Nonlinear Sciences

A nonlinear chain with six-order polynomial on-site potential is used to analyze the evolution of the total to kinetic energy ratio during development of modulational instability of extended nonlinear vibrational modes. For the on-site potential of hard-type (soft-type) anharmonicity, the instability of q=π mode ( q=0 mode) results in the appearance of long-living discrete breathers (DBs) that gradually radiate their energy and eventually the system approaches thermal equilibrium with spatially uniform and temporally constant temperature. In the hard-type (soft-type) anharmonicity case, the total to kinetic energy ratio is minimal (maximal) in the regime of maximal energy localization by DBs. It is concluded that DBs affect specific heat of the nonlinear chain and for the case of hard-type (soft-type) anharmonicity they reduce (increase) the specific heat.

We consider the Cahn-Hilliard (CH) equation with a Burgers-type convective term that is used as a model of coarsening dynamics in laterally driven phase-separating systems. In the absence of driving, it is known that solutions to the standard CH equation are characterized by an initial stage of phase separation into regions of one phase surrounded by the other phase (i.e., clusters or drops/holes or islands are obtained) followed by the coarsening process, where the average size of the structures grows in time and their number decreases. Moreover, two main coarsening modes have been identified in the literature, namely, coarsening due to volume transfer and due to translation. In the opposite limit of strong driving, the well-known Kuramoto-Sivashinsky (KS) equation is recovered, which may produce complicated chaotic spatio-temporal oscillations. The primary aim of the present work is to perform a detailed and systematic investigation of the transitions in the solutions of the convective CH (cCH) equation for a wide range of parameter values, and, in particular, to understand in detail how the coarsening dynamics is affected by an increase of the strength of the lateral driving force. Considering symmetric two-drop states, we find that one of the coarsening modes is stabilized at relatively weak driving, and the type of the remaining mode may change as driving increases. Furthermore, there exist intervals in the driving strength where coarsening is completely stabilized. In the intervals where the symmetric two-drop states are unstable they can evolve, for example, into one-drop states, two-drop states of broken symmetry or even time-periodic two-drop states that consist of two traveling drops that periodically exchange mass. We present detailed stability diagrams for symmetric two-drop states in various parameter planes and corroborate our findings by selected time simulations.

The Peregrine soliton is often considered as a prototype of the rogue waves. After recent advances in the semi-classical limit of the 1-D focusing Nonlinear Schrödinger (NLS) equation this conjecture can be seen from another perspective. In the present paper, connecting deterministic and statistical approaches, we numerically demonstrate the effect of the universal local appearance of Peregrine solitons on the evolution of statistical properties of random waves. Evidences of this effect are found in recent experimental studies in the contexts of fiber optics and hydrodynamics. The present approach can serve as a powerful tool for the description of the transient dynamics of random waves and provide new insights into the problem of the rogue waves formation.

Many dynamical systems exhibit abrupt transitions or tipping as the control parameter is varied. In scenarios where the parameter is varied continuously, the rate of change of control parameter greatly affects the performance of early warning signals (EWS) for such critical transitions.We study the impact of variation of the control parameter with a finite rate on the performance of \textcolor{black}{EWS for critical transitions} in a thermoacoustic system (a horizontal Rijke tube) exhibiting subcritical Hopf bifurcation. There is a growing interest in developing early warning signals for tipping in real systems. Firstly, we explore the efficacy of early warning signals based on critical slowing down and fractal characteristics. From this study, lag-1 autocorrelation (AC) and Hurst exponent H are found to be good measures to predict the transition well-before the tipping point. The warning time, obtained using AC and H , reduces with an increase in the rate of change of the control parameter following an inverse power law relation. Hence, for very fast rates, the warning time may be too short to perform any control action. Furthermore, we report the observation of a hyperexponential scaling relation between the AC and the variance of fluctuations during such dynamic Hopf bifurcation. We construct a theoretical model for noisy Hopf bifurcation wherein the control parameter is continuously varied at different rates to study the effect of rate of change of parameter on EWS. Similar results, including the hyperexponential scaling, are observed in the model as well.

From among the waves whose dynamics are governed by the nonlinear Schrödinger (NLS) equation, we find a robust, spatiotemporally disordered family, in which waves initialized with increasing amplitudes, on average, over long time scales, effectively evolve as ever more weakly coupled collections of plane waves.

A method of windowed spatio-temporal spectral filtering is proposed to segregate different nonlinear wave components, and to calculate the surface of free waves. The dynamic kurtosis (i.e., produced by the free wave component) is shown able to contribute essentially to the abnormally large values of the surface displacement kurtosis, according to the direct numerical simulations of realistic sea waves. In this situation the free wave stochastic dynamics is strongly non-Gaussian, and the kinetic approach is inapplicable. Traces of coherent wave patterns are found in the Fourier transform of the directional irregular sea waves; they may form 'jets' in the Fourier domain which strongly violate the classic dispersion relation.

The influence of a temporal forcing on the pattern formation in Langmuir-Blodgett transfer is studied employing a generalized Cahn-Hilliard model. The occurring frequency locking effects allow for controlling the pattern formation process. In the case of one-dimensional (i.e., stripe) patterns one finds various synchronization phenomena such as entrainment between the distance of deposited stripes and the forcing frequency. In two dimensions, the temporal forcing gives rise to the formation of intricate complex patterns such as vertical stripes, oblique stripes and lattice structures. Remarkably, it is possible to influence the system in the spatial direction perpendicular to the forcing direction leading to synchronization in two spatial dimensions.

Embedded solitons are exceptional modes in nonlinear-wave systems with the propagation constant falling in the system's propagation band. An especially challenging topic is seeking for such modes in nonlinear dynamical lattices (discrete systems). We address this problem for a system of coupled discrete equations modeling the light propagation in an array of tunnel-coupled waveguides with a combination of intrinsic quadratic (second-harmonic-generating) and cubic nonlinearities. Solutions for discrete embedded solitons (DESs) are constructed by means of two analytical approximations, adjusted, severally, for broad and narrow DESs, and in a systematic form with the help of numerical calculations. DESs of several types, including ones with identical and opposite signs of their fundamental-frequency and second-harmonic components, are produced. In the most relevant case of narrow DESs, the analytical approximation produces very accurate results, in comparison with the numerical findings. In this case, the DES branch extends from the propagation band into a semi-infinite gap as a family of regular discrete solitons. The study of stability of DESs demonstrates that, in addition to ones featuring the well-known property of semi-stability, there are linearly stable DESs which are genuinely robust modes.

Emergent phenomena are ubiquitous in nature and refer to spatial, temporal, or spatiotemporal pattern formation in complex nonlinear systems driven out of equilibrium that is not contained in the microscopic descriptions at the single-particle level. Examples range from novel phases of matter in both quantum and classical many-body systems, to galaxy formation or neural dynamics. Two characteristic phenomena are length scales that exceed the characteristic interaction length and spontaneous symmetry breaking. Recent advances in integrated photonics indicate that the study of emergent phenomena is possible in complex coupled nonlinear optical systems. Here we demonstrate that out-of-equilibrium driving of a strongly coupled ("dimer") pair of photonic integrated Kerr microresonators, which at the "single-particle" (i.e. individual resonator) level generate well understood dissipative Kerr solitons, exhibit emergent nonlinear phenomena. By exploring the dimer phase diagram, we find unexpected and therefore unpredicted regimes of soliton hopping, spontaneous symmetry breaking, and periodically emerging (in)commensurate dispersive waves. These phenomena are not included in the single-particle description and related to the parametric frequency conversion between hybridized supermodes. Moreover, by controlling supermode hybridization electrically, we achieve wide tunability of spectral interference patterns between dimer solitons and dispersive waves. Our findings provide the first critical step towards the study of emergent nonlinear phenomena in soliton networks and multimode lattices.

In this work, we develop an analytical framework to explain the influence of dissipation and detuning parameters on the emergence and stability of autoresonance in a strongly nonlinear weakly damped chain subjected to harmonic forcing with a slowly-varying frequency. Using the asymptotic procedures, we construct the evolutionary equations, which describe the behavior of the array under the condition of 1:1 resonance and then approximately compute the slow amplitudes and phases as well as the duration of autoresonance. It is shown that, in contrast to autoresonance in a non-dissipative chain with unbounded growth of energy, the energy in a weakly damped array being initially at rest is growing only in a bounded time interval up to an instant of simultaneous escape from resonance of all autoresonant oscillators. Analytical conditions of the emergence and stability of autoresonance are confirmed by numerical simulations.