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Deflation is an efficient numerical technique for identifying new branches of steady state solutions to nonlinear partial differential equations. Here, we demonstrate how to extend deflation to discover new periodic orbits in nonlinear dynamical lattices. We employ our extension to identify discrete breathers, which are generic exponentially localized, time-periodic solutions of such lattices. We compare different approaches to using deflation for periodic orbits, including ones based on a Fourier decomposition of the solution, as well as ones based on the solution's energy density profile. We demonstrate the ability of the method to obtain a wide variety of multibreather solutions without prior knowledge about their spatial profile.

This chapter discusses the development and implementation of algorithms based on Equation-Free/Dynamic Data Driven Applications Systems (EF/DDDAS) protocols for the computer-assisted study of the bifurcation structure of complex dynamical systems, such as those that arise in biology (neuronal networks, cell populations), multiscale systems in physics, chemistry and engineering, and system modeling in the social sciences. An illustrative example demonstrates the experimental realization of a chain of granular particles (a so-called engineered granular chain). In particular, the focus is on the detection/stability analysis of time-periodic, spatially localized structures referred to as "dark breathers". Results in this chapter highlight, both experimentally and numerically, that the number of breathers can be controlled by varying the frequency as well as the amplitude of an "out of phase" actuation, and that a "snaking" structure in the bifurcation diagram (computed through standard, model-based numerical methods for dynamical systems) is also recovered through the EF/DDDAS methods operating on a black-box simulator. The EF/DDDAS protocols presented here are, therefore, a step towards general purpose protocols for performing detailed bifurcation analyses directly on laboratory experiments, not only on their mathematical models, but also on measured data., Accepted for publication as a chapter in the Handbook of Dynamic Data Driven Applications Systems

The instabilities of the nontrivial phase elliptic solutions in a repulsive Bose-Einstein condensate (BEC) with a periodic potential are investigated. Based on the defocusing nonlinear Schr\"{o}dinger (NLS) equation with an elliptic function potential, the well-known modulational instability (MI), the more recently identified high-frequency instability, and an unprecedented -- to our knowledge -- variant of the MI, the so-called isola instability are identified numerically. Upon varying parameters of the solutions, instability transitions occur through the suitable bifurcations, such as the Hamiltonian Hope one. Specifically, (i) increasing the elliptic modulus $k$ of the solutions, we find that MI switches to the isola instability and the dominant disturbance has twice the elliptic wave's period, corresponding to a Floquet exponent $\mu=\frac{\pi}{2K(k)}$. The isola instability arises from the collision of spectral elements at the origin of the spectral plane. (ii) Upon varying $V_{0}$, the transition between MI and the high-frequency instability occurs. Differently from the MI and isola instability where the collisions of eigenvalues happen at the origin, high-frequency instability arises from pairwise collisions of nonzero, imaginary elements of the stability spectrum; (iii) In the limit of sinusoidal potential, we show that MI occurs from a collision of eigenvalues with $\mu=\frac{\pi}{2K(k)}$ at the origin; (iv) we also examine the dynamic byproducts of the instability in chaotic fields generated by its manifestation. An interesting observation is that, in addition to MI, the isola instability could also lead to dark localized events in the scalar defocusing NLS equation.

Motivated by the work of J.K. Jang et al., Nat. Commun. 6, 7370 (2015), where the authors experimentally tweeze cavity solitons in a passive loop of optical fiber, we study the amenability to tweezing of cavity solitons as the properties of a localized tweezer are varied. The system is modeled by the Lugiato-Lefever equation, a variant of the complex Ginzburg-Landau equation. We produced an effective, localized, trapping tweezer potential by assuming a Gaussian phase-modulation of the holding beam. The potential for tweezing is then assessed as the width, the total (temporal) displacement and speed of the tweezer are varied and corresponding phase diagrams are presented. As the relative speed of the tweezer is increased we find four possible dynamical scenarios: (a) successful tweezing, (b) release and recapture, (c) destruction due to absorption, and (d) release without recapture of the cavity soliton. We also deploy a non-conservative variational approximation (NCVA) based on a Lagrangian description which reduces the original no-Hamiltonian partial differential equation to a set of coupled ordinary differential equations on the cavity soliton parameters. We show how the NCVA is capable of predicting the threshold in the tweezer parameters for successful tweezing. Through the numerical study of the Lugiato-Lefever equation and its NCVA reduction, we have developed a process to identify regions of successful tweezing which can aid in the experimental design and reliability of temporal tweezing used for information processing., 33 pages, 14 figures

We unravel the existence and stability properties of dark soliton solutions as they extend from the regime of trapped quantum droplets towards the Thomas-Fermi limit in homonuclear symmetric Bose mixtures. Leveraging a phase-plane analysis, we identify the regimes of existence of different types of quantum droplets and subsequently examine the possibility of black and gray solitons and kink-type structures in this system. Moreover, we employ the Landau dynamics approach to extract an analytical estimate of the oscillation frequency of a single dark soliton in the relevant extended Gross-Pitaevskii model. Within this framework, we also find that the single soliton immersed in a droplet is stable, while multisoliton configurations exhibit parametric windows of oscillatory instabilities. Our results pave the way for studying dynamical features of nonlinear multisoliton excitations in a droplet environment in contemporary experimental settings., 16 pages, 9 figures, fixed some typos