic waves. In the nonlinear fiber system with third-order dispersion, we illustrate their generation conditions by the modified linear stability analysis, and numerically generate them from initial perturbations on continuous waves. We implement the quantitative control over their dynamical features, including the wave type, velocity, periodicity, and localization. Our results may provide an effective scheme for generating optical solitons on continuous waves, and it can also be applied for wave generations in other various nonlinear systems.

SQUID (Superconducting QUantum Interference Device) metamaterials, subject to a time-independent (dc) flux gradient and driven by a sinusoidal (ac) flux field, support chimera states that can be generated with zero initial conditions. The dc flux gradient and the amplitude of the ac flux can control the number of desynchronized clusters of such a generated chimera state (i.e., its `heads') as well as their location and size. The combination of three measures, i.e., the synchronization parameter averaged over the period of the driving flux, the incoherence index, and the chimera index, is used to predict the generation of a chimera state and its multiplicity on the parameter plane of the dc flux gradient and the ac flux amplitude. Moreover, the full-width half-maximum of the distribution of the values of the synchronization parameter averaged over the period of the ac driving flux, allows to distinguish chimera states from non-chimera, partially synchronized states.

We study the dynamics of Bose-Einstein condensate coupled to a waveguide with parity-time symmetric potential in the presence of quadratic-cubic nonlinearity modelled by Gross-Pitaevskii equation with external source. We employ the self-similar technique to obtain matter wave solutions, such as bright, kinktype, rational dark and Lorentzian-type self-similar waves for this model. The dynamical behavior of self-similar matter waves can be controlled through variation of trapping potential, external source and nature of nonlinearities present in the system.

In our report we consider two weakly coupled Schrödinger equations as a model of the interchain energy transport in the DNA double-helix. We use the reduction of the Yakushevich-type model considering the torsional dynamics of the DNA. In the previous works only small amplitude excitations and stationary dynamics were investigated, while we focus on the non-stationary dynamics of the double-helix. We consider the system as a model of two weakly interacting DNA strands. Supposing that initially only one of the chains is excited in form of breather we demonstrate the existence of invariant which allows to reduce the order of the problem and consider the system of the phase plane. The analysis provided demonstrates analytical tool for prediction of the periodic interchain excitation transitions of its localization on one of the chains. The technique also takes into account the spreading of the excitations with time.

We consider an extension of the classical Fisher-Kolmogorov equation, called the \textit{Fisher-Stefan} model, which is a moving boundary problem on 0<x<L(t) . A key property of the Fisher-Stefan model is the \textit{spreading-vanishing dichotomy}, where solutions with L(t)> L c will eventually spread as t→∞ , whereas solutions where L(t)≯ L c will vanish as t→∞ . In one dimension is it well-known that the critical length is L c =π/2 . In this work we re-formulate the Fisher-Stefan model in higher dimensions and calculate L c as a function of spatial dimensions in a radially symmetric coordinate system. Our results show how L c depends upon the dimension of the problem and numerical solutions of the governing partial differential equation are consistent with our calculations.

Swarming patterns that emerge from the interaction of many mobile agents are a subject of great interest in fields ranging from biology to physics and robotics. In some application areas, multiple swarms effectively interact and collide, producing complex spatiotemporal patterns. Recent studies have begun to address swarm-on-swarm dynamics, and in particular the scattering of two large, colliding swarms with nonlinear interactions. To build on early numerical insights, we develop a mean-field approach that can be used to predict the parameters under which colliding swarms are expected to form a milling state. Our analytical method relies on the assumption that, upon collision, two swarms oscillate near a limit-cycle, where each swarm rotates around the other while maintaining an approximately constant and uniform density. Using this approach we are able to predict the critical swarm-on-swarm interaction coupling, below which two colliding swarms merely scatter, for near head-on collisions as a function of control parameters. We show that the critical coupling corresponds to a saddle-node bifurcation of a stable limit cycle in the uniform, constant density approximation. Our results are tested and found to agree with large multi-agent simulations.

We propose a generic uncertainty relationship for cross-diffusion (quasi-soliton) waves triggered by local instabilities through Thermo-Hydro-Mechano-Chemical (THMC) coupling and cross-scale feedbacks. Cross-diffusion waves nucleate when the overall stress field is incompatible with accelerations from local feedbacks of generalized THMC thermodynamic forces with generalized thermodynamic fluxes of another kind. Cross-diffusion terms in the 4 x 4 THMC diffusion matrix are shown to lead to multiple diffusional P - and S -wave solutions of the coupled THMC equations. Uncertainties in the location of local material instabilities are captured by wave scale correlation of probability amplitudes. Cross-diffusional waves have unusual dispersion patterns and, although they assume a solitary state, do not behave like solitons but have a quasi-elastic particle-like state. Their characteristic wavenumber and constant speed defines mesoscopic internal material time-space relations entirely defined by the coefficients of the coupled THMC reaction-cross-diffusion equations. These coefficients are identified here as material parameters underpinning the criterion for nucleation and speed of diffusional waves. Interpreting patterns in nature as features of standing or propagating diffusional waves offers a simple mathematical framework for analysis of multi-physics instabilities and evaluation of their uncertainties similar to their quantum-mechanical analogues.

We study the formation and propagation of chirped elliptic and solitary waves in cubic-quintic nonlinear Helmholtz (CQNLH) equation. This system describes nonparaxial pulse propagation in a planar waveguide with Kerr-like and quintic nonlinearities along with spatial dispersion originating from the nonparaxial effect that becomes dominant when the conventional slowly varying envelope approximation (SVEA) fails. We first carry out the modulational instability (MI) analysis of a plane wave in this system by employing the linear stability analysis and investigate the influence of different physical parameters on the MI gain spectra. In particular, we show the nonparaxial parameter suppresses the conventional MI gain spectrum and also leads to a nontrivial monotonic increase in the gain spectrum near the tails of the conventional MI band, a qualitatively distinct behaviour from the standard nonlinear Schrödinger (NLS) system. We then study the MI dynamics by direct numerical simulations which demonstrate production of ultra-short nonparaxial pulse trains with internal oscillations and slight distortions at the wings. Following the MI dynamics, we obtain exact elliptic and solitary wave solutions using the integration method by considering physically interesting chirped traveling wave ansatz. In particular, we show that the system features intriguing chirped anti-dark, bright, gray and dark solitary waves depending upon the nature of nonlinearities. We also show that the chirping is inversely proportional to the intensity of the optical wave. Especially, the bright and dark solitary waves exhibit unusual chirping behaviour which will have applications in nonlinear pulse compression process.

On the curved surfaces of living and nonliving materials, planar excitable waves frequently exhibit directional change and subsequently undergo a topological change; that is, a series of wave dynamics from fusion, annihilation to splitting. Theoretical studies have shown that excitable planar stable waves change their topology significantly depending on the initial conditions on flat surfaces, whereas the directional-change of the waves occurs based on the geometry of curved surfaces. However, it is not clear if the geometry of curved surfaces induces this topological change. In this study, we first show the curved surface geometry-induced topological changes in a planar stable wave by numerically solving an excitable reaction-diffusion equation on a bell-shaped surface. We determined two necessary conditions for inducing topological change: the characteristic length of the curved surface (i.e., height of the bell-shaped structure) should be larger than the width of the wave and than a threshold independent of the wave width. As for the geometrical mechanism of the latter, we found that a bifurcation of the globally minimum geodesics (i.e. minimal paths) on the curved surface leads to the topological change. These conditions imply that wave topology changes can be predicted on the basis of curved surfaces, whose structure is larger than the wave width.

In this paper, we establish a new scheme for identification and classification of high intensity events generated by the propagation of light through a photorefractive SBN crystal. Among these events, which are the inevitable consequence of the development of modulation instability, are speckling and soliton-like patterns. The usual classifiers developed on statistical measures, such as the significant intensity, often provide only a partial characterization of these events. Here, we try to overcome this deficiency by implementing the convolution neural network method to relate experimental data of light intensity distribution and corresponding numerical outputs with different high intensity regimes. The train and test sets are formed of experimentally obtained intensity profiles at the crystal output facet and corresponding numerical profiles. The accuracy of detection of speckles reaches maximum value of 100%, while the accuracy of solitons and caustic detection is above 97%. These performances are promising for the creation of neural network based routines for prediction of extreme events in wave media.